A poisson GLM and a quasipoisson regression model will given identical point estimates for the beta parameter of the linear predictor. The quasipoisson model is typically used when there is overdispersion in the poisson GLM fit. Since use of the quasipoisson will not affect point estimates, it will only affect inferential procedures such as calculating standard errors, p-values, confidence intervals etc. Why not just bootstrap the estimator of interest using the poisson model? Overdispersion already gives us reason to be skeptical of model fit and the quasipoisson seems like a hacky adjustment -- I would expect the bootstrap to be more robust and require fewer assumptions. Is there another advantage to using the quasipoisson?
Perhaps someone else will come up with a better answer, but:
- bootstrapping is necessarily many times slower than the original fit (e.g. at least 1000 times slower if you follow the most popular answer here). That's fine if your problem is relatively small, but cumbersome and/or impossible as the problem becomes larger.
- implementing the correct bootstrap can be a little tricky if your responses aren't independent (e.g. spatial/temporal correlation, grouping). Obviously you'd also have to account for that in your original Poisson GLM, but if you have then you might not want to deal with the complexities of structured bootstrapping as well.
- I would often choose to use a solution other than quasi-likelihood to deal with overdispersion (e.g. a negative binomial or lognormal-Poisson model), but it is easy and fast, and could be adequate in many cases.
I agree that you should always check your model to get some idea about the source of the overdispersion. If your model is biased (e.g. there's a quadratic trend in your supposedly linear model), I don't think bootstrapping is going to fix the problem ...