My question would be about Pearson residuals applied to the binary model.
When we build confidence intervals for a proportion, p: we have a sample, and we count the number of individuals who have a certain characteristic in n (sample size), and therefore, p ^ = x / n, where x is the number of individuals with this characteristic. In this case, X is a binomial (n, p), that is, X is the sum of n random variables with Bernoulli distribution. When n is large, we apply the Central Limit theorem and say that X converges to Normal because the sum of many independent random variables converges to Normal.
But if Y has bernoulli distribution (1 or 0), then E(Y)=p and Var(Y)=1-p, so can we say that Z=(Y - E(Y))/sqrt(Var(Y) converges to a standard Normal distribution? Can I apply the Central Limit theorem?
In my study, each individual follows a Bernoulli distribution, that is, a Binomial(1, p). How do we bring together a group of individuals to add several Bernoulli in such a way that we can apply the Central Limit theorem? Or would you have another theory?
Thanks