Date:         Fri, 3 Nov 2006 19:57:18 -0600
Reply-To:     Robin High <robinh@UNLSERVE.UNL.EDU>
Sender:       "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From:         Robin High <robinh@UNLSERVE.UNL.EDU>
Subject:      Re: Negative Binomial Dispersion
Comments: To: Bill Anderson <wnilesanderson@COX.NET>
In-Reply-To:  <023d01c6ff91$f3ea30b0$6601a8c0@Martingale>
Content-Type: TEXT/PLAIN; charset=US-ASCII

> 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.

Are you referring to the lower confidence limit printed on the line for
the dispersion estimate? The "Wald confidence limit" calculations can
produce a negative lower bound when the dispersion estimate is close to 0
or the sample size is small, yet I haven't seen a negative estimate for
the dispersion parameter itself.

Robin High