[Limdep Nlogit List] Deviance for unconditional fixed effects negative binomial regression model

[rjcal at u.washington.edu]

(Quick update: replacing lambda_i with an overdispersion parameter lambda for all observations adjusted the deviance upwards slightly, such that now I'm multiplying the standard errors by .925 instead of .8. But as Allison and Waterman suggest that the multiplier is meant to adjust the standard errors upwards, I think I'm still wrong, most likely concerning the deviance computation.)

Best,

Richard Callahan

On Fri, 14 Aug 2009 rjcal at u.washington.edu wrote:

> Dear LIMDEP users,
>
> I'd like to follow up regarding the code I just posted for calculating adjusted 
> standard errors for the unconditional fixed effects negative binomial 
> regression, as I think I've got something wrong. As I understand it, the 
> multiplier is supposed to always be greater than one, meaning that the deviance 
> should always be greater than the degrees of freedom for the model. I think I'm 
> making a mistake in going from Dr. Greene's notation in the LIMDEP manual to 
> Dr. Allison and Dr. Waterman's notation in their Sociological Methodology 
> article.
>
> I think I'm wrong in one of these ways:
> 1. Allison and Waterman (2002, p. 253) discuss the overdispersion parameter 
> lambda in introducing the NB-2 model (Cameron and Trivedi 1998), and allow 
> lambda to vary by group (hence lambda_i). Elsewhere (see 
> <http://books.google.com/books?id=23fNCTegnywC&pg=PA93&lpg=PA93&dq=Negative+binomial+NB2+mass+function&source=bl&ots=sERwjlX9S-&sig=s9kEg94qxoqcf93HBSbL1YEqSK4&hl=en&ei=yqGFSqOCGo_YsgO9kdyhBw&sa=X&oi=book_result&ct=result&resnum=4#v=onepage&q=Negative%20binomial%20NB2%20mass%20function&f=false>, 
> pardon the long link), Dr. Allison specifies a model where the overdispersion 
> parameter doesn't vary by group, and it doesn't vary by group in their 
> simulation. So perhaps I should use the point estimate of the overdispersion 
> parameter from the LIMDEP output in the deviance calculation for all 
> observations.
>
> 2. I calculated (Allison and Waterman's) lambda = (Greene's) theta = exp(alpha) 
> based on page E26-15 of the LIMDEP manual, but I think I mistakenly based that 
> conversion the NB-1 parameterization and I should use theta = 1/alpha in the 
> model (p. E24-16). But even making this correction, this doesn't entirely solve 
> the problem -- deviance / degrees of freedom becomes greater than one for one 
> of my dependent variables and about 0.8 for a second variable, so I'm still 
> doing something wrong, unless the adjustment could actually revise the standard 
> errors downwards (sounds contrary to the intent that Allison and Waterman had 
> in their article).
>
> 3. I may be computing degrees of freedom for the model incorrectly. I'm using 
> the number of observations actually used in the model (not associated with 
> groups for which fixed effects are undefined), minus the number of parameters 
> in the model (number of groups + number of parameters + 1 for the constant 
> term). Or, N-(p+k+1).
>
> 4. My syntax could be wrong, of course -- particularly when calculating 
> deviance -- although I've checked it a couple of times.
>
> Again, Allison and Waterman's formula:
>
> D = Sum_i{Sum_t{y_it*log(y_it/mu_it)-(y_it+lambda)log[(y_it + 
> lambda)/(mu_it+lambda)]}}.
>
> Thanks, all. Any suggestions? I'm going to try doing (1) and (2) above 
> simultaneously and see if that works, but I'm skeptical.
>
> Best,
>
> Richard Callahan
>
> /We need a federal commission for regulating probability distribution notation 
> and an independent counsel for preserving degrees of freedom
>
>
> ---
> Richard Callahan
> Graduate Student, Department of Sociology
> University of Washington
> (206) 769-9812
> rjcal<at>u.washington.edu
> http://students.washington.edu/rjcal/
>
>
>