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		<title>free range statistics - R</title>
		<description>Posts categorised as 'R'</description>
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				<title>Prime numbers as sums of three squares. by @ellis2013nz</title>
					<description>&lt;p&gt;I was interested by a &lt;a href=&quot;https://www.linkedin.com/posts/fermatslibrary_397-is-conjectured-to-be-the-largest-prime-activity-7242947116719915008-BRz7?utm_source=share&amp;amp;utm_medium=member_desktop&quot;&gt;LinkedIn post about the number 397&lt;/a&gt;:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“397 is conjectured to be the largest prime that can be represented uniquely as the sum of three positive squares”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;That is, 3^2 + 8^2 + 18^2 = 397&lt;/p&gt;

&lt;p&gt;This led to some confusion in the comments as people found other prime numbers that can be created as the sum of three squares. But the wording is sloppy; better wording would be:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“397 is conjectured to be the largest prime that can be represented as the sum of three positive squares of integers in exactly one way”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;Let’s confirm that. The only method I know for something like this is brute force. I can make a data frame with three columns, each with all the squares of positive integers up to some maximum point - so the data frame has every combination of those, discarding duplicate combinations by making the order of the columns strictly non-decreasing. Then we sum those three squares, and store the results:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;primes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;30&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;squares&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;primes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;generate_primes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# This is the part that gets slower with larger k as you make k^3 combinations&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;expand_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;squares&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;squares&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;squares&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum_3_sq&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# example:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum_3_sq&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;397&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This gets us our one combination that adds up to 397:&lt;/p&gt;
&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;     s1    s2    s3 sum_3_sq
  &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt;    &amp;lt;dbl&amp;gt;
1     9    64   324      397
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Next step is to count the number of times each result appears.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;s3s_sum&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum_3_sq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# example:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s_sum&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sum_3_sq&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;So for example we see that 54 (which of course is not a prime - we haven’t yet filtered to primes) can be made 3 ways: as the sum of the squares of 1, 2, 7; of 2, 5, 5; and 3, 3, 6:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;  sum_3_sq number_3_square_sums    s1    s2    s3
     &amp;lt;dbl&amp;gt;                &amp;lt;int&amp;gt; &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt;
1       54                    3     1     4    49
2       54                    3     4    25    25
3       54                    3     9     9    36
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Then it’s a simple matter of joining that summary (of counts of the number of ways to get a given total of three squares) to a data frame of the prime numbers, and drawing a plot of the results:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;primes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s3s_sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;p&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sum_3_sq&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace_na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;point&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;shape&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;397&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;text&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;397&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;500&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Prime number&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of ways&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of ways to make a prime number as sum of three positive squares of integers&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;397 (circled) is the largest with exactly one way, of primes up to {comma(k^2)}.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_3_square_sums&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;panel.grid.minor.y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_blank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;There’s a bit of fiddling there to make the charts look nice for low values of k (where k is the maximum number I square). In my real life code the above is surrounded by a loop of different values of k, with the results saved as SVG and PNG images for use in this blog. All of which gets me this result:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0280-primes-squares-k30.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0280-primes-squares-k30.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;And here we see for some larger results of k:&lt;/p&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0280-primes-squares-k100.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0280-primes-squares-k100.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;As k gets bigger of course the program gets slower to run. This method doesn’t scale well; for primes above about 250,000 the step of making a data frame of all the combinations of three squares starts taking too long. If I wanted to extend this further I’d have to find some ways to do this more efficiently, or put that step into a database that can handle bigger-than-memory data objects.&lt;/p&gt;

&lt;p&gt;I’m sure there’s some interesting maths behind why this is just a “conjecture” and no-one has been able to prove it!&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0280-primes-squares-k300.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0280-primes-squares-k300.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;That’s all for today.&lt;/p&gt;
</description>
				<pubDate>Sat, 21 Sep 2024 00:00:00 +1100</pubDate>
				<link>https://freerangestats.info/blog/2024/09/21/primes-squares</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/09/21/primes-squares</guid>
			</item>
		
			<item>
				<title>Stepwise selection of variables in regression is Evil. by @ellis2013nz</title>
					<description>&lt;p&gt;I’ve recently noticed that stepwise regression is still fairly popular, despite being well and truly frowned upon by well-informed statisticians. By stepwise regression, I mean any modelling strategy that involves adding or subtracting variables from a regression model on the basis that they are “significant”, reduce the Akaike Information Criterion, or increase adjusted R-squared, or in fact any other data-driven statistics.&lt;/p&gt;

&lt;p&gt;This might be an automated variable procedure or it might be a matter of eyeballing the results of the first model you fit and saying (for example) “Let’s take literacy out, it’s p-value is not significant, and it will be a more parsimonious model once we do that.”.&lt;/p&gt;

&lt;p&gt;And then people produce and report t tests, F tests, and so on, as though the end model was the one they always intended to run.&lt;/p&gt;

&lt;p&gt;Let me clear about this. This is wrong. It’s not as disastrously wrong as, say, sorting the data separately one column at a time before you fit your model, but it’s still objectively bad. As my professor once told our class:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“If you choose the variables in your model based on the data and then run tests on them, you are Evil; and you will go to Hell.”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;Why is it wrong? Here are the seven reasons given by Frank Harrell in his must-read classic, &lt;em&gt;Regression Modeling Strategies&lt;/em&gt;:&lt;/p&gt;

&lt;ol&gt;
  &lt;li&gt;The R-squared or even adjusted R-squared values of the end model are biased high.&lt;/li&gt;
  &lt;li&gt;The F and Chi-square test statistics of the final model do not have the claimed distribution.&lt;/li&gt;
  &lt;li&gt;The standard errors of coefficient estimates are biased low and confidence intervals for effects and predictions are falsely narrow.&lt;/li&gt;
  &lt;li&gt;The p values are too small (there are severe multiple comparison problems in addition to problems 2. and 3.) and do not have the proper meaning, and it is difficult to correct for this.&lt;/li&gt;
  &lt;li&gt;The regression coefficients are biased high in absolute value and need shrinkage but this is rarely done.&lt;/li&gt;
  &lt;li&gt;Variable selection is made arbitrary by collinearity.&lt;/li&gt;
  &lt;li&gt;It allows us to not think about the problem.&lt;/li&gt;
&lt;/ol&gt;

&lt;p&gt;At the core of the problem is using statistical inference methods like p values, confidence intervals and ANOVA F tests that were designed and valid for a pre-specified model, but applying them instead to a model we have structured based on the data. The variables are selected partly based on chance, and we are giving ourselves a sneaky headstart in making a variable being significant.&lt;/p&gt;

&lt;p&gt;Basically, this is the sort of thing that leads to the reproducibility crisis in science.&lt;/p&gt;

&lt;p&gt;Some of the problems don’t matter as much if your goal for the model is just prediction, not interpretation of the model and its coefficients. But most of the time that I see the method used (including recent examples being distributed by so-called experts as part of their online teaching), the end model is indeed used for interpretation, and I have no doubt this is also the case with much published science. Further, even when the goal is only prediction, there are better methods like the Lasso, of dealing with a problem of a high number of variables.&lt;/p&gt;

&lt;p&gt;Let’s look at a couple of simulations to show how this is a problem.&lt;/p&gt;

&lt;h2 id=&quot;increases-the-false-positive-rate-even-with-white-noise&quot;&gt;Increases the false positive rate even with white noise&lt;/h2&gt;

&lt;p&gt;First, let’s take a case where we simulate data that is known to have no relation at all to the response variable. In the code below I simulate 1,000 observations with 100 explanatory X variables and 1 response variable y. All of these variables are unrelated to eachother and are just normally distributed with a mean of zero and standard deviation of 1.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;MASS&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;doParallel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#--------------------X not related to y--------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;42&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvrnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mu&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# we get 6 variables that look &apos;significant&apos; - about what&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# we&apos;d expect, about 5% false positives:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficients&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Pr(&amp;gt;|t|)`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;When I fit a regression model of y ~ X, I should expect about five of the columns to appear ‘significant’ by conventional p value of 0.05 or less - because that’s more or less the definition of that critical cut-off value. That is, we tolerate a 1 in 20 false positive rate. In this case we have six variables below the cut-off, about what we’d expect:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;       Estimate Std. Error   t value   Pr(&amp;gt;|t|)
V2  -0.06606672 0.03213918 -2.055644 0.04010514
V38  0.06881778 0.03406473  2.020206 0.04365814
V62  0.06263414 0.03137298  1.996436 0.04618768
V91  0.06826250 0.03302463  2.067018 0.03901824
V94 -0.07923079 0.03423568 -2.314275 0.02087724
V96 -0.07962012 0.03290373 -2.419790 0.01572689
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;Now let’s use stepwise selection, “both” directions (so we can remove variables from the model or add them), using the Akaike Information Criterion to choose a ‘better’ model at each step. This is better than just using p values, and much better than using p values and a low cut-off like 0.05, so I’m giving the stepwise method a fair go here.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;step&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficients&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Pr(&amp;gt;|t|)`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That gets us these variables showing up as ‘significant’:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;       Estimate Std. Error   t value    Pr(&amp;gt;|t|)
V2  -0.06312445 0.03046917 -2.071748 0.038550346
V4  -0.07353379 0.03222313 -2.282019 0.022702106
V32 -0.06120508 0.03094750 -1.977707 0.048241917
V38  0.06383031 0.03227612  1.977633 0.048250238
V90  0.06288076 0.03094938  2.031729 0.042450732
V91  0.07450724 0.03105172  2.399456 0.016605492
V94 -0.06617689 0.03208892 -2.062297 0.039442821
V96 -0.08052606 0.03073565 -2.619957 0.008930218
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;So the net impact of this fancy-looking automated procedure is to worsen our false positive rate from 6% to 8%.&lt;/p&gt;

&lt;p&gt;OK, that’s just one dataset. Let’s try it with a range of others, of different sample sizes, and to make things more interesting let’s let the X variables sometimes be correlated with eachother. The stepwise selection process can be a bit slow so I spread the 700 runs of the simulation below over seven parallel processes:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# set up parallel processing cluster&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;makeCluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# only any good if you have at least 7 processors :)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;registerDoParallel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clusterEvalQ&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;MASS&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;})&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise_results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;700&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.combine&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%dopar%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;400&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;800&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1600&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;runif&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvrnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mu&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepAIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;trace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# count the false positives&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;false_pos1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficients&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Pr(&amp;gt;|t|)`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;false_pos2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficients&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Pr(&amp;gt;|t|)`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;false_pos1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;false_pos2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise_results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;noise_results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`All variables included`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Stepwise selection of variables`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepped&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sample size: {n}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_hline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;gam&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;formula&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of false positives -\nvariables returned as &apos;significant&apos;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Correlation of the X predictor variables&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;False positive rates when using stepwise variable selection&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Models with 100 X explanatory variables that are in truth unrelated to Y; expecting 5 falsely &apos;significant&apos; variables. 
Small sample sizes make the false positive problem for stepwise selection of variables; multicollinearity in the X when no relation to the Y doesn&apos;t matter.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;The average false positive rate of the full model is 5.1%; for the stepwise variable selection it is 9.5%. In the chart below we can see that sample size relative to the number of variables in X matters a lot here:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0279-noisy.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0279-noisy.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;For example, the case with the sample size of only 200 observations gets a false positive rate above 15% from the stepwise method. But even with larger samples, we take a hit in false positives from the stepwise approach. The degree of multicollinearity in the X doesn’t seem to make much difference.&lt;/p&gt;

&lt;p&gt;It might seem unfair to have a model with 100 explanatory variables and only 200 observations, but out there on the internet (I’m not going to link) there are guides telling you it is ok to do this procedure even when you have more variables than observations. In fact I have a horrible fear that this practice might be common in some parts of science. You can imagine how doing &lt;em&gt;that&lt;/em&gt; is basically a machine for generating false, non-reproducible findings.&lt;/p&gt;

&lt;h2 id=&quot;even-the-correctly-retained-variables-coefficients-are-biased-big&quot;&gt;Even the correctly-retained variables’ coefficients are biased big&lt;/h2&gt;

&lt;p&gt;The above simulation was pure noise so everything was a false positive. What does stepwise variable selection do in a more realistic case where some of the variables are correctly in the model and are related to y?&lt;/p&gt;

&lt;p&gt;To explore this I wrote a function (code a little way further down the blog) to simulate data with 15 X correlated variables and 1 y variable. The true model is:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;y = V1 + V2 + V3 + V4 + V5 + V6 + V7 + 0.1 (V8 + V9 + V10) + e
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;That is, the true regression coefficients for variables V1 to V7 are 1; for V8, V9 and V10 they are 0.1; for the the remaining 5 variables there is no structural relationship to y.&lt;/p&gt;

