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<h1 class="title toc-ignore">Linear models</h1>
<h4 class="author"><em>Tad, SR Schachat</em></h4>
</div>
<p><strong>Assigned Reading:</strong></p>
<blockquote>
<p>Appendix A from: Zuur, A. F., Ieno, E. N., Walker, N., Saveliev, A. A. and Smith, G. M. 2009. <em>Mixed Effects Models and Extensions in Ecology with R.</em> Springer. <a href="https://link-springer-com.stanford.idm.oclc.org/book/10.1007%2F978-0-387-87458-6">Here’s the whole book</a> or <a href="https://link.springer.com/content/pdf/bbm%3A978-0-387-87458-6%2F1.pdf">you can download just the appendix</a>.</p>
</blockquote>
<div id="key-points" class="section level3">
<h3>Key Points</h3>
<div id="data-exploration-outliers-and-collinearity" class="section level4">
<h4>Data exploration: outliers and collinearity</h4>
<ul>
<li>Normality not needed for explanatory variables</li>
<li>Correlation - “value of 0.6 (and -0.6) is not large enough to worry us”?</li>
</ul>
</div>
<div id="linear-regression-model-selection" class="section level4">
<h4>Linear regression: model selection</h4>
<ul>
<li>Which interactions to include - opinions vary (p. 537)</li>
<li><code>summary(M1)</code>, <code>drop1(M1, test="F")</code>, <code>anova(M1)</code> - Is <code>drop1()</code> the best method?</li>
<li>Which explanatory variables to drop - opinions vary (p. 541)</li>
<li><code>step(M1)</code> - easy one-line command; AIC-based backwards selection</li>
</ul>
</div>
<div id="linear-regression-model-validation" class="section level4">
<h4>Linear regression: model validation</h4>
<ul>
<li>Mostly done graphically</li>
<li>But not very powerful when n is small (as common in ecological data?)</li>
<li>Model selection, then model validation - Is this always ok?</li>
</ul>
</div>
<div id="linear-regression-model-interpretation" class="section level4">
<h4>Linear regression: model interpretation</h4>
<ul>
<li>Graphical presentation - Data should be plotted.</li>
</ul>
</div>
<div id="additive-modeling" class="section level4">
<h4>Additive modeling</h4>
<ul>
<li>Zuur et al.’s favorite tool? (more in Ch 3)</li>
<li>Try model with all explanatory variables, to look at potential non-linear effect</li>
<li>Difficulties: overfitting and need for large data?</li>
</ul>
<hr />
</div>
</div>
<div id="analysis-example" class="section level3">
<h3>Analysis Example</h3>
<p>In this example, we’ll use linear regression to examine whether there is any sort of relationship beween Damage Type (DT) diversity and herbivorized leaf area on fossil leaves.</p>
<p>First, let’s read in the data:</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="co"># Read in a data file from the data folder on the BIO 202 GitHub repository</span>
data <-<span class="st"> </span><span class="kw">read.csv</span>(<span class="st">"https://raw.githubusercontent.com/FukamiLab/BIO202/master/data/03-B-permian.csv"</span>)
data <-<span class="st"> </span>data[<span class="kw">order</span>(data$dts),]</code></pre></div>
<p>Now, we can simply plot DT diversity against herbivorized leaf area:</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">plot</span>(data$dts~data$rem)</code></pre></div>
<p><img src="images/03-B/unnamed-chunk-3-1.png" width="672" /></p>
<p>Not a particularly clear relationship there! We can try running a simple linear regression:</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">fit <-<span class="st"> </span><span class="kw">lm</span>(dts~<span class="dv">0</span>+rem, <span class="dt">data=</span>data)
<span class="kw">summary</span>(fit)</code></pre></div>
<pre><code>##
## Call:
## lm(formula = dts ~ 0 + rem, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.5759 0.0000 0.6205 1.6602 10.9268
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## rem 19.876 1.255 15.84 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.879 on 1025 degrees of freedom
## Multiple R-squared: 0.1966, Adjusted R-squared: 0.1959
## F-statistic: 250.9 on 1 and 1025 DF, p-value: < 2.2e-16</code></pre>
<p>Very low p-value (< 2.2e-16), very unconvincing R-squared (0.10).</p>
