consul's generalised poisson

[Matt Piercy]

Hi all,
I am interested if anyone has had experience using consul's generalised poisson. I have two questions: 1) can you use it for non-count data (like ordinary poisson) assuming the mean if correctly specified and 2) what are the caveats with the gamma variable? I have been reading Cameron and Trivedi (1999) who say that the ML assumption of asymptotic normality is violated by the gamma variable because it skews the distribution. Has anyone experience with this model? Thanks, Matt

[Dirk Enzmann]

This is no answer to your problem, but a follow-up question triggered by your question:

When reading a paper by Tuenter (2000) I stumbled across the description of the generalized Poissson distribution (GPD) as "It extends the Poisson distribution by its ability to describe situations where the probability of occurrence of a single event does not remain constant (as in a Poisson process), but is affected by previous occurrences." (Tuenter, 2000, p. 374). Therefore, I wonder whether GPD could be used as an alternative to the negative binomial distribution where overdispersion can be the result of dependencies of events. Any clarifying thoughts?

By the way: When referring to authors / literature, it should be standard to give references because you should not require members of the Stata Forum to search for this by themselves. However, in this instance your omission helped to me to learn about a (possible) alternative to model count data.

Reference:
Tuenter, H. J. H. (2000). On the generalized Poisson distribution. Statistica Neerlandica, 54(3), 374–376. https://doi.org/10.1111/1467-9574.00147. [free access paper (preprint?): https://arxiv.org/abs/math/0606238]

[Matt Piercy]

Thanks Dirk.
As I understand it, it is different from the negative binomial. It has its own log likelihood, with the gamma variable controlling dispersion. Obviously my reference is dated, hence why I asked. This is one of the books on the stata manual for Poisson. I use a bit of AI (mostly copilot), and although that says you can't use it for non-count data, it says the same for Poisson, which I know you can if the mean is correctly specified. I can't see a reason why it wouldn't work, bearing in mind the problems with the gamma variable skewing the distribution, so some of the tests might be inaccurate.

[Joao Santos Silva]

Dear Matt Piercy,

As for your first question, I do not think that a distribution that accounts for overdispersion can be used to model non-count data with no natural scale. The reason is that the degree of overdispersion changes with the scale of the data, so the estimates will depend on the units in which the outcome is measured.

As for your second question, could you please post the exact quote you are referring to?

Best wishes,

Joao

[Matt Piercy]

Here it is. From Regression analysis of count data, A.C. Cameron and P.K. Trivedi (1999) Cambridge: CUP Scan2026-08-23_134936(3).pdf

[Matt Piercy]

Thanks Dirk for the article. Very interesting. I have a daughter in Heemskirk at the moment staying for a year in Holland.

[Dirk Enzmann]

Matt Piercy : (referring to #3): Gamma is not a variable but a parameter. And I would not qualify literature as "outdated" only because it has been published many years ago. You would not say that Euler's publication Institutiones calculi differentialis (1755) (if you want I could take the effort finding an English translation) is outdated although published more than 270 years ago.

Joao Santos Silva : The works by Consul are quoted in Tuenter (2000, see #2):

• Consul, P. C. (1989). Generalized Poisson Distributions: Properties and Applications, Volume 99 of Statistics: Textbooks and Monographs. New York: Marcel Dekker Inc. [most likely pp. 117-129]
• Consul, P. C. & Jain, C. G. (1973). A generalization of the Poisson distribution.Technometrics 15 (4), 791–799.

I still would like to know in which situations the GPD can be used as an alternative for the negative binomial distribution -- some thoughts / hints are welcome.

[Dirk Enzmann]

Add-on:

Generalized Poisson regression has been discussed by Harris et al. (2012) for modeling underdispersed count data using gpoisson (see findit gpoisson). They refer to a somewhat more recent article by Consul and Famoye (1992). Any additional thoughts/comments on situations/conditions where the use of gpoisson for modeling overdispersed ccount data may be preferred to negative binomial regression?

References:

• Consul, P. C., & Famoye, F. (1992). Generalized poisson regression model. Communications in Statistics - Theory and Methods, 21(1), 89–109. https://doi.org/10.1080/03610929208830766
• Harris, T., Yang, Z., & Hardin, J. W. (2012). Modeling underdispersed count data with generalized poisson regression. The Stata Journal: Promoting Communications on Statistics and Stata, 12(4), 736–747. https://doi.org/10.1177/1536867X1201200412

[Matt Piercy]

Cool thanks. I didn't know there was a gpoisson command. That's great. I think that's what I was looking for. Cheers, Matt

[Joao Santos Silva]

Dear Dirk Enzmann,

In the 1973 Technometrics paper you mentioned above, the generalised Poisson is presented as the limiting form of the generalised negative binomial distribution. Therefore, it can be used to model overdispersion. I would be careful about using it to model underdispersion because I understand that in that case some probabilities can be negative. See

Nelson, D. L. (1975). "Some Remarks on the Generalizations of the Negative Binomial and Poisson Distributions". Technometrics, 17, 135-136.

[Nick Cox]

Prem Chandra Consul (August 10, 1923 - June 16, 2023) was an Indian-Canadian statistician born in Uttar Pradesh and died in Calgary.

[Matt Piercy]

Hi Joao,
There is a worked example in J.M. Hilbe (2014) Modelling Count Data, Cambridge: CUP, which does use gpoisson for underdispersed count data. He compares it with glm poisson and comments that negative binomial can't model underdispersed count data. Scan2026-08-23_134936.pdf Scan2026-08-23_134936(4).pdf Cheers, Matt

[Joao Santos Silva]

Thanks, Matt Piercy. However, do keep in mind Nelson's result I mentioned above. It all depends on what you want to do: if you just want to estimate a conditional mean (as would be the case with non-count data), then plain Poisson regression is preferable.

[Matt Piercy]

OK. Thanks Joao. Your insights have been very appreciated.