&lt;p&gt;When we simulate 50 data sets of this sort and use stepwise variable selection to regress y on X, here are the coefficients we get. Each point represents the coefficient for one variable from one of those runs.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0279-main-sim.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0279-main-sim.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;The large dots on zero indicate the multiple runs in which that particular variable was not included in the final model. We see:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;Many occasions, variables V11 to V15 were rightly excluded, but a smattering of occasions they do get included in the model.&lt;/li&gt;
  &lt;li&gt;A lot of false negatives - variables V1 to V10 that should be found in the model and aren’t&lt;/li&gt;
  &lt;li&gt;Worse, when one of variables V1 to V10 is correctly included in the final model, the coefficient estimated for it is &lt;em&gt;always&lt;/em&gt; (in this dataset) larger than the true coefficient (which remember should be 1 or 0.1 - the correct values shown by the red crosses).&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;For comparison, here is an equivalent chart for when we fit the full model to these data. There’s a lot of variation in the coefficient estimates, but at least they’re not biased (that is, on average they are correct, their expected value is the true value):&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0279-main-sim-full.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0279-main-sim-full.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Here’s the code for the function simulating that data and drawing the plots:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#---------------------when X is related to y--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param xcm correlations of the X variables with eachother, as a multiplier of&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   their standard deviation (all X variables have the same variance / sd of 1)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param ysdm standard deviation of the y variable, expressed as a multiplier of variance&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   of the X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param n sample size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param k number of columns in X. Only currently works if this is 15 (because of the hard-coded true_coef)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param runs number of simulations to run&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param seed random seed for reproducibility&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_steps&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;runs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;321&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;runs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Sigma, not sigma squared&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mvrnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mu&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Sigma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.data.frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%*%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;step_mod&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stepAIC&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;trace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;csm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;step_mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;csm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;csm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;run&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Stepwise&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                       &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;run&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Full model&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                       &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;factor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;levels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;complete&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;run&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;factor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;levels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef_df&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;variable&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bias&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient.x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient.y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r.squared&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summary&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;step_mod&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r.squared&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bias&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mclabel&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;negligible&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.19&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;mild&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.39&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;medium&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;            &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;strong&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paste&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;multicollinearity&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Stepwise&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;shape&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;12&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_size_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Coefficient value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Variable&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of observations:\n(usually only one, except when coefficient dropped altogether)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Stepwise regression returns coefficient estimates biased away from zero&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Black dots show coefficient estimates from one run of a stepwise (AIC-based) model fitting. Red squares show correct values.
Coefficients of variables left in the model are on average {biases[&apos;Stepwise&apos;]} too large (compared to real value of 0, 0.1 or 1).
Also, real explanatory variables are often dropped. Fake ones are often included.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Simulated data with a model that explains about {percent(summary(step_mod)$r.squared)} of variation in response variable, with {mclabel}, by https://freerangestats.info&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Stepwise&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coefficient&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;true_coef_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;shape&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;12&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Coefficient value&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Variable&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Using the full model returns unbiased coefficient estimates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Black dots show coefficient estimates from one run of a all-variables-in model fitting. Red squares show correct values.
Coefficients of variables left in the model are on average {biases[&apos;Full model&apos;]} too large (compared to real value of 0, 0.1 or 1).&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Simulated data with a model that explains about {percent(summary(mod)$r.squared)} of variation in response variable, with mild multicollinearity, by https://freerangestats.info&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_sim&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_steps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_sim&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# plot results for stepwise selection&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_sim&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# plot results for full model&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Now, you can see that there are a few arbitrary aspects to that simulation - in particular the sample size, the variance of the y relative to the X, and the multicollinearity of the X. The idea of having it as a function is that you can play around with these and see the impacts; for example, if you make the variance of y smaller compared to that of the X, the model gets better at explaining the variance and the stepwise algorithm is less prone to getting things wrong. Rather than include a whole bunch of individual cases, I ran some more simulations covering a range of such values so we can see the relationship to those parameters of the average bias in the estimated regression coefficients remaining in the model.&lt;/p&gt;

&lt;p&gt;So here is the relationship of that bias to the R-squared of the model, at various levels of correlation between the X variables.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0279-rsquared-bias.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0279-rsquared-bias.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;And here is the relationship of the bias to sample size, standard deviation of Y, and correlation between the X all in one chart:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0279-corr-sd-bias.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0279-corr-sd-bias.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Of the two visualisations I probably prefer that last one, the heat map. First, it dramatically shows (all that white) that the regression estimates from the true model aren’t biased at all. Secondly, it nicely shows that the bias in the estimates returned by stepwise regression are worse&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;for smaller samples&lt;/li&gt;
  &lt;li&gt;with higher correlation between the X variables&lt;/li&gt;
  &lt;li&gt;and with high variance of the y variable&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;So, finally, here’s the code for those simulations:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# parameters to run this for:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_params&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;expand_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# export onto the cluster some objects we need to use:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clusterExport&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sim_steps&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;var_params&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# run all the simulations&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;many_params&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.combine&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%dopar%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sim_steps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;runs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;biases_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# linechart plot:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;many_params&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;n = {n}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bias&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.ordered&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_smooth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;se&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;loess&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;formula&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;span&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;R-squared (proportion of Y&apos;s variance explained by model)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Average bias of estimated variable coefficients&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bias in regression coefficients after stepwise selection of variables&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bias is worst with small samples, models with low R-squared, and correlation in the explanatory variables (shown from 0.1 to 0.9).
Model with 15 explanatory &apos;X&apos; variables. Correct values of coefficients are 0, 0.1 or 1;  so a bias of +1 is very serious.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Model fitting procedure&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# heatmap plot:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;many_params&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;n = {n}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xcm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ysdm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;bias&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_grid&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_tile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_fill_gradientn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colours&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;darkred&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Correlation between the X variables&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Standard deviation of the Y variable&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Average bias of estimated variable coefficients:&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bias in regression coefficients after stepwise selection of variables&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Bias is worst with small samples, high variance response, and correlation in the explanatory variables.
Model with 15 explanatory &apos;X&apos; variables. Correct values of coefficients are 0, 0.1 or 1;  so a bias of +1 is very serious.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That’s all folks. Just don’t use stepwise selection of variables. Don’t use automated addition and deletion of variables, and don’t take them out yourself by hand “because it’s not significant”. Use theory-driven model selection if it’s explanation you’re after, Bayesian methods are going to be good too as a complement to that and forcing you to think about the problem; and for regression-based prediction use a lasso or elastic net regularization.&lt;/p&gt;
</description>
				<pubDate>Sat, 14 Sep 2024 00:00:00 +1100</pubDate>
				<link>https://freerangestats.info/blog/2024/09/14/stepwise</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/09/14/stepwise</guid>
			</item>
		
			<item>
				<title>Gender and sexuality in Australian surveys and census by @ellis2013nz</title>
					<description>&lt;h2 id=&quot;the-2021-abs-standard-for-sex-gender-variations-of-sex-characteristics-and-sexual-orientation-variables&quot;&gt;The 2021 ABS Standard for ‘Sex, Gender, Variations of Sex Characteristics and Sexual Orientation Variables’&lt;/h2&gt;

&lt;p&gt;Over the past two weeks there has been &lt;a href=&quot;https://www.abc.net.au/news/2024-09-01/anthony-albanese-census-gender-identity-intersex/104296702&quot;&gt;quite a controversy&lt;/a&gt; relating to questions about sexuality and gender in the next Australian Census of Population and Housing, in 2026. Things hit the news when the testing of “new questions” was withdrawn, then there was an announcement that “the question” will be included; then apparent clarification that one of “the questions” will be but not all of them.&lt;/p&gt;

&lt;p&gt;My work includes understanding risks for these sorts of controversies (in other countries, not Australia) so I spent a bit of time getting my head around it. The Australian Broadcasting Corporation reports that the &lt;a href=&quot;https://www.abc.net.au/news/2024-09-06/2026-census-questions-revealed/104321662&quot;&gt;three originally proposed questions were&lt;/a&gt;:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;What is the person’s gender?&lt;/li&gt;
  &lt;li&gt;How does the person describe their sexual orientation?”&lt;/li&gt;
  &lt;li&gt;Has the person been told they were born with a variation of sex characteristics?&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The are essentially three of &lt;a href=&quot;https://www.abs.gov.au/statistics/standards/standard-sex-gender-variations-sex-characteristics-and-sexual-orientation-variables/latest-release&quot;&gt;the standard questions on sex, gender, sexual orientation&lt;/a&gt; in the relevant Australian Bureau of Statistics (ABS) Standard that was released in early 2021 and has had some minor tweaks and corrections since. A fourth question, apparently not up for debate, is:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;What was [your/the person’s name/their] sex recorded at birth?&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;The standard options for this are “Male”, “Female”, and “Another term (please specify)”.&lt;/p&gt;

&lt;p&gt;At the time of writing, it looks like the sexual orientation and sex at birth questions &lt;em&gt;will&lt;/em&gt; be included in the Census but not those on gender and variation of sex characteristics. I guess this reflects a perception that Australian society / politics has a definite widespread acceptance of variation in sexual orientation - it’s generally seen that people can sleep with and indeed marry whomever they choose. What hasn’t yet become as common apparently is acceptance of a modest degree of complexity in sexual characteristics and gender identity.&lt;/p&gt;

&lt;p&gt;The Australian Standard is well worth a read in full for anyone who has to ask or analyse survey questions relating to sex and gender. It’s a good piece of work, was very thoroughly researched and consulted on. The concepts are explained clearly, most critically the difference between sex relating to physical characteristics and gender as a social and cultural concept.&lt;/p&gt;

&lt;p&gt;The standard isn’t just a list of questions and the wording to use for the answer options (although it does give this, and the carefully defined acceptable variants in limited circumstances, in explicit detail of course). Some of the elements that may not be obvious to people new to this field include, for example, strict rules that the questions have to be asked exactly as written. Interviewers are forbidden from inferring either sex or gender from appearance, names, etc. Or that the preferred word to use for statistics relating to the whole population is “Persons”, not “Total” or “People”, when presenting total population counts for sex.&lt;/p&gt;

&lt;p&gt;As per the Standard, the key variable in the “Cisgender and Trans and Gender Diverse Classification” has to be derived by comparing the answer to “What was this person’s sex recorded at birth” and “What is this person’s gender”.&lt;/p&gt;

&lt;p&gt;There’s a decision table in the Standard on how to do this; it gets complicated when one or both of the questions had “Another Term” selected for sex at birth, or “Prefer not to answer” to gender now. In either of these cases the correct value for Cisgender and Trans and Gender Divsere variable becomes “Inadequately described”.&lt;/p&gt;

&lt;p&gt;If sex at birth differs from gender now, or gender now is “non-binary” or “different term”, the person is classified as “trans”. If sex at birth matches gender now, the person is classified as “cis”. All this of course looks straightforward and simple when set out clearly. It makes sense, and flows naturally from the realisation that cis/trans status is quite different from gender now .&lt;/p&gt;

&lt;p&gt;It is both offensive and leads to poor quality data for example to give people a choice of male, female, trans-male, trans-female in response to a question on gender. This is precisely because the whole point of “transitioning” for many people is that now you are just whatever the final gender is, not an amalgam of previous and current genders - “trans-women &lt;em&gt;are women&lt;/em&gt;”, and so on. So the correct statistical procedure is to ask separate questions on sex at birth and gender now, and combine the two to determine cis / trans status.&lt;/p&gt;

&lt;h2 id=&quot;whats-the-information-for-and-what-we-know&quot;&gt;What’s the information for, and what we know&lt;/h2&gt;

&lt;p&gt;The outcry from the LGBTIQ+ community in response to some of these questions being reported as scrapped from the next Census had it seemed to me two substantive points:&lt;/p&gt;

&lt;ol&gt;
  &lt;li&gt;More detailed information on these issues is needed for government policy and program decision-making, particularly in health and social policy.&lt;/li&gt;
  &lt;li&gt;There is a “right to be counted” and recognised for who people are, and the Census is an important part of the state validating this aspect of people’s existence.&lt;/li&gt;
&lt;/ol&gt;

&lt;blockquote&gt;
  &lt;p&gt;“Without comprehensive and inclusive Australian census data, the full diversity of LGBTQ+ communities remain invisible and marginalised. This unnecessary lack of data hinders proper coordination of public health measures and social policy. We need political leadership.”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;&lt;em&gt;&lt;a href=&quot;https://aus.social/@Lukas/113035241786325182&quot;&gt;Lukas Parker on Mastodon&lt;/a&gt;&lt;/em&gt;&lt;/p&gt;

&lt;p&gt;I think the second of these two arguments - about the rights and symbolism - is actually the stronger.&lt;/p&gt;

&lt;p&gt;Important background on why this is such a sore point is that in the 2021 Census, there was a question on sex that was designed before the 2021 Standard came fully into force (Census questionnaires take years to develop). The very small proportion of people who picked “non-binary” in response to the question on sex was &lt;a href=&quot;https://www.abc.net.au/news/2022-10-07/latham-one-nation-transgender-abs-census-population-problematic/101507074&quot;&gt;unhelpfully picked up by a politician as evidence that there were hardly any transgender people in Australia&lt;/a&gt;. This is clearly an invalid interpretation of a question about sex (not about gender, and certainly not about cis/trans gender status), but one that is unfortunately likely to be repeated in the lack of clearly more appropriate estimates.&lt;/p&gt;

&lt;p&gt;Note that, as described above, the Standard now would require “Another term (please specify)” as the valid third choice for a question on sex at birth. But this still gets one nowhere in terms of counting the number of trans people; a second question on “gender now” is required.&lt;/p&gt;

&lt;p&gt;For health or social policy planning, any question in the Census needs to be justified in terms of the extra benefits gained from making &lt;em&gt;everyone&lt;/em&gt; in the country (not just a survey sample) fill it in. This might be justified, for example, because very fine level of spatial detail is needed that could never be provided by surveys; or a definitive population total is needed for survey design purposes; or we need to know this information on all individuals for later analysis with integrated data.&lt;/p&gt;

&lt;p&gt;I have in fact seen the argument that these population totals are needed for survey calibration and design made in this recent debate but unfortunately have lost the link. But I don’t think it’s particularly compelling, and have yet to see a good technical argument to the contrary. I believe that survey data could give us good enough estimates of the totals for decision-making, down to number of persons per state, and understanding of characteristics, and creating purposive samples for any given relationship to population proportions.&lt;/p&gt;

&lt;p&gt;This is certainly the case for sexual orientation, where we can go (for example) to the latest (2020) &lt;a href=&quot;https://www.abs.gov.au/statistics/people/people-and-communities/general-social-survey-summary-results-australia/2020#data-downloads&quot;&gt;General Social Survey Table 5&lt;/a&gt; and see that 773,000 (our of 20.3m) persons aged 15 or over are estimated to be Gay, Lesbian, Bisexual or ‘Other’.&lt;/p&gt;

