Relationship with Generalized Linear Models

Generalized linear models (Nelder and Wedderburn 1972; McCullagh and Nelder 1989) are a special case of GLMMs. If and , the GLMM reduces to either a generalized linear model (GLM) or a GLM with overdispersion. For example, if is a vector of Poisson variables so that is a diagonal matrix containing on the diagonal, then the model is a Poisson regression model for and overdispersed relative to a Poisson distribution for . Because the Poisson distribution does not have an extra scale parameter, you can model overdispersion by adding the following statement to your GLIMMIX program:

 random _residual_;

If the only random effect is an overdispersion effect, PROC GLIMMIX fits the model by (restricted) maximum likelihood and not by one of the methods specific to GLMMs.