Odds ratios are a common measure of the size of an effect and may be reported in case-control studies, cohort studies, or clinical trials. Increasingly, they are also used to report the findings from systematic reviews and meta-analyses. Odds ratios are hard to comprehend directly and are usually interpreted as being equivalent to the relative risk. Unfortunately, there is a recognised problem that odds ratios do not approximate well to the relative risk when the initial risk (that is, the prevalence of the outcome of interest) is high.1,2 Thus there is a danger that if odds ratios are interpreted as though they were relative risks then they may mislead.
The advice given in many texts is unusually coy on the matter. For example: “The odds ratio is approximately the same as the relative risk if the outcome of interest is rare. For common events, however, they can be quite different.”3 How close is “approximately the same,” how uncommon does an event have to be to qualify as “rare,” and how different is “quite different”?
- If the odds ratio is interpreted as a relative risk it will always overstate any effect size: the odds ratio is smaller than the relative risk for odds ratios of less than one, and bigger than the relative risk for odds ratios of greater than one
- The extent of overstatement increases as both the initial risk increases and the odds ratio departs from unity
- However, serious divergence between the odds ratio and the relative risk occurs only with large effects on groups at high initial risk. Therefore qualitative judgments based on interpreting odds ratios as though they were relative risks are unlikely to be seriously in error
- In studies which show reductions in risk (odds ratios of less than one), the odds ratio will never underestimate the relative risk by a greater percentage than the level of initial risk
- In studies which show increases in risk (odds ratios of greater than one), the odds ratio will be no more than twice the relative risk so long as the odds ratio times the initial risk is less than 100%
This short note quantifies the discrepancy between odds ratios and relative risks in different circumstances, and assesses whether such a discrepancy may seriously mislead if an odds ratio is used as an estimate of the relative risk.