## Bayesian Covariance Selection in Generalized Linear Mixed Models

### Bo Cai and David B. Dunson

### Biostatistics Branch, NIEHS

* January, 2005 *

The generalized linear mixed model (GLMM), which extends the generalized linear model
(GLM) to incorporate random effects characterizing heterogeneity among subjects, is
widely used in analyzing correlated and longitudinal data. Although there is often
interest in identifying the subset of predictors that have random effects, random effects
selection can be challenging, particularly when outcomes distributions are non-normal.
This article proposes a fully Bayesian approach to the problem of simultaneous selection
of fixed and random effects in GLMMs. Integrating out the random effects induces a
covariance structure on the multivariate outcome data, and an important problem which we
also consider is that of covariance selection. Our approach relies on variable
selection-type mixture priors for the components in a special LDU decomposition of the
random effects covariance. A stochastic search MCMC algorithm is developed, which relies
on Gibbs sampling, with Taylor series expansions used to approximate intractable
integrals. Simulated data examples are presented for different exponential family
distributions, and the approach is applied to discrete survival data from a
time-to-pregnancy study.

Keywords: Bayes factor; Latent variables; Marginal likelihood; MCMC algorithm; Random
effects; Stochastic search; Taylor series; Variable selection.

The manuscript is available in
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and
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formats.