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<h1>Causation, Collision, and Confusion</h1>
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<h2> Contents </h2>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#three-million-babies-cant-be-wrong">Three Million Babies Can’t Be Wrong</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#other-groups">Other Groups</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#the-end-of-the-paradox">The End of the Paradox</a></li>
<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#causal-diagrams">Causal diagrams</a></li>
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<section id="causation-collision-and-confusion">
<h1>Causation, Collision, and Confusion<a class="headerlink" href="#causation-collision-and-confusion" title="Permalink to this heading">#</a></h1>
<p><a class="reference external" href="https://colab.research.google.com/github/AllenDowney/ProbablyOverthinkingIt/blob/book/notebooks/birthweight.ipynb">Click here to run this notebook on Colab</a>.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Install empiricaldist if we don't already have it</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">import</span> <span class="nn">empiricaldist</span>
<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
<span class="o">!</span>pip<span class="w"> </span>install<span class="w"> </span>empiricaldist
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># download utils.py</span>
<span class="kn">from</span> <span class="nn">os.path</span> <span class="kn">import</span> <span class="n">basename</span><span class="p">,</span> <span class="n">exists</span>
<span class="k">def</span> <span class="nf">download</span><span class="p">(</span><span class="n">url</span><span class="p">):</span>
<span class="n">filename</span> <span class="o">=</span> <span class="n">basename</span><span class="p">(</span><span class="n">url</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">exists</span><span class="p">(</span><span class="n">filename</span><span class="p">):</span>
<span class="kn">from</span> <span class="nn">urllib.request</span> <span class="kn">import</span> <span class="n">urlretrieve</span>
<span class="n">local</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">urlretrieve</span><span class="p">(</span><span class="n">url</span><span class="p">,</span> <span class="n">filename</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'Downloaded '</span> <span class="o">+</span> <span class="n">local</span><span class="p">)</span>
<span class="n">download</span><span class="p">(</span><span class="s2">"https://github.com/AllenDowney/ProbablyOverthinkingIt/raw/book/notebooks/utils.py"</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">decorate</span>
<span class="c1"># Set the random seed so we get the same results every time</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">17</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>The Low Birthweight Paradox was born in 1971, when Jacob Yerushalmy, a researcher at U.C. Berkeley, published “The relationship of parents’ cigarette smoking to outcome of pregnancy – implications as to the problem of inferring causation from observed associations”.
As the title suggests, the paper is about the relationship between smoking during pregnancy, the weight of babies at birth, and mortality in the first month of life.</p>
<p>Based on data from about 13,000 babies born near San Francisco between 1960 and 1967, Yerushalmy reported that</p>
<ul class="simple">
<li><p>Babies of mothers who smoked were about 6% lighter at birth.</p></li>
<li><p>Smokers were about twice as likely to have babies lighter than 2500 grams, which is considered “low birthweight”.</p></li>
<li><p>Low-birthweight babies were much more likely to die within a month of birth: the mortality rate was 174 per 1000 for low-birthweight babies and 7.8 per 1000 for others.</p></li>
</ul>
<p>These results were not surprising. At that time, it was well known that children of smokers were lighter at birth, and that low-birthweight babies were more likely to die.</p>
<p>In the first part of this notebook, there are several cells like the following that compute percentages based on numbers from tables in Yerushalmy’s paper.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mi">3464</span> <span class="o">-</span> <span class="mi">3255</span><span class="p">)</span> <span class="o">/</span> <span class="mi">3255</span> <span class="o">*</span> <span class="mi">100</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>6.42089093701997
</pre></div>
</div>
</div>
</div>
<p>Putting those results together, you might expect mortality rates to be higher for children of smokers.
