Under what circumstance is the odds ratio a good approximation of the relative risk?

• A. Only in randomized trials
• B. In cohort studies with large sample sizes
• C. For common diseases
• D. For rare diseases

Answer: (D)

The key idea is that the odds ratio and the relative risk align best when the outcome is rare in both groups. When the probability of the disease is small, the odds (P/(1−P)) is numerically close to the probability P itself. Since the odds ratio is the ratio of odds between exposed and unexposed, it becomes close to the risk ratio, which is the ratio of risks (probabilities) in the two groups.

Mathematically, OR = [P1/(1−P1)] ÷ [P0/(1−P0)]. If P1 and P0 are small, then 1−P1 and 1−P0 are near 1, so OR ≈ P1/P0 = RR. For example, with a rare disease where 1% are cases in the unexposed and 2% in the exposed, RR ≈ 2, and OR ≈ (0.02/0.98)/(0.01/0.99) ≈ 2.04—very close. When the disease is common, the odds diverge more from the probability, and OR can overestimate RR (e.g., 20% vs 40% leads to a larger discrepancy).

So, the best circumstance for OR to be a good approximation of RR is when the disease is rare in both groups.