ABSTRACT
Generalized linear models ( G LM) are a standard class of models in contemporary statistical data analysis (McCullagh and Neider 1989). The widely available GLIM software as well as SPlus facilitate computation under these models. Bayesian fitting of GLM's via Gibbs sampling is discussed in Dellaportas and Smith (1993). In GLM's, the underlying distribution of responses is assumed to be of the exponential family form, and a link function transformation of its expectation is modeled as a linear function of observed covariates, assuming that the variance of the response is a specified function of its mean. This allows modeling in various nonnormal situations such as the binomial, Poisson, negative binomial etc. However, in many applications, such a simple functional relationship is inadequate to handle the heterogeneity in the data; a common problem is the so-called overdispersion problem, where the variance of the response exceeds the nominal variance (Cox 1983). For instance, regression analysis of count data in biomedical research areas such as toxicology, epidemiology etc., must handle extra-Poisson variation. Count data analyzed under a Poisson assumption (Breslow 1984; Lawless 1987a,b; McCullagh and Neider 1989, Sec. 6.2) often exhibit overdispersion. In modeling categorical or ordered categorical data in longitudinal studies, toxicity studies, or in biological experimental research, the use of binomial or multinomial distributions with overdispersion is encountered.
