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

Tim
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user45765
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    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 '21 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? – user45765 Sep 10 '21 at 17:57
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    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 https://stats.stackexchange.com/questions/69561/case-control-study-and-logistic-regression/69564#69564 and https://stats.stackexchange.com/questions/67903/does-down-sampling-change-logistic-regression-coefficients/68726#68726 – Sycorax Sep 10 '21 at 18:47
  • @Sycorax Thanks. I think both 1 and 2 are answered by the intro book and the post. – user45765 Sep 10 '21 at 19:20

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