The GLIMMIX Procedure

The Basic Model

Suppose bold upper Y represents the left-parenthesis n times 1 right-parenthesis vector of observed data and bold-italic gamma is a left-parenthesis r times 1 right-parenthesis vector of random effects. Models fit by the GLIMMIX procedure assume that

normal upper E left-bracket bold upper Y vertical-bar bold-italic gamma right-bracket equals g Superscript negative 1 Baseline left-parenthesis bold upper X bold-italic beta plus bold upper Z bold-italic gamma right-parenthesis

where g left-parenthesis dot right-parenthesis is a differentiable monotonic link function and g Superscript negative 1 Baseline left-parenthesis dot right-parenthesis is its inverse. The matrix bold upper X is an left-parenthesis n times p right-parenthesis matrix of rank k, and bold upper Z is an left-parenthesis n times r right-parenthesis design matrix for the random effects. The random effects are assumed to be normally distributed with mean bold 0 and variance matrix bold upper G.

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

bold-italic eta equals bold upper X bold-italic beta plus bold upper Z bold-italic gamma

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

normal upper V normal a normal r left-bracket bold upper Y vertical-bar bold-italic gamma right-bracket equals bold upper A Superscript 1 slash 2 Baseline bold upper R bold upper A Superscript 1 slash 2

The matrix bold upper A 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 bold upper R 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 bold upper R matrix. For example, the gamma distribution with mean mu has the variance function a left-parenthesis mu right-parenthesis equals mu squared and normal upper V normal a normal r left-bracket upper Y vertical-bar bold-italic gamma right-bracket equals mu squared phi. If your model calls for G-side random effects only (see the next section), the procedure models bold upper R equals phi bold upper I, where bold upper I is the identity matrix. Table 22 identifies the distributions for which phi identical-to 1.

Last updated: December 09, 2022