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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?

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In a prototypical logistic regression model, the response variable is distributed as a Bernoulli. The parameter that controls the behavior of the Bernoulli distribution is $p$, the probability of 'success'. So the model is trying to estimate a function that maps the covariate values to how $p$ changes.

The binomial distribution generalizes the Bernoulli. Its behavior is governed by two parameters, $p$ and $n$: The probability of 'success' and the number of Bernoulli trials, respectively. Logistic regression can be used with a binomial outcome just as with a Bernoulli. The number of trials is always given to the model in some form1, so it doesn't need to be estimated. That means that logistic regression with a binomial response is once again just trying to estimate a function that maps the covariate values to how $p$ changes.

1. For example, SAS requires that you provide the counts of events and trials, whereas R requires that you provide either the number of events and non-events, or the proportion of events and the number of events (cf., Difference in output between SAS's proc genmod and R's glm).

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