Odds Ratio (OR)
The odds of an outcome in one group divided by the odds in another — the natural output of case-control studies and logistic regression.
The ratio of the odds of the outcome (or exposure) in one group to the odds in the other. From a 2×2 table with cells a, b, c, d, the cross-product gives OR = (a·d) ÷ (b·c). Odds are events ÷ non-events, not events ÷ total — which is why OR differs from relative risk.
| Outcome + | Outcome − | |
|---|---|---|
| Exposed | a = 90exposed, outcome | b = 10exposed, no outcome |
| Unexposed | c = 60unexposed, outcome | d = 40unexposed, no outcome |
OR = (a·d) ÷ (b·c) = (90×40) ÷ (10×60) = 6.0 — but the RR is only 1.5 (common outcome)
The cross-product of the diagonal cells. Here the outcome is common (90/100 exposed, 60/100 unexposed), so the odds (90:10 = 9 vs 60:40 = 1.5) are inflated relative to the risks (90% vs 60%). The OR of 6.0 sits far further from 1 than the RR of 1.5 — a vivid illustration of OR overstating the effect for common outcomes.
OR > 1 = exposure associated with more outcome, OR < 1 = less, OR = 1 = no association — same direction rules as RR. When the outcome is rare the non-event cells dominate and OR ≈ RR; as the outcome becomes common the two diverge, with OR always the more extreme. Read an OR as an odds ratio, never silently as a risk ratio.
The OR is the only effect measure you can estimate from a case-control study (no true denominator) and the native output of logistic regression, where adjusted ORs are how confounding is handled. That is exactly why exams love it — and why misreading a common-outcome OR as a relative risk systematically exaggerates the apparent effect.
OR = (a·d) ÷ (b·c)(2×2 cross-product)OR ≈ RRonly when the outcome is rare- Native to
case-controlstudies &logistic regression
Case-control & logistic regression — the classic Doll & Hill smoking studies and modern risk models report associations as odds ratios because they sample on outcome or adjust for many covariates, where a relative risk cannot be directly estimated. An adjusted OR from logistic regression is how a study reports an effect “after controlling for” confounders. For a rare outcome the OR closely approximates the RR; for a common one it does not — so an OR from a high-incidence population must not be quoted as if it were a risk ratio.
- OR ≠ RR for common outcomes — the OR exaggerates; only when the outcome is rare are they interchangeable.
- Reciprocal direction — flipping exposure and outcome (or the reference group) inverts the OR (e.g. 4.0 becomes 0.25); state the comparison explicitly.
- Matched vs unmatched analysis — a matched case-control design needs a matched (conditional) analysis; analysing it as unmatched biases the OR.
Quick check
An outcome occurs in 40% of patients — is the OR ≈ the RR?
Answer: No. With a common outcome the OR exaggerates relative to the RR (it sits further from 1). The rare-disease approximation only holds when the outcome is uncommon, roughly under ~10%; at 40% you must report and interpret the OR as an odds ratio, not a risk ratio.
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