Statistics Concepts
Relative Risk vs Odds Ratio: What Is the Difference?
Relative risk (the risk ratio) divides the probability of an outcome in one group by the probability in another. The odds ratio divides the odds instead. The two are close only when the outcome is rare. If 50 of 70 exposed subjects and 30 of 130 unexposed subjects have the outcome, the relative risk is 3.10 and the odds ratio is 8.33: the same data on two different scales.
The two measures side by side
Both come from the same 2×2 table with the counts a, b in the exposed row and c, d in the unexposed row, where a and c are the subjects who have the outcome. Both equal 1 when the outcome is equally likely in the two groups, and both range from 0 to infinity.
| Relative risk (risk ratio) | Odds ratio | |
|---|---|---|
| Compares | Probabilities: events / total | Odds: events / non-events |
| Formula | [a / (a + b)] / [c / (c + d)] | (a × d) / (b × c) |
| Reads as | Times as likely | Times the odds |
| Natural for | Cohort studies, trials, cross-sectional surveys | Case-control studies and logistic regression |
| Swapping the outcome categories | Gives a different number | Gives the reciprocal |
Worked example on one table
In a cohort of 200 people, 70 were exposed and 130 were not. The outcome occurred in 50 of the exposed and 30 of the unexposed subjects, so the table is [[50, 20], [30, 100]].
Risk (exposed) = 50 / 70 = 0.7143; risk (unexposed) = 30 / 130 = 0.2308
Relative risk = 0.7143 / 0.2308 = 3.0952 (95% CI 2.1876 to 4.3795)
Odds (exposed) = 50 / 20 = 2.5; odds (unexposed) = 30 / 100 = 0.3
Odds ratio = 2.5 / 0.3 = 8.3333 (95% CI 4.3079 to 16.1204)
Risk difference = 0.7143 − 0.2308 = 0.4835
Exposed subjects are 3.1 times as likely to have the outcome, and their odds are 8.3 times as high. Both statements are true, but only the first says what most readers understand by "times as likely". Reading the odds ratio of 8.33 as an eightfold risk would exaggerate the effect. Check the numbers with the relative risk calculator and the odds ratio calculator.
Why the odds ratio is further from 1
The odds of an event with risk p are p / (1 − p). At low risks this is almost p, but it grows faster and faster: a risk of 50% is odds of 1, a risk of 80% is odds of 4 and a risk of 90% is odds of 9. Because of this, the odds ratio is always further from 1 than the relative risk, in the same direction, and the gap widens with the baseline risk. The table doubles the risk in every row.
| Risk (unexposed) | Risk (exposed) | Relative risk | Odds ratio |
|---|---|---|---|
| 1% | 2% | 2 | 2.0204 |
| 5% | 10% | 2 | 2.1111 |
| 10% | 20% | 2 | 2.25 |
| 20% | 40% | 2 | 2.6667 |
| 40% | 80% | 2 | 6 |
The same happens for protective effects: halving the risk (a relative risk of 0.5) gives an odds ratio of 0.4975 at a baseline risk of 1%, 0.375 at 40% and 0.1667 at 80%.
When the two agree: rare outcomes
When the outcome is rare, non-events are almost everyone, so odds ≈ risk and OR ≈ RR. Suppose 120 of 10,000 exposed and 90 of 10,000 unexposed subjects have the outcome. The risks are 1.2% and 0.9%, the relative risk is 1.3333 (95% CI 1.0159 to 1.7499) and the odds ratio is 1.3374 (1.0161 to 1.7602). This is why the odds ratio is used to approximate the relative risk in case-control studies of rare diseases, the "rare disease assumption". A common guideline asks for an outcome risk below about 10% (Davies, Crombie and Tavakoli, BMJ 1998).
Which one should you report?
- Cohort study or randomized trial: the relative risk, with the risk difference and the number needed to treat, because these designs observe the risks.
- Case-control study: the odds ratio. The investigators fix the numbers of cases and controls, so risks cannot be estimated, but the odds ratio of exposure equals the odds ratio of disease.
- Logistic regression: the coefficients are log odds ratios. Exponentiate them to get odds ratios, and do not describe them as relative risks unless the outcome is rare.