&lt;p&gt;The relative margin of error for that estimate is 9.0%, so perhaps a confidence interval would be around 700 and 850 thousand persons - between 3.5% and 4.2% of the total persons aged 15 and over. 9% isn’t great, but I’m confident the measurement error is a bigger risk than sampling error here. That is, people who either don’t understand, don’t know, or don’t want to tell an interviewer (or a government form online) their sexual orientation. A census would remove the sampling error but doesn’t help measurement error.&lt;/p&gt;

&lt;p&gt;Relating to the number of trans persons in Australia, an estimate could be derived from the &lt;a href=&quot;https://www.abs.gov.au/methodologies/national-health-survey-methodology/2022#overview&quot;&gt;2022 National Health Survey&lt;/a&gt;, which included all four necessary questions in the sex, gender, etc Standard mentioned above. Surprisingly given the public interest in the issue, I haven’t been able to find such an estimate published anywhere. From eyeballing and back-of-enveloping the size of confidence intervals in one of the reports based on it, the &lt;a href=&quot;https://www.abs.gov.au/statistics/health/mental-health/national-study-mental-health-and-wellbeing/2020-2022&quot;&gt;National Study of Mental Health and Wellbeing&lt;/a&gt;, I infer that trans people were between 1% and 4% of the &lt;em&gt;sample&lt;/em&gt; of the National Health Survey. However, as young people were deliberately oversampled in the survey and we definitely think age is correlated to trans status, this is surely an overestimate of the population compared to what you could do with the original microdata.&lt;/p&gt;

&lt;p&gt;Someone should do this! Or if it exists, let me know in the comments. One hour’s work (for someone with access to the microdata), compared to the very large cost of including a question in the census. Alternatively, if the data isn’t good enough for estimating the number of trans persons in Australia, we should have a discussion of why and what to do about it. Many of the candidate reasons why - to do with measurement or validity error of some kind - would also mean that using the Census for this purpose wouldn’t help.&lt;/p&gt;

&lt;p&gt;Anyway, my blog is about playing around with data and charts and not politics, so let’s go to a visualisation of some of the published results from the 2020 General Social Survey. This survey included the question on sexual orientation and a good range of results are reported from it. After a lot of iteration I came up with this chart:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0278-lollipops.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0278-lollipops.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;You may need to zoom in on this, but it should be readable on most screens. In this chart I have selected the survey questions that had the largest estimated ratio of difference between heterosexuals and gay, lesbian or bisexual choosing that answer, having first filtered out any questions with less than 5 percentage points of absolute difference. There’s a lot packed in but there’s some interesting things here:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;Why are more young people reporting as gay, lesbian or bisexual? Obviously there are several candidate explanations: people become straighter as they get older (unlikely); more people are gay, lesbian or bisexual than before; or more people are &lt;em&gt;admitting to themselves and interviewers&lt;/em&gt; that they are gay, lesbian or bisexual than before. Most likely the answer is some version of this latter point. Cultural norms have changed astonishingly fast, in historical terms, in this space so it’s not surprising that there is a generational difference in reported prevalence.&lt;/li&gt;
  &lt;li&gt;Gay, lesbian and bisexual people apparently less likely to be unemployed and more likely to be in the highest income quintile&lt;/li&gt;
  &lt;li&gt;Gay, lesbian and bisexual people much more likely to be not married or in a de facto marriage than heterosexual people.&lt;/li&gt;
  &lt;li&gt;Sadly but unsurprisingly, gay, lesbian and bisexual people much more likely to have experienced discrimination or disagree with the statement that the police and justice system can be trusted&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;Note that I am very deliberately using the expression “LGB+” here in these charts, not the familiar “LGBTIQ+” expression. LGBTIQ of course does not aspire to be a statistical classification and is more like a social or political coalition; it combines sexual orientation (LGB), sexual characteristics (the “I” for Intersex ) and trans/cis gender status (the “T” for Trans and possibly the “Q” for Queer, although that could really refer to some combination of the three concepts).&lt;/p&gt;

&lt;p&gt;Here’s the code that downloads the data, tidies it up and presents that chart:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readxl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://www.abs.gov.au/statistics/people/people-and-communities/general-social-survey-summary-results-australia/2020/GSS_Table5.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;gss_table5.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mode&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;wb&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;demog_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sex&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Whether currently smokes&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Age group&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Employed&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Level of highest non-school qualification&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Engagement in employment or study&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Family composition of household&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Marital status&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Main Source of Household Income&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Current weekly household equivalised gross income quintiles&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_excel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;gss_table5.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;skip&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sheet&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Table 5.1_Estimate&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;...1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;is.na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;is.na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.direction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;down&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;is.na&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_wrap&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;20&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;demog_cats&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Characteristics&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Experiences and attitudes&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Gay, Lesbian or Bisexual&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Gay, Lesbian, Bisexual or Other&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check no typos in the demography categories&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;all&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;demog_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# some categories have long answers and are difficult to present on a chart&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;difficult_cats&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Community involvement&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Cultural tolerance and discrimination&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Family and community support&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Crime and safety&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Stressors&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# draw basic faceted barchart&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;difficult_cats&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Total persons&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Persons aged 15 years and over&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;var_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;dodge&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;35&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;free&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison of LGB+ and heterosexual attitudes&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Selected questions from Australia&apos;s General Social Survey, 2020&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;axis.text.x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;angle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;45&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sc&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# better visualisation that focuses on results.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# number of questions to pick to highlight:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;22&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;spread&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sexuality&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Gay, Lesbian, Bisexual or Other`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rabs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmax&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Gay, Lesbian, Bisexual or Other`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;category&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Age group&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Marital status&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Employed&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Engagement in employment or study&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
      &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;{category}:\n&apos;{variable}&apos;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lab_seq&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;grepl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;employ&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ignore.case&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-150&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;grepl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;income&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ignore.case&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Characteristics&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.factor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lab_seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rabs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# don&apos;t show any where the absolute difference is less than 5 percentage points:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;diff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.05&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Gay, Lesbian, Bisexual or Other`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-0.026&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.026&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# variable for using to draw colour of segments and arrows:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Gay, Lesbian, Bisexual or Other&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Heterosexual&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# draw lollipop / arrow plot&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Gay, Lesbian, Bisexual or Other`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Heterosexual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cat_type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;free_y&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_segment&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;linewidth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glb2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stage&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;after_scale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prismatic&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clr_lighten&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;space&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;combined&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;angle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.15&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;inches&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Prevalence of response&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Key differences between LGB+ and heterosexual Australians&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Responses from the General Social Survey 2020&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;In case anyone’s interested, I did over 100 iterations of this chart, looking for something that really presents some analytical findings in an interpretable and aesthetically pleasing way. It’s much harder to do presentation charts than the straightforward ones that just show all the numbers at once. One interesting point is that the great majority of the work is in reshaping and tidying data - in this case, different (but obviously derivative) data frames for drawing the points and for drawing the arrow segments. A nice example of how data management and plot polishing become an iterative process. You’d never publish the data in the format for a plot like this, it needs to be up to the end analyst to identify this as the plot they want, and reshape the data accordingly.&lt;/p&gt;

&lt;p&gt;There was also a lot of effort going into decisions like “Do I need the ‘category’ (e.g. Age group) for each label on the charts, or are some self-explanatory, and how do I code this in without retyping all the data?”.&lt;/p&gt;

&lt;p&gt;That code also produced a more straightforward chart which I show here for comparison. It’s the same info (although question categories with long names are excluded for readability), but puts a lot more work on the reader to pick out the key messages. The final plot I used tries to help the reader with this by doing a bunch of analysis ourselves first and presenting in a way that highlights the most important aspects of it.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0278-many-facets.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0278-many-facets.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;That’s all for today. Do read (and follow, in your next survey questionnaire with questions on sex or gender) that standard on sex, gender, etc. - it’s good stuff.&lt;/p&gt;

&lt;h3 id=&quot;late-edits--update&quot;&gt;Late edits / update&lt;/h3&gt;

&lt;ul&gt;
  &lt;li&gt;thanks to @baptnz@social.nz who pointed out on Mastodon a better method of making the arrows appear light rather than the transparency hack I originally used. His method, using the ‘prismatic’ package, I have edited my code above and is now in use.&lt;/li&gt;
  &lt;li&gt;immediately after I wrote all the above I discovered that while I was publishing the original post, the Government has now announced that the questions on sexual orientation, sex at birth, and gender now will all be included in the Census; but not the question on sexual characteristics. So this is a good outcome for those wanting to estimate gender issues and trans/cis status, although those with interest in sexual characteristics are still disappointed.&lt;/li&gt;
&lt;/ul&gt;
</description>
				<pubDate>Sun, 08 Sep 2024 00:00:00 +1100</pubDate>
				<link>https://freerangestats.info/blog/2024/09/08/sex-gender</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/09/08/sex-gender</guid>
			</item>
		
			<item>
				<title>Sampling without replacement with unequal probabilities by @ellis2013nz</title>
					<description>&lt;h2 id=&quot;not-proportional-to-w&quot;&gt;Not proportional to w&lt;/h2&gt;

&lt;p&gt;A week ago I was surprised to read on &lt;a href=&quot;https://notstatschat.rbind.io/2024/08/26/another-way-to-not-sample-with-replacement/&quot;&gt;Thomas Lumley’s Biased and Inefficient&lt;/a&gt; blog that when using R’s &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;sample()&lt;/code&gt; function without replacement and with unequal probabilities of individual units being sampled:&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“What R currently has is sequential sampling: if you give it a set of priorities w it will sample an element with probability proportional to w from the population, remove it from the population, then sample with probability proportional to w from the remaining elements, and so on. This is useful, but a lot of people don’t realise that the probability of element i being sampled is not proportional to w_i”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;This surprised me profoundly - in the way that Bertrand Russell once wrote of surprise in some epistemological context of getting to the bottom of the stairs and finding there was one more stair to go than you unconsciously expected. That is, I hadn’t even been consciously aware of assuming that the probability of an element being sampled is proportional to the weight given to it, but if I &lt;em&gt;had&lt;/em&gt; been asked if this was the case I would have given the wrong answer (‘yes’).&lt;/p&gt;

&lt;p&gt;So I didn’t know this, but I should have.&lt;/p&gt;

&lt;p&gt;Thomas had some examples in his blog relating to streams of data but I wanted to see for myself. So I wrote a function to simulate sampling from a small, finite population, with equal or unequal weights. So mostly, I sampled 10 units at a time from a population of 20; and gave them weights from about 0.005 to 0.095 ( a big range). The weights are forced to add up to 1 so they can be easily compared with that element’s proportion of the eventual sample of samples. And then I take some thousands of these samples of 10, and look at the proportion of the total collection that is each element.&lt;/p&gt;

&lt;p&gt;In the charts the follow, the brown diagonal line would be where the eventual proportion of element i being sampled is proportional to its weight. First, here is the main situation of concern - unequal weights, sampling without replacement, sample size is quite a big proportion of the population (10 out of 20):&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0276-20-10-no-replace.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0276-20-10-no-replace.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;OK! so we see the elements with low weights get sampled a bit more than naively expected, and the elements with high weights a bit less.&lt;/p&gt;

&lt;p&gt;Just to check I’m not dreaming, here’s the same simulation but this time we are sampling &lt;em&gt;with&lt;/em&gt; replacement. Now, everything works as intuitively might be expected, the eventual proportion in our sample of samples is exactly matched to the original weights:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0276-20-10-replace.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0276-20-10-replace.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Here’s the code for that function and those two runs of it.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;compare_ppswor&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;20&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;FUN&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paste0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;unit&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;N&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;factor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;levels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;samples&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;FUN&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;})&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;with replacement&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;without replacement&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;identical&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;FUN&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;R&apos;s native `sample()` function.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;identical&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;FUN&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample_unequal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;experimental function based on Brewer (1975).&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;identical&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;FUN&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample_brewer&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Brewer (1975) as implemented by Tillé/Matei.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;original&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;selected&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;table&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unlist&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;samples&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;original&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;selected&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_abline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;slope&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;intercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;orange&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Original &apos;probability&apos; or weight&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Actual proportion of selections&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Population of {N}, sample size {n}, sampling {s}.\nUsing {m}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Use of `sample()` with unequal probabilities of sampling&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coord_equal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;compare_ppswor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;compare_ppswor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;When the sample is a smaller proportion of a population, the no-replacement discrepancy is less, as seen in these examples with population sizes of 50 and 250.&lt;/p&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0276-50-10-no-replace.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0276-50-10-no-replace.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0276-250-10-no-replace.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0276-250-10-no-replace.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;At the extreme, if the sample size is the same size as the population (20 each in the simulation below), then of course, all samples without replacement are identical to the population, so it is impossible for the end proportion to be anything other than 1/N each:&lt;/p&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0276-20-20-no-replace.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0276-20-20-no-replace.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;h2 id=&quot;brewers-1975-algorithm&quot;&gt;Brewer’s 1975 algorithm&lt;/h2&gt;
&lt;p&gt;Thomas was writing because there are plans to change this behaviour of &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;sample()&lt;/code&gt;, or at least add an option to change it, so instead the eventual proportion of samples containing an element will be proportional to the weight even when sampling with unequal probabilities and without replacement. The method that is going to be implemented in R (written in C of course before being hooked into R - it’s an inherently iterative thing that will only be efficient in compiled code) is based on a method published way back in 1975 by Ken Brewer.&lt;/p&gt;

&lt;p&gt;An aside - I had the privilege and pleasure to be taught sampling by Professor Brewer back in the 1990s at the Australian National Universitiy - at least it was a pleasure at the time (sheer entertainment value as well as the pleasure of witnessing someone who was a virtuoso in the subject matter, master of the theory of sampling), and I now belatedly realise what a privilege it was. I do think he struggled occasionally with some of the students though; one of my vivid memories of my whole education at ANU was of him in confusion when one of the less mathematically inclined asked him to explain what an equation taking up a fair proportion of the blackboard “means” - “Means? means? it means what it says!”&lt;/p&gt;

