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.

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    $\begingroup$ Was the model saturated? $\endgroup$ – Scortchi Mar 30 '16 at 9:03
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    $\begingroup$ to add to @Scortchi : saturated means that all explanatory variables are categorical variables and all (higher level) interactions are included in the model. In that case your model exactly replicates the proportion of "successes" for each combination of your explanatory variables, regardless of which link function you use. $\endgroup$ – Maarten Buis Mar 30 '16 at 9:22

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