Log binomial regression with a case-control sample

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)?

• When I say "log binomial regression", I mean using log instead of logit as the link function, so you model prevalence ratio instead of prevalence odds ratio, as described in Spriegelman, 2005 and Wacholder, 1986. The reason for doing this instead of logistic regression (less biased estimates with high-prevalence outcomes) is described in Zochetti, 1997. May 18 '16 at 2:22
• Spiegelman, D. (2005). Easy SAS Calculations for Risk or Prevalence Ratios and Differences. American Journal of Epidemiology, 162(3). May 18 '16 at 2:25
• Wacholder, S. (1986). Binomial regression in GLIM: estimating risk ratios and risk differences. American Journal of Epidemiology, 123(1). May 18 '16 at 2:26
• Zocchetti, C., Consonni, D., & Bertazzi, P. A. (1997). Relationship between prevalence rate ratios and odds ratios in cross-sectional studies. International Journal of Epidemiology, 26(1). May 18 '16 at 2:26

If you replace your natural link function logit by log, the estimator for $\beta$ will no longer be unbiased under case-control sampling. To see this, we consider a simple binary exposure $X$ and disease status $Y$. Then the model with log link function is
$log(\pi(X=x)) = \alpha + \beta x$ where $\pi(X=x)=P(Y=1|X=x)$, that is, the probability of disease given the exposure $x$. Then, $\beta=log(\pi(X=1))-log(\pi(X=0))=log(\frac{\pi(X=1)}{\pi(X=0)}).$
This is the log risk ratio and you cannot estimate $\pi(X=x)$ i.e. $P(Y=1|X=x)$ because in case-control study, you fix the disease status rather than the exposure so you can only estimate $P(X=1|Y=y)$ or the probability of exposure given the disease status.
As for the intercept $\alpha$:
$\alpha=log(\pi(X=0))$ and for the same reason as above, under case-control sampling, the estimator for $\pi(X=0)$ will be biased.