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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.
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.
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# 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 0.00000000Fitting a model is very similar
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:
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 3.141636VarCorr(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 1.0000000If 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:
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 3.177518And, you can always operate on any single element of the list:
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 -0.228summary(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 = 5.970fastdisp(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.
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 1.0000000Let’s apply this to our model list.
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 5.973125summary(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 = 5.970modelFixedEff(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 166028.05ranef(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 0.0331380964Often 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.
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