<p>We can try plotting a different line for each site (as in Fig. A.3 from the reading, on page 544).</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">fit <-<span class="st"> </span><span class="kw">lm</span>(dts~<span class="dv">0</span>+rem+site, <span class="dt">data=</span>data)
<span class="kw">summary</span>(fit)</code></pre></div>
<pre><code>##
## Call:
## lm(formula = dts ~ 0 + rem + site, data = data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.8717 -1.2556 -0.2946 0.6452 9.7026
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## rem 11.35979 1.08391 10.480 <2e-16 ***
## siteCCP 1.25560 0.05452 23.031 <2e-16 ***
## siteMCF 0.25235 0.12777 1.975 0.0485 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.525 on 1023 degrees of freedom
## Multiple R-squared: 0.4717, Adjusted R-squared: 0.4702
## F-statistic: 304.5 on 3 and 1023 DF, p-value: < 2.2e-16</code></pre>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">D1 <-<span class="st"> </span>data[data$site==<span class="st">"CCP"</span>,]
D2 <-<span class="st"> </span>data[data$site==<span class="st">"MCF"</span>,]
P1 <-<span class="st"> </span><span class="kw">predict</span>(fit, <span class="dt">newdata =</span> D1)
P2 <-<span class="st"> </span><span class="kw">predict</span>(fit, <span class="dt">newdata =</span> D2)
<span class="kw">plot</span>(data$dts~data$rem)
<span class="kw">lines</span>(D1$rem, P1, <span class="dt">col=</span><span class="st">"blue"</span>)
<span class="kw">lines</span>(D2$rem, P2, <span class="dt">col=</span><span class="st">"red"</span>)</code></pre></div>
<p><img src="images/03-B/unnamed-chunk-5-1.png" width="672" /></p>
<p>We can try plotting a different line for each species instead of each site.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">fit <-<span class="st"> </span><span class="kw">lm</span>(dts~<span class="dv">0</span>+rem+sp, <span class="dt">data=</span>data)
D1 <-<span class="st"> </span>data[data$sp==<span class="st">"cau"</span>,]
D2 <-<span class="st"> </span>data[data$sp==<span class="st">"cev"</span>,]
D3 <-<span class="st"> </span>data[data$sp==<span class="st">"cta"</span>,]
D4 <-<span class="st"> </span>data[data$sp==<span class="st">"mgi"</span>,]
D5 <-<span class="st"> </span>data[data$sp==<span class="st">"mta"</span>,]
P1 <-<span class="st"> </span><span class="kw">predict</span>(fit, <span class="dt">newdata =</span> D1)
P2 <-<span class="st"> </span><span class="kw">predict</span>(fit, <span class="dt">newdata =</span> D2)
P3 <-<span class="st"> </span><span class="kw">predict</span>(fit, <span class="dt">newdata =</span> D3)
P4 <-<span class="st"> </span><span class="kw">predict</span>(fit, <span class="dt">newdata =</span> D4)
P5 <-<span class="st"> </span><span class="kw">predict</span>(fit, <span class="dt">newdata =</span> D5)
<span class="kw">plot</span>(data$dts~data$rem)
<span class="kw">lines</span>(D1$rem, P1, <span class="dt">co=</span><span class="st">"red"</span>)
<span class="kw">lines</span>(D2$rem, P2, <span class="dt">col=</span><span class="st">"blue"</span>)
<span class="kw">lines</span>(D3$rem, P3, <span class="dt">col=</span><span class="st">"green"</span>)
<span class="kw">lines</span>(D4$rem, P4, <span class="dt">col=</span><span class="st">"purple"</span>)
<span class="kw">lines</span>(D5$rem, P5, <span class="dt">col=</span><span class="st">"yellow"</span>)</code></pre></div>
<p><img src="images/03-B/unnamed-chunk-6-1.png" width="672" /></p>
<p>To be thorough, we can try adding all possible predictor variables, dropping them, and seeing what happens.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">fit.full <-<span class="st"> </span><span class="kw">lm</span>(dts~<span class="dv">0</span>+rem+sp+site+area, <span class="dt">data=</span>data)
<span class="kw">drop1</span>(fit.full, <span class="dt">test=</span><span class="st">"F"</span>)</code></pre></div>
<pre><code>## Single term deletions
##
## Model:
## dts ~ 0 + rem + sp + site + area
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 1882.4 636.66
## rem 1 188.86 2071.3 732.76 102.236 < 2.2e-16 ***
## sp 4 506.35 2388.8 873.08 68.526 < 2.2e-16 ***
## site 0 0.00 1882.4 636.66
## area 1 470.84 2353.2 863.71 254.878 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1</code></pre>
<p>Now a run-of-the-mill GAM:</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(mgcv)
