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<blockquote>
<p>Ordinal logistic regression can be used to model a ordered factor response.</p>
</blockquote>
<p>The <code>polr()</code> function from the <code>MASS</code> package can be used to build the <em>proportional odds logistic regression</em> and predict the class of multi-class ordered variables. One such use case is described below.</p>
<h2>Example: Predict Cars Evaluation</h2>
<p>Below is a example on how we can use ordered logistic regression to predict the cars evaluation based on <a href="http://archive.ics.uci.edu/ml/datasets/Car+Evaluation">cars evaluation dataset</a>. The cars are evaluated as one amongst <code>very good</code>, <code>good</code>, <code>acceptable</code> or <code>unacceptable</code>. The attributes of the cars available to use to predict this decision are:</p>
<ol style="list-style-type: decimal">
<li>buying : v-high, high, med, low</li>
<li>maint : v-high, high, med, low</li>
<li>doors : 2, 3, 4, 5-more</li>
<li>persons : 2, 4, more</li>
<li>lug_boot : small, med, big</li>
<li>safety : low, med, high</li>
</ol>
<p>Also, it is worthwhile to note that about 70% of the cars are evaluated as <em>unacceptable</em>. The class distribution of the ordered multi class <span class="math inline"><em>Y</em></span> is as follows:</p>
<table>
<thead>
<tr class="header">
<th align="left">class</th>
<th align="left">N</th>
<th align="left">N[%]</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left">unacc</td>
<td align="left">1210</td>
<td align="left">(70.023 %)</td>
</tr>
<tr class="even">
<td align="left">acc</td>
<td align="left">384</td>
<td align="left">(22.222 %)</td>
</tr>
<tr class="odd">
<td align="left">good</td>
<td align="left">69</td>
<td align="left">(3.993 %)</td>
</tr>
<tr class="even">
<td align="left">v-good</td>
<td align="left">65</td>
<td align="left">(3.762 %)</td>
</tr>
</tbody>
</table>
<p>Lets being the modeling process by first importing the data and assigning the correct orders to the factor variables.</p>
<h2>Import the data</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">carsdata <-<span class="st"> </span><span class="kw">read.csv</span>(<span class="st">"http://archive.ics.uci.edu/ml/machine-learning-databases/car/car.data"</span>, <span class="dt">header=</span>F, <span class="dt">stringsAsFactors=</span>F) <span class="co"># import string variables as characters.</span>
<span class="kw">colnames</span>(carsdata) <-<span class="st"> </span><span class="kw">c</span>(<span class="st">"buying"</span>, <span class="st">"maint"</span>, <span class="st">"doors"</span>, <span class="st">"persons"</span>, <span class="st">"lug_boot"</span>, <span class="st">"safety"</span>, <span class="st">"class"</span>)</code></pre></div>
<h4>Reorder the levels of factors</h4>
<p>In order logistic regression, the order of the levels in the factor variables matters. So, lets define them explicitly. This is an critical step, otherwise, predictions could go worng easily.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="co"># Reorder</span>
carsdata$buying <-<span class="st"> </span><span class="kw">factor</span>(carsdata$buying, <span class="dt">levels=</span><span class="kw">c</span>(<span class="st">"low"</span>, <span class="st">"med"</span>, <span class="st">"high"</span>, <span class="st">"vhigh"</span>), <span class="dt">ordered=</span><span class="ot">TRUE</span>)
carsdata$maint <-<span class="st"> </span><span class="kw">factor</span>(carsdata$maint, <span class="dt">levels=</span><span class="kw">c</span>(<span class="st">"low"</span>, <span class="st">"med"</span>, <span class="st">"high"</span>, <span class="st">"vhigh"</span>), <span class="dt">ordered=</span><span class="ot">TRUE</span>)
carsdata$doors <-<span class="st"> </span><span class="kw">factor</span>(carsdata$doors, <span class="dt">levels=</span><span class="kw">c</span>(<span class="st">"2"</span>, <span class="st">"3"</span>, <span class="st">"4"</span>, <span class="st">"5more"</span>), <span class="dt">ordered=</span><span class="ot">TRUE</span>)
carsdata$persons <-<span class="st"> </span><span class="kw">factor</span>(carsdata$persons, <span class="dt">levels=</span><span class="kw">c</span>(<span class="st">"2"</span>, <span class="st">"4"</span>, <span class="st">"more"</span>), <span class="dt">ordered=</span><span class="ot">TRUE</span>)
carsdata$lug_boot <-<span class="st"> </span><span class="kw">factor</span>(carsdata$lug_boot, <span class="dt">levels=</span><span class="kw">c</span>(<span class="st">"small"</span>, <span class="st">"med"</span>, <span class="st">"big"</span>), <span class="dt">ordered=</span><span class="ot">TRUE</span>)
carsdata$safety <-<span class="st"> </span><span class="kw">factor</span>(carsdata$safety, <span class="dt">levels=</span><span class="kw">c</span>(<span class="st">"low"</span>, <span class="st">"med"</span>, <span class="st">"high"</span>), <span class="dt">ordered=</span><span class="ot">TRUE</span>)
carsdata$class <-<span class="st"> </span><span class="kw">factor</span>(carsdata$class, <span class="dt">levels=</span><span class="kw">c</span>(<span class="st">"unacc"</span>, <span class="st">"acc"</span>, <span class="st">"good"</span>, <span class="st">"vgood"</span>), <span class="dt">ordered=</span><span class="ot">TRUE</span>)</code></pre></div>
