# Relationship between quasi-likelihood and "GLM-conjugate" models?

Suppose we have a response variable that represents proportions with a poorly defined denominator. Two ways to handle this (1) a quasibinomial model, which assumes only that the variance is proportional to $$p(1-p)$$ (where $$p$$ is the mean/expected proportion for a particular response) and (2) a Beta regression, which uses a full likelihood specification. According to Wikipedia (yes I'm lazy), if the shape parameters are $$\alpha$$ and $$\beta$$ (so $$p=\alpha/(\alpha+\beta)$$ and $$(1-p)=\beta/(\alpha+\beta)$$ then the variance is

$$\textrm{Var} = \frac{\alpha\beta}{(\alpha+\beta)^2 (\alpha+\beta + 1)} = \frac{p(1-p)}{\alpha+\beta+1}$$

which does not quite match (at least in the standard parameterization). However, we also don't necessarily have a reason to believe that this assumed mean-variance specification is actually better than the quasi-binomial version, for a particular data set. More generally, if we're working in a general MLE framework (i.e. not restricted to GLM-based solutions) and want to reparameterize the distribution we set any mean-variance relationship we want.

This is all a bit vague, but: does anyone have references/ideas about places where people have pursued these connections and the pros and cons of different approaches? (e.g.: quasi-binomial models can be efficiently fitted by IRLS methods; Beta models have a proper likelihood [allowing model comparison, profile CIs, etc.])

This question generalizes to other families, e.g. quasi-Poisson vs Gamma (in that case the Gamma is actually in the exponential family ...)

I'm admittedly asking this question out of laziness, I may be able to figure out if I think/dig hard enough.