[R-meta] Questions about the use of metaprop for the pooling of proportions

Happy to see other people spending time at 11pm thinking about this kind of stuff :)

If we want to be really precise, the MLE of the logit-transformed true proportion is qlogis((sum r_i)/(sum n_i)) for the logistic regression model with a logit link, but since MLEs are invariant under transformations, so plogis(qlogis((sum r_i)/(sum n_i))) = (sum r_i)/(sum n_i)) is the MLE of the true proportion. In fact, this is neatly demonstrated by fitting the logistic regression with an identity link (do we even call this 'logistic' regression?!?):

coef(glm(out1/n ~ 1, weights = n, family = binomial(link = "identity")))

That all of this happens 'automagically' is really a neat feature of logistic regression.