Inferential procedures for random effects in generalized linear mixed models

We study three commonly applied measures of uncertainty for random effects prediction in generalized linear mixed models (GLMMs), namely the unconditional and conditional mean squared errors of prediction (UMSEP and CMSEP, respectively), and the unconditional variance of the prediction gap used by the popular R package for glmmTMB. We demonstrate that, although the three theoretical measures differ in how they quantify uncertainty, the resulting estimators all turn out to be very similar in form. We derive asymptotic results regarding the consistency of the three measures of uncertainty, and in doing so resolve a contradiction between theoretical and empirical results for the glmmTMB variance estimator by re-interpreting it conditionally on a finite subset of the random effects. Our results have important implications for predictive inference in GLMMs, particularly around the legitimacy and implications of coupling these measures with a normality assumption to construct prediction intervals for the random effects.