---
title: "Analyzing Imputed Data with Multilevel Models and merTools"
author: "Jared Knowles"
date: "2020-06-22"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Imputation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---




## Introduction

Multilevel models are valuable in a wide array of problem areas that involve
non-experimental, or observational data. In many of these cases the data on
individual observations may be incomplete. In these situations, the analyst may
turn to one of many methods for filling in missing data depending on the specific
problem at hand, disciplinary norms, and prior research.

One of the most common cases is to use multiple imputation. Multiple imputation
involves fitting a model to the data and estimating the missing values for
observations. For details on multiple imputation, and a discussion of some of
the main implementations in R, look at the documentation and vignettes for the
`mice` and `Amelia` packages.

The key difficulty multiple imputation creates for users of multilevel models
is that the result of multiple imputation is K replicated datasets corresponding
to different estimated values for the missing data in the original dataset.

For the purposes of this vignette, I will describe how to use one flavor of
multiple imputation and the function in `merTools` to obtain estimates from a
multilevel model in the presence of missing and multiply imputed data.


## Missing Data and its Discontents

To demonstrate this workflow, we will use the `hsb` dataset in the `merTools`
package which includes data on the math achievement of a wide sample of students
nested within schools. The data has no missingness, so first we will simulate
some missing data.


```r
data(hsb)

# Create a function to randomly assign NA values

add_NA <- function(x, prob){
  z <- rbinom(length(x), 1, prob = prob)
  x[z==1] <- NA
  return(x)
}

hsb$minority <- add_NA(hsb$minority, prob = 0.05)
table(is.na(hsb$minority))
#> 
#> FALSE  TRUE 
#>  6868   317

hsb$female <- add_NA(hsb$female, prob = 0.05)
table(is.na(hsb$female))
#> 
#> FALSE  TRUE 
#>  6802   383

hsb$ses <- add_NA(hsb$ses, prob = 0.05)
table(is.na(hsb$ses))
#> 
#> FALSE  TRUE 
#>  6803   382

hsb$size <- add_NA(hsb$size, prob = 0.05)
table(is.na(hsb$size))
#> 
#> FALSE  TRUE 
#>  6825   360
```


```r
# Load imputation library
library(Amelia)
# Declare the variables to include in the imputation data
varIndex <- names(hsb)
# Declare ID variables to be excluded from imputation
IDS <- c("schid", "meanses")
# Imputate
impute.out <- amelia(hsb[, varIndex], idvars = IDS,
                         noms = c("minority", "female"),
                         m = 5)
#> -- Imputation 1 --
#> 
#>   1  2  3  4
#> 
#> -- Imputation 2 --
#> 
#>   1  2  3
#> 
#> -- Imputation 3 --
#> 
#>   1  2  3
#> 
#> -- Imputation 4 --
#> 
#>   1  2  3
#> 
#> -- Imputation 5 --
#> 
#>   1  2  3
summary(impute.out)
#> 
#> Amelia output with 5 imputed datasets.
#> Return code:  1 
#> Message:  Normal EM convergence. 
#> 
#> Chain Lengths:
#> --------------
#> Imputation 1:  4
#> Imputation 2:  3
#> Imputation 3:  3
#> Imputation 4:  3
#> Imputation 5:  3
#> 
#> Rows after Listwise Deletion:  5853 
#> Rows after Imputation:  7185 
#> Patterns of missingness in the data:  14 
#> 
#> Fraction Missing for original variables: 
#> -----------------------------------------
#> 
#>          Fraction Missing
#> schid          0.00000000
#> minority       0.04411969
#> female         0.05330550
#> ses            0.05316632
#> mathach        0.00000000
#> size           0.05010438
#> schtype        0.00000000
#> meanses        0.00000000
```


```r
# Amelia is not available so let's just boostrap resample our data
impute.out <- vector(mode = "list", 5)

for (i in 1:5) {
  impute.out[[i]] <- hsb[sample(nrow(hsb), nrow(hsb), replace = TRUE), ]
}

# Declare the variables to include in the imputation data
summary(impute.out)
```


## Fitting and Summarizing a Model List

Fitting a model is very similar


```r
fmla <- "mathach ~ minority + female + ses + meanses + (1 + ses|schid)"
mod <- lmer(fmla, data = hsb)
if(amelia_eval) {
  modList <- lmerModList(fmla, data = impute.out$imputations)
} else {
  # Use bootstrapped data instead
  modList <- lmerModList(fmla, data = impute.out)
}
```


The resulting object `modList` is a list of `merMod` objects the same length as the number
of imputation datasets. This object is assigned the class of `merModList` and
`merTools` provides some convenience functions for reporting the results of
this object.

