# How can I get confidence intervals for fixed effects using the rlmer function (robustlmm package)?

I need to fit a linear mixed model but my dependent variable is right-skewed with some big outliers. Thus I used the rlmer function of the robustlmm package. it works quite nicely, however what I am missing now is confidence intervals for the fixed effects. Is anyone working with this package and has some tips for me? Otherwise I would also be interested in other packages to fit robust linear mixed models.

• Wald confidence intervals: these assume that the sampling distribution of the parameters is multivariate Normal (a much weaker assumption than that the conditional distribution of the residuals is Normal). They are relatively easily to compute (for the fixed-effects parameters) by extracting the parameter values (fixef()) and the standard errors (sqrt(diag(vcov()))) and computing $\beta \pm z \cdot \sigma$ ...

Example:

library(robustlmm)
r <- rlmer(Reaction ~ Days + (1|Subject), sleepstudy)

confint.rlmerMod <- function(object,parm,level=0.95) {
beta <- fixef(object)
if (missing(parm)) parm <- names(beta)
se <- sqrt(diag(vcov(object)))
z <- qnorm((1+level)/2)
ctab <- cbind(beta-z*se,beta+z*se)
colnames(ctab) <- stats:::format.perc(c((1-level)/2,(1+level)/2),
digits=3)
return(ctab[parm,])
}
confint(r)
##                  2.5 %    97.5 %
## (Intercept) 235.575485 269.27845
## Days          9.211197  12.04305

• Thank you for your answer. I already did that myself, but I am not so happy with Wald confidence intervals in this situation, as the normality assumption is obviously violated and my distribution is not symmetric... – Steffi Sep 8 '16 at 11:12
• see edits. A non-symmetric response distribution doesn't imply that the Wald CIs are bad ... – Ben Bolker Sep 8 '16 at 14:27
• Thank you for expanding your answer! I was also thinking about bootstrapping, but I was/am unsure how to do that correctly in this situation, incorporating the random effects etc. I will have a closer look at your links and see if I manage! It would be interesting also for comparison with the Wald confidence intervals. About your suggestion to calculate the CIs on another scale: There I see the problem with the interpretation. The model is for a surgery trial and the surgeons need to be able to interpret the results - many of them do not have an in-depth knowledge about statistics. – Steffi Sep 9 '16 at 6:32

It is also now possible to obtain confidence intervals of rlmerMod objects using the effects package.

Example:

library(robustlmm)
library(effects)

r <- rlmer(Reaction ~ Days + (1|Subject), sleepstudy)
as.data.frame(effect("Days",r))

###   Days      fit       se    lower    upper
###      0 252.4270 8.597854 235.4601 269.3938
###      2 273.6812 8.161895 257.5747 289.7877
###      4 294.9355 7.967757 279.2120 310.6589
###      7 326.8168 8.161895 310.7103 342.9234
###      9 348.0711 8.597854 331.1042 365.0379