How different from one does a dispersion ratio have to be to be considered significant?

Question

I am in the process of conducting zero-inflated generalised mixed effects models with Poisson distributions and have been using the testDispersion() function of the DHARMa package in R to determine if my data is significantly over/underdispersed. Whilst this function does provide a p-value to indicate significance of the dispersion I was wondering if there was a particular dispersion ratio above or below 1 that is used to recommend using a different distribution (e.g. negative binomial) to account for the dispersion.

Answer 1

No, there is no universal rule of thumb for this, and you should definitey not rely on p values. If you have reason to think that you have over or under dispersion, then go ahead and fit a model that accounts for this and compare the model fit with the Poisson model.