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<h1 class="title">Multiple Imputation</h1>

</div>


<div id="required-packages-and-datasets" class="section level2">
<h2>Required Packages and Datasets</h2>
<p>In this lesson we will use the <code>Amelia</code> package and a subset of Beatrice Magistro’s <a href="https://www.dropbox.com/s/521gqn9neemmsd3/ess_sub.csv?dl=0">dataset</a>, with data from the <code>European Social Survey</code>.</p>
<pre class="r"><code># install.packages(&quot;Amelia&quot;)
library(Amelia)
library(ggplot2)
library(dplyr)
ess_sub &lt;- read.csv(&quot;data/ess_sub.csv&quot;)</code></pre>
<p>Each row in the dataset is an indivdual’s response to the survey and it has the following variables:</p>
<table>
<thead>
<tr class="header">
<th align="left">variable</th>
<th align="left">description</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td align="left"><code>stfdem</code></td>
<td align="left">Satisfaction with democracy: {1-10 scale} (ordinal)</td>
</tr>
<tr class="even">
<td align="left"><code>year</code></td>
<td align="left">Year: {2002, 2003, …, 2012} (time variable)</td>
</tr>
<tr class="odd">
<td align="left"><code>cntry</code></td>
<td align="left">Country: {DE, GB, …, NL} (cross section variable)</td>
</tr>
<tr class="even">
<td align="left"><code>crisis</code></td>
<td align="left">Crisis: {post, pre} (ordinal)</td>
</tr>
<tr class="odd">
<td align="left"><code>age_gr</code></td>
<td align="left">Age group: {18-34, 35-64, +65} (ordinal)</td>
</tr>
<tr class="even">
<td align="left"><code>edulvla</code></td>
<td align="left">Education level: {low, medium, high} (ordinal)</td>
</tr>
<tr class="odd">
<td align="left"><code>gndr</code></td>
<td align="left">Gender: {Men, Women} (categorical)</td>
</tr>
<tr class="even">
<td align="left"><code>peripherial</code></td>
<td align="left">Peripherial countries: {core, peri} (categorical)</td>
</tr>
</tbody>
</table>
</div>
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>Often the datasets we use to test our theories and hypotheses have some, and sometimes numerous, missing values (<code>NA</code>). What do we do?</p>
<p>One option would be to drop the rows that have a missing value for one of our covariates and/or the variable of interest. For example, when we estimate a linear model, <code>R</code> automatically gets rid of the rows with missing values for the variables in the model (“listwise deletion”). In this case 7,344 observations.</p>
<pre class="r"><code>model &lt;- lm(stfdem ~ crisis + age_gr + edulvla + gndr + peripherial,
            data = ess_sub)
summary(model)</code></pre>
<pre><code>## 
## Call:
## lm(formula = stfdem ~ crisis + age_gr + edulvla + gndr + peripherial, 
##     data = ess_sub)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.0930 -1.6938  0.1299  1.8842  5.6123 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(&gt;|t|)    
## (Intercept)      5.87012    0.01879 312.469  &lt; 2e-16 ***
## crisispre        0.22285    0.01177  18.931  &lt; 2e-16 ***
## age_gr18-34     -0.10679    0.01754  -6.087 1.15e-09 ***
## age_gr35-64     -0.20238    0.01500 -13.488  &lt; 2e-16 ***
## edulvlalow      -0.71168    0.01522 -46.752  &lt; 2e-16 ***
## edulvlamedium   -0.62842    0.01382 -45.470  &lt; 2e-16 ***
## gndrWomen       -0.24200    0.01134 -21.349  &lt; 2e-16 ***
## peripherialperi -0.32635    0.01316 -24.800  &lt; 2e-16 ***
## ---
## Signif. codes:  0 &#39;***&#39; 0.001 &#39;**&#39; 0.01 &#39;*&#39; 0.05 &#39;.&#39; 0.1 &#39; &#39; 1
## 
## Residual standard error: 2.414 on 182566 degrees of freedom
##   (7344 observations deleted due to missingness)
## Multiple R-squared:  0.02418,    Adjusted R-squared:  0.02414 
## F-statistic: 646.1 on 7 and 182566 DF,  p-value: &lt; 2.2e-16</code></pre>
<p>However, despite not having a value for a particular variable or set of variables, thoses row may provide information about the other variables. Moreover, dropping those observation may result in biased and inefficient estimates.</p>
