On the need for the rare disease assumption in case-control studies

The conditions under which matched and unmatched odds ratios are consistent estimators of the incidence-density ratio in case-control studies are examined. Under "incidence-density" sampling, in which controls are selected from those at risk at the time of onset of each case, the matched estimator is shown to be consistent. In contrast, the unmatched estimator is biased unless the proportion exposed in the population at risk is constant over the study period; the bias is, however, negligible unless there is very large variation in the proportion exposed. No assumption of rarity of the disease is needed for these results. On the other hand, when the risk ratio is the parameter of interest, the assumption of rarity is needed for the odds ratio to be a consistent estimator. In such situations, the odds ratio obtained under "incidence-density" sampling will in general provide a better approximation to the risk ratio than will the odds ratio obtained under "cumulative-incidence" sampling, in which controls are selected from those still unaffected at the end of the study period. Even if the disease is rare, however, the odds ratio obtained under cumulative sampling need not consistently estimate any parameter of interest unless the proportion exposed is approximately constant.