Odds Ratio & Risk Ratio
Odds ratio and risk ratio from a 2×2 table, each with a 95% confidence interval.
95% CI 1.417 to 5.02. The interval excludes 1, so the association is significant at the 5% level.
Risk in the exposed group 40% versus 20% unexposed. 95% CI 1.263 to 3.167. Only meaningful when the design measures risk, not for case-control studies.
In a case-control study only the odds ratio is interpretable, because the sampling fixes the number of cases and controls rather than measuring how often the outcome occurs.
The odds ratio is the cross-product of a 2×2 table: OR = ad ÷ bc. With 40 and 60 in the exposed group and 20 and 80 in the unexposed, OR = (40 × 80) ÷ (60 × 20) = 2.67, with a 95% confidence interval of 1.42 to 5.02 — the interval excludes 1, so the association is significant.
Odds and risk are not the same thing
Risk is the proportion of a group that experiences the outcome: 40 out of 100 exposed people is a risk of .40. Odds compare those who did against those who did not: 40 against 60, or 0.67. The two coincide when the outcome is rare and diverge sharply when it is common.
That divergence matters for interpretation. In the example above the risk ratio is 2.00 — exposed people are twice as likely to have the outcome — while the odds ratio is 2.67. Reporting an odds ratio of 2.67 as “nearly three times more likely” overstates the effect, and this misreading is common enough to have its own literature.
Why the odds ratio persists
Given the risk ratio is easier to explain, the odds ratio survives for two solid reasons. In a case-control study the investigator fixes how many cases and controls to recruit, so the data cannot estimate risk at all — but the odds ratio remains valid. And logistic regression models the log-odds directly, so its coefficients exponentiate into odds ratios rather than risk ratios.
The interval is computed on the log scale, where the sampling distribution is approximately normal, then transformed back — which is why it is asymmetric around the odds ratio.
Worked example: 40 / 60 / 20 / 80
Cross-multiply, then build the interval on the log scale:
- 1 Lay out the 2×2 table. Exposed with the outcome 40, exposed without 60, unexposed with 20, unexposed without 80.
- 2 Cross-multiply. (40 × 80) ÷ (60 × 20) = 3200 ÷ 1200 = 2.67.
- 3 Find the standard error on the log scale. √(1/40 + 1/60 + 1/20 + 1/80) = √0.1042 = 0.323.
- 4 Build the interval in logs. ln(2.67) = 0.981, so the interval is 0.981 ± 1.96 × 0.323, from 0.348 to 1.613.
- 5 Exponentiate back. exp(0.348) = 1.42 and exp(1.613) = 5.02, giving a 95% CI of 1.42 to 5.02.
- 6 Interpret. The interval excludes 1, so the association is significant at the 5% level. Compare with the risk ratio of 2.00.
Reading an odds ratio
The confidence interval carries more information than the point estimate — a wide interval means the study cannot pin the effect down, whatever the centre value.
| Result | Meaning |
|---|---|
| OR = 1 | No association — the odds are the same in both groups |
| OR > 1 | The outcome is more likely in the exposed group |
| OR < 1 | The outcome is less likely in the exposed group — a protective association |
| CI excludes 1 | Significant at the 5% level |
| CI includes 1 | Not significant at the 5% level |
| Very wide CI | Little precision, usually from small cell counts |
Zero cells, rarity, and what an odds ratio cannot do
A zero in any cell makes the odds ratio undefined or infinite, and the standard error breaks down because it divides by each cell. The common remedy is the Haldane-Anscombe correction, adding 0.5 to every cell before computing — a fix that should be disclosed when used, since it shrinks the estimate towards 1.
The odds ratio approximates the risk ratio well when the outcome is rare, conventionally under about 10% in the unexposed group. Above that the two separate quickly, and quoting an odds ratio in the language of risk becomes actively misleading.
Finally, the interval here is a crude one: it reflects sampling error alone, not confounding. A 2×2 table cannot adjust for age, sex, or anything else, so an unadjusted odds ratio describes an association in this data rather than a causal effect. Adjusted estimates come from regression, not from a four-cell table.