## Reference Bayesian Methods for Generalized Linear Mixed Models

July, 1998

Tech Report

### Author(s)

Ranjini Natarajan and Robert E. Kass

### Abstract

Bayesian methods furnish an attractive approach to inference in generalized linear mixed models. In the absence of subjective prior information for the 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 both strategies, and we show here that, as in normal mixed models, the standard invariant prior leads to an improper posterior distribution for generalized linear mixed models. The purpose of this paper is to propose and investigate two alternate reference (i.e., "objective" or "noninformative") priors: an approximate uniform shrinkage prior and an approximate Jeffreys's prior. We give conditions 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 logit-normal model reveal that the approximate uniform shrinkage prior improves substantially upon a plug-in empirical Bayes rule and fully Bayesian methods using diffuse conjugate specifications. The methodology is illustrated on a seizure data set.

(Revised 08/99)