And you would be right, but the difference was not very big. For White mothers, the mortality rate was 11.3 per 1000 for children of smokers, compared to 11.0 for children of nonsmokers.</p>
<p>That’s strange, but it gets even stranger. If we select only the low-birthweight (LBW) babies, we find:</p>
<ul class="simple">
<li><p>For LBW babies of nonsmokers, the mortality rate was 218 per 1000;</p></li>
<li><p>For LBW babies of smokers, it was only 114 per 1000, about 48% lower.</p></li>
</ul>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mi">218</span> <span class="o">-</span> <span class="mi">114</span><span class="p">)</span> <span class="o">/</span> <span class="mi">218</span> <span class="o">*</span> <span class="mi">100</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>47.706422018348626
</pre></div>
</div>
</div>
</div>
<p>Yerushalmy also compared rates of congenital anomalies (birth defects).</p>
<ul class="simple">
<li><p>For LBW babies of nonsmokers, the rate was 147 per 1000,</p></li>
<li><p>For LBW babies of smokers, it was 72 per 1000, about 53% lower.</p></li>
</ul>
<p>These results make maternal smoking seem <em>beneficial</em> for low-birthweight babies, somehow protecting them from birth defects and mortality.
Yerushalmy concluded:</p>
<blockquote>
<div><p>These paradoxical findings raise doubts and argue against the proposition that cigarette smoking acts as an exogenous factor which interferes with intrauterine development of the fetus.</p>
</div></blockquote>
<p>In other words, maybe maternal smoking isn’t bad for babies after all.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mi">147</span> <span class="o">-</span> <span class="mi">72</span><span class="p">)</span> <span class="o">/</span> <span class="mi">142</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.528169014084507
</pre></div>
</div>
</div>
</div>
<p>But it was a mistake.
At the risk of giving away the ending, the Low Birthweight Paradox is a statistical artifact.
In fact, maternal smoking is harmful to babies, regardless of birthweight.
It only seems beneficial because the analysis is misleading.</p>
<p>An explanation came in 2006 from epidemiologists at Harvard University and the National Institutes of Health (NIH), based on data from 3 million babies born in 1991.
Using the same dataset, which is available from the National Center for Health Statistics (NCHS), I will replicate their results and summarize their explanation.
Then I’ll repeat the analysis with data from 2018, and we’ll see what has changed.</p>
<section id="three-million-babies-cant-be-wrong">
<h2>Three Million Babies Can’t Be Wrong<a class="headerlink" href="#three-million-babies-cant-be-wrong" title="Permalink to this heading">#</a></h2>
<p>The data are originally from the <a class="reference external" href="https://www.cdc.gov/nchs/data_access/vitalstatsonline.htm">National Center for Health Statistics (NCHS)</a>.</p>
<p>I selected the columns we need and stored them in a compressed HDF file.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">DATA_PATH</span> <span class="o">=</span> <span class="s2">"https://github.com/AllenDowney/ProbablyOverthinkingIt/raw/book/data/"</span>
<span class="n">download</span><span class="p">(</span><span class="n">DATA_PATH</span> <span class="o">+</span> <span class="s2">"nchs.hdf"</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">vs1991</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">read_hdf</span><span class="p">(</span><span class="s2">"nchs.hdf"</span><span class="p">,</span> <span class="s2">"vs1991"</span><span class="p">)</span>
<span class="n">vs1991</span><span class="o">.</span><span class="n">shape</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(4115493, 5)
</pre></div>
</div>
</div>
</div>
<p>If age of death is NaN, that means the baby survived.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">vs1991</span><span class="p">[</span><span class="s2">"mort"</span><span class="p">]</span> <span class="o">=</span> <span class="n">vs1991</span><span class="p">[</span><span class="s2">"aged"</span><span class="p">]</span><span class="o">.</span><span class="n">notnull</span><span class="p">()</span>
<span class="n">vs1991</span><span class="p">[</span><span class="s2">"mort"</span><span class="p">]</span><span class="o">.</span><span class="n">value_counts</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>mort
False 4079973
True 35520
Name: count, dtype: int64
</pre></div>
</div>
</div>
</div>
<p>Recode the tobacco variable.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">vs1991</span><span class="p">[</span><span class="s2">"tobacco"</span><span class="p">]</span><span class="o">.</span><span class="n">replace</span><span class="p">([</span><span class="mi">9</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span> <span class="n">inplace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">vs1991</span><span class="p">[</span><span class="s2">"tobacco"</span><span class="p">]</span><span class="o">.</span><span class="n">value_counts</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>tobacco
