How can I get pooled random effects for lmer after multiple imputation?

I am using mice to multiple impute a dataframe. And lme4 for a mixed model with random intercept and random slope. Pooling lmer goes fine, except that it doesn't pool the random effects. I have searched a lot for a solution with out any luck. I tried the mi package, however I only see pooled output for the estimate and std.error. I've tried exporting mice object to spss without any luck. I saw some discussion on Zelig. I thought that might solve my problem. I was however unable to figure out how to use the package with imputed data for lmer.

I know the mice package only supports pooling the fixed effects. Is there a work around?

Multiple imputation:

Data <- subset(Data0, select=c(id, faculty, gender, age, age_sqr, occupation, degree, private_sector, overtime, wage))
ini <- mice(Data, maxit=0, pri=F) #get predictor matrix
pred <- ini$pred
    pred[,"id"] <- 0 #don't use id as predictor
    meth <- ini$meth
meth[c("id", "faculty", "gender", "age", "age_sqr", "occupation", "degree", "private_sector", "overtime", "wage")] <- "" #don't impute these variables, use only as predictors.
imp <- mice(Data, m=22, maxit=10, printFlag=TRUE, pred=pred, meth=meth) #impute Data with 22 imputations and 10 iterations. 

Multilevel model:

    fm1 <- with(imp, lmer(log(wage) ~ gender + age + age_sqr + occupation + degree + private_sector + overtime + (1+gender|faculty))) #my multilevel model
    summary(est <- pool(fm1)) #pool my results

Update Results from pooled lmer:

> summary(est <- pool(fm1))
                                est           se            t       df     Pr(>|t|)         lo 95         hi 95 nmis       fmi    lambda
(Intercept)   7,635148e+00 0,1749178710 43,649905006 212,5553 0,000000e+00  7,2903525425  7,9799443672   NA 0,2632782 0,2563786
Gender        -1,094186e-01 0,0286629154 -3,817427078 117,1059 2,171066e-04 -0,1661834550 -0,0526537238   NA 0,3846276 0,3742069
Occupation1   1,125022e-01 0,0250082538  4,498601518 157,6557 1,320753e-05  0,0631077322  0,1618966049   NA 0,3207350 0,3121722
Occupation2   2,753089e-02 0,0176032487  1,563966385 215,6197 1,192919e-01 -0,0071655902  0,0622273689   NA 0,2606725 0,2538465
Occupation3   1,881908e-04 0,0221992053  0,008477365 235,3705 9,932433e-01 -0,0435463305  0,0439227120   NA 0,2449795 0,2385910
Age           1,131147e-02 0,0087366178  1,294719230 187,0021 1,970135e-01 -0,0059235288  0,0285464629    0 0,2871640 0,2795807
Age_sqr       -7,790476e-05 0,0001033263 -0,753968159 185,4630 4,518245e-01 -0,0002817508  0,0001259413    0 0,2887420 0,2811131
Overtime      -2,376501e-03 0,0004065466 -5,845581504 243,3563 1,614693e-08 -0,0031773002 -0,0015757019    9 0,2391179 0,2328903
Private_sector  8,322438e-02 0,0203047665  4,098760934 371,9971 5,102752e-05  0,0432978716  0,1231508962   NA 0,1688478 0,1643912

This information is missing, which I get when running lmer without multiple imputation:

Random effects:
 Groups   Name        Variance Std.Dev. Corr
 Faculty  (Intercept) 0,008383 0,09156      
          Genderfemale0,002240 0,04732  1,00
 Residual             0,041845 0,20456      
Number of obs: 698, groups:  Faculty, 17
  • $\begingroup$ Is your problem that you don't know how to characterize RE uncertainty after MI? I don't get what procedures your code is trying to do. $\endgroup$ Commented Oct 2, 2014 at 11:27
  • $\begingroup$ (1+gender|faculty) : gender as random slope, faculty as random intercept. I'm trying to get pooled results from all 22 imputations for the random effects (gender and faculty) $\endgroup$ Commented Oct 2, 2014 at 11:38
  • $\begingroup$ Small update. When I multiple impute in SPSS and run a mixed model; SPSS only pools the fixed effects, not the random effects. Same goes for the mi package for R. I am starting to think that this is not possible to do. $\endgroup$ Commented Oct 3, 2014 at 8:45
  • 2
    $\begingroup$ In reply to Helgi: It is statistically possible to do - Stata provides pooled estimates of variance components estimates after using multiple imputation. The only difficulty is obtaining the estimates and standard errors of the variance components, and that the pooling should be done on a scale for which the posterior is approximately normal. I believe Stata does the pooling on the log standard deviation scale to make the approximation more reasonable. $\endgroup$ Commented Sep 13, 2015 at 8:55

2 Answers 2


You can do this somewhat by hand if by taking advantage of the lapply functionality in R and the list-structure returned by the Amelia multiple imputation package. Here's a quick example script.

library(plyr) # for collapsing estimates

Amelia is similar to mice so you can just substitute your variables in from the mice call here -- this example is from a project I was working on.

 a.out <- amelia(dat[sub1, varIndex], idvars = "SCH_ID", 
            noms = varIndex[!varIndex %in% c("SCH_ID", "math12")], 
            m = 10)

a.out is the imputation object, now we need to run the model on each imputed dataset. To do this, we use the lapply function in R to repeat a function over list elements. This function applies the function -- which is the model specification -- to each dataset (d) in the list and returns the results in a list of models.

 mods <- lapply(a.out$imputations,
           function(d) lmer((log(wage) ~ gender + age + age_sqr + 
            occupation + degree + private_sector + overtime + 
             (1+gender|faculty), data = d)

Now we create a data.frame from that list, by simulating the values the fixed and random effects using the functions FEsim and REsim from the merTools package

imputeFEs <- ldply(mods, FEsim, nsims = 1000)
imputeREs <- ldply(mods, REsim, nsims = 1000)

The data.frames above include separate estimates for each dataset, now we need to combine them using a collapse like argument collapse

imputeREs <- ddply(imputeREs, .(X1, X2), summarize, mean = mean(mean), 
               median = mean(median), sd = mean(sd), 
               level = level[1])

imputeFEs <- ddply(imputeFEs, .(var), summarize, meanEff = mean(meanEff), 
               medEff = mean(medEff), sdEff = mean(sdEff))

Now we can also extract some statistics on the variance/covariance for the random effects across the imputed values. Here I have written two simple extractor functions to do this.

