Quantifying overdispersion effects in count regression data

The Poisson regression model is often used as a first model for
count data with covariates. Since this model is a GLM with
canonical link, regression parameters can be easily fitted using
standard software. However the model requires equidispersion, which
might not be valid for the data set under consideration. There have
been many models proposed in the literature to allow for
overdispersion. One such model is the negative binomial regression
model. In addition, score tests have been commonly used to detect
overdispersion in the data. However these tests do not allow to
quantify the effects of overdispersion. In this paper we propose
easily interpretable discrepancy measures which allow to quantify
the overdispersion effects when comparing a negative binomial
regression to Poisson regression. We propose asymptotic
$\alpha$-level tests for testing the size of overdispersion effects
in terms of the developed discrepancy measures. A graphical display
of p-values curves can then be used to allow for an exact
quantification of the overdispersion effects. This can lead to a
validation of the Poisson regression or a discrimination of the
Poisson regression with respect to the negative binomial
regression. The proposed asymptotic tests are investigated in small
samples using simulation and applied to two examples.