One thing I've been struggling with for a while is this:
When the binomial logistic regression model includes different number of trials across the observations, what are we estimating at the end of the day?
Are we estimating the "average" proportion of successes across the trials which share the same characteristics defined by the predictors included in the model? (In which case it seems to me that we are somehow discounting the fact that some of these trials may have different sizes.)
The only explanation I was able to find is that we are estimating the probability of success in any one trial, but that seems vague to me.
I want to understand this especially since another alternative to this type of model is a beta regression model and in that context the interpretation I suggested above seems to make more sense.
I also wonder why people choose to do more than one trial in certain settings? Is that because they want to get a more accurate estimate of whatever it is we are estimating? Does anyone think in advance about criteria to decide how many trials would be sufficient for that purpose? Is there a formal way to determine that?