Bootstrap Methods for Generalized Linear Mixed Models With Applications to Small Area Estimation

Generalized linear mixed models (GLMMs) provide a unified framework for analyzing relationships between binary, count or continuous response variables and predictors with either fixed or random effects. Recent advances in approximate fitting procedures and Markov Chain Monte Carlo techniques, as well as the widespread availability of high speed computers suggest that GLMM software will soon be a standard feature of many statistical packages. Although the difficulty of fitting of GLMMs has to a large extent been overcome, there are still many unresolved problems, particularly with regards to inference. For example, analytical formulas for standard errors and confidence intervals for linear combinations of fixed and random effects are often unreliable or not available, even in the classical case with normal errors. In this paper we propose the use of the parametric bootstrap as a practical tool for addressing problems associated with inference from GLMMs. The power of the bootstrap approach is illustrated in two small area estimation examples. In the first example, it is shown that the bootstrap reproduces complicated analytical formulas for the standard errors of estimates of small area means based on a normal theory mixed linear model. In the second example, involving a logistic-normal model, the bootstrap produces sensible estimates for standard errors, even though no analytical formulas are available.