2.0 2471563
1.0 533202
Name: count, dtype: int64
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">empiricaldist</span> <span class="kn">import</span> <span class="n">Pmf</span>
<span class="n">Pmf</span><span class="o">.</span><span class="n">from_seq</span><span class="p">(</span><span class="n">vs1991</span><span class="p">[</span><span class="s2">"tobacco"</span><span class="p">])</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_html"><div>
<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>probs</th>
</tr>
<tr>
<th>tobacco</th>
<th></th>
</tr>
</thead>
<tbody>
<tr>
<th>1.0</th>
<td>0.177452</td>
</tr>
<tr>
<th>2.0</th>
<td>0.822548</td>
</tr>
</tbody>
</table>
</div></div></div>
</div>
<p>Getting some numbers from the paper into a PMF.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">white</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">6067</span><span class="p">,</span> <span class="mi">3726</span><span class="p">])</span>
<span class="n">black</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">2219</span><span class="p">,</span> <span class="mi">1071</span><span class="p">])</span>
<span class="n">pmf</span> <span class="o">=</span> <span class="n">Pmf</span><span class="p">(</span><span class="n">white</span> <span class="o">+</span> <span class="n">black</span><span class="p">,</span> <span class="n">index</span><span class="o">=</span><span class="p">[</span><span class="s2">"Nonsmoker"</span><span class="p">,</span> <span class="s2">"Smoker"</span><span class="p">])</span>
<span class="n">pmf</span><span class="o">.</span><span class="n">normalize</span><span class="p">()</span>
<span class="n">pmf</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_html"><div>
<style scoped>
.dataframe tbody tr th:only-of-type {
vertical-align: middle;
}
.dataframe tbody tr th {
vertical-align: top;
}
.dataframe thead th {
text-align: right;
}
</style>
<table border="1" class="dataframe">
<thead>
<tr style="text-align: right;">
<th></th>
<th>probs</th>
</tr>
</thead>
<tbody>
<tr>
<th>Nonsmoker</th>
<td>0.633341</td>
</tr>
<tr>
<th>Smoker</th>
<td>0.366659</td>
</tr>
</tbody>
</table>
</div></div></div>
</div>
<p>Checking the distribution of birthweights.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">empiricaldist</span> <span class="kn">import</span> <span class="n">Cdf</span>
<span class="n">vs1991</span><span class="p">[</span><span class="s2">"birthweight"</span><span class="p">]</span><span class="o">.</span><span class="n">replace</span><span class="p">([</span><span class="mi">7777</span><span class="p">,</span> <span class="mi">9999</span><span class="p">],</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span><span class="p">,</span> <span class="n">inplace</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">Cdf</span><span class="o">.</span><span class="n">from_seq</span><span class="p">(</span><span class="n">vs1991</span><span class="p">[</span><span class="s2">"birthweight"</span><span class="p">])</span><span class="o">.</span><span class="n">plot</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><Axes: xlabel='birthweight'>
</pre></div>
</div>
<img alt="_images/cf5f1c317f960b5f63e6d4f9ca9a2e5dd4be05bb174cb21c9c35c862d57f8ba3.png" src="_images/cf5f1c317f960b5f63e6d4f9ca9a2e5dd4be05bb174cb21c9c35c862d57f8ba3.png" />
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">vs1991</span><span class="p">[</span><span class="s2">"birthweight"</span><span class="p">]</span><span class="o">.</span><span class="n">nlargest</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>1282965 8164.0
2328658 8164.0
2345800 8147.0
3078492 8108.0
1407733 7966.0
3037159 7965.0
2395933 7910.0
3046903 7890.0
2513239 7889.0
2582863 7880.0
Name: birthweight, dtype: float64
</pre></div>
</div>
</div>
</div>
<p>Flagging low birthweight babies.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">vs1991</span><span class="p">[</span><span class="s2">"lbw"</span><span class="p">]</span> <span class="o">=</span> <span class="n">vs1991</span><span class="p">[</span><span class="s2">"birthweight"</span><span class="p">]</span> <span class="o"><</span> <span class="mi">2500</span>
</pre></div>
</div>
</div>
</div>
<p>Dividing birthweights into bins.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">bins</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">1000</span><span class="p">,</span> <span class="mi">5000</span><span class="p">,</span> <span class="mi">250</span><span class="p">)</span>