Using this, we can directly compare the model fit with missing data excluded to
the aggregate from the imputed models:


```r
fixef(mod) # model with dropped missing
#> (Intercept)    minority      female         ses     meanses 
#>   14.149102   -2.868687   -1.318437    2.067309    2.833490
fixef(modList)
#> (Intercept)    minority      female         ses     meanses 
#>   14.028792   -2.680352   -1.213086    1.966725    3.141636
```


```r
VarCorr(mod) # model with dropped missing
#>  Groups   Name        Std.Dev. Corr  
#>  schid    (Intercept) 1.54204        
#>           ses         0.52515  -0.765
#>  Residual             5.98842
VarCorr(modList) # aggregate of imputed models
#> $stddev
#> $stddev$schid
#> (Intercept)         ses 
#>   1.5183804   0.6468874 
#> 
#> 
#> $correlation
#> $correlation$schid
#>             (Intercept)        ses
#> (Intercept)   1.0000000 -0.5247666
#> ses          -0.5247666  1.0000000
```

If you want to inspect the individual models, or you do not like taking the
mean across the imputation replications, you can take the `merModList` apart
easily:


```r
lapply(modList, fixef)
#> $imp1
#> (Intercept)    minority      female         ses     meanses 
#>   13.976636   -2.587948   -1.170291    1.984663    3.170845 
#> 
#> $imp2
#> (Intercept)    minority      female         ses     meanses 
#>   14.070484   -2.673140   -1.294932    1.959564    3.143996 
#> 
#> $imp3
#> (Intercept)    minority      female         ses     meanses 
#>   14.040516   -2.728450   -1.215497    1.958265    3.134720 
#> 
#> $imp4
#> (Intercept)    minority      female         ses     meanses 
#>   14.030150   -2.698588   -1.214679    1.997264    3.081103 
#> 
#> $imp5
#> (Intercept)    minority      female         ses     meanses 
#>   14.026175   -2.713636   -1.170030    1.933870    3.177518
```

And, you can always operate on any single element of the list:


```r
fixef(modList[[1]])
#> (Intercept)    minority      female         ses     meanses 
#>   13.976636   -2.587948   -1.170291    1.984663    3.170845
fixef(modList[[2]])
#> (Intercept)    minority      female         ses     meanses 
#>   14.070484   -2.673140   -1.294932    1.959564    3.143996
```

