Appendix C: The Odds Ratio as an Estimator of the Incidence Rate Ratio

To understand further the design of a case–control study, and why the odds ratio from a case–control study estimates the incidence rate ratio, consider first the occurrence of a particular disease within an underlying population that gives rise to the cases. This underlying cohort is sometimes called the source population. At the start of follow-up, all subjects are disease-free; P people are exposed and Q are not. After following the cohort for t years, A of the P exposed and B of the Q unexposed subjects develop disease (Table C–1).

To calculate incidence rates for this cohort, as discussed in Chapter 2: Epidemiologic Measures, first calculate the person-years of observation (py), which is given by

This equation simplifies if few people develop disease during the follow-up period and the population undergoes no major demographic shifts, a situation termed the “steady state.” If the steady state holds, the size of the source population is nearly constant and the equation above simplifies to

Application of the latter equation to the hypothetical cohort yields P × t and Q × t person-years of observation for the exposed and unexposed subjects, respectively (Table C–1). Thus, the incidence rate in the exposed subjects is A/(P × t) and in the unexposed subjects is B/(Q × t), and the incidence rate ratio (IRR) is

Now, turning to the case–control study design, cases arise from a clearly defined source population and the investigator then chooses controls from this same population. Thus, to conduct a case–control study using the source population described in Table C–1, subjects with disease (cases) and subjects without disease (controls) are sampled and their respective exposure histories are then determined. Excluded from both the case and control groups are potential subjects known to have had disease when the study began—only newly diagnosed or “incident” cases are included. In practice, it is possible to contact newly diagnosed cases and a sample from the general population, ask each subject about prior disease, and exclude those who were diagnosed prior to the study period. The data can be summarized as in Table C–2.

Cases and controls are selected without regard to exposure. For example, it is possible to randomly sample cases from all those in the population who develop disease during the study period and randomly select controls from all those without disease. Sampling is “blind” to exposure, so that the proportion of cases in the study that was exposed should, on average, equal the proportion of cases in the full cohort. Similarly, the proportion of controls in the study that was exposed should, on average, equal the proportion of those without disease in the full cohort. Thus, the exposure odds among cases, a/b, is an estimate of A/B, the corresponding odds among new cases arising from the full cohort. In a parallel manner, the exposure odds among controls, c/d, is an estimate of P/Q, the corresponding odds among persons without disease in the full cohort. The cross-product or odds ratio (OR) is the odds that a case is exposed, a/b, divided by the odds that a control is exposed, c/d:

The odds ratio from the case–control study is therefore an estimate of the incidence rate ratio in the full cohort, which was shown above to equal (A × Q)/(B × P).

If exposure increases risk, more cases than controls will be exposed, so that a/b > c/d, or OR > 1. On the other hand, if exposure decreases risk, fewer cases than controls will be exposed, so that a/b < c/d, or OR < 1. Thus, an OR greater than one suggests that exposure is associated with higher incidence rates or risk, an OR less than one suggests that exposure is associated with lower incidence rates or risk, and an OR near one suggests that exposure is not associated with higher or lower than average risk.