&lt;p&gt;I don’t have easy access to the 1975 paper but I found this very interesting &lt;a href=&quot;https://stats.stackexchange.com/questions/110178/brewers-method-for-sampling-with-unequal-probabilities-with-n2&quot;&gt;question relating to it on Cross-Validated&lt;/a&gt;. From that discussion and reading the part of the SAS manual referred to, it sounds like SAS implements the algorithm only for when the sample in a given strata is 2, but Brewer had generalised his method to larger samples. The answer given by the ever-reliable StasK seems pretty clear on how this is done, so presuming he got it right I had a go at implementing this method into R, with code below in the form of two functions &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;P()&lt;/code&gt; and &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;sample_unequal()&lt;/code&gt;:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param p probabilities of remaining units&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param n total sample size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param k which sequence of the sample this is for the sampling of&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;D&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;((&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;D&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample_unequal&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;keep&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sample size cannot be larger than the population of units&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Only sampling without replacement implemented at this point&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Sample size ({size}) cannot be larger than 1 / max(prob) (which is {1 / max(prob)})&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_sample&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NULL&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remnants&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remnant_p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;P&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remnant_p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;keep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remnants&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paste0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;k&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;k&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;x&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;warning&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Some negative probabilities returned&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmax&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;latest_sample&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remnants&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;new_p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;which_chosen&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;which&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remnants&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;latest_sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remnants&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remnants&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;which_chosen&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remnant_p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remnant_p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;which_chosen&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_sample&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;latest_sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
      &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;keep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_sample&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;else&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;I’m not guaranteeing I got it right - feedback is welcome.&lt;/p&gt;

&lt;p&gt;Of course, I designed this to be plugged into my function I made earlier for comparing eventual sample proportions to the weights given, so with &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;compare_ppswor(FUN = sample_unequal, reps = 10000)&lt;/code&gt; I can get this comparison:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0276-20-10-no-replace-brewer.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0276-20-10-no-replace-brewer.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Interestingly, it’s still not exactly ‘right’ in the sense that the eventual proportions in the sample aren’t exactly those given in the form of weights. But it’s much better at doing this than the current &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;sample()&lt;/code&gt; method is.&lt;/p&gt;

&lt;p&gt;And if we have a sample that is say 25% of the population instead of 50% with &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;compare_ppswor(n = 5, FUN = sample_unequal, reps = 10000)&lt;/code&gt;, then we get results where the proportions and weights are pretty indistinguishable:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0276-20-5-no-replace-brewer.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0276-20-5-no-replace-brewer.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;OK that’s all. I played around with this a bit more, particularly trying to understand the limitations on the algorithm which breaks down as the sample size gets bigger in relation to the population, but none of it was really interesting enough to write up. A full script with a bit more in it is available in the &lt;a href=&quot;https://github.com/ellisp/blog-source/blob/master/_working/0276-sampling-unequal-p.R&quot;&gt;source code for this blog&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;I guess the meta-lesson is to watch out for things you’re assuming without noticing you are.&lt;/p&gt;

&lt;h2 id=&quot;late-addition&quot;&gt;Late addition!&lt;/h2&gt;

&lt;p&gt;When I posted about this on Mastodon, Thomas Lumley drew my attention to the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;sampling&lt;/code&gt; library by Yves Tillé and Alina Matei, which contains multiple algorithms for sampling with unequal probabilities without replacement. I hadn’t even stopped to look because really this was about me learning, but yes, of course there’s an R package for that. So I wrapped their &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;UPbrewer&lt;/code&gt; function in a little function to work with my comparison. Unlike me they had read Brewer’s actual article, and their implementation of course works perfectly:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0276-20-10-brewer-better.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0276-20-10-brewer-better.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;So the version I hacked together based on a Cross-Validated answer clearly doesn’t do justice to the method proposed by Brewer (or more likely was some other method he proposed separately).&lt;/p&gt;

&lt;p&gt;For the record here’s the little convenience function to test the Tillé/Matei implementation:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#--------------using sampling library------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sampling&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample_brewer&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;replace&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;keep&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pik&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;UPbrewer&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pik&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_sample&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;which&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;s&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
 &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
 &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;compare_ppswor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;FUN&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample_brewer&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;I’ts &lt;em&gt;much&lt;/em&gt; faster too!&lt;/p&gt;
</description>
				<pubDate>Sat, 31 Aug 2024 00:00:00 +1100</pubDate>
				<link>https://freerangestats.info/blog/2024/08/31/ppswor</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/08/31/ppswor</guid>
			</item>
		
			<item>
				<title>Ratios of indexed line charts by @ellis2013nz</title>
					<description>&lt;p&gt;A few months Michael Read of the Australian Financial Review published a chart showing real household income in Australia plummetting, relative to the OECD, since late 2021. A version of this &lt;a href=&quot;https://mastodon.social/@andyjennings@aus.social/113020422083231612&quot;&gt;drifted across my social media feed&lt;/a&gt; and interested me.&lt;/p&gt;

&lt;p&gt;First I wanted to re-create the chart, which was straightforward enough as it was using data published by the OECD, accessible in SDMX format from their dot Stat tool. After a bit of experimenting I decided to also look at household expenditure and GDP per capita; because I wanted to understand how much this was just an income issue or a broader one translating into expenditure and GDP. That gives me this chart; the middle panel is identical in substance to Read’s original:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0275-inc-exp-gdp-2007.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0275-inc-exp-gdp-2007.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;There’s some interesting things here!&lt;/p&gt;
&lt;ul&gt;
  &lt;li&gt;Yes, the story is bad for Australia household income since late 2021, as per the story of the original chart.&lt;/li&gt;
  &lt;li&gt;It’s not just an income story - expenditure and GDP have both declined as well, although not as dramatically (implying that some of the lost income is translating into lower savings / wealth).&lt;/li&gt;
  &lt;li&gt;The Covid spike upwards in income is matched with a downwards spike in expenditure and GDP.&lt;/li&gt;
  &lt;li&gt;Putting aside recent declines, all three series (household income, household expenditure, GDP per capita) show for both Australia and the OECD average that people are better off than in 2007, when the index is set to 100.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;It’s worth noting before we go on that “real gross disposable income” means it has been adjusted for inflation, it is income after subtracting taxes, social security contributions, change in net equity in pension funds, and interest on financial liabilities (including, I believe, mortgages). So an increase in interest rates that hits households with mortgages is one thing that would show up here.&lt;/p&gt;

&lt;p&gt;Here’s the R code that produced that first chart:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rsdmx&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;janitor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lubridate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;metadata&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tribble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;B6GS1M_R_POP&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Real gross disposable income per capita of households and NPISH&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;P3S1M_R_POP&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Real final consumption expenditure per capita of households and NPISH&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                    &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;B1GQ_R_POP&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Real gross domestic product per capita&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: OECD dot Stat P3S1M_R_POP, B6GS1M_R_POP, B1GQ_R_POP. &apos;NPISH&apos; means &apos;non-profit institutions serving households&apos;.&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readSDMX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://sdmx.oecd.org/public/rest/data/OECD.SDD.NAD,DSD_HHDASH@DF_HHDASH_INDIC,1.0/Q.OECD+AUS.B1GQ_R_POP+B6GS1M_R_POP+P3S1M_R_POP.?dimensionAtObservation=AllDimensions&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clean_names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;metadata&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;measure&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ref_area&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ref_area&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;AUS&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Australia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ref_area&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;OECD&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;OECD average&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;palette&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Australia&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`OECD average`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;blue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ref_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;40&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_manual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;palette&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Index (2007 = 100)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Growth in household expenditure and income, Australia and OECD average&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Indexes set to 100 in 2007. Comparison shows relative growth rates, not absolute difference.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Now, I’ve long been interested in the impact of choice of year for indexing series like this. I have argued before &lt;a href=&quot;/blog/2016/08/18/dualaxes&quot;&gt;on this blog&lt;/a&gt; that many of the criticisisms of dual axes timeseries plots can be mitigated by treating the lines on the page as indexed time series and choosing scales accordingly; but related to this is the risk that choosing a different index year for your series can have significant difference on the visual impression of the chart.&lt;/p&gt;

&lt;p&gt;In the case of the income chart, I was struck with how Australia’s real household income grew faster than the OECD average for some years from 2007 and then, from around 2014, the OECD started to catch up. But to see this you have to pay attention to the gap between the lines, whereas the crossing of the lines in 2022 is visually dramatic and appears to really mean something. But it doesn’t! or at least, not as much as the naive viewer might think; what the crossing of the lines means is just that the OECD average growth since 2007 has caught up with the Australian growth since 2007, it doesn’t mean the OECD average has caught up in absolute terms.&lt;/p&gt;

&lt;p&gt;If we’re not particularly interested in 2007 as a starting point, we can choose some other period. In the next chart I’ve set it to be 2014 which seems to me to be the time that OECD average growth sped up, and Australian growth slowed down. And I think this chart really helps to show that the poor relative performance of Australia on these indicators isn’t just the dramatic decline in the last couple of years, but a longer lasting phenomenon:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0275-inc-exp-gdp-2014.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0275-inc-exp-gdp-2014.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Interesting, huh? that chart drawn with this:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ref_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;2014-Q1&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ref_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;40&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_manual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;palette&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Index (2014 = 100)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Growth in household expenditure and income, Australia and OECD average&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Indexes set to 100 in first quarter of 2014. Comparison shows relative growth rates, not absolute difference.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Finally, I wanted to help the viewer make the comparison I was doing for myself in my head. I realised that one way to do this is to turn the two lines into a single line that is the ratio between them. That gives me this chart. I like this; I think the analytical step of dividing one of the indexed indicators by the other really is adding value here, if the intent is to compare growth in Australian indicators relative to the OECD’s. Now the turning point in around 2014 is much more obvious, and the shape of the line is robust to choice of index year, which is a very desirable property:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0275-aus-oecd-ratios.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0275-aus-oecd-ratios.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;That final chart drawn with this code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ref_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;2014-Q1&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time_period&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aus_oecd_ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ref_area&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Australia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ref_area&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;OECD average&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aus_oecd_ratio&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;full_measure&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;40&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Australia | OECD average relative growth in various measures&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Ratio set to equal 100 in first quarter of 2014. Indicator shows a relative comparison of growth rates, not an absolute comparison.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_caption&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Index (2014 = 100)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;So, that’s all for today. I’m not going to try to explain structural challenges in the Australian economy or look to possible causes of this relative decline since about 2014; that would be well out of scope for today’s blog. But I’m still interested in anyone who believes they have the answer!&lt;/p&gt;
</description>
				<pubDate>Fri, 30 Aug 2024 00:00:00 +1100</pubDate>
				<link>https://freerangestats.info/blog/2024/08/30/australian-economy</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/08/30/australian-economy</guid>
			</item>
		
			<item>
				<title>Polar-centred maps by @ellis2013nz</title>
					<description>&lt;p&gt;I stumbled across &lt;a href=&quot;https://brilliantmaps.com/no-settlement-further-north/?fbclid=IwY2xjawE2h7NleHRuA2FlbQIxMAABHdcWMCUYztUNNvniU4XJvlfREJo24ulRyp4qks8c_cWKOpwspzjYCXa4uQ_aem_hy058OxYQsPH_Ay_SX8t5Q&quot;&gt;this page by Brilliant Maps&lt;/a&gt; showing settlements with no larger settlement to their north. The author noted that Helsinki was omitted by error. I wanted to address that error, and as a resident of the southern half of the globe myself, to produce a similar map with relation to the south pole.&lt;/p&gt;

&lt;p&gt;First there’s a matter of sourcing data on cities’ and towns’ populations. There are several possible sources but I chose &lt;a href=&quot;https://public.opendatasoft.com/explore/dataset/geonames-all-cities-with-a-population-1000/table/?disjunctive.cou_name_en&amp;amp;sort=name&quot;&gt;this dataset on opendatasoft, of all cities with a population &amp;gt; 1000&lt;/a&gt;. This doesn’t go to the tiny settlements in the original Brilliant Maps image, but it’s good enough for me.&lt;/p&gt;

&lt;p&gt;Here’s my prep stage, downloading that data and getting a few things ready for later use in spatial transformations (coordinate reference systems that are centred on the norht and south poles) and plot polishing (abbreviating numbers, fonts, etc).&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-----------------prep----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extrafont&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnaturalearth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggforce&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for geom_circle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Function for summarising numbers&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;{round(x / 1e6, 1)}m&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;{round(x / 1e3, 0)}k&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;signif&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Font to use&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ff&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Roboto&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#-------------------download cities data-------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# obtained polar-centred coordinate reference systems (CRS) for later use, from &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# https://spatialreference.org/ref/epsg/?search=polar&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;south_crs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_crs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3031&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;north_crs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_crs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3411&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# 27MB download:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;file.exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cities.txt&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://public.opendatasoft.com/api/explore/v2.1/catalog/datasets/geonames-all-cities-with-a-population-1000/exports/csv?lang=en&amp;amp;timezone=UTC&amp;amp;use_labels=true&amp;amp;delimiter=%3B&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cities.txt&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cities&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cities&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_delim&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;cities.txt&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;delim&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cities&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;separate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Coordinates&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sep&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;,&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;into&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lat&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;long&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lat&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_squish&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lat&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;long&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;str_squish&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;long&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Name&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Country name EN`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Country Code`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Population&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lat&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;long&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Next is the job of identifying cities that are bigger than any others to their north. This was a bit of a fiddly problem which I addressed by sorting all the cities by distrance from the north pole (highest latitude first), making a new column which is the cumulative maximum population so far, and retaining only those cities that are equal to or higher than that maximum population. This gets me a list of just 20 cities.&lt;/p&gt;

