Meaning of alpha in xtpoisson

[Bram Hogendoorn]

Dear all,

I am estimating a random-effects Poisson regression with Gamma distribution, using -xtpoisson, re-. The output shows the parameters alpha and /lnalpha. What are these exactly?

The help-file of -xtpoisson- mentions that "the exponentiated random effects are distributed as gamma with mean one and variance alpha". It later mentions that "follows a gamma distribution with mean one and variance 1/theta". This confuses me, because there are many ways of parametrizing Gamma distributions, including the parameters alpha, beta, lambda, mu, theta, and k. Any explanation would be welcomed.

[William Lisowski]

Have you followed the link at the top of the output from help xtpoisson to "View complete manual PDF entry" and reviewed the Methods and formulas section, and perhaps followed the link to the documentation for xtgee which is used by xtpoisson?

[Bram Hogendoorn]

Thank you, William. I have followed the link and read that -xtpoisson, pa- can be estimated with -xtgee-. But this does not seem to answer my question regarding the meaning of alpha.
To my knowledge, the mean of a Gamma distribution is usually written as alpha/beta and its variance as alpha/(beta)². Stata deviates from this in ways that I don't understand. It appears that alpha [in stata] is 1/beta [conventional] and that theta [stata] is the same as beta [conventional], but again, I am not sure.

[William Lisowski]

The Wikipedia discussion of the Gamma distribution found at

https://en.wikipedia.org/wiki/Gamma_distribution

discusses the three common parameterizations of the Gamma distribution, and the interpretation of the parameters.

[Bram Hogendoorn]

I am aware of the Wikipedia page, but I am unable to reconcile it with Stata's notation. The notation of -xtpoisson- also appears inconsistent with -rgamma()-, which uses parameters "a" and "b". For now, I will just live with the idea that Stata distributes the exponentiated random effects as a Gamma distribution with variance "alpha".

[Rich Goldstein]

The following is from the manual - if this does not cover what you need, please clarify your question:

The output also includes a likelihood-ratio test of = 0, which compares the panel estimator with
the pooled (Poisson) estimator.
We find that the incidence rate for accidents is significantly different for the periods of construction
and operation of the ships and that the random-effects model is significantly different from the pooled
model.