<span class="n">bins</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([1000, 1250, 1500, 1750, 2000, 2250, 2500, 2750, 3000, 3250, 3500,
3750, 4000, 4250, 4500, 4750])
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">vs1991</span><span class="p">[</span><span class="s2">"bin"</span><span class="p">]</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">cut</span><span class="p">(</span><span class="n">vs1991</span><span class="p">[</span><span class="s2">"birthweight"</span><span class="p">],</span> <span class="n">bins</span><span class="p">)</span>
<span class="n">vs1991</span><span class="p">[</span><span class="s2">"bin"</span><span class="p">]</span><span class="o">.</span><span class="n">value_counts</span><span class="p">()</span><span class="o">.</span><span class="n">sort_index</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>bin
(1000, 1250] 12374
(1250, 1500] 14661
(1500, 1750] 21393
(1750, 2000] 34816
(2000, 2250] 61218
(2250, 2500] 123119
(2500, 2750] 246404
(2750, 3000] 426903
(3000, 3250] 696739
(3250, 3500] 817652
(3500, 3750] 712945
(3750, 4000] 482449
(4000, 4250] 240867
(4250, 4500] 122762
(4500, 4750] 46395
Name: count, dtype: int64
</pre></div>
</div>
</div>
</div>
<p>Checking the codes for congenital birth defects.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">vs1991</span><span class="p">[</span><span class="s2">"congenit"</span><span class="p">]</span><span class="o">.</span><span class="n">value_counts</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>congenit
2222222222222222222222 3556241
9999999999999999999999 498038
2222222222222222222221 23448
2222222222222222221222 5843
2222221222222222222222 3771
...
2222211112112122221222 1
2222121222122222222222 1
2212122222221221222222 1
2222222122222211222222 1
2222122222122222122222 1
Name: count, Length: 1008, dtype: int64
</pre></div>
</div>
</div>
</div>
<p>Each code is a vector that indicates the presence or absence of a particular condition. All 2’s means no anomalies.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">vs1991</span><span class="p">[</span><span class="s2">"anomaly"</span><span class="p">]</span> <span class="o">=</span> <span class="n">vs1991</span><span class="p">[</span><span class="s2">"congenit"</span><span class="p">]</span> <span class="o">!=</span> <span class="s2">"2222222222222222222222"</span>
<span class="n">vs1991</span><span class="p">[</span><span class="s2">"anomaly"</span><span class="p">]</span><span class="o">.</span><span class="n">value_counts</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>anomaly
False 3556241
True 559252
Name: count, dtype: int64
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">no_defect</span> <span class="o">=</span> <span class="n">vs1991</span><span class="p">[</span><span class="s2">"congenit"</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"2222222222222222222222"</span>
<span class="n">no_defect</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>3556241
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">unknown</span> <span class="o">=</span> <span class="n">vs1991</span><span class="p">[</span><span class="s2">"congenit"</span><span class="p">]</span> <span class="o">==</span> <span class="s2">"9999999999999999999999"</span>
<span class="n">vs1991</span><span class="o">.</span><span class="n">loc</span><span class="p">[</span><span class="n">unknown</span><span class="p">,</span> <span class="s2">"anomaly"</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">nan</span>
<span class="n">unknown</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>498038
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">known_defect</span> <span class="o">=</span> <span class="p">(</span><span class="o">~</span><span class="n">no_defect</span><span class="p">)</span> <span class="o">&</span> <span class="p">(</span><span class="o">~</span><span class="n">unknown</span><span class="p">)</span>
<span class="n">known_defect</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>61214
</pre></div>
</div>
</div>
</div>
<p>If we select cases with known tobacco use and birthweight, we’re down to about three million live births.</p>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">subset</span> <span class="o">=</span> <span class="n">vs1991</span><span class="o">.</span><span class="n">dropna</span><span class="p">(</span><span class="n">subset</span><span class="o">=</span><span class="p">[</span><span class="s2">"tobacco"</span><span class="p">,</span> <span class="s2">"birthweight"</span><span class="p">])</span>
<span class="n">subset</span><span class="o">.</span><span class="n">shape</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(3001607, 9)
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</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">subset</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s2">"tobacco"</span><span class="p">):</span>
