Parameter _Alpha in COUNTREG with DIST=NEGBIN(P=1)

PGStats — Mon, 27 Feb 2012 17:59:03 GMT

I am using PROC COUNTREG to look at time trends in fish capture data. As I really don’t expect the fish counts to be from a Poisson distribution, I request the regression to consider the counts as negative binomial variates. 
PROC COUNTREG offers two ways to relate the variance to the mean when doing negative binomial regression. Either σ²= µ + α₂µ², in which case, observations with mean µ are assumed to follow the distribution (Eq. 1):
 
P(x=m) = PDF(“NEGB”, m, α₂µ/(1+ α₂µ), 1/α₂), 
or σ²=µ + α₁µ, which implies the distribution (Eq. 2):
 
P(x=m) = PDF(“NEGB”, m, α₁/(1+ α₁), µ/α₁).
 
(I added a subscript to alpha because both values are not, and should not be, the same) 
 
I wanted to compare the fit of both alternatives to my data. I was able to generate a confidence interval around the fitted curves using Eq. 1, but when tried to do the same with Eq. 2, the values made no sense. 
 
I would like to know: what does the estimated parameter _Alpha, as reported in the OUTEST= dataset, represent when DIST=NEGBIN(P=1) is requested?
PG

Re: Parameter _Alpha in COUNTREG with DIST=NEGBIN(P=1)

PGStats — Tue, 28 Feb 2012 22:41:30 GMT

Too late to retract this question… The answer was that Eq. 1 and Eq. 2 were in error. They should have read (Eq. 1) : P(x=m) = PDF(“NEGB”, m, 1/(1+ α2µ), 1/α2), and (Eq. 2) : P(x=m) = PDF(“NEGB”, m, 1/(1+ α1), µ/α1). The values of _Alpha produced by PROC COUNTREG in negative binomial regressions effectively correspond to the factor in the mean to variance relationships. - PG

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Parameter _Alpha in COUNTREG with DIST=NEGBIN(P=1)

Re: Parameter _Alpha in COUNTREG with DIST=NEGBIN(P=1)