[ADMB Users] What is the tau in the negative binomial and why is tau>1?

dave fournier otter at otter-rsch.com
Sat Aug 22 14:44:46 PDT 2009

A previous question to the list was "why is tau restricted to be >1"
in the negative binomial.  This is actually an interesting point.

The log of the negative binomial density is most easily parametreized in
terms of mu, the mean and r where r>0.  It looks like


Now the variance of this is equal to

           mu * (1 + r*mu)

and as r --> 0 it approaches a log poisson dist.

If we consider the factor 1 + r*mu to be the overdispersion
then the overdispersion depends on the mean.  Often in mixed models
there are a collection of different means being fitted, say mu_i.

so that for a fixed value of r we get varying overdispersion

     1 + r * mu_i

there is nothing wrong with that per se but one might want to
investigate a model where the overdispersion is a constant independent
of the mean  Call this constant tau.

         tau = 1 + r*mu_i

which implies that

        r_i = (tau -1)/ mu_i

where I have written r_i since r_i depends on both tau and mu_i

Thus tau must be restricted to be >1.0

David A. Fournier
P.O. Box 2040,
Sidney, B.C. V8l 3S3
Phone/FAX 250-655-3364

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