Risk ratio vs odds ratio: compare the right fractions
The same two groups can give a risk ratio of 2.00 and an odds ratio of 2.67. The difference is in what goes below the line.
Risk ratio and odds ratio both compare groups, but they compare different quantities. Reading an odds ratio as a risk ratio can exaggerate the impression of an association when an outcome is common.
The film above uses two fictional cohorts. Here is the complete calculation, including the interpretation that should accompany each answer.
Two groups, the same observation period
Imagine two groups of 100 people, followed over the same period. Forty people in group A experience the outcome; 20 in group B do.
| Group | Outcome | No outcome | Total |
|---|---|---|---|
| A | 40 | 60 | 100 |
| B | 20 | 80 | 100 |
These counts are made up to explain the arithmetic. The labels do not identify a treatment, and the example does not establish a cause of the outcome.
Risk uses everyone in each group
Group A has an observed risk of 40/100 = 40%. Group B has an observed risk of 20/100 = 20%. Divide the first risk by the second:
Risk ratio (RR) = (40/100) / (20/100) = 2.00.
In this example, group A has twice the observed risk of group B over the stated period. This is a comparison of risks, not proof that group membership caused the difference.
Odds use people without the outcome below the line
For odds, compare the number with the outcome to the number without it. Group A has odds of 40/60. Group B has odds of 20/80.
Odds ratio (OR) = (40/60) / (20/80) = 2.666… ≈ 2.67.
The odds in group A are approximately 2.67 times the odds in group B. That is not a claim that its risk is 2.67 times as high. The risk ratio in the very same data is 2.00.
Also keep an odds value separate from a probability: 40/60 is approximately 0.667 in odds, while 40/100 is a 0.40 probability. These are different denominators.
Why rare outcomes can make the numbers look similar
Change the fictional counts to two outcomes among 100 people in group A and one among 100 in group B. The risk ratio is still 2.00. The odds ratio becomes (2/98)/(1/99) ≈ 2.02.
| Example | Risk in A | Risk in B | Risk ratio | Odds ratio |
|---|---|---|---|---|
| Rare outcome | 2% | 1% | 2.00 | 2.02 |
| Our original example | 40% | 20% | 2.00 | 2.67 |
The approximation is close in the rare-outcome example. It is much less close when the outcome occurs in 40% and 20% of the groups. Read the named measure rather than assuming RR and OR are interchangeable.
The study design still matters
Our example supplies cohort totals, so risks can be calculated directly. A conventional case–control sample selects people by outcome status; the selected cases and controls do not supply the population risks directly. Odds ratios are commonly used for that design. The CDC references below explain this distinction.
Before interpreting the answer
- Identify whether the reported measure is a risk ratio or an odds ratio.
- Check the comparison direction: A divided by B is not B divided by A.
- For risk, use all people in each group; for odds, use those without the outcome below the line.
- Interpret the number using the same measure named in the calculation.
- In real research, consider uncertainty and possible bias before drawing conclusions. A point estimate alone is not evidence of precision or causality.
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