Weighted Beta-Binomial Bayesian model

I’m not really familiar with incorporating survey weights in analyses, but I don’t see a reason why this particular model couldn’t be implemented in brms, which would save you from having to wonder whether the Stan code you come up with is correct. It seems like this would just be:

fit <- brm(y  | weights(weights) + trials(...) ~ 1,
           family = beta_binomial(link = "logit", link_phi = "log"),
           ...)

Also, just as another discussion of survey weights in Stan, this thread from the forum may have useful information for you: Survey weights in brms/stan - Simulation based on design effect, feedback sought!

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