Monte Carlo Likelihood Approximation for Generalized Linear Mixed Models

Abstract

Frequentist likelihood-based inference for generalized linear mixed models is often difficult to perform. Because the likelihood cannot depend on unobserved data (such as random effects), the likelihood for a generalized linear mixed model is an integral that is often high-dimensional and intractable. The method of Monte Carlo likelihood approximation (MCLA) approximates the entire likelihood function using random effects simulated from an importance sampling distribution. The resulting Monte Carlo likelihood approximation can be used for any frequentist likelihood-based inference. Due to the challenge of finding an importance sampling distribution that works well in practice, very little publicly-available MCLA software existed prior to 2015. I present an importance sampling distribution to be used in implementing MCLA for generalized linear mixed models; establish its theoretical validity; implement it in the R package glmm; and demonstrate how to use the package to perform maximum likelihood, test hypotheses, and calculate confidence intervals.