Obsidian | Level 7

## Fitting Overdispersed Poisson by MLE vs. MQLE

Hi!

If I understand correctly, PROC GENMOD fits overdispersed Poisson models by maximum quasi-likelihood estimation (Generalized Linear Models Theory :: SAS/STAT(R) 12.1 User's Guide); they claim that the regression parameter estimates are not affected by the value of the dispersion / scale parameter, so I'm not really sure what role the dispersion / scale parameter plays in the quasi-likelihood function.  It looks like there are four possible ways of incorporating the dispersion / scale parameter, either fit it or fix it one of three ways.  You can fix it to a number, you can fix it to the chi-square divided by the degrees of freedom, or you can fix it to the deviance divided by the degrees of freedom.  In each of these cases, it appears that "his fixes the scale parameter at a value of 1 in the estimation procedure" and then uses the presumed value to adjust standard errors and likelihood ratio statistics.  If the scale parameter needs to be fit, it says that "for the normal, inverse Gaussian, and gamma distributions, the scale parameter is estimated by maximum likelihood."  It doesn't mention how the fit is done in Poisson, but I imagine that the scale parameter is chosen as the ratio of the variance over the mean.  Is this true?

It is known that if Y is overdispersed Poisson with scale parameter Gamma, then (Y / Gamma) has a Poisson distribution (http://fa319.voila.net/Schmidt.pdf).  So it would be possible in theory to do a full MLE fit of the Poisson regression and the scale parameter together, but this would require keeping the factorial denominator of the Poisson formula in the likelihood function, and actually using its derivative.  I wonder whether anyone has had any experience trying to fit an overdispersed Poisson by direct MLE this way?

Thanks!

2 REPLIES 2
Super User

## Re: Fitting Overdispersed Poisson by MLE vs. MQLE

Maybe you should check proc genmod .

## Re: Fitting Overdispersed Poisson by MLE vs. MQLE

I think the closest you could do in GENMOD is to iteratively estimate the SCALE parameter, feeding it back in to successive runs until a stable estimate was obtained.

But the easy way would be to use GLIMMIX.  Look at example 43.12 Fitting a Marginal (GEE-Type) Model, for a method that directly accommodates overdispersion.  Granted, the fit is through pseudo-likelihood, but that is the price you have to pay to get the extra scale parameter estimated directly.

Steve Denham

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