<p>The other option is to <strong>impute</strong> (“fill in”) the missing values. In this lesson we will see one method and <code>R package</code> to do so: <code>Amelia</code>. This method assumes that the data:</p>
<ol style="list-style-type: lower-alpha">
<li>follows a multivariate normal distribution</li>
<li>is missing at random -MAR- (which means that the <em>missingness</em> depends only on the observed data)</li>
</ol>
<p>Then the <code>Amelia</code> uses a bootstrap-EM algorithm (EMB) to estimate/impute the missing values.</p>
</div>
<div id="exploring-missing-values" class="section level2">
<h2>Exploring missing values</h2>
<p>The <code>summary()</code> function provides you with information about the number of missing values per variable. This dataset has missing values for the outcome variable (<code>stfdem</code>) and the covariates <code>edulvla</code> and <code>gndr</code>.</p>
<pre class="r"><code>summary(ess_sub)</code></pre>
<pre><code>##      stfdem            year          cntry         crisis      
##  Min.   : 0.000   Min.   :2002   DE     : 16145   post: 68948  
##  1st Qu.: 4.000   1st Qu.:2004   GB     : 12521   pre :120970  
##  Median : 5.000   Median :2008   IE     : 12324                
##  Mean   : 5.189   Mean   :2007   PT     : 11774                
##  3rd Qu.: 7.000   3rd Qu.:2010   FI     : 11260                
##  Max.   :10.000   Max.   :2012   NL     : 11048                
##  NA&#39;s   :6405                    (Other):114846                
##    age_gr         edulvla         gndr        peripherial  
##  +65  : 40674   high  :55013   Men  : 88053   core:137997  
##  18-34: 46887   low   :61545   Women:101752   peri: 51921  
##  35-64:102357   medium:72443   NA&#39;s :   113                
##                 NA&#39;s  :  917                               
##                                                            
##                                                            
## </code></pre>
<p>The <code>Amelia</code> package has a function that helps visualizing the missing data in a dataset: <code>missmap()</code>. Warning: takes few minutes to run, depending on the size of the dataset.</p>
<pre class="r"><code>missmap(ess_sub)</code></pre>
<p><img src="lessons_imputation_files/figure-html/unnamed-chunk-5-1.png" title="" alt="" width="672" /></p>
</div>
<div id="imputation" class="section level2">
<h2>Imputation</h2>
<p>The <code>amelia()</code> function takes the following parameters:</p>
<ul>
<li><code>x</code>: the dataset (e.g. ess_sub)</li>
<li><code>m</code>: number of imputed datasets to create (e.g. 5, see Fox, p.564)</li>
<li><code>logs</code>: a vector with variables that are <code>log</code> transformations</li>
<li><code>logstc</code>: a vector with variables that are <code>logistic</code> transformations</li>
<li><code>noms</code>: a vector with variables that are nominal (e.g. <code>gnder</code>, <code>peripherial</code>)</li>
<li><code>ords</code>: a vector with variables that are ordinal (e.g. <code>stfdem</code>)</li>
