Semiparametric Bayesian Methods for Random Effects Models

Abstract

The linear mixed effects model with normal errors is a popular model for the analysis of repeated measures and longitudinal data. The generalized linear model is useful for data that has non-normal errors but where the errors are uncorrelated. A descendant of these two models generates a model for correlated data with non-normal errors, called the generalized linear mixed model (GLMM). Frequentist attempts to fit these models generally rely on approximate results and inference relies on asymptotic assumptions. Recent methodological and technological advances have made Bayesian approaches to this class of models feasible. Markov chain Monte Carlo methods can be used to obtain ‘exact’ inference for these models, as demonstrated by Zeger and Karim (1991). In the linear or generalized linear mixed model, the random effects are typically taken to have a fully parametric distribution, such as the normal distribution. In tins chapter, we extend the normal random effects model and the GLMM by allowing the random effects to have a nonparametric prior distribution. We do this using a Dirichlet Process prior for the general distribution of the random effects. The approach can be easily extended to more general population models. Computations for the models are accomplished using the Gibbs sampler.