# Choosing between overdispersed Poisson and a negative binomial models

I have a Poisson glmm (using glmer) that is slightly over-dispersed at 1.854. I tried re-fitting it as a negative binomial model (using glmer.nb). On inspection of some of the diagnostic plots, I'm stuck with choosing which model is the better fit.

The qq-normal plot for the Poisson model (bottom left) looks better than that for the NB model (top left), while the fitted vs. residual plot looks somewhat better for the NB model (top right). I think that the structure that is appearing in the fitted vs. residual plot for the Poisson model is due to the fact that the data set is somewhat zero-inflated.

My gut tells me to go with the slightly over-dispersed Poisson. Can anyone suggest a way to choose? Or a way to adjust for the over-dispersion in the Poisson model?

The data are from a paired design.

• Why are you making a normal QQ plot of the residuals from a Poisson or a negative binomial model??? – GoF_Logistic May 3 '17 at 18:34
• stackoverflow.com/questions/22842017/… – JKO May 3 '17 at 18:35
• Having variance larger than the mean (the definition of overdispersion in a Poisson model) has absolutely nothing to do with the residuals (the pearson residuals I guess?) being normally distributed. – GoF_Logistic May 3 '17 at 18:46
• Right. I am trying to assess which model is a better fit. If it is NB, then I don't need to worry about the dispersion parameter. If it is Poisson, then I need to correct for the over-dispersion. Thus, what I need is a way to distinguish between the two models. – JKO May 3 '17 at 20:00