<li><code>ts</code>: name of the variable indicating time (for time series data) (e.g. <code>year</code>)</li>
<li><code>cs</code>: name of the cross section variable (for cross section data) (e.g. <code>cntry</code>)</li>
<li><code>idvars</code>: name of a variablde indicating ID, so <code>Amelia</code> can ignore it</li>
</ul>
<p>The <code>amelia()</code> function will create <code>m</code> (so 5) new datasets with imputed values for the all missing values in<code>ess_sub</code>.</p>
<pre class="r"><code>m &lt;- 5
amelia_output &lt;- amelia(ess_sub, m = m, ts = &quot;year&quot; , cs = &quot;cntry&quot;,
                      ords = c(&quot;stfdem&quot;, &quot;crisis&quot;, &quot;age_gr&quot;,&quot;edulvla&quot;),
                      noms = c(&quot;gndr&quot;, &quot;peripherial&quot;))</code></pre>
<pre><code>## -- Imputation 1 --
## 
##   1  2
## 
## -- Imputation 2 --
## 
##   1  2
## 
## -- Imputation 3 --
## 
##   1  2
## 
## -- Imputation 4 --
## 
##   1  2
## 
## -- Imputation 5 --
## 
##   1  2</code></pre>
<p>The imputed datasets are in the <code>imputations</code> list within the <code>amelia_output</code>. You can combine them into one dataset using the following loop:</p>
<pre class="r"><code>ess_all &lt;- NULL
for (i in 1:length(amelia_output$imputations)) {
  imp &lt;- as.data.frame(amelia_output$imputations[i])
  colnames(imp) &lt;- colnames(ess_sub)
  imp$imp &lt;- paste0(&quot;imp&quot;, i)
  ess_all &lt;- rbind(ess_all, imp)
}</code></pre>
<p>You can also access a matrix indicating the original missing values in the <code>amelia_output</code>. This will be useful to compare differences between actual and imputed data and to make judgements about the quality/plausibility of the imputed data.</p>
<pre class="r"><code>missMatrix &lt;- amelia_output$missMatrix</code></pre>
</div>
<div id="diagnostics" class="section level2">
<h2>Diagnostics</h2>
<p>For the variables with missing values (e.g. <code>stfdem</code>, <code>edulvla</code>), explore differences across imputed datasets.</p>
<pre class="r"><code>ggplot(ess_all %&gt;%
         group_by(imp,stfdem)%&gt;%
         summarize(n = n()), aes(y = n, x = factor(stfdem))) +
  geom_bar(stat = &quot;identity&quot;) +
  geom_text(aes(label = n), vjust = 0.5, y = 1000, hjust = 0,position = ,
            angle = 90, color = &quot;white&quot;) +
  facet_wrap(~ imp)</code></pre>
<p><img src="lessons_imputation_files/figure-html/unnamed-chunk-9-1.png" title="" alt="" width="672" /></p>
<pre class="r"><code>ggplot(ess_all %&gt;%
         group_by(imp,edulvla)%&gt;%
         summarize(n = n()), aes(y = n, x = factor(edulvla))) +
  geom_bar(stat = &quot;identity&quot;) +
  geom_text(aes(label = n), vjust = 0.5, y = 20000, hjust = 0.5,position = ,
            color = &quot;white&quot;) +
  facet_wrap(~ imp)</code></pre>
<p><img src="lessons_imputation_files/figure-html/unnamed-chunk-10-1.png" title="" alt="" width="672" /></p>
<p>Check Jeff Arnold’s lesson on <a href="https://uw-pols503.github.io/pols_503_sp16/missing_data_imputation.html#models">Multiple Imputation</a> to see other ways of evaluating the plausibility of the imputated data.</p>
</div>
<div id="combining-imputated-datasets-in-your-analysis" class="section level2">
<h2>Combining imputated datasets in your analysis</h2>
<p>See Fox p.656 for details about how to combine the coefficients from different imputated datasets.</p>
<p>The <code>Amelia</code> package has a function to do so: <code>mi.meld()</code>. The function takes 2 paramters:</p>
