We have count data, but it appears that this is overdispersed. Therefore the assumed Poisson distribution should be replaced by a quasi-Poisson or a negative binomial. Although there is some literature around this topic (for instance see http://fisher.utstat.toronto.edu/reid/sta2201s/QUASI-POISSON.pdf), it is rather technical, and we were wondering if there is a pragmatic approach in R to determine whether to use Poisson, quiasi-Poisson or negative binomial as the underlying distribution of the response data?
You might check if the Vuong test will tell you what you need.
Also, you might consider Hermite regression, which is a more-general distribution, but not as well supported in R.
I'm not a statistician, but the examples and references in the "Hermite and Poisson Regression for Count Data" chapter of my book (free online) might be helpful.
You may want to read a recent paper in Methods in Ecology and Evolution, 8(7):882-890 Three points to consider when choosing a LM or GLM test for count data (David Warton et al). It may give you some thoughts for consideration.
