It is my understanding that log binomial regression involves a direct comparison of prevalence ratios ("% cases among the exposed" vs. "% cases among the unexposed"), rather than using prevalence odds ratios. Does this mean that it cannot be used with a case-control study, where the % cases was manipulated as part of study design?
From what I understand, logistic regression can handle a case-control design by giving you a screwed-up intercept. That is, the intercept would normally represent the baseline risk in a population when all covariates were 0, but when you do logistic regression in a case-control sample it instead represents the baseline risk in your sample (which is useless because you manipulated that). Logistic regression with a case-control study then manages to still produce useful results, because each beta represents "increase in log odds over baseline", so it doesn't really matter what your intercept is because what you are really interested in is how your covariates change things from that starting point.
So with a case-control sample, would log binomial then do the same thing, give you a screwed-up intercept (representing log prevalence in the SAMPLE when all covariates=0), but good betas (because the change in log prevalence is the same regardless of whether your intercept represents the sample or the population)?