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Suppose that I have a dependent variable which is the proportion of persons infected with a certain disease out of the total number tested in different locations. Assuming the difference in geographical areas is negligible, would it be appropriate to use binomial GLM and include the weights (Total number tested) or beta regression, keeping in mind that the N (total number tested) vary across the regions

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Provided that there are enough cases so that the proportions can be considered to be sampled from continuous distributions, the choice depends on your understanding of the nature of the sampling. Beta regression, at least as implemented in the R betareg package, also allows for case weights.

Quoting from this answer by the author of that package:

The binomial is for modeling Bernoulli variables (i.e., binary) or binomial variables (i.e., the number of successes from a certain number of independent trials)

The beta regression model, on the other hand, is intended for situations where you only have a direct rate that does not correspond to success rates from a known number of independent trials. It uses a different likelihood and hence can lead to different results. Specifically, it has an additional precision parameter which is related to the variance of the observations.

Thus, if your proportions above come from a known number of independent trials, then supply this information and use a binomial GLM. Otherwise you can consider beta regression.

The choice thus depends on whether you can consider the sampling of disease/not status to be a set of independent binary outcomes.

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