&lt;p&gt;I then turn them into simple features objects with geometry (based on the latitude and longitude), and transform them to coordinates that will represent the world flattened out centred around the north pole, using the coordinate reference system I calculated before.&lt;/p&gt;

&lt;p&gt;Then, I source country boundaries from the invaluable Natural Earth repository of map data (accessed via the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;rnaturalearth&lt;/code&gt; R package), and give them the same transformation, and turn both data sets into a single map.&lt;/p&gt;

&lt;p&gt;Here’s the code doing those things:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-----------------most north-------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_north&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lat&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;max_pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;cummax&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Population&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;max_pop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_as_sf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coords&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;long&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lat&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remove&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_crs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WGS84&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_north&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_transform&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_north&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;north_crs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_north&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_north&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_coordinates&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_north&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# borders of countries north of Shanghai, from Natural Earth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;north_world&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ne_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label_y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;30&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_transform&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;north_crs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_north&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zero&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;radius&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;{Name}, {nf(Population)}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# draw the sea as background:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_circle&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;8.5e6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
              &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_sf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;north_world&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey90&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_circle&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zero&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zero&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;radius&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;radius&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_sf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_text_repel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# these limits are in the transformed coordinates, were chosen by hand / trial and error:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coord_sf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ylim&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-3000000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7000000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lims_method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;orthogonal&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xlim&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-6000000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6000000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme_void&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;base_family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_viridis_c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;direction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Settlements that have no larger settlement further north of them&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;none&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;And that results in this map:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0274-most-north.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0274-most-north.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;It’s a little different from the original. As expected, I am missing settlements with less than 1,000 people, and I have correctly inserted Helsinki. But there are other differences which come from ambiguities in the population size of cities. For example, if you google “Tokyo 37 million” you are prompted by Google with the frequently searched question “Is Tokyo’s population 14 or 37 million?”. It turns out the “Tokyo Metropolis” is 14m and “Tokyo metropolitan area” is 37 million.&lt;/p&gt;

&lt;p&gt;Obviously these are numbers that depend on decisions about boundaries, which as soon as you start thinking them through are revealed as being at least to a degree arbitrary. My data source opted on the 14m side for Tokyo. New York is also defined as a smaller area than in the Brilliant Maps original. I’m not particularly concerned with (or qualified to talk about) these urban definitional matters, so I’m just going to stick to the definitions used by my data source.&lt;/p&gt;

&lt;p&gt;My first effort at drawing these maps did the transformation on the fly using &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;coord_map(projection=&apos;orthographic&apos;)&lt;/code&gt; which worked ok, although it was harder (not impossible) to colour in the sea, and the constant-latitude circles I tried to draw for each city were crude and polygons. But when I tried to do the map centred on the south pole this method tripped up, and noting that &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;coord_map()&lt;/code&gt; is now deprecated I went the route of doing the transformation explicitly and in advance first. This gave a nicer result, particularly for the circles, and control over the bounding box of the map. So it was worth the half hour or so of cursing as I worked to understand the method. R’s power in geocomputation is growing so fast that anything on the web older than a couple of years is likely to be out of date - not necessarily wrong, but likely to miss a simpler and more powerful way of doing what you are trying.&lt;/p&gt;

&lt;p&gt;Anyway, the “explicit transformation done before drawing the map” method worked well for any arbitrary projection, and here is the south pole version:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#------------most south---------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_south&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lat&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;max_pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;cummax&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Population&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;max_pop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_as_sf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coords&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;long&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;lat&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;remove&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_crs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WGS84&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_south&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_transform&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_south&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;south_crs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_south&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_south&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_coordinates&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_south&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# borders of countries south of Shanghai&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;south_world&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ne_countries&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label_y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;35&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;st_transform&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;crs&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;south_crs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;m2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;most_south&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zero&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;radius&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sqrt&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;{Name}, {nf(Population)}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;rect&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xmin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xmax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_sf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;south_world&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey90&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_circle&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zero&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;zero&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;r&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;radius&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;radius&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_sf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_text_repel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;X&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# these limits are in the transformed coordinates, were chosen by hand / trial and error:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coord_sf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ylim&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-11000000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;11000000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lims_method&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;orthogonal&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xlim&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-7000000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;20000000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme_void&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;base_family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ff&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_viridis_c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;direction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Settlements that have no larger settlement further south of them&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;none&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;…which gets me this map. Nice.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0274-most-south.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0274-most-south.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;This map goes well over the equator and in some ways gives a distorted view of the world; the sphere has been badly flattened and warped to show Shanghai and the south pole on the same rectangle. But I’m not going to worry about it for now.&lt;/p&gt;

&lt;p&gt;OK that’s all folks.&lt;/p&gt;
</description>
				<pubDate>Sat, 24 Aug 2024 00:00:00 +1100</pubDate>
				<link>https://freerangestats.info/blog/2024/08/24/polar-maps</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/08/24/polar-maps</guid>
			</item>
		
			<item>
				<title>Perturbing a non-symmetrical probability distribution by @ellis2013nz</title>
					<description>&lt;p&gt;So I will fess up, I was one of the 34% of people who chose the first, incorrect answer on this quiz on Mastodon:
&lt;img src=&quot;https://freerangestats.info/img/0272-toot-1.png&quot; width=&quot;95%&quot; /&gt;&lt;/p&gt;

&lt;p&gt;&lt;a href=&quot;https://mastodon.social/@tslumley@fediscience.org/112974355500253935&quot;&gt;Original Toot, and explanatory responses, is available here&lt;/a&gt;.&lt;/p&gt;

&lt;p&gt;My reasoning - and my only excuse is I didn’t think it through much - was simplistic. I imagined a distribution that had the same mean and median, and in my head all the two or three examples of such were symmetrical, like a normal or a uniform distribution. And I  thought correctly that if you add some random noise to that, then the mean and median will stay the same.&lt;/p&gt;

&lt;p&gt;But even when Thomas gave the correct answer (“not necessarily”) and explained that if there is a bit of local skew in the original distribution around the mean / median, I struggled to intuit why this is the case.&lt;/p&gt;

&lt;blockquote&gt;
  &lt;p&gt;“Basically, to keep mean and median equal you need at least local symmetry around the mean. Otherwise the mean stays the same and the median moves”&lt;/p&gt;
&lt;/blockquote&gt;

&lt;p&gt;Eventually I decided this way of thinking about it:&lt;/p&gt;

&lt;ul&gt;
  &lt;li&gt;First, adding random noise with mean zero leaves the mean of the original distribution unchanged - this is basic probability theory.&lt;/li&gt;
  &lt;li&gt;When you add the noise, half the observations you are adding some value to were originally below the median and half are above it. On average the size of the noise is the same. But if your original distribution has some local skew, then observations on one side of the median will have a different chance of being perturbed by enough distance to “cross” the median (and hence change the median of the resulting distribution). So the median of the new distribution will be changed, in a direction that depends on the direction and level of the skew in the vicinity of the original mean/median.&lt;/li&gt;
&lt;/ul&gt;

&lt;p&gt;To help me understand this I simulated some data. I built on Thomas’ suggestion in the Mastodon thread that (with a continuous distribution) “the examples tend to look a bit more contrived, but nothing essential changes. Take a positively skewed distribution, add a small bump far off to the left to move the mean down to equal the median. X+E will have median higher than mean.”&lt;/p&gt;

&lt;p&gt;First, there’s the job of taking a skewed distribution and adding “a small bump on the left”. I chose to do this with a mixture distribution, 10 parts standard log-normal (ie &lt;em&gt;e&lt;/em&gt; to the power of a N(0, 1) normal distribution) and 1 part normal with parameters chosen to make the mean and median identical. Now, there would be a way to analytically calculate the right parameters, but it’s much easier (for me) to find them numerically, so I made a function to generate the mixture given a set of parameters and used the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;optim()&lt;/code&gt; function for Nelder-Mead general-purpose numerical optimisation:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# ------------------perturbing a skewed continues distribution&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Make a mixture distribution, 1 part normal and 10 parts standard exponential&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# normal, and return the difference between the median and mean&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mixture&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;round&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;params&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;abs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;median&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Find the set of parameters for the Normal part of a mixture&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# that results in median and mean being as close as possible:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;best&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;optim&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mixture&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;best&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;This gives us:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; best
$par
[1] -6.421733  6.757088

$value
[1] 3.841277e-09
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;We can now generate our data from that - in this case 100,000 observations, to be safe - and we see that the results are in fact very close for mean and medain&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# Generate data from that mixture&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;best&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;par&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;best&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;par&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;exp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Mean and median are very close&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;median&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; c(mean(x), median(x))
[1] 0.9121877 0.9124763
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;All very good. Now I just want to add some random noise - say a standard N(0, 1) normal distribution - to each observation…&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# perturb it a little&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;… and see what it does to the mean and median:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;&amp;gt; # now the median has shifted but mean has stayed the same:
&amp;gt; c(mean(y), median(y))
[1] 0.9128076 1.0989388
&amp;gt; # compared to original:
&amp;gt; c(mean(x), median(x))
[1] 0.9121877 0.9124763
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;OK, as predicted. Can a visualisation help? Here’s one showing the original distribution and, superimposed, the version that has been perturbed with white noise. In this version, the horizontal axis has been given a modulus transform (one of the &lt;a href=&quot;/blog/2015/09/07/transforming-breaks-in-a-scale&quot;&gt;very first things I blogged about&lt;/a&gt; - a great way to visualise data that feels like it needs a logarithm or Cox-Box transform, but inconveniently includes negative values). This transform is good helping see the “bump” on the left, present in both the original and perturbed distribution:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0272-both.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0272-both.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;However, the modulus transform makes it harder to understand the skew, so here’s the same chart with an untransformed x axis. This time we can see the original skew around the original mean and median, and perhaps this helps us understand what’s going on.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0272-both-untransformed.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0272-both-untransformed.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Seeing it like this, in the original scale, is useful for me at least, helping imagine adding random noise to the original distribution and seeing how doing that to the points just to the left of the original median pushes them over it (dragging the median up), whereas at least some of the points to the right of the median don’t get perturbed enough leftwards to push them over the median and drag it down.&lt;/p&gt;

&lt;p&gt;Here’s the code for drawing those plots:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;124&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Original skewed variable`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`With extra jitter`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample_n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#geom_rug() +&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_density&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;transform&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;modulus_trans&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-40&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-20&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;20&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;40&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_vline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_vline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;median&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;text&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Post-jitter median&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;text&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.86&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Original equal mean and median, also post-jitter mean&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Value (modulus transformed scale)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Density&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Adding jitter to a mixture of a skewed and symmetrical distributions&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;The mean stays the same with the jitter, but the median moves if the distribution wasn&apos;t symmetrical around the original mean.&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Based on an idea in a toot by Thomas Lumley&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# with transform:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;       

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# without transform:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Value (untransformed, axis truncated)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;That’s all folks. What out for those tricky distributions. Actually, although the simulated example above with exactly matching mean and median is obviously contrived, a distibution that is a mix of a skewed log-normal with a lump of something far to the left isn’t unusual in some economic areas - like firms’ profits or individuals’ incomes (mostly they are shaped log normal and strictly positive, but a small selection make some degree of a loss, and it’s definitely a mixture of two distributions not some easily-described single mathematical function).&lt;/p&gt;
</description>
				<pubDate>Tue, 20 Aug 2024 00:00:00 +1100</pubDate>
				<link>https://freerangestats.info/blog/2024/08/20/symmetry</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/08/20/symmetry</guid>
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			<item>
				<title>Snakes and ladders by @ellis2013nz</title>
					<description>&lt;p&gt;More than five years ago I started a &lt;a href=&quot;https://freerangestats.info/blog/2018/10/27/dice-games&quot;&gt;series of posts on games of chance&lt;/a&gt; and for one reason or another never got around to finishing them. Partially redressing that, here is a long-lost and now tidied up post on Snakes and Ladders. Read it before you next &lt;em&gt;gaze across the smoke filled casino table at your opponent, a sinister operative of SMERSH, and take a sip at your vodka martini while deciding whether to take his proferred wager or not with him on square 90 and you on 75…&lt;/em&gt;&lt;/p&gt;

&lt;p&gt;OK, maybe that scenario is unlikely. But one of my themes in these posts is that even games of pure chance will become skill if you gamble on them. Choices like whether to bet, what odds to offer or accept, and (in some games) whether to offer, accept or decline a doubling cube all turn even a pure chance game like &lt;a href=&quot;/blog/2018/12/23/persian-monarchs&quot;&gt;Persian Monarchs&lt;/a&gt; into one where the best player will win (in the long run).&lt;/p&gt;

&lt;p&gt;First, a reminder of how &lt;a href=&quot;https://en.wikipedia.org/wiki/Snakes_and_ladders&quot;&gt;Snakes and Ladders&lt;/a&gt; works. Here is a stylised version of a common Snakes and Ladders board, the Milton Bradley 1952 “Chutes and Ladders” (American children are frightened of snakes, apparently) depicted on the Wikipedia page. I’ve used arrows rather than snakes or chutes and ladders, and of course omitted all the extra imagery, which while interesting for a historian of morality and games, isn’t relevant to gameplay. I was actually in Sri Lanka back in 2018 when I started this Snakes and Ladders work, so had a particular interest in its origins, but I don’t have time to explain that now.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0130-s-and-l-board.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0130-s-and-l-board.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;You start at a virtual square 0, roll a six-sided dice and move your counter that number of squares. If you end in a position with a ladder, you move up to where it finishes. So if your first roll is a 1,you move to square 1 and immediately climb the ladder to square 38. If you end on a snake, you move down to the tail of the snake.&lt;/p&gt;

