The phenomenon of ‘over-dispersion’ in a GLM arises whenever we use a model that restricts the variance of the response variable, and the data exhibits greater variance than the model restriction allows. This occurs commonly when modelling count data using a Poisson GLM, and it can be diagnosed by well-known tests. If tests show that there is statistically significant evidence of over-dispersion then we usually generalise the model by using a broader family of distributions that free the variance parameter from the restriction occurring under the original model. In the case of a Poisson GLM it is common to generalise either to a negative-binomial or quasi-Poisson GLM.

This situation is pregnant with an obvious objection. Why start with a Poisson GLM at all? One can start directly with the broader distributional forms, which have a (relatively) free variance parameter, and allow the variance parameter to be fit to the data, ignoring over-dispersion tests completely. In other situations when we are doing data analysis we almost always use distributional forms that allow freedom of at least the first two-moments, so why make an exception here?

My Question:Is there any good reason to start with a distribution that fixes the variance (e.g., the Poisson distribution) and then perform an over-dispersion test? How does this procedure compare with skipping this exercise completely and going straight to the more general models (e.g., negative-binomial, quasi-Poisson, etc.)? In other words, why notalwaysuse a distribution with a free variance parameter?

**Answer**

In principle, I actually agree that 99% of the time, it’s better to just use the more flexible model. With that said, here are two and a half arguments for why you might not.

(1) Less flexible means more efficient estimates. Given that variance parameters tend to be less stable than mean parameters, your assumption of fixed mean-variance relation *may* stabilize standard errors more.

(2) Model checking. I’ve worked with physicists who believe that various measurements can be described by Poisson distributions due to theoretical physics. If we reject the hypothesis that mean = variance, we have evidence against the Poisson distribution hypothesis. As pointed out in a comment by @GordonSmyth, if you have reason to believe that a given measurement *should* follow a Poisson distribution, if you have evidence of over dispersion, you have evidence that you are missing important factors.

(2.5) Proper distribution. While the negative binomial regression comes from a valid statistical distribution, it’s my understanding that the Quasi-Poisson does not. That means you can’t really simulate count data if you believe Var[y]=αE[y] for α≠1. That might be annoying for some use cases. Likewise, you can’t use probabilities to test for outliers, etc.

**Attribution***Source : Link , Question Author : Ben , Answer Author : Cliff AB*