2
$\begingroup$

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

$\endgroup$
4
  • 2
    $\begingroup$ 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. $\endgroup$
    – Sycorax
    Sep 10, 2021 at 17:54
  • $\begingroup$ @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? $\endgroup$
    – user45765
    Sep 10, 2021 at 17:57
  • 1
    $\begingroup$ 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/… $\endgroup$
    – Sycorax
    Sep 10, 2021 at 18:47
  • $\begingroup$ @Sycorax Thanks. I think both 1 and 2 are answered by the intro book and the post. $\endgroup$
    – user45765
    Sep 10, 2021 at 19:20

0