&lt;p&gt;Here’s the R code to draw that board:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;doParallel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# add in the snakes and ladders:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# see https://en.wikipedia.org/wiki/Snakes_and_Ladders &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# &quot;In the original game the squares of virtue are: Faith (12), Reliability (51), Generosity (57), &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Knowledge (76), and Asceticism (78). The squares of vice or evil are: Disobedience (41), &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Vanity (44), Vulgarity (49), Theft (52), Lying (58), Drunkenness (62), Debt (69), Murder (73), &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Rage (84), Greed (92), Pride (95), and Lust (99).[8]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Milton Bradley version, from image on Wikipedia.  This is identical&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# to the 10 snakes and 9 ladders listed in Althoen et al:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;snakes_and_ladders&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;starting&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;16&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;21&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;28&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;36&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;47&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;49&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;51&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;56&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;62&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;64&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;71&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;80&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;87&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;93&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;95&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;98&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ending&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;38&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;14&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;31&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;42&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;84&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;44&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;26&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;67&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;53&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;19&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;60&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;91&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;24&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;73&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;75&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;78&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ending&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;starting&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Snake&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Ladder&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# the board is a vector of numbers that basically show where you end up&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;99&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;95&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;snakes_and_ladders&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;starting&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;snakes_and_ladders&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ending&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#------------visualise board-------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrows&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_row&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrows&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrows&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrows&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrows&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_col&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ifelse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_row&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                            &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_row&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sl_df&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;snakes_and_ladders&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sequence&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;board_row&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;board_col&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;starting&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sequence&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%&amp;gt;%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sequence&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;board_row&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;board_col&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;ending&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sequence&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_row&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_tile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_curve&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sl_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_row.x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_col.x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                                 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_row.y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_col.y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;closed&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;inches&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;curvature&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_manual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Ladder&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Snake&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;orange&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme_void&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;base_family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Roboto&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;none&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot.title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;heading_font&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coord_equal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggtitle&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Snakes and ladders board&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Now, the Wikipedia page claims that Snakes and Ladders can be represented exactly as an absorbing Markov chain, because the transition probability from each square to any other square is fixed and easy to define, and not dependent on the path the counter took to that square. However, this model is only a useful one if you omit the common rule that rolling a six gets you another roll; and rolling three sixes sends you “back to square one” (in fact, Snakes and Ladders is the origin of this common English phrase).&lt;/p&gt;

&lt;p&gt;I want a realistic simulation of the game as actually played, so after mucking around with Markov chains for a while realised that it was going to be much simpler to write code that mimics how the players go about it. That’s what I used to simulate 10,000 solo games to create this animation of a hundred Snakes and Ladders games:&lt;/p&gt;

&lt;p&gt;&lt;img src=&quot;https://freerangestats.info/img/0130-snakes-and-ladders.gif&quot; width=&quot;90%&quot; /&gt;&lt;/p&gt;

&lt;p&gt;The code for which is below. All the heavy work is done by the &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;sl_game()&lt;/code&gt; function, which takes as an argument the current position of the player (defaults to zero) - we’ll be using this more later.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#---play the game---------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sl_game&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NULL&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;while&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dice&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dice&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dice&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dice&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dice&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sample&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dice&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;else&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dice&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;position&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rolls&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sl_game&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# should be NULL as it&apos;s not possible to finish a turn on 1 so can&apos;t start there either:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sl_game&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sl_game&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# simulate 10000 games starting from square zero&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;set.seed&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sims_0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sl_game&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# make an animation out of 100 of them&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dir.create&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;0130-tmp&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_sim&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data_frame&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;square&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sims_0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;positions&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;square&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sequence&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_row&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_tile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_curve&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sl_df&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_row.x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_col.x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                                 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_row.y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xend&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board_col.y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                                 &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;type&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;closed&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unit&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;inches&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
               &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;curvature&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sequence&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_path&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_sim&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_manual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Ladder&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Snake&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;orange&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme_void&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;base_family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Roboto&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;none&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
          &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot.title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;heading_font&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coord_equal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggtitle&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Simulated snakes and ladders games&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;png&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;paste0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;0130-tmp/&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;.png&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;200&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dev.off&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# knit the images together into an animation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pd&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;setwd&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;0130-tmp&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;system&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s1&quot;&gt;&apos;magick -loop 0 -delay 40 *.png &quot;0130-snakes-and-ladders.gif&quot;&apos;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;setwd&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pd&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# a bit of analysis of the number of turns it takes to win:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_turns&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sims_0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rolls&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sapply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sims_0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rolls&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_turns&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;median&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;number_turns&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_density&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;blue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of moves before winning&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Distribution of length of one player snakes and ladders games&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;6 gets you another role, three 6s is back to square 1.
&apos;Bounce back&apos; rule applies when trying to finish exactly on 100&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Here’s the output from that bit of analysis of the number of turns and rolls (remembering you can get more rolls than turns, if you get sixes)&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;  variable `mean(value)` `median(value)`
  &amp;lt;chr&amp;gt;            &amp;lt;dbl&amp;gt;           &amp;lt;dbl&amp;gt;
1 rolls             42.7              34
2 turns             35.0              28
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;It’s a bit more complicated than the mean 39.2 dice throws that can be analytically calculated as the average number of rolls needed under the simplified version modelled with a Markov chain. The expected number of rolls is higher because of ‘back to square one’; the number of turns is fewer because of the bonus rolls you get when you have a six.&lt;/p&gt;

&lt;p&gt;The distribution of the number of rolls required is very much squewed with a long tail - it is in fact possible for a game to go on forever (although extremely improbable):&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0130-lengths.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0130-lengths.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;OK, so where does the gambling come in? Gambling becomes interesting when there is a choice about the &lt;em&gt;timing&lt;/em&gt; or &lt;em&gt;odds&lt;/em&gt; of a wager. If all gambling was an even odds stake put down before the game, then Snakes and Ladders would stay a game of pure chance. But if it is possible, say at the beginning of your turn, to size up the board and say “I’ll bet you on even odds that I’ll win”, you’re making a choice based on your knowledge of the game. A naive observer might think that anyone who is at a more advanced square is more likely to win, and therefore could be trapped into taking a bet that would only be fair if there were favourable odds given to them.&lt;/p&gt;

&lt;p&gt;To explore this with Snakes and Ladders, I simulated ten thousand solo games with the counter starting from each square that is a valid starting position for a turn (e.g. excluding squares that are at the bottom of a ladder - you can’t end a turn on that square because if you land there you would go straight up the ladder). This gives us a probability distribution for how many turns it is expected to take to win from that point. If we do a full join of this distribution to itself, we will get a joint probability for how many turns it will take player 1 to win from that position and every combination of player 2’s starting points.&lt;/p&gt;

&lt;p&gt;We can visualise the result in a chart like this one. The highlight boxes are where Player 1, who is about to roll, has a surprisingly good chance of winning (higher than 0.55), despite being behind Player 2 in the race. These are the probable opportunities to offer a 50:50 bet to your opponent; only an unusually disciplined or knowledgeable player would think they were losing in this position.&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0130-unusual-winning-chances.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0130-unusual-winning-chances.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;
&lt;p&gt;.&lt;/p&gt;

&lt;p&gt;Here are some of those positions:&lt;/p&gt;

&lt;div class=&quot;language-plaintext highlighter-rouge&quot;&gt;&lt;div class=&quot;highlight&quot;&gt;&lt;pre class=&quot;highlight&quot;&gt;&lt;code&gt;   start_p1 start_p2    p1    p2 unusual how_surprising
      &amp;lt;int&amp;gt;    &amp;lt;int&amp;gt; &amp;lt;dbl&amp;gt; &amp;lt;dbl&amp;gt; &amp;lt;lgl&amp;gt;            &amp;lt;dbl&amp;gt;
 1       65       81 0.611 0.389 TRUE             0.761
 2       66       81 0.607 0.393 TRUE             0.746
 3       65       82 0.590 0.410 TRUE             0.744
 4       67       81 0.615 0.385 TRUE             0.743
 5       68       81 0.622 0.378 TRUE             0.740
 6       69       81 0.626 0.374 TRUE             0.735
 7       63       81 0.570 0.430 TRUE             0.733
 8       22       29 0.555 0.445 TRUE             0.731
 9       74       81 0.667 0.333 TRUE             0.730
10       66       82 0.587 0.413 TRUE             0.729
&lt;/code&gt;&lt;/pre&gt;&lt;/div&gt;&lt;/div&gt;

&lt;p&gt;The &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;how_surprising&lt;/code&gt; column is a metric I made up that tries to incorporate both the high probability of Player 1 winning and how far they currently &lt;em&gt;seem&lt;/em&gt; to be behind. So if you are Player 1 on square 65, and see your opponent on 81, now is the time to offer them a bet.&lt;/p&gt;

&lt;p&gt;Intuitively, why are 81 and 82 bad squares? It’s because you’ve missed the ladders from 80 straight to 100 (instant win) and from 71 to 90. You still have snakes at 93, 95 and 98 that might trip you up whereas your opponent might skip them altogether if they land on square 80 - which is a non-trivial 1/6 chance for them.&lt;/p&gt;

&lt;p&gt;Similarly, square 29 is a bad one because you’ve just missed the big ladder starting at square 28, whereas your apparently-behind opponent still has a chance to get on it.&lt;/p&gt;

&lt;p&gt;Here’s the code for running those simulations and drawing the chart:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-----------------chance of winning from different positions------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# set up parallel processing cluster&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;makeCluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# only any good if you have at least 7 processors :)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;registerDoParallel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clusterEvalQ&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;})&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# how many times to play the game from each starting position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e4&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# export onto each cluster some objects we need to use:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clusterExport&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cluster&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;board&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;sl_game&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;reps&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# define the starting positions, which should only be valid ones:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;starts&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;99&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;starts&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;starts&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;starts&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;board&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# simulate all the games from the different starting positions&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# takes a while to run even with the parallel processing&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;foreach&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;starts&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;.combine&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%dopar%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;these_results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_game&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sl_game&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_game&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;these_results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;these_results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;res&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;these_results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;dim&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# average time to win from each starting position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;avg_turns&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# distribution of results - count of number of turns it takes to win&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# from each starting position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distrib&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;results&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;count&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;n&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;link&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;link&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# for each combination of starting positions and number of turns to win,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# who wins with one player at one position and another player at another&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# (caution this makes a very big object because of the full join)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;who_wins&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distrib&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;distrib&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;link&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;suffix&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_p1&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;_p2&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;relationship&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;many-to-many&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns_p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns_p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prob&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns_p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;turns_p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unusual&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.55&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Visualise the chance of winning from various positions&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;who_wins&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_tile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_tile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;who_wins&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unusual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_fill_gradientn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colours&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;blue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Player one current square&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Player 2 current square&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Probability of player one winning&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Rules: 6 gets you another roll. Three 6s is back to square 1. &apos;Bounce back&apos; rule applies if you don&apos;t land
exactly on square 100. Player 1 has the dice.
Black squares indicate situations where it is worth betting on Player 1 even though they are behind&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Chance of winning a standard snakes and ladders game at different positions&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# some examples where it would be worth putting money on player 1 even&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# though they are behind&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;who_wins&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unusual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;how_surprising&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;desc&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;how_surprising&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Finally, because of my backgammon interests, I wondered about the use of a doubling cube. In backgammon, at the beginning of your turn (before throwing the dice), you have the option of offering your opponent the doubling cube. They can accept, in which case the game is now being played for twice as many points / stakes; or refuse in which case they lose the game instantly at the current stake. Once they have accepted the cube, only they can offer it again (presumably if the fortunes change their way).&lt;/p&gt;

&lt;p&gt;Generally speaking - putting aside some backgammon-specific complications - it makes sense to accept the doubling cube if you have a 1 in 4 or better chance of winning. If you have a 0.75 probability of winning and have the cube available, you should definitely offer the double; and your opponent should refuse. Now, I’ve never heard of Snakes and Ladders with a doubling cube but actually I am sure it has happened (or will happen in the future). So it’s worth highlighting what are the points at which Player 2 should decline the cube if offered and accept the loss at the current stake? This chart answers this for us (although probably a table would be more useful for actual use):&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0130-doubling-cube.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0130-doubling-cube.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Produced with this little snipped of extra code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;who_wins&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_tile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_tile&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;who_wins&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.25&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;black&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_fill_gradientn&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colours&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;red&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;blue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Player one current square&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Player 2 current square&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Probability of player one winning&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Rules: 6 gets you another roll. Three 6s is back to square 1. &apos;Bounce back&apos; rule applies if you don&apos;t land
exactly on square 100. Player 1 has the dice.
Black squares indicate situations where Player 2 should decline an offered doubling cube&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Chance of winning a standard snakes and ladders game at different positions&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;OK folks, that’s all. Remember to gamble responsibly. In particular, if you use this post to rip off any kids, their sadness will be on your conscience, and I absolutely disclaim all responsibility for any misuse of any kind of any of the above material.&lt;/p&gt;
</description>
				<pubDate>Sat, 10 Aug 2024 00:00:00 +1100</pubDate>
				<link>https://freerangestats.info/blog/2024/08/10/snakes-and-ladders</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/08/10/snakes-and-ladders</guid>
			</item>
		
			<item>
				<title>Population age changes in the Pacific by @ellis2013nz</title>
					<description>&lt;p&gt;The UN has released the &lt;a href=&quot;https://population.un.org/wpp/&quot;&gt;2024 Revision of World Population Prospects&lt;/a&gt; and I wanted to see the latest estimates of how Pacific Island Country and Territories (PICTs) populations will age in coming years.&lt;/p&gt;

&lt;h2 id=&quot;aging-populations&quot;&gt;Aging populations&lt;/h2&gt;

&lt;p&gt;After quite a bit of experimentation and iteration through different variants, I came up with this chart of the proportion of elderly people in the different PICTs:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0271-aging-linechart.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0271-aging-linechart.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;The choice of “percentage of population that is aged 65 and over” seemed straightforward enough giving my aims to understand population aging. The harder questions were about what to map colour to. I tried an individual colour for each country - this was probably the easiest to read, and maybe what I’d use if the chart was for a more general readership. I also tried mapping colour to subregion (Melanesia, Polynesia and Micronesia), to population growth rates, and population sizes, before settling on GDP per capita in 2023 as a genuinely interesting additional variable that highlights some relationships between aging and income.&lt;/p&gt;