<span class="nb">print</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">group</span><span class="p">[</span><span class="s2">"anomaly"</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="o">*</span> <span class="mi">1000</span><span class="p">)</span>
</pre></div>
</div>
</div>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>1.0 20.599922088040515
2.0 18.1934659988842
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</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">percent_diff</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">rel</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Percentage difference.</span>
<span class="sd"> x: the number of interest</span>
<span class="sd"> rel: the number it is relative to</span>
<span class="sd"> returns: float percent</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="n">rel</span><span class="p">)</span> <span class="o">/</span> <span class="n">rel</span> <span class="o">*</span> <span class="mi">100</span>
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">percent_diff</span><span class="p">(</span><span class="mf">20.6</span><span class="p">,</span> <span class="mf">18.2</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>13.186813186813199
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</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">subset_no_defect</span> <span class="o">=</span> <span class="n">subset</span><span class="p">[</span><span class="n">subset</span><span class="p">[</span><span class="s2">"anomaly"</span><span class="p">]</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span>
<span class="n">subset_no_defect</span><span class="o">.</span><span class="n">shape</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(2816996, 9)
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">weights</span> <span class="o">=</span> <span class="n">subset</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s2">"tobacco"</span><span class="p">)[</span><span class="s2">"birthweight"</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
<span class="n">weights</span>
</pre></div>
</div>
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<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>tobacco
1.0 3145.189278
2.0 3370.030102
Name: birthweight, dtype: float64
</pre></div>
</div>
</div>
</div>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">percent_diff</span><span class="p">(</span><span class="mi">3145</span><span class="p">,</span> <span class="mi">3370</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>-6.6765578635014835
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</div>
</div>
</div>
<p>In the 1991 data from NCHS, about 18% of the mothers reported smoking during pregnancy, down from 37% in Yerushalmy’s dataset from the 1960s.
Babies of smokers were lighter on average than babies of nonsmokers by about 7%, which is comparable to the difference in the 1960s data.</p>
<p>The following figure shows the distribution of weights for the two groups. The vertical line is at 2500 grams, the threshold for low birthweight.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">utils</span> <span class="kn">import</span> <span class="n">kdeplot</span>
<span class="n">labels</span> <span class="o">=</span> <span class="p">{</span><span class="mi">1</span><span class="p">:</span> <span class="s2">"Smokers"</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span> <span class="s2">"Nonsmokers"</span><span class="p">}</span>
<span class="n">ls</span> <span class="o">=</span> <span class="p">{</span><span class="mi">1</span><span class="p">:</span> <span class="s2">"-"</span><span class="p">,</span> <span class="mi">2</span><span class="p">:</span> <span class="s2">"--"</span><span class="p">}</span>
<span class="n">xs</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="n">subset</span><span class="p">[</span><span class="s2">"birthweight"</span><span class="p">]</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span> <span class="n">subset</span><span class="p">[</span><span class="s2">"birthweight"</span><span class="p">]</span><span class="o">.</span><span class="n">max</span><span class="p">(),</span> <span class="mi">101</span><span class="p">)</span>
<span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">subset</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s2">"tobacco"</span><span class="p">):</span>
<span class="n">kdeplot</span><span class="p">(</span><span class="n">group</span><span class="p">[</span><span class="s2">"birthweight"</span><span class="p">],</span> <span class="n">xs</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="n">labels</span><span class="p">[</span><span class="n">name</span><span class="p">],</span> <span class="n">ls</span><span class="o">=</span><span class="n">ls</span><span class="p">[</span><span class="n">name</span><span class="p">])</span>
