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<h1>Missing Value Treatment</h1>
<blockquote>
<p>Missing values in data is a common phenomenon in real world problems. Knowing how to handle missing values effectively is a required step to reduce bias and to produce powerful models. Lets explore various options of how to deal with missing values and how to implement them.</p>
</blockquote>
<h2>Data prep and pattern</h2>
<p>Lets use the <code>BostonHousing</code> dataset in <code>mlbench</code> package to discuss the various approaches to treating missing values. Though the original <code>BostonHousing</code> data doesn’t have missing values, I am going to randomly introduce missing values. This way, we can validate the imputed missing values against the actuals, so that we know how effective are the approaches in reproducing the actual data. Lets begin by importing the data from <code>mlbench</code> pkg and randomly insert missing values (NA).</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">data</span> (<span class="st">"BostonHousing"</span>, <span class="dt">package=</span><span class="st">"mlbench"</span>) <span class="co"># initialize the data # load the data</span>
original <-<span class="st"> </span>BostonHousing <span class="co"># backup original data</span>
<span class="co"># Introduce missing values</span>
<span class="kw">set.seed</span>(<span class="dv">100</span>)
BostonHousing[<span class="kw">sample</span>(<span class="dv">1</span>:<span class="kw">nrow</span>(BostonHousing), <span class="dv">40</span>), <span class="st">"rad"</span>] <-<span class="st"> </span><span class="ot">NA</span>
BostonHousing[<span class="kw">sample</span>(<span class="dv">1</span>:<span class="kw">nrow</span>(BostonHousing), <span class="dv">40</span>), <span class="st">"ptratio"</span>] <-<span class="st"> </span><span class="ot">NA</span>
<span class="kw">head</span>(BostonHousing)
<span class="co">#> crim zn indus chas nox rm age dis rad tax ptratio b lstat medv</span>
<span class="co">#> 1 0.00632 18 2.31 0 0.538 6.575 65.2 4.0900 1 296 15.3 396.90 4.98 24.0</span>
<span class="co">#> 2 0.02731 0 7.07 0 0.469 6.421 78.9 4.9671 2 242 17.8 396.90 9.14 21.6</span>
<span class="co">#> 3 0.02729 0 7.07 0 0.469 7.185 61.1 4.9671 2 242 17.8 392.83 4.03 34.7</span>
<span class="co">#> 4 0.03237 0 2.18 0 0.458 6.998 45.8 6.0622 3 222 18.7 394.63 2.94 33.4</span>
<span class="co">#> 5 0.06905 0 2.18 0 0.458 7.147 54.2 6.0622 3 222 18.7 396.90 5.33 36.2</span>
<span class="co">#> 6 0.02985 0 2.18 0 0.458 6.430 58.7 6.0622 3 222 18.7 394.12 5.21 28.7</span></code></pre></div>
<p>The missing values have been injected. Though we know where the missings are, lets quickly check the ‘missings’ pattern using <code>mice::md.pattern</code>.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(mice)
<span class="kw">md.pattern</span>(BostonHousing) <span class="co"># pattern or missing values in data.</span>
<span class="co">#> crim zn indus chas nox rm age dis tax b lstat medv rad ptratio </span>
<span class="co">#> 431 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0</span>
<span class="co">#> 35 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1</span>
<span class="co">#> 35 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1</span>
<span class="co">#> 5 1 1 1 1 1 1 1 1 1 1 1 1 0 0 2</span>
<span class="co">#> 0 0 0 0 0 0 0 0 0 0 0 0 40 40 80</span></code></pre></div>
<h4>There are really four ways you can handle missing values:</h4>
<h2>1. Deleting the observations</h2>
<p>If you have large number of observations in your dataset, where all the classes to be predicted are sufficiently represented in the training data, then try deleting (or not to include missing values while model building, for example by setting <code>na.action=na.omit</code>) those observations (rows) that contain missing values. Make sure after deleting the observations, you have:</p>
<ol style="list-style-type: decimal">
<li>Have sufficent data points, so the model doesn’t lose power.</li>
<li>Not to introduce bias (meaning, disproportionate or non-representation of classes).</li>
</ol>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="co"># Example</span>
<span class="kw">lm</span>(medv ~<span class="st"> </span>ptratio +<span class="st"> </span>rad, <span class="dt">data=</span>BostonHousing, <span class="dt">na.action=</span>na.omit) <span class="co"># though na.omit is default in lm()</span></code></pre></div>
<h2>2. Deleting the variable</h2>
<p>If a paricular variable is having more missing values that rest of the variables in the dataset, and, if by removing that one variable you can save many observations, then you are better off without that variable unless it is a really important predictor that makes a lot of business sense. It is a matter of deciding between the importance of the variable and losing out on a number of observations.</p>