## Output of a Model List


```r
print(modList)
#> $imp1
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: mathach ~ minority + female + ses + meanses + (1 + ses | schid)
#>    Data: d
#> 
#> REML criterion at convergence: 46328.3
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.2652 -0.7199  0.0371  0.7614  2.9108 
#> 
#> Random effects:
#>  Groups   Name        Variance Std.Dev. Corr 
#>  schid    (Intercept)  2.2763  1.5087        
#>           ses          0.3676  0.6063   -0.61
#>  Residual             35.7568  5.9797        
#> Number of obs: 7185, groups:  schid, 160
#> 
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)  13.9766     0.1724  81.089
#> minority     -2.5879     0.1994 -12.978
#> female       -1.1703     0.1576  -7.425
#> ses           1.9847     0.1182  16.787
#> meanses       3.1708     0.3537   8.966
#> 
#> Correlation of Fixed Effects:
#>          (Intr) minrty female ses   
#> minority -0.324                     
#> female   -0.482  0.012              
#> ses      -0.234  0.140  0.036       
#> meanses  -0.102  0.126  0.023 -0.237
#> 
#> $imp2
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: mathach ~ minority + female + ses + meanses + (1 + ses | schid)
#>    Data: d
#> 
#> REML criterion at convergence: 46308.7
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.2162 -0.7183  0.0385  0.7576  2.9117 
#> 
#> Random effects:
#>  Groups   Name        Variance Std.Dev. Corr 
#>  schid    (Intercept)  2.286   1.5118        
#>           ses          0.443   0.6656   -0.47
#>  Residual             35.611   5.9675        
#> Number of obs: 7185, groups:  schid, 160
#> 
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)  14.0705     0.1727  81.485
#> minority     -2.6731     0.1985 -13.467
#> female       -1.2949     0.1578  -8.205
#> ses           1.9596     0.1202  16.299
#> meanses       3.1440     0.3574   8.797
#> 
#> Correlation of Fixed Effects:
#>          (Intr) minrty female ses   
#> minority -0.326                     
#> female   -0.482  0.019              
#> ses      -0.204  0.140  0.038       
#> meanses  -0.094  0.127  0.023 -0.231
#> 
#> $imp3
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: mathach ~ minority + female + ses + meanses + (1 + ses | schid)
#>    Data: d
#> 
#> REML criterion at convergence: 46302.4
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.2651 -0.7164  0.0325  0.7615  2.9216 
#> 
#> Random effects:
#>  Groups   Name        Variance Std.Dev. Corr 
#>  schid    (Intercept)  2.3422  1.5304        
#>           ses          0.4413  0.6643   -0.46
#>  Residual             35.5652  5.9637        
#> Number of obs: 7185, groups:  schid, 160
#> 
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)  14.0405     0.1738  80.763
#> minority     -2.7284     0.1990 -13.709
#> female       -1.2155     0.1578  -7.702
#> ses           1.9583     0.1198  16.345
#> meanses       3.1347     0.3595   8.719
#> 
#> Correlation of Fixed Effects:
#>          (Intr) minrty female ses   
#> minority -0.325                     
#> female   -0.481  0.022              
#> ses      -0.209  0.143  0.044       
#> meanses  -0.092  0.126  0.021 -0.226
#> 
#> $imp4
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: mathach ~ minority + female + ses + meanses + (1 + ses | schid)
#>    Data: d
#> 
#> REML criterion at convergence: 46302
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.2610 -0.7229  0.0305  0.7612  2.9166 
#> 
#> Random effects:
#>  Groups   Name        Variance Std.Dev. Corr 
#>  schid    (Intercept)  2.3036  1.5178        
#>           ses          0.3951  0.6286   -0.62
#>  Residual             35.6111  5.9675        
#> Number of obs: 7185, groups:  schid, 160
#> 
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)  14.0302     0.1728  81.179
#> minority     -2.6986     0.1985 -13.592
#> female       -1.2147     0.1573  -7.721
#> ses           1.9973     0.1190  16.784
#> meanses       3.0811     0.3544   8.693
#> 
#> Correlation of Fixed Effects:
#>          (Intr) minrty female ses   
#> minority -0.326                     
#> female   -0.481  0.021              
#> ses      -0.246  0.140  0.040       
#> meanses  -0.104  0.126  0.023 -0.235
#> 
#> $imp5
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: mathach ~ minority + female + ses + meanses + (1 + ses | schid)
#>    Data: d
#> 
#> REML criterion at convergence: 46324.3
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.2703 -0.7181  0.0316  0.7649  2.9098 
#> 
#> Random effects:
#>  Groups   Name        Variance Std.Dev. Corr 
#>  schid    (Intercept)  2.3200  1.5231        
#>           ses          0.4484  0.6696   -0.46
#>  Residual             35.6782  5.9731        
#> Number of obs: 7185, groups:  schid, 160
#> 
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)  14.0262     0.1734  80.890
#> minority     -2.7136     0.1982 -13.689
#> female       -1.1700     0.1577  -7.417
#> ses           1.9339     0.1204  16.060
#> meanses       3.1775     0.3594   8.842
#> 
#> Correlation of Fixed Effects:
#>          (Intr) minrty female ses   
#> minority -0.329                     
#> female   -0.480  0.026              
#> ses      -0.200  0.141  0.036       
#> meanses  -0.095  0.126  0.026 -0.228
```


```r
summary(modList)
#> [1] "Linear mixed model fit by REML"
#> Model family: 
#> lmer(formula = mathach ~ minority + female + ses + meanses + 
#>     (1 + ses | schid), data = d)
#> 
#> Fixed Effects:
#>             estimate std.error statistic         df
#> (Intercept)   14.029     0.174    80.566  99310.593
#> female        -1.213     0.160    -7.574  16493.051
#> meanses        3.142     0.358     8.769 259740.570
#> minority      -2.680     0.202   -13.289  18540.839
#> ses            1.967     0.120    16.372 166028.049
#> 
#> Random Effects:
#> 
#> Error Term Standard Deviations by Level:
#> 
#> schid
#> (Intercept)         ses 
#>       1.518       0.647 
#> 
#> 
#> Error Term Correlations:
#> 
#> schid
#>             (Intercept) ses   
#> (Intercept)  1.000      -0.525
#> ses         -0.525       1.000
#> 
#> 
#> Residual Error = 5.970 
#> 
#> ---Groups
#> number of obs: 7185, groups: schid, 160
#> 
#> Model Fit Stats
#> AIC = 46331.1
#> Residual standard deviation = 5.970
```


```r
fastdisp(modList)
#> lmer(formula = mathach ~ minority + female + ses + meanses + 
#>     (1 + ses | schid), data = d)
#>             estimate std.error
#> (Intercept)    14.03      0.17
#> female         -1.21      0.16
#> meanses         3.14      0.36
#> minority       -2.68      0.20
#> ses             1.97      0.12
#> 
#> Error terms:
#>  Groups   Name        Std.Dev. Corr  
#>  schid    (Intercept) 1.52           
#>           ses         0.65     -0.61 
#>  Residual             5.97           
#> ---
#> number of obs: 7185, groups: schid, 160
#> AIC = 46331.1---
```

