I fit a logistic regression and calculated the expected difference in probabilities of my outcome between two treatment levels holding all other variables constant. I obtained confidence intervals using the delta method. Then, I fit modeled the probabilities directly with a binomial model and an identity link and directly obtained the confidence intervals. I was surprised to find that I got the exact same point estimates and confidence intervals regardless of the link function I used. Can someone give me some intuition on what's going on here? I would have thought that difference link functions would have produced at least slightly different estimates.