<h2>3. Imputation with mean / median / mode</h2>
<p>Replacing the missing values with the mean / median / mode is a crude way of treating missing values. Depending on the context, like if the variation is low or if the variable has low leverage over the response, such a rough approximation is acceptable and could possibly give satisfactory results.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(Hmisc)
<span class="kw">impute</span>(BostonHousing$ptratio, mean) <span class="co"># replace with mean</span>
<span class="kw">impute</span>(BostonHousing$ptratio, median) <span class="co"># median</span>
<span class="kw">impute</span>(BostonHousing$ptratio, <span class="dv">20</span>) <span class="co"># replace specific number</span>
<span class="co"># or if you want to impute manually</span>
BostonHousing$ptratio[<span class="kw">is.na</span>(BostonHousing$ptratio)] <-<span class="st"> </span><span class="kw">mean</span>(BostonHousing$ptratio, <span class="dt">na.rm =</span> T) <span class="co"># not run</span></code></pre></div>
<p>Lets compute the accuracy when it is imputed with mean</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(DMwR)
actuals <-<span class="st"> </span>original$ptratio[<span class="kw">is.na</span>(BostonHousing$ptratio)]
predicteds <-<span class="st"> </span><span class="kw">rep</span>(<span class="kw">mean</span>(BostonHousing$ptratio, <span class="dt">na.rm=</span>T), <span class="kw">length</span>(actuals))
<span class="kw">regr.eval</span>(actuals, predicteds)
<span class="co">#> mae mse rmse mape </span>
<span class="co">#> 1.62324034 4.19306071 2.04769644 0.09545664</span></code></pre></div>
<h2>4. Prediction</h2>
<h2>4.1. kNN Imputation</h2>
<p><code>DMwR::knnImputation</code> uses k-Nearest Neighbours approach to impute missing values. What kNN imputation does in simpler terms is as follows: For every observation to be imputed, it identifies ‘k’ closest observations based on the euclidean distance and computes the weighted average (weighted based on distance) of these ‘k’ obs.</p>
<p>The advantage is that you could impute all the missing values in all variables with one call to the function. It takes the whole data frame as the argument and you don’t even have to specify which variabe you want to impute. But be cautious not to include the response variable while imputing, because, when imputing in test/production environment, if your data contains missing values, you won’t be able to use the unknown response variable at that time.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(DMwR)
knnOutput <-<span class="st"> </span><span class="kw">knnImputation</span>(BostonHousing[, !<span class="kw">names</span>(BostonHousing) %in%<span class="st"> "medv"</span>]) <span class="co"># perform knn imputation.</span>
<span class="kw">anyNA</span>(knnOutput)
<span class="co">#> FALSE</span></code></pre></div>
<p>Lets compute the accuracy.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">actuals <-<span class="st"> </span>original$ptratio[<span class="kw">is.na</span>(BostonHousing$ptratio)]
predicteds <-<span class="st"> </span>knnOutput[<span class="kw">is.na</span>(BostonHousing$ptratio), <span class="st">"ptratio"</span>]
<span class="kw">regr.eval</span>(actuals, predicteds)
<span class="co">#> mae mse rmse mape </span>
<span class="co">#> 1.00188715 1.97910183 1.40680554 0.05859526 </span></code></pre></div>
<p>The mean absolute percentage error (mape) has improved by ~ 39% compared to the imputation by mean. Good.</p>
<h2>4.2 rpart</h2>
<p>The limitation with <code>DMwR::knnImputation</code> is that it sometimes may not be appropriate to use when the missing value comes from a factor variable. Both <code>rpart</code> and <code>mice</code> has flexibility to handle that scenario. The advantage with <code>rpart</code> is that you just need only one of the variables to be non NA in the predictor fields.</p>
<p>The idea here is we are going to use <code>rpart</code> to predict the missing values instead of <code>kNN</code>. To handle factor variable, we can set the <code>method=class</code> while calling <code>rpart()</code>. For numerics, we use, <code>method=anova</code>. Here again, we need to make sure not to train <code>rpart</code> on response variable (<code>medv</code>).</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(rpart)
class_mod <-<span class="st"> </span><span class="kw">rpart</span>(rad ~<span class="st"> </span>. -<span class="st"> </span>medv, <span class="dt">data=</span>BostonHousing[!<span class="kw">is.na</span>(BostonHousing$rad), ], <span class="dt">method=</span><span class="st">"class"</span>, <span class="dt">na.action=</span>na.omit) <span class="co"># since rad is a factor</span>
anova_mod <-<span class="st"> </span><span class="kw">rpart</span>(ptratio ~<span class="st"> </span>. -<span class="st"> </span>medv, <span class="dt">data=</span>BostonHousing[!<span class="kw">is.na</span>(BostonHousing$ptratio), ], <span class="dt">method=</span><span class="st">"anova"</span>, <span class="dt">na.action=</span>na.omit) <span class="co"># since ptratio is numeric.</span>
