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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
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    $\begingroup$ 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 $\endgroup$ – Gavin Simpson Feb 22 '16 at 15:26
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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.

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  • $\begingroup$ @Rui, in light of the points made in Gavin's answer, you may want to edit & update your Q. If this has resolved your concern, please consider accepting it by clicking the check mark to its left under the vote total. $\endgroup$ – gung - Reinstate Monica Feb 22 '16 at 15:30
  • $\begingroup$ Hi @Gavin!Thank you for the answer! I must probably use the Gamma distribution because the abundance is in CPUE (catch per unit effort) which means that the values can be 0.658(example)! The output of summary is: 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 param for poisson family taken to be 1) Null devia: 22.079 on 18 degrees of freedom Residual devia: 22.079 on 18 degrees of freedom AIC: Inf $\endgroup$ – Rui Feb 22 '16 at 15:39
  • $\begingroup$ @gung, thank you for the editing the question! Is this information enough? $\endgroup$ – Rui Feb 22 '16 at 15:44
  • $\begingroup$ The Gamma distributions, apparently, fits well because I can get a AIC value in the null model! But is it normal that in 12 variables only 1 affect the model? $\endgroup$ – Rui Feb 22 '16 at 15:53
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    $\begingroup$ @gung, like this? $\endgroup$ – Rui Feb 22 '16 at 18:00

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