Model Selection- Poisson and Negative Binomial I have run 2 GLM models with same variable specifications except that one was run with the response variable following a generalized poisson distribution and the other with a negative binomial distribution. I ran these in R using the glmmTMB package. genpois was used for the generalized poisson method and negbinom2 for the negative binomial.
The generalized poisson model saw an overdispersion of 290; while the negative binomial model saw a much lower overdispersion of 3.8. The AIC of the generalized poisson is 2464 and that of the negative binomial is 2466. On running a likelihood ratio test, the genpois method is preferred.
I'm looking for some perspective on on reconciling these different outcomes- where the negative binomial greatly reduces the overdispersion yet is not preferable to the generalized poisson given the LRT.Thanks.
 A: It is a different parameterization and does not mean less overdispersion. What you see off the summary is a dispersion parameter and by typing ?family_glmmTMB:
the dispersion model uses a log link. Denoting the variance as V, the dispersion parameter as phi=exp(eta) (where eta is the linear predictor from the dispersion model), and the predicted mean as mu:
nbinom2 Negative binomial distribution: quadratic parameterization
       (Hardin & Hilbe 2007). V=mu*(1+mu/phi) = mu+mu^2/phi.

genpois Generalized Poisson distribution (Consul & Famoye 1992).
    V=mu*exp(eta). (Note that Consul & Famoye (1992) define phi differently.)

So we fit an intercept only model using rnbinom, where the variance is mu + mu^2/size, meaning if we simulate with size 1, we expect phi (dispersion) close to 1:
library(glmmTMB)
set.seed(111)
y = rnbinom(100,mu=10,size=1)

Based on the 

Formula:          y ~ 1
      AIC       BIC    logLik  df.resid 
 667.1145  672.3248 -331.5572        98 

Number of obs: 100

Overdispersion parameter for nbinom2 family (): 1.09 

Fixed Effects:

Conditional model:
(Intercept)  
      2.267  

Now if we fit the pois:
glmmTMB(y~1,family="genpois")
Formula:          y ~ 1
      AIC       BIC    logLik  df.resid 
 670.6158  675.8262 -333.3079        98 

Number of obs: 100

Overdispersion parameter for genpois family (): 12.2 

Fixed Effects:

Conditional model:
(Intercept)  
      2.267  

And the dispersion roughly works out to be var(y)/mean(y).. Hence you get very different dispersion but if you work back the variance, it's should be roughly the same in your example, hence very similar AIC.
