Poisson regression with both response and explanatory variables as counting

Question

I’ve got the following variables:

Response: number of quota units leased (in and out) by fishers.

Explanatory: number of quota units own by fishers.

I fitted a GLM (Poisson), but I’m not totally sure if it’s right, considering that the explanatory variable is count as well. I’ve found examples of Poisson regression just with categorical and continuous explanatory variables, but not with counting variables.

So:

Am I right using Poisson with my data? If not so, what alternative do I have? The residuals variances of my model are not homogeneous. I understand that Poisson regression allows face this problem, or should I pay attention to this issue and solve it (using weights, for example)?

Any help would much appreciated, Cheers

Answer 1

The process is not a Poisson process; first there's a limited number of quota units that can be leased (that's what a quota is about, after all). Secondly, the earlier units and the later units may not be equally sought after, nor will they be independent (e.g. if someone buys all the quota units they want, they won't take more, so demand may tend to decrease somewhat).

The potential heterogeneity might push us toward the negative binomial while the quota suggests perhaps that a binomial might do slightly better. On the other hand a heterogenous mixture might suggest a beta-binomial or something.

On the other, other hand, a heterogeneous mixture of binomials can sometimes be well approximated by a Poisson, so we're back where we started, and we haven't talked about dependence yet.

To be honest, I'd probably try a Poisson first up, but I wouldn't be surprised if something else worked better. My choice might also be influenced by what information I was trying to get out of it.

The number of leases available would typically be used as an exposure in a Poisson model, but would more typically be the $n$ in a binomial model, suggesting in either case a constant-parameter (but not constant mean) model, unless you have further predictors.

Answer 2

1. As to your concern about using count data as predictors, see the most popular comment on this question: there is nothing wrong in doing that.

2. As to your comment on the variance of residuals, for Poisson regression and other nonlinear models, raw residuals are always heteroscedastic and asymmetric (Cameron & Trivedi's book Regression Analysis for Count Data, 1998, sec. 5.2.1). You may consider alternative types of residuals, e.g., Pearson residuals, which in large samples have zero mean and are homoskedastic (although not symmetric).

3. As to your question on the potentially useful alternatives, Efron's (1986) two-parameter double Poisson model can be considered if the Poisson fit is unsatisfactory. The Poisson model is nested in the double Poisson model. The double Poisson model allows for both overdispersion and underdispersion. With the double Poisson model, the mean and the dispersion parameters may depend on the observed values of the predictor separately, allowing for extra flexibility. If interested, take a look at this analysis of the applicability of the double Poisson (DP) generalised linear model for over- and under dispersed data.

4. In addition, you could search for Neyman's type A, Counsil's generalised Poisson and Conway–Maxwell–Poisson distributions as well as regression models based on them. There is also some literature on bivariate Poisson regression models, but be aware of Ehrenberg (1963), as commented by D.R. Cox (1968) here.

References:

Efron, B. (1986), Double Exponential Families and Their Use in Generalized Linear Regressions, Journal of the American Statistical Association, 81, 709-721.

Ehrenberg, A. S. C. (1963) Bivariate Regression Analysis is Useless Journal of the Royal Statistical Society. Series C (Applied Statistics), 12, 161-179