The GLIMMIX Procedure

Scale and Dispersion Parameters

The parameter $\phi $ in the log-likelihood functions is a scale parameter. McCullagh and Nelder (1989, p. 29) refer to it as the dispersion parameter. With the exception of the normal distribution, $\phi $ does not correspond to the variance of an observation, the variance of an observation in a generalized linear model is a function of $\phi $ and $\mu $. In a generalized linear model (GLM mode), the GLIMMIX procedure displays the estimate of $\phi $ is as "Scale" in the "Parameter Estimates" table. Note that for some distributions this scale is different from that reported by the GENMOD procedure in its "Parameter Estimates" table. The scale reported by PROC GENMOD is sometimes a transformation of the dispersion parameter in the log-likelihood function. Table 45.21 displays the relationship between the "Scale" entries reported by the two procedures in terms of the $\phi $ (or k) parameter in the GLIMMIX log-likelihood functions.

Table 45.21: Scales in Parameter Estimates Table

Distribution

GLIMMIX Reports

GENMOD Reports

Beta

$\widehat{\phi }$

N/A

Gamma

$\widehat{\phi }$

$\widehat{\phi }$

Inverse Gaussian

$\widehat{\phi }$

$\sqrt {\widehat{\phi }}$

Negative binomial

$\widehat{k}$

$\widehat{k}$

Normal

$\widehat{\phi } = \widehat{\mr{Var}}[Y]$

$\sqrt {\widehat{\phi }}$


Note that for normal linear models, PROC GLIMMIX by default estimates the parameters by restricted maximum likelihood, whereas PROC GENMOD estimates the parameters by maximum likelihood. As a consequence, the scale parameter in the "Parameter Estimates" table of the GLIMMIX procedure coincides for these models with the mean-squared error estimate of the GLM or REG procedures. To obtain maximum likelihood estimates in a normal linear model in the GLIMMIX procedure, specify the NOREML option in the PROC GLIMMIX statement.