<span class="n">plt</span><span class="o">.</span><span class="n">axvline</span><span class="p">(</span><span class="mi">2500</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"gray"</span><span class="p">,</span> <span class="n">ls</span><span class="o">=</span><span class="s2">":"</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)</span>
<span class="n">decorate</span><span class="p">(</span>
<span class="n">xlabel</span><span class="o">=</span><span class="s2">"Birthweight (grams)"</span><span class="p">,</span>
<span class="n">ylabel</span><span class="o">=</span><span class="s2">"Relative likelihood"</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="s2">"Distributions of birthweights"</span><span class="p">,</span>
<span class="n">xlim</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">6000</span><span class="p">],</span>
<span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/4e66c00a1104ef2b0947b880cc16905940e4bbdbaa15762e08869cb9c772f51e.png" src="_images/4e66c00a1104ef2b0947b880cc16905940e4bbdbaa15762e08869cb9c772f51e.png" />
</div>
</div>
<p>The shapes of the distributions are similar, but for smokers it is shifted to the left.
For mothers who smoked, the fraction of babies below 2500 grams is about 11%; for nonsmokers it is only 6%.</p>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">subset</span><span class="p">[</span><span class="s2">"lbw"</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.07262476400141657
</pre></div>
</div>
</div>
</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">subset</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s2">"tobacco"</span><span class="p">):</span>
<span class="n">lbw_rate</span> <span class="o">=</span> <span class="n">group</span><span class="p">[</span><span class="s2">"lbw"</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">lbw_rate</span><span class="p">)</span>
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>1.0 0.11421285960264155
2.0 0.06365429478663773
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</div>
</div>
</div>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">percent_diff</span><span class="p">(</span><span class="mf">11.4</span><span class="p">,</span> <span class="mf">6.4</span><span class="p">)</span>
</pre></div>
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</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>78.125
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</div>
</div>
</div>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="p">(</span><span class="mi">9793</span> <span class="o">*</span> <span class="mf">11.1</span> <span class="o">+</span> <span class="mi">3290</span> <span class="o">*</span> <span class="mf">18.5</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mi">9793</span> <span class="o">+</span> <span class="mi">3290</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>12.960888175494917
</pre></div>
</div>
</div>
</div>
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<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">subset</span><span class="p">[</span><span class="s2">"mort"</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="o">*</span> <span class="mi">1000</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>8.449473898481713
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</div>
</div>
</div>
<p>Overall infant mortality was substantially lower in 1991.
In the 1960s dataset, about 13 per 1000 babies died within the first <em>month</em> of life; in 1991, about 8.5 per 1000 died in the first <em>year</em>.</p>
<p>In 1991, the mortality rate was higher for babies of smokers, almost 12 per 1000, than babies of nonsmokers, 7.7 per 1000.
So the risk of mortality was 54% higher for babies of mothers who smoked.</p>
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">name</span><span class="p">,</span> <span class="n">group</span> <span class="ow">in</span> <span class="n">subset</span><span class="o">.</span><span class="n">groupby</span><span class="p">(</span><span class="s2">"tobacco"</span><span class="p">):</span>
<span class="n">rate</span> <span class="o">=</span> <span class="n">group</span><span class="p">[</span><span class="s2">"mort"</span><span class="p">]</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span> <span class="o">*</span> <span class="mi">1000</span>
<span class="nb">print</span><span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">rate</span><span class="p">)</span>
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<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>1.0 11.874562261649707
2.0 7.71068917473998
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</div>