The standard errors reported for the model list include a correction, Rubin's
correction (see documentation), which adjusts for the within and between imputation
set variance as well.

## Specific Model Information Summaries


```r
modelRandEffStats(modList)
#>                        term    group   estimate   std.error
#> 1 cor_(Intercept).ses.schid    schid -0.5247666 0.084101895
#> 2      sd_(Intercept).schid    schid  1.5183804 0.008713530
#> 3   sd_Observation.Residual Residual  5.9703034 0.006244066
#> 4              sd_ses.schid    schid  0.6468874 0.028062351
modelFixedEff(modList)
#>          term  estimate std.error  statistic        df
#> 1 (Intercept) 14.028792 0.1741275  80.566201  99310.59
#> 2      female -1.213086 0.1601572  -7.574345  16493.05
#> 3     meanses  3.141636 0.3582833   8.768580 259740.57
#> 4    minority -2.680352 0.2017037 -13.288566  18540.84
#> 5         ses  1.966725 0.1201239  16.372467 166028.05
VarCorr(modList)
#> $stddev
#> $stddev$schid
#> (Intercept)         ses 
#>   1.5183804   0.6468874 
#> 
#> 
#> $correlation
#> $correlation$schid
#>             (Intercept)        ses
#> (Intercept)   1.0000000 -0.5247666
#> ses          -0.5247666  1.0000000
```

### Diagnostics of List Components


```r
modelInfo(mod)
#>   n.obs n.lvls      AIC   sigma
#> 1  6160      1 39764.15 5.98842
```


Let's apply this to our model list.


```r
lapply(modList, modelInfo)
#> $imp1
#>   n.obs n.lvls      AIC    sigma
#> 1  7185      1 46346.34 5.979699
#> 
#> $imp2
#>   n.obs n.lvls      AIC    sigma
#> 1  7185      1 46326.72 5.967532
#> 
#> $imp3
#>   n.obs n.lvls      AIC    sigma
#> 1  7185      1 46320.43 5.963655
#> 
#> $imp4
#>   n.obs n.lvls      AIC    sigma
#> 1  7185      1 46319.96 5.967506
#> 
#> $imp5
#>   n.obs n.lvls      AIC    sigma
#> 1  7185      1 46342.27 5.973125
```

