The GLIMMIX Procedure

Implied Variance Functions

While link functions are not unique for each distribution (see Table 41.13 for the default link functions), the distribution does determine the variance function $a(\mu )$ . This function expresses the variance of an observation as a function of the mean, apart from weights, frequencies, and additional scale parameters. The implied variance functions $a(\mu )$ of the GLIMMIX procedure are shown in Table 41.20 for the supported distributions. For the binomial distribution, n denotes the number of trials in the events/trials syntax. For the negative binomial distribution, k denotes the scale parameter. The multiplicative scale parameter $\phi$ is not included for the other distributions. The last column of the table indicates whether $\phi$ has a value equal to 1.0 for the particular distribution.

Table 41.20: Variance Functions in PROC GLIMMIX

		Variance function
DIST=	Distribution	$a(\mu )$	$\phi \equiv 1$
BETA	beta	$\mu (1-\mu )/(1+\phi )$	No
BINARY	binary	$\mu (1-\mu )$	Yes
BINOMIAL \| BIN \| B	binomial	$\mu (1-\mu )/n$	Yes
EXPONENTIAL \| EXPO	exponential	$\mu ^2$	Yes
GAMMA \| GAM	gamma	$\mu ^2$	No
GAUSSIAN \| G \| NORMAL \| N	normal	1	No
GEOMETRIC \| GEOM	geometric	$\mu + \mu ^2$	Yes
INVGAUSS \| IGAUSSIAN \| IG	inverse Gaussian	$\mu ^3$	No
LOGNORMAL \| LOGN	lognormal	1	No
NEGBINOMIAL \| NEGBIN \| NB	negative binomial	$\mu +k\mu ^2$	Yes
POISSON \| POI \| P	Poisson	$\mu$	Yes
TCENTRAL \| TDIST \| T	t	$\nu /(\nu -2)$	No

To change the variance function, you can use SAS programming statements and the predefined automatic variables, as outlined in the following section. Your definition of a variance function will override the DIST= option and its implied variance function. This has the following implication for parameter estimation with the GLIMMIX procedure. When a user-defined link is available, the distribution of the data is determined from the DIST= option, or the respective default for the type of response. In a GLM, for example, this enables maximum likelihood estimation. If a user-defined variance function is provided, the DIST= option is not honored and the distribution of the data is assumed unknown. In a GLM framework, only quasi-likelihood estimation is then available to estimate the model parameters.