AM1 <-<span class="st"> </span><span class="kw">gam</span>(dts ~<span class="st"> </span><span class="kw">s</span>(rem) +<span class="st"> </span>sp +<span class="st"> </span>site +<span class="st"> </span><span class="kw">s</span>(area), <span class="dt">data =</span> data)
<span class="kw">anova</span>(AM1)</code></pre></div>
<pre><code>##
## Family: gaussian
## Link function: identity
##
## Formula:
## dts ~ s(rem) + sp + site + s(area)
##
## Parametric Terms:
## df F p-value
## sp 4 6.303 5.2e-05
## site 1 2.061 0.151
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## s(rem) 8.368 8.887 41.91 <2e-16
## s(area) 5.344 6.455 58.11 <2e-16</code></pre>
<p>All predictors except “site” are significant.</p>
<p>Now a fancy GAM with cubic regression spline with shrinkage.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">AM2 <-<span class="st"> </span><span class="kw">gam</span>(dts ~<span class="st"> </span><span class="kw">s</span>(rem, <span class="dt">bs =</span> <span class="st">"cs"</span>) +<span class="st"> </span>sp +<span class="st"> </span>site +<span class="st"> </span><span class="kw">s</span>(area, <span class="dt">bs =</span> <span class="st">"cs"</span>), <span class="dt">data =</span> data)
<span class="kw">anova</span>(AM2)</code></pre></div>
<pre><code>##
## Family: gaussian
## Link function: identity
##
## Formula:
## dts ~ s(rem, bs = "cs") + sp + site + s(area, bs = "cs")
##
## Parametric Terms:
## df F p-value
## sp 4 4.75 0.000845
## site 1 0.29 0.590265
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## s(rem) 8.355 9.000 56.61 <2e-16
## s(area) 2.781 9.000 37.70 <2e-16</code></pre>
<p>Values change but no difference in significance at p = 0.05. So let’s get rid of “site.”</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">AM3 <-<span class="st"> </span><span class="kw">gam</span>(dts ~<span class="st"> </span><span class="kw">s</span>(rem, <span class="dt">bs =</span> <span class="st">"cs"</span>) +<span class="st"> </span>sp +<span class="st"> </span><span class="kw">s</span>(area, <span class="dt">bs =</span> <span class="st">"cs"</span>), <span class="dt">data =</span> data)
<span class="kw">anova</span>(AM3)</code></pre></div>
<pre><code>##
## Family: gaussian
## Link function: identity
##
## Formula:
## dts ~ s(rem, bs = "cs") + sp + s(area, bs = "cs")
##
## Parametric Terms:
## df F p-value
## sp 4 4.75 0.000845
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## s(rem) 8.355 9.000 57.1 <2e-16
## s(area) 2.781 9.000 37.7 <2e-16</code></pre>
<p>We can plot the smoothing function.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">plot</span>(AM3)</code></pre></div>
<p><img src="images/03-B/unnamed-chunk-11-1.png" width="672" /><img src="images/03-B/unnamed-chunk-11-2.png" width="672" /></p>
</div>
<div id="discussion-questions" class="section level3">
<h3>Discussion Questions</h3>
<ol style="list-style-type: decimal">
<li>Page 537: Of the 7 possible ways to treat interaction terms, which do you prefer and why?</li>
<li>Page 537: What are “main terms” and why do they matter?</li>
<li>Page 537: Why are 4-way interactions a bad thing? What are “Cook distance values” and “non-convergence”?</li>
<li>Page 539: Review: What is RSS (residual sum of squares) and what does it tell us?</li>
<li>Page 540: What are “nested models” in this context?</li>
<li>Page 541: Of the 3 possible ways to treat non-significant explanatory variables, which do you prefer and why?</li>
<li>Page 542: How does AIC “judge” model fit?</li>
<li>Page 546: What is it about the fit of the lines in figure A.3 (pg. 544) that suggests that a linear model may not be the best fit?</li>
<li>Page 546: What is a Generalized Additive Model and how does it work?</li>
<li>Page 546: What is a thin plate spline and how is it relevant here?</li>
<li>Page 551: When might the Aikake weight be useful?</li>
</ol>
<hr />
</div>
<div id="after-class-follow-up" class="section level3">
<h3>After-class follow-up</h3>
<ul>
<li>df = 0 for site in the <code>drop1()</code> analysis above because some species were observed only in one of the two sites.</li>
</ul>
</div>
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