Changing from Poisson to NB distribution fixes overdispersion and improves model

Question

I have count data (seconds of behaviour, n=145) with one explanatory categorical variable (Treatment) and 2 interaction terms (Sex and Time). I began by running a GLM with Poisson distribution:

model <- glm(Vigilance ~ Treatment * Sex * Time, family = "poisson", data = fallow)

The output gave significant results for all coefficients but one, and two were not defined because of singularities (so I will remove). Residual deviance = 2412.8 on 135 df and AIC was 2849.4. I checked for overdispersion with:

theta <- model$deviance / model$df.residual

which showed model is overdispersed - theta = ~17.8.

When I ran the same using quasipoisson, none of the coefficients were significant, yet when I ran the same with negative binomial, 2 coefficients were significant, residual deviance reduced to 161.7 on 135 df, AIC dropped to 950, and there was overdispersion - theta = 1.18.

Other options outlined by Zuur et al. (Mixed Effects Models and Extensions in Ecology with R), namely using drop1, are unavailable because my variables are included as interaction terms.

Is there anything else I should try, or can I safely use the NB distribution?

Answer 1

If you have overdispersion, NB is always preferable over a simple Poisson.

However, better doesn't mean good. There are two things to consider

• First of all, there are several possible reasons for overdispersion, but one is simply that your model is misspecified. Before you move to NB (which adjusts the dispersion, but doesn't solve the problem), you should plot residuals against all your predictors to see if you have any systematic errors.

• Once you have moved to NB, you won't have general overdispersion. But you can still have many other problems, e.g. heteroscedasticity, or again model misspecification. Thus, check your fitted model!

Procedures for model diagnostics in both cases are explained, e.g., here.

p.s.: Treatment * Sex * Time is hardly ever a good idea. (Treatment + Sex + Time)^2 is probably preferable, but are you sure you need all the 2-way interactions?