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

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02-27-2012 12:59 PM

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

σ^{2}=µ + α_{2}µ^{2}, in which case, observations with meanµare assumed to follow the distribution (Eq. 1):

P(x=m) = PDF(“NEGB”, m,

α/(1+_{2}µα), 1/_{2}µα),_{2}or

σ^{2}=µ + αwhich implies the distribution (Eq. 2):_{1}µ,

P(x=m) = PDF(“NEGB”, m,

α/(1+_{1}α),_{1}µ/α)._{1}

(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

PG_Alpha, as reported in the OUTEST= dataset, represent when DIST=NEGBIN(P=1) is requested?

PG

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Solution

02-28-2012
05:41 PM

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02-28-2012 05:41 PM

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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Solution

02-28-2012
05:41 PM

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02-28-2012 05:41 PM

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