Modelling catch rates: difference between Gamma and poisson distribution

I am confused about the difference between modelling counts (i.e. catch) using a poisson distribution with an offset (i.e. effort) and modelling catch per unit effort using a gamma(link=log) distribution.

I have always used poisson models for count data, but I have lately analysed data that are highly dispersed and since the model did not converge (I have also tried ZIP, OLRE, negative binomial...) , I modelled catch per unit effort using a Gamma model, that gave me a very good fit.

I wonder if the Gamma model is a correct alternative approach and why it converges while the poisson does not.

(OP is probably long gone by now, but ...)

tl;dr you probably want to use a negative binomial model with an offset to account for exposure.

Offsets have a separate purpose from the choice of distribution; they model variation in the outcome based on sampling effort (exposure). They are necessary for count-data models (Poisson, negative binomial) where changing the scale or units of the response will mess up the mean-variance relationship, and merely convenient for continuous-data models (where they typically change the value of the estimated parameter but not the overall model fit/inference).

A Gamma distribution might give you a roughly equivalent fit to a negative binomial; for example, the variance of the Gamma for a sample with mean $$\mu$$ is $$\mu^2/a$$ (where $$a$$ is the shape parameter), while the variance of the negative binomial is $$\mu + \mu^2/k$$. (For low values of the mean, the variance of the negative binomial is a bit higher because of the effects of sampling a discrete variable.)

The negative binomial probably makes more sense because it is intended for modeling a discrete outcome.

For count data where the absolute size of the count is very large you'll probably get similar results from Gamma, negative binomial, and log-Normal models.

• Thanks Ben, OP still around :) Apr 7 at 20:28