- Cross-sectional survey of a common outcome: the prevalence ratio, which is a relative risk, rather than the odds ratio.
- Testing the association: use the chi-square test of independence or Fisher's exact test; both give the same p-value whichever measure you report. To compare two risks directly see the two-proportion z-test.
Converting an odds ratio to a relative risk
If you know the risk in the unexposed group, p₀, the relative risk that corresponds to an odds ratio is (Zhang and Yu, JAMA 1998):
RR = OR / (1 − p₀ + p₀ × OR)
For the odds ratio of 2.25 at a baseline risk of 10%, RR = 2.25 / (1 − 0.1 + 0.1 × 2.25) = 2.25 / 1.125 = 2. For the worked table, OR = 8.3333 and p₀ = 30 / 130 = 0.2308 return the relative risk of 3.0952. The conversion is only as good as the baseline risk: take it from the population you are reporting on, and treat the interval with care unless that risk is known.
Common mistakes
- Reading an odds ratio of 3 as "three times as likely" when the outcome is common.
- Comparing odds ratios from studies with different baseline risks as if they were the same effect.
- Reporting only the relative measure. A relative risk of 2 is a very different public-health fact at a baseline risk of 0.01% and at 10%; give the absolute risks or the risk difference too.
- Forgetting the interval. The point estimate of the worked example is 3.10, but its 95% interval runs from 2.19 to 4.38; see confidence intervals and the p-value explained.
- Choosing a test by the measure. Which test fits your data is a separate question; see which statistical test to use.
Try the Relative Risk Calculator
Risk ratio with its confidence interval, risk difference, relative risk reduction and number needed to treat.
Try the Odds Ratio Calculator
Odds ratio with a Woolf confidence interval and the chi-square and Fisher p-values.
Frequently Asked Questions
What is the difference between relative risk and odds ratio?
Relative risk divides the probability of the outcome in one group by the probability in the other; the odds ratio divides the odds, events over non-events, in one group by the odds in the other. For 50 of 70 exposed and 30 of 130 unexposed subjects with the outcome, the relative risk is 3.0952 and the odds ratio 8.3333. Both are 1 when the groups do not differ.
When are the odds ratio and the relative risk similar?
When the outcome is rare in both groups, because then the odds, events / (total − events), are almost equal to the risks, events / total. With 120 events in 10,000 exposed subjects and 90 in 10,000 unexposed subjects the relative risk is 1.3333 and the odds ratio 1.3374. A common guideline asks for an outcome risk below about 10%.
Why is the odds ratio further from 1 than the relative risk?
Odds grow faster than risk as the risk rises: a risk of 50% is odds of 1, a risk of 80% is odds of 4. Whenever the risk differs between the groups the odds ratio is therefore further from 1 than the relative risk, in the same direction. Doubling a baseline risk of 10% gives an odds ratio of 2.25, and doubling a baseline risk of 40% gives 6.
Can I report an odds ratio from a cohort study or a trial?
It is valid, but the relative risk answers the question more directly, because a cohort study or trial observes the risks. If you report an odds ratio for a common outcome, do not describe it as 'times as likely': that wording is only approximately right for a rare outcome. Report the risk difference or the number needed to treat as well.
Why do case-control studies use the odds ratio?
In a case-control study the investigators choose how many cases and controls to sample, so the share of exposed subjects who are cases is set by the design and the risks cannot be estimated. The odds ratio of exposure between cases and controls equals the odds ratio of disease between exposed and unexposed subjects, so it can be estimated from either sampling scheme.
How do I convert an odds ratio to a relative risk?
With the risk in the unexposed group p0, RR = OR / (1 − p0 + p0 × OR). For OR = 2.25 and p0 = 0.10 this gives 2.25 / 1.125 = 2. The conversion needs a credible baseline risk for the population and outcome you care about, and its interval is only valid if that risk is known rather than estimated.
Does an odds ratio of 2 mean twice the risk?
Only when the outcome is rare. If the baseline risk is 1%, an odds ratio of 2 is a relative risk of about 1.98. If the baseline risk is 40%, an odds ratio of 2 is a relative risk of about 1.43, and a doubled risk of 80% would show up as an odds ratio of 6.