Limits of correction for over dispersion with Poisson / binomial data

Question

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.

Answer 1

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.