The GLIMMIX Procedure

Relationship with Generalized Linear Models

Generalized linear models (Nelder and Wedderburn 1972; McCullagh and Nelder 1989) are a special case of GLMMs. If $\bgamma = \mb{0}$ and $\mb{R} = \phi \mb{I}$, the GLMM reduces to either a generalized linear model (GLM) or a GLM with overdispersion. For example, if $\mb{Y}$ is a vector of Poisson variables so that $\mb{A}$ is a diagonal matrix containing $\mr{E}[\mb{Y}]=\bmu $ on the diagonal, then the model is a Poisson regression model for $\phi = 1$ and overdispersed relative to a Poisson distribution for $\phi > 1$. 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.