Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 1 / f / Solution

Because models 1 and 3 are full likelihood models for the same response, one can compare their Akaike information criterion values; that favors model 1. One can also compare held-out count prediction by K-fold cross-validation, using a common loss such as Poisson deviance or negative log predictive density.

The AIC comparison cannot include model 2 because a Quasi-Poisson fit specifies only mean and variance and has no full likelihood. Cross-validation can compare model 2 with model 3 if all predictions are scored by the same proper out-of-sample loss.