just a basic question, but is there a difference between generalized linear model with binomial distribution and binary logistic regression? Or it´s the same?
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Binary logistic regression is a generalized linear model with the Bernoulli distribution. The Bernoulli distribution is just a special case of the binomial distribution. So they're not "the same" necessarily, but one is a special case of the other.
This is why in R there is no Bernoulli option in the glm() function. You use binomial for both binary data and binomial count data.