Collapsibility in case-control studies

It is well-established that the odds ratio (OR) is non-collapsible and that adjustment of the OR for a risk factor can produce a change in the main effect estimate even when there is no confounding. In contrast, the risk ratio (RR) is collapsible, whereas the rate ratio is non-collapsible, but usually to a lesser extent than is the case with ORs. However, almost all discussions of non-collapsibility have been conducted in the context of effect measures in cohort studies. In this paper, I consider non-collapsibility in case-control studies where the OR inherits the collapsibility or non-collapsibility of the ratio effect measure that the study was designed to estimate. This is most clearly illustrated in the situation of a closed cohort with a fixed follow-up time. In this situation, choosing controls by cumulative incidence sampling estimates the OR in the corresponding cohort study, and therefore the resulting case-control OR is non-collapsible. Choosing controls by case-cohort sampling estimates the cohort study RR, and the resulting case-control OR is collapsible. Choosing controls by density sampling (perhaps the most common approach) estimates the cohort study rate ratio, and the resulting case-control OR is non-collapsible, but to a lesser extent than for cumulative incidence sampling. Moreover, if the disease under study is rare (in the cohort study), all three methods produce case-control ORs that are approximately collapsible.