&lt;p&gt;What I like about the chart I eventually chose is how it clearly shows the three clusters of countries, in terms of projected aging to 2050; and the top two (most aging) clusters of four countries each are clearly richer (in terms of GDP per capita in 2023) than the larger, bottom (least aging) cluster of poorer countries. Tiny Niue and Nauru being exceptions as countries with less predicted aging despite relatively high GDP; but Niue is a particularly difficult country to model because of its tiny size and ease of movement to New Zealand, and Nauru’s economic status is complex for all sorts of interesting historical reasons.&lt;/p&gt;

&lt;p&gt;I did feel the sub-regional aspect was an important one too, despite not choosing it in the end for colour. So I made a faceted version of the same chart to show that, which I think works quite nicely too:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0271-aging-linechart-facet.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0271-aging-linechart-facet.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;All the code for these charts is in one chunk at the bottom of the blog. It’s all straightforward stuff, importing data from two data sources and then some moderately finicky plot polishing. The power of the grammar of graphics really shone through in making it easy to iterate through different versions of these before coming to the final production charts; this isn’t so obvious in just reading the code except for when we see the simple one-liner conversion of the main chart into the faceted one.&lt;/p&gt;

&lt;h2 id=&quot;absolute-size-of-population-by-age&quot;&gt;Absolute size of population by age&lt;/h2&gt;

&lt;p&gt;I wanted a sense of the absolute magnitudes too. Proportions of populations by PICT are all very well, but how many millions of people are we talking about in different age groups? It turns out this is an important thing to visualise because of how easy it is to forget that, in numbers of people terms, the Pacific is dominated by Melanesia, and Melanesia in turn is mostly Papua New Guinea. Again after a bit of iteration and experimentation I came up with this chart to show that:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0271-aging-areachart.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0271-aging-areachart.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;h2 id=&quot;r-code&quot;&gt;R code&lt;/h2&gt;

&lt;p&gt;Well that’s all for today very straightforward stuff but some nice charts. Big thanks to the UN for making the detailed data in the World Population Projections so nicely accessible. Here’s the code to import that data, plus GDP per capita from the Pacific Data Hub at my own work, and turn it into those charts.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readxl&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;janitor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggrepel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rsdmx&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extrafont&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#-------------------Download and prep data-----------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# GDP per capita of Pacific Island countries and terrritories--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# From the Pacific Data Hub&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;readSDMX&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://stats-sdmx-disseminate.pacificdata.org/rest/data/SPC,DF_POCKET,3.0/A..GDPCPCUSD?startPeriod=2023&amp;amp;endPeriod=2023&amp;amp;dimensionAtObservation=AllDimensions&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as_tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clean_names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geo_pict&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_per_capita_2023&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;obs_value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# UN World Population Projections 2024&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# a) population by 5-year age group and sex:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;options&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;timeout&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;600&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;pp24_agegrp.csv&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;file.exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://population.un.org/wpp/Download/Files/1_Indicator%20(Standard)/CSV_FILES/WPP2024_Population1JanuaryByAge5GroupSex_Medium.csv.gz&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mode&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;wb&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# b) standard basic indicators (for some reason the CSV version is missing):&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;pp24_standard.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;file.exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unlink&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://population.un.org/wpp/Download/Files/1_Indicator%20(Standard)/EXCEL_FILES/1_General/WPP2024_GEN_F01_DEMOGRAPHIC_INDICATORS_COMPACT.xlsx&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mode&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;wb&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;standard&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_excel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;skip&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;16&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rename&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Total Population, as of 1 July (thousands)`&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;agegrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;df1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;clean_names&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;picts&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Fiji&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;New Caledonia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Papua New Guinea&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Solomon Islands&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Vanuatu&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Guam&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Kiribati&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Marshall Islands&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Micronesia (Fed. States of)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Nauru&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Northern Mariana Islands&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Palau&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;American Samoa&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Cook Islands&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;French Polynesia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Niue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Samoa&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Tokelau&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Tonga&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Tuvalu&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Wallis and Futuna Islands&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subregion&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Melanesia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Micronesia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Polynesia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;times&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;7&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;9&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check we have all 21 SPC members (other than Pitcairn)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;picts&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;21&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;         
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check we have the same spelling of names as in the UN World Population Projections:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;all&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;picts&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;agegrp&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Some age categories for use later&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;old_grps&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;65-69&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;70-74&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;75-79&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;80-84&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;85-89&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;90-94&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;95-99&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;100+&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yng_grps&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;0-4&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;5-9&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;10-14&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;all&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;old_grps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yng_grps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;agegrp&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age_grp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; Compound annual growth rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param x1 starting observation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param x2 finishing observation&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param years number of years between x1 and x2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @examples&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; cagr(100, 110, 5)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cagr&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;^&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# make a summary of the PICTs values for use later&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pict_sum&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;agegrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;inner_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;picts&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;location&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;iso2_code&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;geo_pict&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;old&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age_grp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;old_grps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;young&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age_grp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yng_grps&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subregion&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_per_capita_2023&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_total&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_old&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_total&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;old&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_young&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_total&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;young&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
            &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_old&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_old&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop2024&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2024&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yoy_growth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dplyr&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;::&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lag&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;coming_growth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;cagr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop2024&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2050&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;26&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ungroup&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop2024_rank&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.factor&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop2024&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1950&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2050&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# filter(time %in% seq(from = 1950, to = 2050, by = 5))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#----------------------compare aging to GDP-----------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;myfont&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Roboto&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pict_sum&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_old&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gdp_per_capita_2023&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;rect&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xmin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2011&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xmax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2023.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey90&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;rect&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xmin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2023.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xmax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2050.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey70&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_point&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pict_sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1950&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2024&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2050&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
             &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_text_repel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pict_sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2050&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;direction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;y&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;hjust&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nudge_x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seed&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;123&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;min.segment.length&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;family&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;myfont&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2.9&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;force&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.001&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;force_pull&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_colour_viridis_c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;option&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;D&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;trans&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;log10_trans&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dollar_format&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;largest_with_cents&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;percent&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;seq&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1950&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2050&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;25&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;limits&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1950&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2070&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_size_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma_format&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;suffix&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;m&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;20&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;max_size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;12&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.62&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_text&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;10&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Proportion of population that is 65 or older&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;GDP per capita in 2023 (USD)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;size&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot; Population&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Aging populations in the Pacific&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Shaded areas indicate increasingly dependent on projections&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: UN World Population Prospects 2024 (population); SPC Pocket Summary (GDP)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;                  
                  
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# faceted version                  &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subregion&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.position&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.8&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
        &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;legend.direction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;horizontal&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#------------------population by age group in absolute numbers-----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# colour palette for the sub-regions&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sr_pal&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;palegreen&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;brewer_pal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;palette&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Accent&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)[&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p3&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pict_sum&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Papua New Guinea&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subregion&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Melanesia&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Melanesia other than PNG&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subregion&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_working_age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_old&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_young&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subregion&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_old&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_young&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_working_age&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subregion&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;pop_old&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;65 and older&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;pop_young&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Younger than 15&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;pop_working_age&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;15-64&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_relevel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Papua New Guinea&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                                              &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Melanesia other than PNG&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_relevel&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Younger than 15&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;group_by&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;summarise&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;location2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;rect&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xmin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2011&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xmax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2023.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey90&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;annotate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;rect&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xmin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2023.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xmax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2050.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymin&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ymax&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;Inf&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;grey70&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;alpha&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_area&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_fill_manual&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sr_pal&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma_format&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;suffix&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;m&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;People (millions)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Rapidly growing working age population in Melanesia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Shaded areas indicate increasingly dependent on projections&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: UN World Population Prospects 2024&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;print&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p3&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

</description>
				<pubDate>Sat, 27 Jul 2024 00:00:00 +1100</pubDate>
				<link>https://freerangestats.info/blog/2024/07/27/aging-pacific</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/07/27/aging-pacific</guid>
			</item>
		
			<item>
				<title>Reproducing and adapting the UN Population Projections by @ellis2013nz</title>
					<description>&lt;p&gt;I set out to see if it was possible to reproduce the UN’s &lt;a href=&quot;https://population.un.org/wpp/&quot;&gt;2022 Revision of World Population Prospects&lt;/a&gt; for a given country by cohort component projection from the fertility, mortality and immigration rates and population starting point published as part of their projection. The motivation is to make small changes to some of those parameters - for example by substituting in a recent census result for the population and a given year - and “re-run” the projections to see the impact of changes, or to get a more up-to-date version with data that wasn’t available to the UN at the time of their projection.&lt;/p&gt;

&lt;p&gt;It turns out this wasn’t too hard (one morning’s work for the modelling, then a few hours of write-up), particularly in cases where migration is small in the projection period. I was able to reproduce almost exactly population, birth and death totals to 2100 for Vanuatu, and demonstrate the impact of updating the 2020 year for their recent census population totals and fertility rates, and getting a slightly lower projection as a result.&lt;/p&gt;

&lt;p&gt;Here’s my reproduction of Vanuatu’s population projection from 2020 to 2100, using just the 2020 population totals and the forecast fertility, mortality and migration rates. As you can see it’s basically identical to the UN totals:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-vanuatu-pop.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-vanuatu-pop.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;And here’s the same method tweaked for the actual 2020 census total and with a rough adjustment made to fertility rates based on what was observed at the 2020 census:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-vanuatu-revised.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-vanuatu-revised.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;Of course, this method delivers a full set of projections by age and sex, and we could construct life tables or any indicators we want from it. Here are population pyramids comparing the published UN projections for 2050 with my revised set. Not visually stunning in its comparison, but enough to prove that it’s possible:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-vanuatu-pyramid.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-vanuatu-pyramid.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;h2 id=&quot;reproducing-un-projections&quot;&gt;Reproducing UN projections&lt;/h2&gt;

&lt;p&gt;So here’s how I went about that.&lt;/p&gt;

&lt;p&gt;First, downloading all the data. The UN recommend bulk downloads of their CSV files. For my purposes I first need the fertility, mortality and population by sex and one year age groups. Of the population original data I am only going to use the 2020 year, and then project it forward myself based on fertility, mortality and migration; but I want the full set for comparison purposes. For migration, I couldn’t see in my hasty look at the UN site a dataset of migration projections by age and sex, so I just use the much simpler net migration rate (per thousand people) per year in the projection period. Here’s code to download all this UN data:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tidyverse&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scales&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;library&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;patchwork&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dir.create&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;showWarnings&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;FALSE&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#------------------download and import data for all countries from existing projections----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;list.files&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;files&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_Fertility_by_Age1.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_DeathsBySingleAgeSex_Medium_1950-2021.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_DeathsBySingleAgeSex_Medium_2022-2100.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_Population1JanuaryBySingleAgeSex_Medium_1950-2021.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_Population1JanuaryBySingleAgeSex_Medium_2022-2100.zip&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;WPP2022_Demographic_Indicators_Medium.zip&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# let downloads take up to 10 minutes rather than 1 minute max, as files are&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# large (largest is fertility for single age, 78MB)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;options&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;timeout&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;600&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;file.exists&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_Demographic_Indicators_Medium.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# download the zip files:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;files&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;download.file&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;https://population.un.org/wpp/Download/Files/1_Indicators%20(Standard)/CSV_FILES/{files[i]}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                   &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;destfile&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/{files[i]}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mode&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;wb&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# unzip them&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;files&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unzip&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;glue&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/{files[i]}&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;exdir&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fert_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;    &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_Fertility_by_Age1.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_past&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_DeathsBySingleAgeSex_Medium_1950-2021.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_future&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_DeathsBySingleAgeSex_Medium_2022-2100.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_past&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;    &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_Population1JanuaryBySingleAgeSex_Medium_1950-2021.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_future&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_Population1JanuaryBySingleAgeSex_Medium_2022-2100.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;read_csv&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;data-pop-proj-2022/WPP2022_Demographic_Indicators_Medium.csv&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_past&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_future&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_past&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_future&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# clean up&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_past&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_future&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_past&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_future&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Next, I wrote my own cohort component population projection function. I wanted to do this from scratch rather than using an existing demography package to make sure I understood what was happening (I’m not a demographer) and could make tweaks as necessary to match the UN approach. I used &lt;a href=&quot;https://farid-flici.github.io/tuto.html&quot;&gt;this tutorial&lt;/a&gt; by Farid Flici as my starting point; abstracted his code into a function for easy use with multiple countries, and added a net migration component.&lt;/p&gt;

&lt;p&gt;Because migrants’ ages tend to be dissimilar from the country they are migrating too - they are more likely to be in the prime of their working / family life I believe - I needed a way to set the ages of migrants. In the function below I defaulted to a normal distribution of ages, mean 28 and standard deviation 11, which worked well to get similar results to the UN in a few countries I tried. This was the hardest and most discretionary part of the exercise.&lt;/p&gt;

