# Overdispersion tests dependence on used covariates in Poisson model

One of the shortcomings of the Poisson regression model is that the mean, conditional on the independent variables should equal the conditional variance. If the observed variance is larger (smaller) than this theoretical variance ( which is the observed conditional mean), there is overdispersion (underdispersion).

Possible causes of overdispersion are

• Omitted variables
• Excess zero counts
• Correlation between individual responses
• Cluster sampling
• More...

Now there are several methods of testing for overdispersion including

• Auxiliary regression (in R)
• Likelihood ratio test (in R)

These involve testing for overdispersion in a fitted model. My question now is, since I have around 100 possible covariates, do I first select covariates and then test for overdispersion or the other way around?

Basically, what I want to know is:

• Does including other covariates affect the result of an overdispersion test (I would say yes, but don't know in what way)
• Would finding a 'good' subset of covariates in a Poisson model, correspond to a 'good' set of covariates in a Negative Binomial model?

I was wondering if it might be the case that when selecting a subset of possible covariates I deem 'correct' for the Poisson model and then switch over to a Negative Binomial model due to overdispersion, I might have to do the covariate selection again?

However, if a negative binomial (NB) model fits your data well enough, I simply would go with that rather than the Poisson. The worst thing that can happen is that the true $\theta = \infty$ (= Poisson case) and you have estimated one parameter too much. Thus, you have lost a little bit of efficiency but are still consistent.
On the other hand, if you would really need the NB (with finite $\theta$) and use a Poisson model, then your parameter estimates are still consistent but all standard errors are too small. And if you were to do model selection based on significance tests, you would get spuriously significant regressors.