# Why it is not possible to estimate $\beta$ in binary response model with links other than logit? [duplicate]

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.

1. 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?

2. 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 $$P(X|y)=P(Y|x)$$

• The key phrase in that sentence is "With case control studies" -- you're correct that non-logit models can be estimated with MLE in general.
– Sycorax
Sep 10, 2021 at 17:54
• @Sycorax Sorry for being dumb. Would you mind pointing out why case-control studies implies that $\beta$ cannot be estimated for links other than logit? Sep 10, 2021 at 17:57
• I don't think you're being dumb at all; it's a subtle distinction. Agresti is talking about the particulars of odds ratios -- see stats.stackexchange.com/questions/69561/… and stats.stackexchange.com/questions/67903/…
– Sycorax
Sep 10, 2021 at 18:47
• @Sycorax Thanks. I think both 1 and 2 are answered by the intro book and the post. Sep 10, 2021 at 19:20