<h2>Prepare training and test data</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="co"># Prepare Training and Test Data</span>
<span class="kw">set.seed</span>(<span class="dv">100</span>)
trainingRows <-<span class="st"> </span><span class="kw">sample</span>(<span class="dv">1</span>:<span class="kw">nrow</span>(carsdata), <span class="fl">0.7</span> *<span class="st"> </span><span class="kw">nrow</span>(carsdata))
trainingData <-<span class="st"> </span>carsdata[trainingRows, ]
testData <-<span class="st"> </span>carsdata[-trainingRows, ]</code></pre></div>
<h2>Build the ordered logistic regression model</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">### Build ordered logistic regression model
<span class="kw">options</span>(<span class="dt">contrasts =</span> <span class="kw">c</span>(<span class="st">"contr.treatment"</span>, <span class="st">"contr.poly"</span>))
polrMod <-<span class="st"> </span><span class="kw">polr</span>(class ~<span class="st"> </span>safety +<span class="st"> </span>lug_boot +<span class="st"> </span>doors +<span class="st"> </span>buying +<span class="st"> </span>maint, <span class="dt">data=</span>trainingData)
<span class="kw">summary</span>(polrMod)
<span class="co">#> Call:</span>
<span class="co">#> polr(formula = class ~ safety + lug_boot + doors + buying + maint, </span>
<span class="co">#> data = trainingData)</span>
<span class="co">#> </span>
<span class="co">#> Coefficients:</span>
<span class="co">#> Value Std. Error t value</span>
<span class="co">#> safety.L 19.9443 0.06145 324.5411</span>
<span class="co">#> safety.Q -10.6548 0.10088 -105.6189</span>
<span class="co">#> lug_boot.L 1.0119 0.14011 7.2224</span>
<span class="co">#> lug_boot.Q -0.3197 0.13355 -2.3940</span>
<span class="co">#> doors.L 0.5415 0.15573 3.4774</span>
<span class="co">#> doors.Q -0.2787 0.15466 -1.8018</span>
<span class="co">#> doors.C -0.1096 0.15372 -0.7132</span>
<span class="co">#> buying.L -2.0945 0.18137 -11.5480</span>
<span class="co">#> buying.Q -0.1369 0.15659 -0.8746</span>
<span class="co">#> buying.C 0.5219 0.15318 3.4069</span>
<span class="co">#> maint.L -1.8209 0.17533 -10.3856</span>
<span class="co">#> maint.Q -0.4768 0.15811 -3.0153</span>
<span class="co">#> maint.C 0.3319 0.15518 2.1388</span>
<span class="co">#> </span>
<span class="co">#> Intercepts:</span>
<span class="co">#> Value Std. Error t value </span>
<span class="co">#> unacc|acc 9.4557 0.0740 127.8297</span>
<span class="co">#> acc|good 11.8726 0.1345 88.2882</span>
<span class="co">#> good|vgood 13.1331 0.1997 65.7533</span>
<span class="co">#> </span>
<span class="co">#> Residual Deviance: 1300.15 </span>
<span class="co">#> AIC: 1332.15</span></code></pre></div>
<h2>Predict on test data</h2>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">### Predict
predictedClass <-<span class="st"> </span><span class="kw">predict</span>(polrMod, testData) <span class="co"># predict the classes directly</span>
<span class="kw">head</span>(predictedClass)
<span class="co">#> [1] unacc unacc unacc unacc unacc unacc</span>
<span class="co">#> Levels: unacc acc good vgood</span>
predictedScores <-<span class="st"> </span><span class="kw">predict</span>(polrMod, testData, <span class="dt">type=</span><span class="st">"p"</span>) <span class="co"># predict the probabilites</span>
<span class="kw">head</span>(predictedScores)
<span class="co">#> unacc acc good vgood</span>
<span class="co">#> 3 0.9774549 2.049194e-02 1.470224e-03 5.829671e-04</span>
<span class="co">#> 6 0.9347665 5.904708e-02 4.424660e-03 1.761744e-03</span>
<span class="co">#> 12 0.9774549 2.049194e-02 1.470224e-03 5.829671e-04</span>
<span class="co">#> 13 1.0000000 3.574918e-14 2.664535e-15 8.881784e-16</span>
<span class="co">#> 14 0.9762376 2.159594e-02 1.551314e-03 6.151902e-04</span>
<span class="co">#> 18 0.9120030 7.946377e-02 6.099087e-03 2.434191e-03</span>
## Confusion matrix and misclassification error
<span class="kw">table</span>(testData$class, predictedClass) <span class="co"># confusion matrix</span>
<span class="co">#> predictedClass</span>
<span class="co">#> unacc acc good vgood</span>
<span class="co">#> unacc 305 45 0 4</span>
<span class="co">#> acc 60 60 0 0</span>
<span class="co">#> good 0 17 0 0</span>
<span class="co">#> vgood 0 18 0 10</span>
<span class="kw">mean</span>(<span class="kw">as.character</span>(testData$class) !=<span class="st"> </span><span class="kw">as.character</span>(predictedClass)) <span class="co"># misclassification error</span>
<span class="co">#> 0.277</span></code></pre></div>
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<style type="text/css">
/* reduce spacing around math formula*/
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margin: 0em 0em;
}
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font-family: 'Helvetica Neue', Roboto, Arial, sans-serif;
font-size: 16px;
line-height: 27px;
font-weight: 400;
}
blockquote p {
line-height: 1.75;
color: #717171;
}
.well li{
line-height: 28px;
}
li.dropdown-header {
display: block;
padding: 0px;
font-size: 14px;
}
</style>
</body>
</html>