Date:         Fri, 3 Nov 2006 13:49:35 -0800
Reply-To:     Bill Anderson <wnilesanderson@COX.NET>
Sender:       "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From:         Bill Anderson <wnilesanderson@COX.NET>
Subject:      Negative Binomial Dispersion
Comments: cc: wnilesanderson@aol.com
Content-Type: text/plain; charset="iso-8859-1"

When GENMOD is used to fit a model with the negative binomial distribution, SAS gives an estimate for the intercept and the dispersion.  As I read the manual, the dispersion must be greater than zero, since its log appears in the log likelihood formula.  This situation is also described in the textbook of Cameron and Trivedi, where it is shown that the negative binomial becomes the Poisson in the limit as the dispersion goes to zero.

But SAS is quite happy to print a negative maximum likelihood estimator for the dispersion.  It is easy to generate such a data set.  Just generate random Poisson variates, until you get a set where the variance is less than the mean (happens about half the time).  Most such sets result in a negative estimate for the dispersion.

It seems to me that such a dispersion would give an imaginary value for the log-likelihood, but SAS prints a perfectly respectable real number.  


What is going on here?  

At first I thought it might be related  to the link, since SAS gives different intercept estimates with the identity or log link.  But the dispersion estimates are the same. So I need a better theory.

Thanks in advance for any help you can give me.