Compare hazard ratios in case-control setup quick disclaimer: first time poster; i have searched through the forums extensively but please bear with me if this is a silly question or point me towards the solution if it has been answered before.
anyways, I have received two datasets:
A contains a number of cases, i.e. affected by a disease. 
B contains 4 times as many controls. they are similar in age and sex. outcome and covariates are identically recorded in the two datasets.
i have then constructed two separate cox-ph models with the same dichotomized covariate and it appears that this covariate is protective, i.e. HR 0.8, in the one group and a borderline-harmful, i.e. HR ~1.02, in the other. both results have very narrow 95% CIs and are (highly) significant owing to the large sample size in both groups.
is there a valid way to compare these HRs and obtain a p-value for such testing? I have seen simple comparisons with x² in (high ranking) journals, but surely that can't be appropriate.
thank you!
 A: Welcome to the site.
One simple way to proceed would be to combine the two datasets into a single dataset that annotates whether each case came from group A or group B. Then you could include your covariate of interest, the group membership, and the interaction between group membership and that covariate as predictors in a multiple-regression Cox model. A significant interaction term would document that the covariate has different implications for outcome depending on whether a case is in group A or group B.
Two warnings. First, you say that your covariate is "dichotomized." If this is truly a two-valued covariate that's fine, but if you have taken a numeric covariate and broken it into two classes based on some cutoff you could be getting into trouble. You would probably be better off modeling with the numeric value directly (perhaps after some transformation). See this page for one discussion of this issue on this site. Second, although this is a case-control design, you might still want to incorporate the other clinical covariates into the Cox model as case matching doesn't always work as well as one might hope.
