The GLIMMIX Procedure

Effects of Adding Overdispersion

You can add a multiplicative overdispersion parameter to a generalized linear model in the GLIMMIX procedure with the statement

random _residual_;

For models in which $\phi \equiv 1$, this effectively lifts the constraint of the parameter. In models that already contain a $\phi $ or k scale parameter—such as the normal, gamma, or negative binomial model—the statement adds a multiplicative scalar (the overdispersion parameter, $\phi _ o$) to the variance function.

The overdispersion parameter is estimated from Pearson’s statistic after all other parameters have been determined by (restricted) maximum likelihood or quasi-likelihood. This estimate is

\[  \widehat{\phi }_ o = \frac{1}{\phi ^ p m} \sum _{i=1}^{n} f_ i w_ i \frac{(y_ i - \mu _ i)^2}{a(\mu _ i)}  \]

where $m = f-\mr {rank}\{ \mb {X}\} $ if the NOREML option is in effect, and $m=f$ otherwise, and f is the sum of the frequencies. The power p is –1 for the gamma distribution and 1 otherwise.

Adding an overdispersion parameter does not alter any of the other parameter estimates. It only changes the variance-covariance matrix of the estimates by a certain factor. If overdispersion arises from correlations among the observations, then you should investigate more complex random-effects structures.