Generalized Linear Mixed Models

Generalized linear mixed models can be fit with the GLIMMIX and NLMIXED procedures in SAS/STAT software. The GLIMMIX procedure is specifically designed to fit this class of models and offers syntax very similar to the syntax of other linear modeling procedures, such as the MIXED procedure. Consider a generalized linear model with linear predictor and link function

\[  \mr {E}[\bY |\bgamma ] = g^{-1}(\bm {\eta }) = g^{-1}(\bX \bbeta + \bZ \bgamma )  \]

and distribution in the exponential family. The fixed-effects design matrix $\bX $ is specified in the MODEL statement of the GLIMMIX procedure, and the random-effects design matrix $\bZ $ is specified in the RANDOM statement, along with the covariance matrix of the random effects and the covariance matrix of R-side random variables. The link function and (conditional) distribution are determined by defaults or through options in the MODEL statement.

The GLIMMIX procedure can fit heterocatanomic multivariate data—that is, data that stem from different distributions. For example, one measurement taken on a patient might be a continuous, normally distributed outcome, whereas another measurement might be a binary indicator of medical history. The GLIMMIX procedure also provides capabilities for mixed model smoothing and mixed model splines.

The GLIMMIX procedure offers an extensive array of postprocessing features to produce output statistics and to perform linear inference. The ESTIMATE and LSMESTIMATE statements support multiplicity-adjusted p-values for the protection of the familywise Type-I error rate. The LSMEANS statement supports the slicing of interactions, simple effect differences, and ODS statistical graphs for group comparisons.

The default estimation technique in the GLIMMIX procedure depends on the class of models fit. For linear mixed models, the default technique is restricted maximum likelihood, as in the MIXED procedure. For generalized linear mixed models, the estimation is based on linearization methods (pseudo-likelihood) or on integral approximation by adaptive quadrature or Laplace methods.

The NLMIXED procedure facilitates the fitting of generalized linear mixed models through several built-in distributions from the exponential family (binary, binomial, gamma, negative binomial, and Poisson). You have to code the linear predictor and link function with SAS programming statements and assign starting values to all parameters, including the covariance parameters. Although you are not required to specify starting values with the NLMIXED procedure (because the procedure assigns a default value of 1.0 to every parameter not explicitly given a starting value), it is highly recommended that you specify good starting values. The default estimation technique of the NLMIXED procedure, an adaptive Gauss-Hermite quadrature, is also available in the GLIMMIX procedure through the METHOD=QUAD option in the PROC GLIMMIX statement. The Laplace approximation that is available in the NLMIXED procedure by setting QPOINTS=1 is available in the GLIMMIX procedure through the METHOD=LAPLACE option.