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A Bayesian Approach for Assessing Heterogeneity in Generalized Linear Models

Generalized linear mixed models (GLMMs) are used routinely for analyzing clustered data arising in a broad variety of applications. In Bayesian analyses, inverse Wishart or inverse gamma priors are almost always used for the covariance of the random effects, for computational convenience and to enforce the positive definite constraint on the covariance matrix. In this article, we propose a new class of prior distributions based on a Gaussian structure for variance component parameters underlying the random effects covariance. The proposed prior assigns positive probability not only to the full model but also to reduced models that exclude one or more of the random effects. This structure facilitates Bayesian inferences about the covariance structure, while also accounting for uncertainty in the random effects model in estimating the population parameters. A Markov chain Monte Carlo algorithm is proposed for posterior computation, and the approach is illustrated using data on prenatal exposure to PCBs and psychomotor development.