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In this article the author talks about fitting beta-binomial models to data when the there data is over-dispersed relative to the assumptions of a model with binomial errors. Near the end they present two options for testing whether the additional parameter accounting for the over-dispersion is necessary.

One of these options is:

enter image description here

However, from what I have read about likelihood theory, you can't compare likelihoods when the underlying distribution is different (see this link).

Can you use likelihood to compare these two distributions if all else is held constant?

I have googled around and searched this site but I can't seem to find any answers on the subject.

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The likelihood ratio test supposes nested hypothesis, that is, the smaller model is a special case of the larger model. And, that is the case for the binomial and beta-binomial models.

See parametrizations for the beta distribution, you could reparametrize the beta binomial distribution using the mean/variance parametrization for the beta distribution, and then you recover the binomial distribution by letting the beta variance be zero.

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