rad_pred <-<span class="st"> </span><span class="kw">predict</span>(class_mod, BostonHousing[<span class="kw">is.na</span>(BostonHousing$rad), ])
ptratio_pred <-<span class="st"> </span><span class="kw">predict</span>(anova_mod, BostonHousing[<span class="kw">is.na</span>(BostonHousing$ptratio), ])</code></pre></div>
<p>Lets compute the accuracy for <code>ptratio</code></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">actuals <-<span class="st"> </span>original$ptratio[<span class="kw">is.na</span>(BostonHousing$ptratio)]
predicteds <-<span class="st"> </span>ptratio_pred
<span class="kw">regr.eval</span>(actuals, predicteds)
<span class="co">#> mae mse rmse mape </span>
<span class="co">#> 0.71061673 0.99693845 0.99846805 0.04099908 </span></code></pre></div>
<p>The mean absolute percentage error (mape) has improved additionally by another ~ 30% compared to the <code>knnImputation</code>. Very Good.</p>
<p>Accuracy for <code>rad</code></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">actuals <-<span class="st"> </span>original$rad[<span class="kw">is.na</span>(BostonHousing$rad)]
predicteds <-<span class="st"> </span><span class="kw">as.numeric</span>(<span class="kw">colnames</span>(rad_pred)[<span class="kw">apply</span>(rad_pred, <span class="dv">1</span>, which.max)])
<span class="kw">mean</span>(actuals !=<span class="st"> </span>predicteds) <span class="co"># compute misclass error.</span>
<span class="co">#> 0.25 </span></code></pre></div>
<p>This yields a mis-classification error of 25%. Not bad for a factor variable!</p>
<h2>4.3 mice</h2>
<p><code>mice</code> short for <a href="http://www.jstatsoft.org/article/view/v045i03/v45i03.pdf">Multivariate Imputation by Chained Equations</a> is an R package that provides advanced features for missing value treatment. It uses a slightly uncommon way of implementing the imputation in 2-steps, using <code>mice()</code> to build the model and <code>complete()</code> to generate the completed data. The <code>mice(df)</code> function produces multiple complete copies of <code>df</code>, each with different imputations of the missing data. The <a href="http://www.inside-r.org/packages/cran/mice/docs/complete"><code>complete()</code></a> function returns one or several of these data sets, with the default being the first. Lets see how to impute ‘rad’ and ‘ptratio’:</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r"><span class="kw">library</span>(mice)
miceMod <-<span class="st"> </span><span class="kw">mice</span>(BostonHousing[, !<span class="kw">names</span>(BostonHousing) %in%<span class="st"> "medv"</span>], <span class="dt">method=</span><span class="st">"rf"</span>) <span class="co"># perform mice imputation, based on random forests.</span>
miceOutput <-<span class="st"> </span><span class="kw">complete</span>(miceMod) <span class="co"># generate the completed data.</span>
<span class="kw">anyNA</span>(miceOutput)
<span class="co">#> FALSE</span></code></pre></div>
<p>Lets compute the accuracy of <code>ptratio</code>.</p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">actuals <-<span class="st"> </span>original$ptratio[<span class="kw">is.na</span>(BostonHousing$ptratio)]
predicteds <-<span class="st"> </span>miceOutput[<span class="kw">is.na</span>(BostonHousing$ptratio), <span class="st">"ptratio"</span>]
<span class="kw">regr.eval</span>(actuals, predicteds)
<span class="co">#> mae mse rmse mape </span>
<span class="co">#> 0.36500000 0.78100000 0.88374204 0.02121326</span></code></pre></div>
<p>The mean absolute percentage error (mape) has improved additionally by ~ 48% compared to the <code>rpart</code>. Excellent!.</p>
<p>Lets compute the accuracy of <code>rad</code></p>
<div class="sourceCode"><pre class="sourceCode r"><code class="sourceCode r">actuals <-<span class="st"> </span>original$rad[<span class="kw">is.na</span>(BostonHousing$rad)]
predicteds <-<span class="st"> </span>miceOutput[<span class="kw">is.na</span>(BostonHousing$rad), <span class="st">"rad"</span>]
<span class="kw">mean</span>(actuals !=<span class="st"> </span>predicteds) <span class="co"># compute misclass error.</span>
<span class="co">#> 0.15</span></code></pre></div>
<p>The mis-classification error reduced to 15%, which is 6 out of 40 observations. This is a good improvement compared to rpart’s 25%.</p>
<p>If you’d like to dig in deeper, here is the <a href="http://www.stefvanbuuren.nl/publications/MICE%20V1.0%20Manual%20TNO00038%202000.pdf">manual</a>.</p>
<p>Though we have an idea of how each method performs, there is not enough evidence to conclude which method is better or worse. But these are definitely worth testing out the next time you impute missing values.</p>
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