### Model List Generics


```r
summary(modList)
#> [1] "Linear mixed model fit by REML"
#> Model family: 
#> lmer(formula = mathach ~ minority + female + ses + meanses + 
#>     (1 + ses | schid), data = d)
#> 
#> Fixed Effects:
#>             estimate std.error statistic         df
#> (Intercept)   14.029     0.174    80.566  99310.593
#> female        -1.213     0.160    -7.574  16493.051
#> meanses        3.142     0.358     8.769 259740.570
#> minority      -2.680     0.202   -13.289  18540.839
#> ses            1.967     0.120    16.372 166028.049
#> 
#> Random Effects:
#> 
#> Error Term Standard Deviations by Level:
#> 
#> schid
#> (Intercept)         ses 
#>       1.518       0.647 
#> 
#> 
#> Error Term Correlations:
#> 
#> schid
#>             (Intercept) ses   
#> (Intercept)  1.000      -0.525
#> ses         -0.525       1.000
#> 
#> 
#> Residual Error = 5.970 
#> 
#> ---Groups
#> number of obs: 7185, groups: schid, 160
#> 
#> Model Fit Stats
#> AIC = 46331.1
#> Residual standard deviation = 5.970
```


```r
modelFixedEff(modList)
#>          term  estimate std.error  statistic        df
#> 1 (Intercept) 14.028792 0.1741275  80.566201  99310.59
#> 2      female -1.213086 0.1601572  -7.574345  16493.05
#> 3     meanses  3.141636 0.3582833   8.768580 259740.57
#> 4    minority -2.680352 0.2017037 -13.288566  18540.84
#> 5         ses  1.966725 0.1201239  16.372467 166028.05
```


```r
ranef(modList)
#> $schid
#>       (Intercept)           ses
#> 1224 -0.157795533  0.0451127840
#> 1288 -0.044476754  0.0191957958
#> 1296 -0.126472259  0.0218757135
#> 1308  0.064357632 -0.0167977336
#> 1317  0.088861755 -0.0350837887
#> 1358 -0.301385760  0.1053888143
#> 1374 -0.350736225  0.1064976917
#> 1433  0.307310844 -0.0444663946
#> 1436  0.284513686 -0.0602282100
#> 1461 -0.045882842  0.0719067703
#> 1462  0.348424677 -0.1562366964
#> 1477  0.042686687 -0.0406549686
#> 1499 -0.293156885  0.0838236409
#> 1637 -0.097080749  0.0324268391
#> 1906  0.048446937 -0.0150064112
#> 1909 -0.052969237  0.0205894104
#> 1942  0.209581012 -0.0525053879
#> 1946 -0.042287233  0.0350616964
#> 2030 -0.429112816  0.0588461805
#> 2208 -0.024593477  0.0228554436
#> 2277  0.309800057 -0.1834173408
#> 2305  0.550610497 -0.2049548526
#> 2336  0.142313348 -0.0290535691
#> 2458  0.245993091 -0.0255602587
#> 2467 -0.222494935  0.0640753511
#> 2526  0.449997476 -0.1312121315
#> 2626  0.027751982  0.0238061610
#> 2629  0.335613322 -0.0942540137
#> 2639  0.094386542 -0.0820201077
#> 2651 -0.393517983  0.1350175898
#> 2655  0.640384122 -0.1435806679
#> 2658 -0.243275105  0.0607205634
#> 2755  0.135787228 -0.0631922841
#> 2768 -0.268666958  0.0917130815
#> 2771  0.033436716  0.0272030521
#> 2818 -0.018785461  0.0214043728
#> 2917  0.152738008 -0.0762445189
#> 2990  0.448844959 -0.0935887501
#> 2995 -0.235287167  0.0148768819
#> 3013 -0.106680710  0.0516779815
#> 3020  0.090727137 -0.0308716386
#> 3039  0.243996619 -0.0435977108
#> 3088 -0.042231336 -0.0122411932
#> 3152 -0.034103349  0.0356155581
#> 3332 -0.259777846  0.0305681683
#> 3351 -0.461248418  0.0996270996
#> 3377  0.142496875 -0.1211102758
#> 3427  0.841386693 -0.2339682964
#> 3498  0.024887322 -0.0537205006
#> 3499 -0.119817169  0.0080680143
#> 3533 -0.149220939  0.0010719643
#> 3610  0.297746069 -0.0014053243
#> 3657 -0.069261452  0.0633533767
#> 3688 -0.061555723  0.0315302117
#> 3705 -0.427141188  0.0523408834
#> 3716  0.061285137  0.0757199239
#> 3838  0.485386271 -0.1598435378
#> 3881 -0.309537022  0.0860578519
#> 3967 -0.056525049  0.0445060296
#> 3992  0.075297122 -0.0637600889
#> 3999 -0.055817277  0.0457642823
#> 4042 -0.197812746  0.0313570583
#> 4173 -0.082777595  0.0432272733
#> 4223  0.266360906 -0.0698408106