<ul>
<li><code>q</code>: a dataset/matrix with <em>k</em> rows (<em>k</em> = number of imputated datasets) and <em>v</em> columns (<em>v</em> = number of covariates in the model). This dataset contains the coefficients for all covariates across imputated datasets.</li>
<li><code>se</code>: a dataset/matrix with <em>k</em> rows (<em>k</em> = number of imputated datasets) and <em>v</em> columns (<em>v</em> = number of covariates in the model). This dataset contains the standard errors for all covariates across imputated datasets.</li>
</ul>
<p>Calculate <code>q</code> and <code>se</code></p>
<pre class="r"><code>q &lt;- NULL
se &lt;- NULL
form &lt;- formula(stfdem ~ + crisis + age_gr + edulvla + gndr + peripherial)
for (i in 1:m) {
  model &lt;- lm(form, data = amelia_output$imputations[[i]])
  q &lt;- rbind(q, coef(model))
  se &lt;- rbind(se, coef(summary(model))[,2])
}
q</code></pre>
<pre><code>##      (Intercept) crisispre age_gr18-34 age_gr35-64 edulvlalow
## [1,]    5.874327 0.2180225 -0.11535668  -0.2099701 -0.6956348
## [2,]    5.867108 0.2221274 -0.11024628  -0.2062861 -0.6938102
## [3,]    5.867050 0.2222165 -0.09941342  -0.2009758 -0.6930619
## [4,]    5.868573 0.2184105 -0.10471958  -0.2029573 -0.6904310
## [5,]    5.864418 0.2221892 -0.10514010  -0.1998890 -0.6960684
##      edulvlamedium  gndrWomen peripherialperi
## [1,]    -0.6297112 -0.2389228      -0.3270229
## [2,]    -0.6276297 -0.2380649      -0.3295379
## [3,]    -0.6298683 -0.2419523      -0.3363500
## [4,]    -0.6307592 -0.2403058      -0.3271767
## [5,]    -0.6299122 -0.2383574      -0.3305466</code></pre>
<pre class="r"><code>se</code></pre>
<pre><code>##      (Intercept)  crisispre age_gr18-34 age_gr35-64 edulvlalow
## [1,]  0.01855185 0.01160933  0.01719708  0.01468296 0.01498298
## [2,]  0.01855029 0.01161053  0.01719835  0.01468398 0.01498446
## [3,]  0.01854366 0.01160459  0.01719172  0.01467780 0.01497655
## [4,]  0.01854561 0.01160602  0.01719158  0.01467775 0.01498019
## [5,]  0.01855177 0.01160988  0.01719767  0.01468360 0.01498452
##      edulvlamedium  gndrWomen peripherialperi
## [1,]    0.01372982 0.01118389      0.01293630
## [2,]    0.01372948 0.01118494      0.01293854
## [3,]    0.01372459 0.01117896      0.01292999
## [4,]    0.01372468 0.01118066      0.01293270
## [5,]    0.01372955 0.01118407      0.01293638</code></pre>
<pre class="r"><code>cb_model &lt;- mi.meld(q = q, se = se)
cb_model_df &lt;- data.frame(varname = colnames(cb_model[[1]]),
                          coef = as.numeric(cb_model[[1]]),
                          se = as.numeric(cb_model[[2]]),
                          coef.lwr = as.numeric(cb_model[[1]] - (2 * as.numeric(cb_model[[2]]))),
                          coef.upr = as.numeric(cb_model[[1]] + (2 * as.numeric(cb_model[[2]]))))
cb_model_df</code></pre>
<pre><code>##           varname       coef         se   coef.lwr    coef.upr
## 1     (Intercept)  5.8682952 0.01898387  5.8303274  5.90626290
## 2       crisispre  0.2205932 0.01184989  0.1968934  0.24429300
## 3     age_gr18-34 -0.1069752 0.01842968 -0.1438346 -0.07011585
## 4     age_gr35-64 -0.2040157 0.01536028 -0.2347362 -0.17329510
## 5      edulvlalow -0.6938013 0.01518468 -0.7241706 -0.66343191
## 6   edulvlamedium -0.6295761 0.01378656 -0.6571492 -0.60200299
## 7       gndrWomen -0.2395207 0.01132061 -0.2621619 -0.21687945
## 8 peripherialperi -0.3301268 0.01358620 -0.3572992 -0.30295442</code></pre>
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