Estimation of a Generalized Linear Mixed‐Effects Model with a Finite‐Support Random‐Effects Distribution via Gibbs Sampling

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Abstract
We discuss a Bayesian hierarchical generalized linear mixed‐effects model with a finite‐support random‐effects distribution and show how Gibbs sampling can be used for estimating the posterior distribution of the parameters and for clustering on the basis of longitudinal data. When directly sampling from the conditional distributions is laborious, the adaptive rejection sampling (ARS, Gilks and Wild, 1992; Gilks, 1992) algorithm or adaptive rejection Metropolis sampling (ARMS, Gilks et al., 1995) algorithm is used. Log‐concavity, a prerequisite of ARS, of the conditional distributions is examined. We also discuss a Bayesian solution to the uncertainty of the support size of the random‐effects distribution in statistical inference. The methodology is illustrated with an analysis of data from a study of regulation of serum parathyroid hormone secretion.

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