&lt;p&gt;My observation is that demographers seem to think in terms of matrices of numbers rather than database-oriented tidy data, and I have kept that matrix approach in this function.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#----------------------projections for one country-------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# adapting the implementation of the method for cohort component projection at &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# https://farid-flici.github.io/tuto.html&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param start_pop_m is a vector of population of males at single year age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   periods beginning of year starting age 0&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param start_pop_f as start_pop_m but for females&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param start_year first year ie the year for which the actual population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   number refers to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param end_year end year for the population projection&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param fertility a matrix with rows for female agegroups 15 to 49 and a&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   column for each year, values are births per woman (not per thousand women)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   that year and age. Must have 35 rows and number of columns equal to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   end_year minus start_year plus one.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param mort_m a matrix with rows for age 0 to at least 50 and columns for&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   each year, for males. Must have number of columns equal to end_year minus&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   start_year plus one.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param mort_f as mort_m but for females. Must have same number of rows and&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   columns as mort_m.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param sex_ratio_birth number of boys born for every girl born, vector with&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   length of years to be projected.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param net_migration vector of proportions of the population that migrate to&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   the country, net of those who leave. Defaults to a vector of zeroes.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param female_share_migration vector of proportions of net migrants that are&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   female. Defaults to a vector of 0.5s.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @param migration_age_weights vector of relative distribution of the age of&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   net migrants. Must be 121 values representing relative proportion of net&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   migrants aged zero to 120. Defaults to a normal distribution of mean 28 and&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   standard deviation 11. Note that this distribution currently must be the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   same through the projection period (unlike eg fertility and mortality,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   which have individual values for each age group and sex for each year).&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   Only the total net migration proportion can change over time, not its age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   make-up.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos; @author Peter Ellis, expanding on a less-featured example by Farid Flici&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;cd&quot;&gt;#&apos;   https://farid-flici.github.io/tuto.html&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1.045&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;net_migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0.5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;dnorm&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mean&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;28&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sd&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;11&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                     &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;35&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;net_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# fill in with high probability of death for ages beyond where we have mortality data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extra_deaths_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extra_deaths_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extra_deaths_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rbind&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;extra_deaths_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Create matrices for male and female population for the full projections&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# each row of the matrix is an age group, each column is a year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deaths&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rownames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;rownames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;                &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# ages&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# first year gets populated with the actual population numbers that we have:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# subsequent years get modified by people aging one year, deaths, births, and net migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;for&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;in&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncols&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;{&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Age one and above:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deaths&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# migration. &lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_migrants&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;net_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_migrants&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;total_migrants&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;female_share_migration&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migration_age_weights&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migm&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;migf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Age zero ie births. Fertility rate by the number of women in the middle of the previous year.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Note that PopF rows 16:50 equates to women age 15:49&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reproducing_women&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;16&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;16&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;50&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_boys&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;          
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_girls&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_boys&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reproducing_women&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%*%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_boys&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;reproducing_women&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%*%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;i&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;prop_girls&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Return a list of the two matrices&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deaths&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deaths&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Next, I wrote a function to extract the necessary fertility, mortality, migration rates and 2020 starting population from the UN data, feed it into my &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;pop_proj()&lt;/code&gt; function and return the result. Unlike &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;pop_proj()&lt;/code&gt;, which is to some degree fully portable, this function is very much specific to this particular project and is really just a convenience function for grabbing the data and turning it into the right units and shapes (vectors and matrices) needed for &lt;code class=&quot;language-plaintext highlighter-rouge&quot;&gt;pop_proj()&lt;/code&gt;.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;repeat_un_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;if&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;!&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;unique&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stop&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Country not found&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_fert&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fert_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Age-Specific Fertility rate&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ASFR&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# turn into proportions, not rates per 1000:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
           &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ASFR&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ASFR&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pivot_wider&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id_cols&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;names_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ASFR&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# should be 35 rows ie fertilities for ages 15 to 49&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;nrow&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_fert&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;35&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# should be 81 columns, 1 column for each year from 2020 to 2100&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_fert&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;length&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Mortality is in numbers not a ratio so we need to join to the population data to turn it into a ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_years&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopMale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopFemale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;left_join&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;by&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Time&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;AgeGrp&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# next step important because we will be sorting by AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;100+&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;suppressWarnings&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
    
    
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# sometimes more deaths than people (eg 1 death, 0 people) so cap the death ratio at 1&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathMale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmin&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathMale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopMale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathMale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pivot_wider&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id_cols&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;names_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathMale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathFemale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pmin&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathFemale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopFemale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathFemale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pivot_wider&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;id_cols&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;names_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;values_from&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;DeathFemale&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;as.matrix&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# check the years are correct, didn&apos;t get mangled or reordered&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;all&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;all&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colnames&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_all&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# next step important because we will be sorting by AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;case_when&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;100+&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
      &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;TRUE&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;suppressWarnings&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;AgeGrp&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# convert to units, not thousands of people:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopMale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopFemale&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# reality check&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Population in millions; should be about 0.3 if the_country is Vanuatu, about 1400 if India:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1e6&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# net migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_cnmr&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;CNMR&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# sex ratio at birth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_srb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;arrange&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pull&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;SRB&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_proj&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;min&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;max&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_years&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_fert&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;net_migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_cnmr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_srb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    
  &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;list&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_proj&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_fert&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_fert&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_mort_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_mort_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_cnmr&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_cnmr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_srb&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;this_srb&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;Note that this function returns, in addition to the results of the population projection, the various inputs in their correct units and shape. This will be useful later when we want to modify some of those inputs.&lt;/p&gt;

&lt;p&gt;Now that we’ve got these functions, using them to do projections from 2020 and compare those projections to the published numbers is pretty straight forward. Here’s the code to do that for Vanuatu, which produces the first chart at the top of this blog post:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Vanuatu&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;repeat_un_proj&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# total population&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`UN original`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;TPopulation1Jan&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Reproduction`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;apply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;apply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# First year should be an exact match:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;stopifnot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`UN original`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Reproduction&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;label&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comma&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Attempt to re-create the UN population projections from population in 2020, fertility and mortality rates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Population&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-vanuatu-pop.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-vanuatu-pop.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;As we can see the results are pretty much identical. Let’s look at my projected births and deaths based on fertility and mortality rates, and compare them to the published projected numbers&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;# births&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`UN original`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Births&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Reproduction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Attempt to re-create the UN population projections from population in 2020, fertility and mortality rates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Births (thousands)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# deaths&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`UN original`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Deaths&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Reproduction&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;my_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;deaths&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Attempt to re-create the UN population projections from population in 2020, fertility and mortality rates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Deaths (thousands)&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-vanuatu-births.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-vanuatu-births.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-vanuatu-deaths.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-vanuatu-deaths.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;I’m pretty happy with those. There’s definitely some discrepancies and a lag in the births which I suspect come down to how one treats populations of mothers - numbers on 1 January v 1 July, that sort of thing. But in the scheme of things these are very small.&lt;/p&gt;

&lt;p&gt;Now, Vanuatu is relatively easy because the UN assumed net zero migration in the projection period. We can see this by comparing the net migration rates in their Indicators dataset for a few countries, with this code:&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;n&quot;&gt;plot_mig&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;k&quot;&gt;function&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;){&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;CNMR&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_vline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2022&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_hline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;yintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;lty&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;steelblue&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
    &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;the_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Net migration rate per thousand people&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;caption&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Source: UN World Population Prospects 2022&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;return&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;}&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_mig&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Vanuatu&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_mig&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Fiji&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_mig&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Australia&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_mig&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;India&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;plot_layout&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ncol&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-migration.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-migration.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;For countries that have good data on it, migration is a big deal in the projections; but forecasting is hard, particularly of the future.&lt;/p&gt;

&lt;p&gt;My first few goes at reproducing the projections for Australia and India tended to be badly out because I had added in net migration evenly across the whole age distribution. My eventual solution, making net migration bell curved with an average age of 28, is a bit of a hack with the parameters chosen to make Australia’s projections come out right. Definitely a better method would be to have actual age-specific net migration forecasts. Whether such things are possible will very much depend on the country; it’s probably possible for Australia, but not for most of the countries I work with.&lt;/p&gt;

&lt;p&gt;Here’s the final result comparing UN projections with mine for a few interesting countries:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-Australia-pop.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-Australia-pop.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-China-pop.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-China-pop.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;
&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-India-pop.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-India-pop.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;h2 id=&quot;adjusting-the-starting-point&quot;&gt;Adjusting the starting point&lt;/h2&gt;

&lt;p&gt;Now, the whole point of this exercise was to see if we can plausibly adjust the starting point - say the population totals in 2020, or the forecast fertility rates - and say we are building on the UN’s projections to get our own.  Here’s my rough demo of how we might do that, again using the case of Vanuatu. Vanuatu’s 2020 census wasn’t available at the time of the UN’s 2022 population projections, so the actual population and fertility numbers for 2020 differ somewhat (of course) from what was projected.&lt;/p&gt;

&lt;p&gt;For the below, I am relying on &lt;a href=&quot;https://sdd.spc.int/digital_library/vanuatu-2020-national-population-and-housing-census-analytical-report-volume-2&quot;&gt;the analytical report of the Vanuatu census&lt;/a&gt;. On a quick glance I didn’t see a table of the actual total population by single year age and sex, so I just adjusted the UN’s projected totals by a factor to make them add up to the correct 2020 total males and females. Of course if we were doing this for real we’d get the real numbers straight from the census; I actually can do this easily enough at work but obviously for this blog wanted to use only easily accessible public data.&lt;/p&gt;

&lt;figure class=&quot;highlight&quot;&gt;&lt;pre&gt;&lt;code class=&quot;language-r&quot; data-lang=&quot;r&quot;&gt;&lt;span class=&quot;c1&quot;&gt;#-----------------changing one or two factors while keeping the rest the same--------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Say we had the 2020 Vanuatu census so we knew the real population then, and wanted&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# to redo the population projections with everything staying as they were before&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# file:///C:/Users/Peter/Downloads/Vanuatu_2020_Census_Analytical_report_Vol_2.pdf&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revision_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Vanuatu&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;repeat_un_proj&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revision_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# rough adjustment of starting population in 2020 to make totals match the Census totals.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# In principle could of course use the actual numbers by single age category.&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;151597&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;148422&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# calculate the crude birth rate in the current projection&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_cbr_2020&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; 
  &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;])&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# make a rough adjustment of future fertility rates assuming they are &quot;out&quot; by the&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# same proportion that crude birth rate was out in 2020&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;adj_ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;28.2&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_cbr_2020&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_fert&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_fert&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;adj_ratio&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;# Refit the projection with the above rough adjustments:&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_proj&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;pop_proj&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_pop_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_pop_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_pop_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;start_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;end_year&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fertility&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_fert&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_m&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_mort_m&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mort_f&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_mort_f&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex_ratio_birth&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_srb&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;net_migration&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_cnmr&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;projected_pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;apply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;apply&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sum&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#-----------------Compare total population----------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;indicators&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Location&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revision_country&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Variant&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Medium&quot;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;amp;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;%in%&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2020&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;select&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`2022 UN projections`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;TPopulation1Jan&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;`Revised with 2020 census`&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.numeric&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;projected_pop&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;/&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;


&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;p5&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;comp_data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2050&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;gather&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;Time&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;variable&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_line&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;()&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revision_country&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Attempt to re-create the UN population projections from population in 2020, fertility and mortality rates&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Population&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;c1&quot;&gt;#--------------Compare population pyramids-------------------------&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;-&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;tibble&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;31&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;revised_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;31&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;   
                  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopM&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;31&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;],&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;van_orig&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;un_proj&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;$&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;PopF&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;[,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;31&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;]),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Male&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Female&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Male&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Female&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;121&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Revised projections with 2020 census, made in 2024&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;UN projections, made in 2022&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;each&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;242&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;rep&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;120&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;mutate&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;agef&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_reorder&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;as.character&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
         &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fct_rev&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;age&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;&amp;lt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;100&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;

&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Female&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;|&amp;gt;&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;ggplot&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;agef&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;facet_wrap&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;~&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;model&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;brown&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_col&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;data&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;filter&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;d&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;sex&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;==&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Male&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;aes&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;-&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;value&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;fill&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;orange&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;kc&quot;&gt;NA&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;geom_vline&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;xintercept&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;colour&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;white&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labs&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;x&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Number of people&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;y&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;title&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Comparison of UN original and revised population projections&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
       &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;subtitle&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Population age distribution in 2050&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_x_continuous&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-4000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;-2000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;2000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;4000&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;),&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;labels&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;nf&quot;&gt;c&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;4,000&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Male&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;0&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;Female&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;,&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;s2&quot;&gt;&quot;4,000&quot;&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;))&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;theme&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;panel.grid&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;element_blank&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;())&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;+&lt;/span&gt;&lt;span class=&quot;w&quot;&gt;
  &lt;/span&gt;&lt;span class=&quot;n&quot;&gt;scale_y_discrete&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;(&lt;/span&gt;&lt;span class=&quot;n&quot;&gt;breaks&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;=&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;1&lt;/span&gt;&lt;span class=&quot;o&quot;&gt;:&lt;/span&gt;&lt;span class=&quot;m&quot;&gt;20&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;o&quot;&gt;*&lt;/span&gt;&lt;span class=&quot;w&quot;&gt; &lt;/span&gt;&lt;span class=&quot;m&quot;&gt;5&lt;/span&gt;&lt;span class=&quot;p&quot;&gt;)&lt;/span&gt;&lt;/code&gt;&lt;/pre&gt;&lt;/figure&gt;

&lt;p&gt;And that’s what gets us these results:&lt;/p&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-vanuatu-revised.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-vanuatu-revised.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;object type=&quot;image/svg+xml&quot; data=&quot;https://freerangestats.info/img/0262-vanuatu-pyramid.svg&quot; width=&quot;100%&quot;&gt;&lt;img src=&quot;https://freerangestats.info/img/0262-vanuatu-pyramid.png&quot; width=&quot;100%&quot; /&gt;&lt;/object&gt;

&lt;p&gt;I’m hoping this might be actually useful for pragmatic updates of the UN population projections when more current data is available, without having to revise everything from scratch.&lt;/p&gt;

&lt;p&gt;OK, that’s all for today.&lt;/p&gt;
</description>
				<pubDate>Sun, 05 May 2024 00:00:00 +1100</pubDate>
				<link>https://freerangestats.info/blog/2024/05/05/pop-projections</link>
				<guid isPermaLink="true">https://freerangestats.info/blog/2024/05/05/pop-projections</guid>
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