chapter  6
30 Pages

Hierarchical GLMs

In this Chapter, we introduce HGLMs as a synthesis of two widely-used existing model classes: GLMs (Chapter 2) and normal linear mixed models (Chapter 5). In an unpublished technical report, Pierce and Sands (Oregon State University, 1975) introduced generalized linear mixed models (GLMMs), where the linear predictor of a GLM is allowed to have, in addition to the usual fixed effects, one or more random components with assumed normal distributions. Although the normal distribution is convenient for specifying correlations among the random effects, the use of other distributions for the random effects greatly enriches the class of models. Lee and Nelder (1996) extended GLMMs to hierarchical GLMs (HGLMs), in which the distribution of random components are extended to conjugates of arbitrary distributions from the GLM family.

6.1 HGLMs