Theoretical justification for quasi- models not reporting AIC
According to this source, the reason why quasi- models don't report AIC scores is "because the R Core team (or at least the ones who wrote glm, glmmPQL, etc.) are purists and don’t believe that quasi- models should report a likelihood." I'm interested in understanding this from a theoretical perspective. Given that, to calculate an AIC, all you need is the model's likelihood function, why couldn't an AIC be calculated for a quasi-binomial model? Would the likelihood function not just be parameterized by your model's terms plus Phi?
Posted Dec 11, 2021, 10:44 PM UTC
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The quasi-likelihood does not correspond to a bona-fide likelihood most of the time. It happens to have a lot of properties associated with likelihood functions, but not all of them; for example, QL doesn't tell you what the right "normalizing constant" should be with respect to (phi, data) - you only get the log-likelihood up-to an arbitrary constant of integration. This matters for AIC, since the normalizing constant appears in AIC. The arbitrariness of the constant goes away with true likelihoods because the likelihood has to integrate to 1 across the data.
About quasi-binomial (since that can be made to correspond to a genuine likelihood), I don't see a problem in principle with computing AIC for that model, but it is unclear to me that you even want to look at AIC there since QL methods usually only care about whether the moment conditions are valid. AIC will be sensitive to model misspecification even if the moment conditions hold, which I don't care about because I don't care if the likelihood really is quasi-binomial to begin with.
This makes a lot of sense — now it's clicked for me. Thanks!