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Editor,—The article by Dr Kopjar in a recent issue of the journal discussing the use of the population preventable fraction concerning bicycle related head injuries and helmet use was very interesting.1 Dr Kopjar uses the odds ratio (OR), obtained from case-control studies of the effectiveness of bicycle helmet use to prevent head injury, to provide an estimate of the relative risk (RR) required in the formula for the population attributable fraction. In the article, Dr Kopjar stated that “Incidence of head injuries is low, suggesting that these ORs can be used as a valid proxy for the RR”.1 This point deserves further comment. There are two distinct types of case-control studies: the incidence density type and the cumulative incidence type. We may assume an underlying cohort for each type from which cases and controls are sampled. The primary difference between the incidence density and cumulative incidence case-control studies is how we view the cohort and what information the control group provides. The incidence density case-control study views the underlying cohort as being stable and dynamic. The control group in an incidence density case-control study is intended to provide an estimate of the fraction of population time exposed and unexposed. The OR, then, is a ratio of pseudorates and provides an unbiased estimate of the incidence rate ratio, with no rare disease assumption (table 1).2 Thus it does not matter whether the disease is rare, only that controls be selected independently of exposure status to be representative of the distribution of the exposure in the source population which produced the cases.3
The cumulative incidence case-control study is where the rare disease assumption is important. The cohort underlying the cumulative incidence case-control study should be thought of as closed and fixed. Incident cases are sampled throughout a defined time period and controls are residual non-cases (that is, those individuals at risk who did not become cases over this period). In this situation, the control group does not provide a representation of person time. Instead, the relationship between the odds and the risk is what is key. That is, when the disease is rare, the odds of disease (cases/non-cases) and the risk of disease (cases/total at risk) are approximately equal (keeping in mind that the odds of disease is not available from a case-control study, only the OR):
Risk = 10 cases/1000 total at risk ≈10 cases /(1000 total at risk−10 cases) ≈ 10 cases/990 non-cases ≈ odds
The case-control studies that provide the OR estimates used in Dr Kopjar's article could be seen as the incidence density type. The OR would then provide an unbiased estimate of the incidence rate ratio, with no rare disease assumption.