REsdExtract <- function(model){
  out <- unlist(lapply(VarCorr(model), attr, "stddev"))

REcorrExtract <- function(model){
  out <- unlist(lapply(VarCorr(model), attr, "corre"))

And now we can apply them to the models and store them as a vector:

modStats <- cbind(ldply(mods, REsdExtract), ldply(mods, REcorrExtract))


The functions below will get you much closer to the output provided by arm::display by operating on the list of lmer or glmer objects. Hopefully this will be incorporated into the merTools package in the near future:

# Functions to extract standard deviation of random effects from model
REsdExtract <- function(model){
  out <- unlist(lapply(VarCorr(model), attr, "stddev"))

#slope intercept correlation from model
REcorrExtract <- function(model){
  out <- unlist(lapply(VarCorr(model), attr, "corre"))

modelRandEffStats <- function(modList){
  SDs <- ldply(modList, REsdExtract)
  corrs <- ldply(modList, REcorrExtract)
  tmp <- cbind(SDs, corrs)
  names(tmp) <- c("Imp", "Int", "Slope", "id", "Corr")
  out <- data.frame(IntSD_mean = mean(tmp$Int), 
                        SlopeSD_mean = mean(tmp$Slope), 
                    Corr_mean = mean(tmp$Corr), 
                        IntSD_sd = sd(tmp$Int),
                    SlopeSD_sd = sd(tmp$Slope), 
                        Corr_sd = sd(tmp$Corr))

modelFixedEff <- function(modList){
  fixEst <- ldply(modList, tidy, effects = "fixed")
  # Collapse
  out <- ddply(fixEst, .(term), summarize,
               estimate = mean(estimate), 
               std.error = mean(std.error))
  out$statistic <- out$estimate / out$std.error

print.merModList <- function(modList, digits = 3){
  len <- length(modList)
  form <- modList[[1]]@call
  cat("\nFixed Effects:\n")
  fedat <- modelFixedEff(modList)
  dimnames(fedat)[[1]] <- fedat$term
  pfround(fedat[-1, -1], digits)
  cat("\nError Terms Random Effect Std. Devs\n")
  cat("and covariances:\n")
  ngrps <- length(VarCorr(modmathG[[1]]))
  errorList <- vector(mode = 'list', length = ngrps)
  corrList <- vector(mode = 'list', length = ngrps)
  for(i in 1:ngrps){
    subList <- lapply(modList, function(x) VarCorr(x)[[i]])
    subList <- apply(simplify2array(subList), 1:2, mean)
    errorList[[i]] <- subList
    subList <- lapply(modList, function(x) attr(VarCorr(x)[[i]], "corre"))
    subList <- min(unique(apply(simplify2array(subList), 1:2, function(x) mean(x))))
    corrList[[i]] <- subList
  errorList <- lapply(errorList, function(x) {
    diag(x) <- sqrt(diag(x))

  lapply(errorList, pfround, digits)
  cat("\nError Term Correlations:\n")
  lapply(corrList, pfround, digits)
  residError <- mean(unlist(lapply(modList, function(x) attr(VarCorr(x), "sc"))))
  cat("\nResidual Error =", fround(residError,
                                             digits), "\n")
  ngrps <- lapply(modList[[1]]@flist, function(x) length(levels(x)))
  modn <- getME(modList[[1]], "devcomp")$dims["n"]
  cat(sprintf("number of obs: %d, groups: ", modn))
  cat(paste(paste(names(ngrps), ngrps, sep = ", "),
            collapse = "; "))
  cat("\nModel Fit Stats")
  mAIC <- mean(unlist(lapply(modList, AIC)))
  cat(sprintf("\nAIC = %g", round(mAIC, 1)))
  moDsigma.hat <- mean(unlist(lapply(modmathG, sigma)))
  cat("\nOverdispersion parameter =", fround(moDsigma.hat,
                                             digits), "\n")
  • 1
    $\begingroup$ This functionality -- most of it -- is baked into the development version of the merTools package. This version will be pushed to CRAN in the coming week. $\endgroup$
    – jknowles
    Commented Feb 18, 2016 at 18:17
  • $\begingroup$ could you say what function to look for to do this with the merTools package? I couldn't find anything. $\endgroup$
    – smillig
    Commented Jul 6, 2017 at 14:19
  • 1
    $\begingroup$ It's not fully documented in the current version, but check out lmerModList and the print method, which combines the results from the model lists. $\endgroup$
    – jknowles
    Commented Jul 7, 2017 at 17:59
  • 1
    $\begingroup$ One could also exploit miceadds::mids2datlist() to get a list of multiply imputed data sets and then iterate over the elements in that list. $\endgroup$ Commented Jan 22, 2022 at 13:49
  • 1
    $\begingroup$ Brilliant - very helpful addition @Dion $\endgroup$
    – jknowles
    Commented Jan 26, 2022 at 18:22

You can also use the testEstimates function after imputation using mice with the following code: testEstimates(as.mitml.result(fm1), var.comp = T)$var.comp


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