Random Effects in Generalized Linear Mixed Models

In this chapter, we examine the use of special forms of correlated random e ects in the generalized linear mixed model (GLMM) setting. A special feature of our GLMM is the inclusion of random residual e ects to account for lack of t due to extra variation, outliers and other unexplained sources of variation. For random e ects, we consider, in particular, the correlation structure and improper priors associated with the autoregressive (AR) model of Ord (1975) and the conditional autoregressive (CAR) model of Besag (1974). We give conditions for the propriety of the posterior distribution of the GLMM when the xed e ects have a constant improper prior and the random e ects have a possibly improper conditional autoregressive prior. Several examples of exponential families as well as computational details for Markov chain Monte Carlo simulation are also presented.