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<blockquote>
<p>Beta regression is commonly used when you want to model <span class="math inline"><em>Y</em></span> that are probabilities themselves.</p>
</blockquote>
<p>This is evident when the value of <span class="math inline"><em>Y</em></span> is a <em>proportion</em> that ranges between 0 to 1. The data points of <span class="math inline"><em>Y</em></span> variable typically represent a proportion of events that form a subset of the total population (assuming that it follows a beta distribution).</p>
<h2>Use Cases</h2>
<ol style="list-style-type: decimal">
<li>From GasolineYield data: Proportion of crude oil converted to gasoline after distillation and fractionation</li>
<li>Proportion of individuals infected with ‘xyz’ when exposed to various levels of artificial preservative agent.</li>
</ol>
<h2>Example: Gasoline Yield</h2>
<p>Lets predict the gasoline <code>yield</code> as a function of <code>batch</code> and temperature. The example below shows an example implementation of beta regression using the GasolineYield data from <code>betareg</code> package.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span> (betareg)
<span class="kw">data</span>(<span class="st">"GasolineYield"</span>, <span class="dt">package =</span> <span class="st">"betareg"</span>) <span class="co"># initialize data</span>
inputData <-<span class="st"> </span>GasolineYield <span class="co"># plug-in your data here</span>
trainingIndex <-<span class="st"> </span><span class="kw">c</span>(<span class="dv">1</span>:(<span class="kw">nrow</span>(inputData)-<span class="dv">1</span>)) <span class="co"># create row indices of training data</span>
trainingData <-<span class="st"> </span>inputData[trainingIndex, ] <span class="co"># training data</span>
testData <-<span class="st"> </span>inputData[-trainingIndex, ] <span class="co"># test data</span>
betaMod <-<span class="st"> </span><span class="kw">betareg</span>(yield ~<span class="st"> </span>batch +<span class="st"> </span>temp, <span class="dt">data =</span> trainingData) <span class="co"># train model. Tune var names.</span>
<span class="kw">summary</span> (betaMod) <span class="co"># model summary</span>
<span class="kw">predict</span> (betaMod, testData) <span class="co"># predict on test data (0.19 vs actual 0.18)</span>
<span class="co">#> Call:</span>
<span class="co">#> betareg(formula = yield ~ batch + temp, data = GasolineYield)</span>
<span class="co">#> </span>
<span class="co">#> Standardized weighted residuals 2:</span>
<span class="co">#> Min 1Q Median 3Q Max </span>
<span class="co">#> -2.8750 -0.8149 0.1601 0.8384 2.0483 </span>
<span class="co">#> </span>
<span class="co">#> Coefficients (mean model with logit link):</span>
<span class="co">#> Estimate Std. Error z value Pr(>|z|) </span>
<span class="co">#> (Intercept) -6.1595710 0.1823247 -33.784 < 2e-16 </span>
<span class="co">#> batch1 1.7277289 0.1012294 17.067 < 2e-16 </span>
<span class="co">#> batch2 1.3225969 0.1179020 11.218 < 2e-16 </span>
<span class="co">#> batch3 1.5723099 0.1161045 13.542 < 2e-16 </span>
<span class="co">#> batch4 1.0597141 0.1023598 10.353 < 2e-16 </span>
<span class="co">#> batch5 1.1337518 0.1035232 10.952 < 2e-16 </span>
<span class="co">#> batch6 1.0401618 0.1060365 9.809 < 2e-16 </span>
<span class="co">#> batch7 0.5436922 0.1091275 4.982 6.29e-07 </span>
<span class="co">#> batch8 0.4959007 0.1089257 4.553 5.30e-06 </span>
<span class="co">#> batch9 0.3857930 0.1185933 3.253 0.00114 ** </span>
<span class="co">#> temp 0.0109669 0.0004126 26.577 < 2e-16 </span>
<span class="co">#> </span>
<span class="co">#> Phi coefficients (precision model with identity link):</span>
<span class="co">#> Estimate Std. Error z value Pr(>|z|) </span>
<span class="co">#> (phi) 440.3 110.0 4.002 6.29e-05 </span>
<span class="co">#> </span>
<span class="co">#> Signif. codes: 0 '' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 </span>
<span class="co">#> </span>
<span class="co">#> Type of estimator: ML (maximum likelihood)</span>
<span class="co">#> Log-likelihood: 84.8 on 12 Df</span>
<span class="co">#> Pseudo R-squared: 0.9617</span>
<span class="co">#> Number of iterations: 51 (BFGS) + 3 (Fisher scoring)</span></code></pre></div>
<p>This page is based on the examples available in <a href="http://cran.r-project.org/web/packages/betareg/vignettes/betareg.pdf">Beta regression vignette</a>.</p>
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