# Limits of correction for over dispersion with Poisson / binomial data

Reading many help pages, I see that a common way to correct for overdispersion is to fit models using quasibinomial or betabinomial distributions. However, I can't find help concerning how far correction is achieved nor how to be confident that a reliable model has been fitted.

I am interested in mutation data from many genes that fall into 10 ordered classes. Each observation comprises a number of bases surveyed (N) and the number of substitutions observed (S). Overdispersion is great (~22!) because the genes vary considerably in how fast they are evolving. There is also the issue that N also varies a lot from <100 up to >30,000.

I want to test the hypothesis that average substitution rate (S/N) increases with class number. I have tried the following options:

1. (awful!) Ignore the fact the data are binomial, set a minimum N>=1000, and fit a Gaussian model with S/N as the response. Fit is poor, though arguably not dreadful.

2. Use cbind(S, N-S) and fit a model with a quasibinomial error structure. Seems to give a similar answer that avoids the crazily low P-values I get when using family=binomial, but how do I know if the degree of correction is valid?

This seems a relatively simple problem, but I want to make sure the model is fitted properly and is, if anything, conservative.

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 it is better to set a minimum success and fail counts rather than sample size. It doesn't need to be too high either. Say $min(S,N-S)\geq 10$ should be fine for normal distribution to work well. – probabilityislogic Aug 8 '12 at 21:39 But, even if you do that, you should use a variance-stabilizing transformation, as I said in my response below. That may be an "old tecnology" from the times when generalized linear models where not readily available in software, but it is still a good approach, especially for simple models. If you have many regressors, a GLM might be better, because the parameters are easier to inyetrpret. – kjetil b halvorsen Oct 7 '12 at 17:25

Just one idea: You can use a variance-stabilizing transformation. If you have $X \sim \text{Bin}(n,p)$, then the variance-stabilizing transformation is $Y=\arcsin(\sqrt(x/n))$, and now you can use (possibly weighted, with the differents $n$'s giving the weights) usual linear regression. This is often effective, and the overdispersion is automatically taken care of.