Questions: Odds Ratio and Case-Control Study Analysis

Question 1

A case-control study of a rare cancer (prevalence ~0.5%) finds an OR of 4.5 for heavy smoking. A researcher concludes 'smokers have 4.5 times the risk of this cancer.' This interpretation is:

• A. Correct, because OR always equals RR in case-control studies
• B. Approximately correct, because the rare disease assumption allows OR to closely approximate RR
• C. Incorrect, because OR must be converted to RR by multiplying by the baseline risk
• D. Impossible to evaluate without incidence data from a cohort study

Answer: (B)

When disease prevalence is below ~10%, the OR closely approximates the RR mathematically. For a very rare cancer (~0.5%), the approximation is tight, so interpreting OR ≈ RR is reasonable. This does NOT mean OR always equals RR — the approximation degrades as disease becomes more common. Option A is wrong because the approximation is never exact. Option C misunderstands the relationship between OR and RR. Option D is wrong because the rare disease assumption provides a valid basis for interpretation.

Question 2

A cross-sectional study of hypertension (prevalence 35%) uses logistic regression and reports an OR of 3.0 for sedentary lifestyle. What is the most accurate interpretation?

• A. Sedentary individuals have approximately 3 times the risk of hypertension
• B. The OR of 3.0 overestimates the true relative risk because the rare disease assumption is violated
• C. The OR of 3.0 underestimates the true relative risk for common diseases
• D. An OR of 3.0 corresponds to an attributable risk of 67%

Answer: (B)

When disease is common (~35%), the OR is always farther from 1.0 than the RR — it overestimates the strength of association. An OR of 3.0 for a 35% prevalence outcome corresponds to an RR substantially less than 3. This is why log-binomial or Poisson regression with robust variance is preferred for common outcomes: they estimate RR directly rather than the inflated OR. Option A is the classic error the rare disease assumption is meant to prevent.

Question 3

In a case-control study, it is impossible to directly calculate incidence rates in the exposed and unexposed groups.

• T. True
• F. False

Answer: True

Case-control studies sample by disease status: you enroll a fixed number of cases and controls, then look back at exposures. The ratio of cases to controls is determined by the researcher, not by the actual disease incidence in the population. This means you cannot calculate 'how many per 100 exposed people developed disease' — that requires knowing the size of the exposed population at risk, which a case-control design does not provide. This is exactly why the odds ratio (comparing odds of exposure among cases vs. controls) was developed as a substitute measure of association.

Question 4

An odds ratio of 2.0 from a case-control study means that the exposed group has twice the risk of developing the disease.

• T. True
• F. False

Answer: False

This is the most common misinterpretation of the OR. An OR of 2.0 means the *odds* of exposure are twice as high among cases as among controls — it is not a ratio of risks (probabilities). The OR approximates the RR only under the rare disease assumption. For a common outcome, OR = 2.0 could correspond to an RR of 1.5 or even less. Risks and odds are different: odds = p/(1-p). They are similar only when p is small.

Question 5

Why does the odds ratio approximate relative risk when disease is rare, and why does this approximation break down for common diseases? Answer in terms of the 2×2 table.

Model answer: In a 2×2 table with cells a (exposed cases), b (exposed controls), c (unexposed cases), d (unexposed controls), the OR = (a×d)/(b×c) and the RR = [a/(a+b)] / [c/(c+d)]. When disease is rare, a is very small relative to b, and c is small relative to d, so (a+b) ≈ b and (c+d) ≈ d. This makes RR ≈ (a/b)/(c/d) = (a×d)/(b×c) = OR. When disease is common, a is not negligible relative to b, and the approximation fails — the OR diverges from the RR, always in the direction of overestimating the strength of association.

The algebraic convergence of OR to RR under the rare disease assumption is why epidemiologists can usefully report ORs from case-control studies of rare diseases. But it also explains why, with logistic regression in a cohort study of a common outcome, the OR from logistic regression overstates the association and should be replaced with a direct RR estimate.