# Negative Binomial or Poisson regression?

My dependent variable (count) shows signs of overdispersion (mean 2.50, Variance 6.60), which led me to use a negative binomial model. This seems to fit better compared to the Poisson regression (lower AIC). However, few of the significant effects I found when using the Poisson regression become highly non-significant when using the negative binomial model.

Is that normal? should I stick with the Negative Binomial regardless?

Thank you

## 1 Answer

You can run both and perform a likelihood ratio test to test whether the negative binomial is to be preferred. In almost all cases, it is. Remember that overdispersion matters for the conditional outcome, not the outcome itself, so examining the mean and variance in the outcome prior to fitting the model does not tell you which model will be preferred.

One reason you have significance in the Poisson and not in the negative binomial is that in the presence of overdispersion, Poisson standard errors are too small. You may be tempted to go with the model that proves your hypothesis, but you should do everything you can to ensure that that is a valid choice.

• thank you for your answer. When deciding the best model based on the likelihood ratio or AIC, should I run them with intercept only (thus, dependent variable only) or including the independent variables (which are quite a few - 8 to be precise)? Thanks – John Walk Jun 16 '20 at 11:03
• With all your predictors. You want to compare two models, and unless comparing intercept-only models is relevant to your research question, there's no use in doing it. Because the models are nested, you should use the LR test, not just compare AIC. Honestly you should just use the NB model. If a Poisson is appropriate, then you've wasted one parameter. Not a big deal. If you were to use a Poisson model and the NB was appropriate, you would make a huge statistical error. It's perfectly fine to just choose the NB without dmonstrating it's better than the Poisson for your data. – Noah Jun 16 '20 at 18:39
• Thank you for the clarification. That was very helpful! – John Walk Jun 17 '20 at 10:21