#> 4253 -0.002838943 -0.0732012994
#> 4292  0.495110532 -0.1764400335
#> 4325  0.021047068  0.0103006817
#> 4350 -0.262817422  0.1005502052
#> 4383 -0.234756733  0.0855789496
#> 4410 -0.063023118  0.0284242048
#> 4420  0.205737288 -0.0273245989
#> 4458 -0.043787877 -0.0105867355
#> 4511  0.216198981 -0.0590666506
#> 4523 -0.253392354  0.0623924215
#> 4530  0.061007622 -0.0141412262
#> 4642  0.120939515 -0.0012115746
#> 4868 -0.225562808  0.0092349324
#> 4931 -0.151489897 -0.0105474646
#> 5192 -0.244884720  0.0662313861
#> 5404 -0.267282666  0.0289963481
#> 5619 -0.088591305  0.1050668069
#> 5640  0.066352031  0.0263435429
#> 5650  0.496007374 -0.1520751279
#> 5667 -0.291090712  0.0849233773
#> 5720  0.091591369 -0.0101163734
#> 5761  0.134959735  0.0032009015
#> 5762 -0.090505308  0.0088358929
#> 5783 -0.093105251  0.0419784658
#> 5815 -0.180032189  0.0567256485
#> 5819 -0.324949316  0.0664861258
#> 5838 -0.038168235  0.0005292275
#> 5937  0.040928181 -0.0176469977
#> 6074  0.361576085 -0.1098990853
#> 6089  0.230329688 -0.0455594013
#> 6144 -0.272422991  0.0809874046
#> 6170  0.279563058 -0.0545497420
#> 6291  0.181117957 -0.0356960554
#> 6366  0.193708113 -0.0594649551
#> 6397  0.183418370 -0.0437084542
#> 6415 -0.082399227  0.0577125726
#> 6443 -0.098586726 -0.0413265591
#> 6464 -0.006930839 -0.0110530398
#> 6469  0.342855296 -0.0923368634
#> 6484  0.099185197 -0.0332806845
#> 6578  0.317864661 -0.0765973348
#> 6600 -0.226249834  0.1266724638
#> 6808 -0.331443100  0.0659644663
#> 6816  0.197569880 -0.0620211170
#> 6897  0.032147952  0.0304756664
#> 6990 -0.298601140  0.0257587587
#> 7011  0.061065847  0.0284790004
#> 7101 -0.108095935  0.0111424320
#> 7172 -0.200642122  0.0236336161
#> 7232 -0.031354643  0.0561977605
#> 7276 -0.071317368  0.0498968187
#> 7332  0.036955530  0.0115037701
#> 7341 -0.284857609  0.0196994369
#> 7342  0.071738535 -0.0234087825
#> 7345 -0.246456373  0.0990572950
#> 7364  0.281626879 -0.0887844808
#> 7635  0.067695672 -0.0045773702
#> 7688  0.594207877 -0.1591233000
#> 7697  0.094743826 -0.0012484228
#> 7734  0.033326916  0.0503135537
#> 7890 -0.289921123  0.0298758440
#> 7919 -0.149007142  0.0495897913
#> 8009 -0.244371368  0.0271887171
#> 8150  0.064657992 -0.0398760061
#> 8165  0.175619037 -0.0474879689
#> 8175  0.106248119 -0.0365013872
#> 8188 -0.114131805  0.0573622366
#> 8193  0.542176501 -0.1395995719
#> 8202 -0.224686594  0.0855379047
#> 8357  0.189677518 -0.0218034980
#> 8367 -0.753895035  0.1525305352
#> 8477  0.074297200  0.0168614134
#> 8531 -0.205339027  0.0413324032
#> 8627 -0.378034984  0.0380125197
#> 8628  0.607613395 -0.1688034840
#> 8707 -0.085939080  0.0478432971
#> 8775 -0.201067311  0.0092501597
#> 8800 -0.001740915  0.0111088913
#> 8854 -0.559785941  0.1509853643
#> 8857  0.264207656 -0.0929013046
#> 8874  0.185982681 -0.0115522511
#> 8946 -0.167392474  0.0227325069
#> 8983 -0.141209027  0.0250288618
#> 9021 -0.240450945  0.0425264700
#> 9104 -0.041255449  0.0031660145
#> 9158 -0.281158323  0.1121016974
#> 9198  0.321737680 -0.0485854075
#> 9225  0.003967024  0.0297600149
#> 9292  0.236024371 -0.0736002233
#> 9340 -0.017193371  0.0168080235
#> 9347 -0.055089446  0.0615863493
#> 9359 -0.048633702 -0.0193351608
#> 9397 -0.475984110  0.1004098564
#> 9508  0.106191420 -0.0031993549
#> 9550 -0.265395980  0.0857421977
#> 9586 -0.141583246  0.0331380964
```


## Cautions and Notes

Often it is desirable to include aggregate values in the level two or level three
part of the model such as level 1 SES and level 2 mean SES for the group. In
cases where there is missingness in either the level 1 SES values, or in the
level 2 mean SES values, caution and careful thought need to be given to how to
proceed with the imputation routine.