The GLIMMIX Procedure

Implied Variance Functions

While link functions are not unique for each distribution (see Table 45.13 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 45.20 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 45.20: 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.