This is related to an assertion made in Agresti's Categorical Data Analysis pg 169.
"With case-control studies, it is not possible to estimate $\beta$ binary response models with links other than logit. Unlike the odds ratio, the effect for conditional distribution of X given y does not equal that for Y given x."
I cannot parse the first statement. I think second statement is true but I do not have a proof and I am not sure.
Every GLM's coefficients are estimated through ML. Thus I could not imagine why $\beta$ cannot be estimated in non-logit links. Replace link by inverse of standard normal Gaussian's CDF and that will yield a GLM for binary response as well. Why $\beta$ cannot be estimated here?
How do I see that "Unlike the odds ratio, the effect for conditional distribution of X given y does not equal that for Y given x"? It seems that causality is reverse here. There is no particular reason to expect $P(X|y)=P(Y|x)$