Odds Ratio or Relative Risk? When the Two Numbers Differ

"Doubles the risk." "Twice the odds." Headlines use these phrases as if they meant the same thing. They do not. The odds ratio and the relative risk are different measures, and the gap between them can turn a modest effect into an impressive-sounding one.

Risk and odds in one minute

Risk is the proportion of people who have the outcome, such as 40 out of 100. Odds compare those who have the outcome with those who do not, so 40 out of 100 is odds of 40 to 60. The relative risk divides one group's risk by the other's. The odds ratio divides one group's odds by the other's.

When the outcome is rare, they agree

Suppose 3% of an exposed group and 2% of an unexposed group have an outcome. The relative risk is 1.5. The odds ratio is 1.55. The odds ratio calculator shows how close they are, and this is why textbooks say the odds ratio approximates the relative risk when the outcome is rare, typically below about 10% of the study population.

When the outcome is common, they drift apart

Now suppose 60% of one group and 40% of another have the outcome. The relative risk is still 1.5, but the odds ratio is 2.25. At 30% against 20%, the relative risk is again 1.5 and the odds ratio is 1.71.

The same real effect looks bigger as the baseline risk rises.

The BMJ authors who studied this noted that an odds ratio above one always overstates the effect when read as a relative risk, and one below one understates it, with an example of an odds ratio of 0.66 that corresponded to a relative risk of 0.81 (Davies, Crombie and Tavakoli, 1998).

Converting one into the other

If you know the baseline risk in the unexposed group, a simple correction recovers the relative risk: RR = OR / ((1 − P0) + P0 × OR). With a baseline risk of 0.40 and an odds ratio of 2.25, the result is 2.25 / (0.60 + 0.90) = 1.5. This method was proposed for cohort studies with common outcomes (Zhang and Yu, 1998).

The difference of two proportions calculator gives the risk difference directly, and the Fisher exact test is the safer choice when counts are small.

Which one should you report?

In cohort studies and randomized trials you can calculate the relative risk directly, so report it, and consider the absolute difference too. The odds ratio is natural in case-control studies, where risks cannot be estimated, and it comes out of logistic regression. If you report one, label it clearly, and do not describe an odds ratio as "risk."

This matters for readers. A reported "odds ratio of 2.25" for a common outcome is not a doubling of risk, and calling it that exaggerates the finding. Our guide to reading a clinical trial explains how to look for the absolute numbers behind any ratio. Always look for the baseline risk too. Doubling a risk of 1 in 10,000 and doubling a risk of 1 in 10 are very different stories, even though both are a relative risk of 2. Ratios without baselines are incomplete, and a percentage change taken from an odds ratio, such as a "125% increase" from 2.25, can overstate a true 50% increase in risk.