Take a 2 × 2 table: a exposed with the outcome, b exposed without, c unexposed with, d unexposed without. The relative risk divides the two risks, (a / (a + b)) / (c / (c + d)). The odds ratio divides the two odds, (a / b) / (c / d) = ad / bc. Calculate both, with confidence intervals, in the Relative Risk Calculator; see Odds Ratio for the definition.
Worked example
Of 100 exposed people 40 have the outcome, and of 100 unexposed 20 do. Risks are 0.40 and 0.20, so the relative risk is 2.0. The odds are 40/60 and 20/80, so the odds ratio is 2.667.
When they agree and when they do not
| Baseline risk | Exposed risk (RR = 2) | Odds ratio |
|---|---|---|
| 1 % | 2 % | 2.02 |
| 5 % | 10 % | 2.11 |
| 10 % | 20 % | 2.25 |
| 20 % | 40 % | 2.67 |
| 40 % | 80 % | 6.00 |
- When the outcome is rare (under about 10 %), the odds ratio is a good approximation of the relative risk.
- When it is common, the odds ratio overstates the effect, and readers who treat it as a risk ratio will be misled.
- Both equal 1 when the exposure makes no difference, and both have the same direction (above 1 means more).
Which to report
- Cohort studies and trials, where risks are observed: report the relative risk (and the absolute difference).
- Case-control studies, logistic regression: report the odds ratio, and say it is an odds ratio.
- For decisions, absolute numbers (risk difference, number needed to treat) usually say more than either ratio.