The GLIMMIX Procedure

The Basic Model

Suppose $\mb {Y}$ represents the $(n \times 1)$ vector of observed data and $\bgamma $ is a $(r \times 1)$ vector of random effects. Models fit by the GLIMMIX procedure assume that

\[  \mr {E}[\mb {Y} | \bgamma ] = g^{-1}(\mb {X}\bbeta + \mb {Z}\bgamma )  \]

where $g(\cdot )$ is a differentiable monotonic link function and $g^{-1}(\cdot )$ is its inverse. The matrix $\bX $ is an $(n \times p)$ matrix of rank k, and $\bZ $ is an $(n \times r)$ design matrix for the random effects. The random effects are assumed to be normally distributed with mean $\mb {0}$ and variance matrix $\bG $.

The GLMM contains a linear mixed model inside the inverse link function. This model component is referred to as the linear predictor,

\[  \bm {\eta } = \bX \bbeta + \bZ \bgamma  \]

The variance of the observations, conditional on the random effects, is

\[  \mr {Var}[\bY | \bgamma ] = \bA ^{1/2}\mb {R}\bA ^{1/2}  \]

The matrix $\bA $ is a diagonal matrix and contains the variance functions of the model. The variance function expresses the variance of a response as a function of the mean. The GLIMMIX procedure determines the variance function from the DIST= option in the MODEL statement or from the user-supplied variance function (see the section Implied Variance Functions). The matrix $\bR $ is a variance matrix specified by the user through the RANDOM statement. If the conditional distribution of the data contains an additional scale parameter, it is either part of the variance functions or part of the $\bR $ matrix. For example, the gamma distribution with mean $\mu $ has the variance function $a(\mu ) = \mu ^2$ and $\mr {Var}[Y|\bgamma ] = \mu ^2 \phi $. If your model calls for G-side random effects only (see the next section), the procedure models $\bR = \phi \mb {I}$, where $\bI $ is the identity matrix. Table 41.20 identifies the distributions for which $\phi \equiv 1$.