Comparison
Odds Ratio vs Risk Ratio — Differences & When to Use
The odds ratio and risk ratio (relative risk) both compare an outcome between two groups, but they are not the same: the risk ratio compares probabilities directly and is easy to interpret, while the odds ratio compares odds and only approximates the risk ratio when the outcome is rare.
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Open the workspace →Risk vs odds
The two measures are built from different ideas of likelihood. Risk is a probability: the number of events divided by the total number of people (events / total). Odds compare events to non-events instead: the number of events divided by the number of non-events (events / non-events).
The risk ratio (also called relative risk, RR) divides the risk in the exposed group by the risk in the unexposed group. The odds ratio (OR) divides the odds in the exposed group by the odds in the unexposed group. Because risk and odds are not the same quantity, the ratios of those quantities also differ.
Formula
Definition
OR = (a × d) / (b × c)
RR = [ a / (a + b) ] / [ c / (c + d) ]
- [[a, b], [c, d]]
- = a 2×2 table where rows are exposed / unexposed and columns are outcome yes / no
- a, b
- = exposed with and without the outcome
- c, d
- = unexposed with and without the outcome
- OR
- = odds ratio = cross-product of the cells, (a×d) / (b×c)
- RR
- = risk ratio = risk in exposed [a/(a+b)] divided by risk in unexposed [c/(c+d)]
Worked example
Worked example
Take a 2×2 table where rows are exposed / unexposed and columns are outcome yes / no:
[[30, 10], [10, 60]] → a=30, b=10, c=10, d=60
Odds ratio: OR = (a × d) / (b × c) = (30 × 60) / (10 × 10) = 1800 / 100 = 18.0.
Risk ratio: risk in exposed = 30 / 40 = 0.75; risk in unexposed = 10 / 70 = 0.143; RR = 0.75 / 0.143 = 5.25.
Notice that the OR (18.0) is much larger than the RR (5.25) here. That is because the outcome is common — risks of 0.75 and 0.143 are far from zero. The odds ratio only approximates the risk ratio when the outcome is rare; for common outcomes it sits much farther from 1 and exaggerates the apparent effect.
When to use it
Use it when
- Use the risk ratio when you have a cohort or prospective study where you can follow groups forward in time.
- Use the risk ratio when the outcome risk is directly estimable from your data.
- Use the risk ratio when you want a plain, intuitive interpretation ("twice as likely").
Not the right tool when
- Use the odds ratio instead when you have a case-control study, where you sample on the outcome and cannot estimate risk directly.
- Use the odds ratio instead when your model is logistic regression, which estimates odds ratios rather than risk ratios.
- Use the odds ratio instead when the outcome is rare — then the odds ratio closely approximates the risk ratio anyway.
How to interpret it
Rule of thumb
Both equal 1 under no association. With a common outcome the odds ratio overstates the risk ratio (it sits farther from 1), so do not read an odds ratio as if it were relative risk — an OR of 18 does not mean the exposed group is 18 times as likely to have the outcome.
Frequently asked questions
- Is the odds ratio the same as relative risk?
- No. The risk ratio (relative risk) compares probabilities — risk in exposed divided by risk in unexposed — while the odds ratio compares odds. They agree only when the outcome is rare; for common outcomes the odds ratio is farther from 1 than the risk ratio.
- When does the odds ratio approximate the risk ratio?
- When the outcome is rare. If the probability of the outcome is small in both groups, odds and risk are nearly equal, so the odds ratio closely approximates the risk ratio. As the outcome becomes common, the two diverge.
- Why is my odds ratio bigger than my risk ratio?
- Because your outcome is common rather than rare. For common outcomes the odds ratio sits farther from 1 than the risk ratio, exaggerating the effect. In the worked example the odds ratio is 18.0 while the risk ratio is only 5.25, because the risks (0.75 and 0.143) are far from zero.
References & further reading
- Cornfield, J. (1951). A method of estimating comparative rates from clinical data. Journal of the National Cancer Institute.
- Odds ratio — Wikipedia
- Relative risk — Wikipedia
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