# GLM - which probability distribution to use for abundance data?

I'm fitting a generalized linear model to try to understand how the abundance of a species of freshwater fish varies in response to some environmental variables. I'm using the AIC to choose between models. My main question is which family of probability distribution to use, Poisson or Gamma?

When I use Poisson, I can't get an AIC value for the null model. The message that appears is: AIC: Inf.

The summary output is this:

Call:
glm(formula = Lampetra ~ 1, family = poisson(link = log), data = cont)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-1.3833  -1.0154  -0.3811   0.1948   2.4742

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)  0.08295    0.22009   0.377    0.706

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 22.079  on 18  degrees of freedom
Residual deviance: 22.079  on 18  degrees of freedom
AIC: Inf

Number of Fisher Scoring iterations: 5

• Can you show the output from summary(mod) where mod is your fitted Poisson GLM. (FYI, you are using R, you just happen to be using R through RStudio). I've partially answered your question but difficult to say what is causing the infinite AIC without additional info – Gavin Simpson Feb 22 '16 at 15:26

The Gamma distribution has support on non-negative real numbers, i.e. it is for continuous data between 0 and $+\infty$. The Poisson has support on the the non-negative integer numbers. We normally record abundance as numbers of individuals, a count, and therefore a discrete distribution like the Poisson would be a reasonable starting point for modelling. The Gamma would be unsuitable as it is for continuous data and we often don't see 2.4 fish.
The infinite AIC suggests there may have been a problem with the fitting of the model or perhaps your data are not Poisson distributed (conditional upon the values of $X$). It's difficult to diagnose potential problems with so little information to go on.