There is more than one solution for the problem of overdispersed count data. One is to use a quasipoisson model. One is to use a negative binomial model. One is to use a mixed-level model with subject-level random intercepts. Is there a rational and non-arbitrary way to choose among these? I ask because of a specific behavior I discovered for some overdispersed data. I have laid out all the details for this behavior in the following Kaggle notebook:
I am adding an update to summarize information given so far.
The quasipoisson, according to the linked thread, has the trait of making fewer assumptions than do likelihood-based methods. It can be somewhat emulated by an "NB1 parameterization" of the negative binomial, but glm.nb uses the "NB2 parameterization.
Negative binomial glm and poisson glmer with subject-level intercepts are the same types of model, if you ignore random intercepts of the glmer and only look at the fixed effects estimates. The major difference is that negative binomial assumes a gamma distribution of individual effects and glmer assumes a gaussian distribution.
So it comes down to how much you know about your data, or such is my guess. Do you know enough about your data to assume that the subject-level effects that contribute to overdispersion can be (at least roughly) parameterized with either a gaussian or gamma approximation? If no, then use quasipoisson. Assign to negative binomial or glmer based on your willingnesss to presume about the specific distribution. If you don't know these things or at least have good reason to presume these things, use quasipoisson.
Is this a good summary of useful principles?