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 identical-to 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 Subscript 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

ModifyingAbove phi With caret Subscript o Baseline equals StartFraction 1 Over phi Superscript p Baseline m EndFraction sigma-summation Underscript i equals 1 Overscript n Endscripts f Subscript i Baseline w Subscript i Baseline StartFraction left-parenthesis y Subscript i Baseline minus mu Subscript i Baseline right-parenthesis squared Over a left-parenthesis mu Subscript i Baseline right-parenthesis EndFraction

where m equals f minus normal r normal a normal n normal k StartSet bold upper X EndSet if the NOREML option is in effect, and m equals 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.

Last updated: December 09, 2022