Reference Bayesian methods for generalized linear mixed models

Bayesian methods furnish an attractive approach to inference in generalized linear mixed models. In the absence of subjective prior information for th e random-effect variance components, these analyses are typically conducted using either the standard invariant prior for normal responses or diffuse conjugate priors. Previous work has pointed out serious difficulties with b oth strategies, and we show here that as in normal mixed models, the standa rd invariant prior leads to an improper posterior distribution for generali zed linear mixed models. This article proposes and investigates two alterna tive reference (i.e., "objective" or "noninformative") priors: an approxima te uniform shrinkage prior and an approximate jeffreys's prior. We give con ditions for the existence of the posterior distribution under any prior for the variance components in conjunction with a uniform prior for the fixed effects. The approximate uniform shrinkage prior is shown to satisfy these conditions for several families of distributions, in some cases under mild constraints on the data. Simulation studies conducted using a legit-normal model reveal that the approximate uniform shrinkage prior improves substant ially on a plug-in empirical Bayes rule and fully Bayesian methods using di ffuse conjugate specifications. The methodology is illustrated on a seizure dataset.