Validity of AIC When Comparing Models with Varying Dispersion Parameters

Question

I'm currently making a binomial model with a logit link, which is parameterised as a quasibinomial since I'm allowing it to calculate the dispersion parameter. I was wondering, since changes to the model structure (adding/removing predictors) changes the model, and therefore the dispersion parameter, does that influence the validity of the AIC in comparing them? For example, say I've got a model with 5 parameters, and then I add a 6th parameter. The dispersion parameter has now changed, so can I use the AIC to decide between them?

I believe the way that the program I'm using calculates the scaled AIC is like so: $-2*\frac{\textrm{loglikelihood}(\theta)}{\phi}+2k$

My lecturer said that as long as the error structure is unchanged, AIC can be used, but my thinking here is that the dispersion parameter would change the variance of the errors, and therefore their distribution. So, can I use the AIC in making decisions about my model, or do I have to set a fixed scale parameter in order to make comparisons?

Answer 1

Technically speaking, Akaike's information criterion $$\mathrm{AIC} = 2p - 2\log(\mathcal{L})$$ is undefined for quasi-models, because they are not fitted by likelihood, but by quasilikelihood.$^\dagger$

The scaled AIC you mention, also called the quasi-AIC, or qAIC $$\mathrm{qAIC} = 2p - 2\frac{\log(\mathcal{L})}{\phi}$$

is not AIC anymore.

However, for single-parameter GLMs like the binomial, the dispersion parameter $\phi = 1$, and hence AIC and qAIC are identical. In that sense, comparing AIC to scaled AIC is valid for your case, although it might be more precise to say that you're comparing qAIC.

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$^\dagger$: A workaround for R users is to 'borrow' the likelihood from a binomial GLM, and then scaling it with the dispersion parameter obtained from the quasibinomial, as Ben Bolker describes here.