## Why OR and RR Are Nearly Identical ### The Rare Disease Assumption **Key Point:** When an outcome is rare in the population (typically <10% incidence), the odds ratio mathematically approximates the relative risk, regardless of study design. ### Mathematical Basis The relationship between OR and RR depends on the prevalence of the outcome: $$\text{OR} = \frac{\text{odds of outcome in exposed}}{\text{odds of outcome in unexposed}} = \frac{a/b}{c/d}$$ $$\text{RR} = \frac{\text{risk in exposed}}{\text{risk in unexposed}} = \frac{a/(a+b)}{c/(c+d)}$$ When the outcome is **rare** (a << b and c << d): - The denominators (b and d) become much larger than numerators (a and c) - Therefore: $a/b \approx a/(a+b)$ and $c/d \approx c/(c+d)$ - **Result: OR ≈ RR** When the outcome is **common** (a and b are comparable): - The denominators do not dominate - **Result: OR > RR** (OR overestimates the association) ### Application to Lung Cancer **High-Yield:** Lung cancer incidence in the general population is approximately 50–60 per 100,000 per year (~0.05–0.06%), which is rare. Even in smokers, the cumulative incidence over a lifetime is <20% in most cohorts. Given: - RR = 8.0 - OR = 8.2 - Difference = 0.2 (only 2.5% higher) This tiny difference is consistent with a rare outcome where the rare disease assumption holds. ### Why Other Options Are Wrong **Warning:** Do NOT confuse: - ~~"Cohort studies always produce similar OR and RR"~~ — False. If the outcome is common (e.g., >20% incidence), OR will be substantially larger than RR in a cohort study. - ~~"OR is always slightly larger than RR in prospective studies"~~ — Misleading. The direction and magnitude depend on outcome frequency, not study timing. ### Mnemonic **RARE:** **R**are outcome → **A**pproximation → **R**R ≈ **E**xpected from OR
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