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

`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`
PG

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‎02-28-2012 05:41 PM
Posts: 5,056

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

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

PG

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‎02-28-2012 05:41 PM
Posts: 5,056

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

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

PG
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