Odds ratio versus risk ratio

Two ways to compare risks

Suppose a binary outcome, such as an event that either happens or does not, is recorded in two groups. The risk of the event in a group is the number of people with the event divided by the number in the group. The odds are the number with the event divided by the number without it. A risk of 25 percent is the same as odds of 1 to 3, which is 0.33.

The risk ratio (also called relative risk) divides the risk in one group by the risk in the other. The odds ratio divides the odds in one group by the odds in the other. The risk difference subtracts one risk from the other and is an absolute measure. All three describe the same two groups, and none of them is wrong. They answer slightly different questions and behave differently when combined across studies.

The risk ratio answers the question people usually ask: how many times more likely is the event in one group than the other? The odds ratio is harder to put into words, because odds are not a quantity most people think in. Its attraction is statistical, as explained below, and it is the natural output of logistic regression, which is why it appears in so many observational studies.

A worked example

In a simulated comparison, 30 of 100 people in the exposed group and 20 of 100 in the comparison group have the event.

• Risks: 0.30 and 0.20.
• Risk ratio: 0.30 / 0.20 = 1.50.
• Odds: 30/70 = 0.429 and 20/80 = 0.250.
• Odds ratio: 0.429 / 0.250 = 1.71.
• Risk difference: 0.30 - 0.20 = 0.10, or 10 percentage points.

The same data give a risk ratio of 1.50 and an odds ratio of 1.71. Someone who reads the odds ratio as a risk ratio would say the event is 71 percent more likely when the true figure is 50 percent. The gap is not an error in either calculation. It is a property of the scale, and it grows as the outcome becomes more common.

Why the two diverge

When an event is rare, almost everyone in each group does not have it, so the number without the event is close to the group size. The odds, events divided by non-events, are then close to the risk, events divided by group size, and the two ratios nearly coincide. When the event is common, the denominator of the odds shrinks and the odds grow faster than the risk. A risk of 50 percent gives odds of 1, and a risk of 80 percent gives odds of 4.

The consequence is systematic. If the ratio is above 1, the odds ratio is further above 1 than the risk ratio. If it is below 1, the odds ratio is further below 1. The odds ratio always exaggerates the risk ratio in the direction of the effect, and the exaggeration increases with the baseline risk and with the size of the effect.

A common rule of thumb is that the two are close enough when the event occurs in fewer than about ten percent of people in both groups. That is a rough guide only, and the table in the next section shows how quickly the gap opens with a large effect.

Converting between the measures

Given the baseline risk in the comparison group, written p₀, an odds ratio can be converted to a risk ratio with a standard formula: risk ratio = odds ratio / (1 - p₀ + p₀ x odds ratio). The baseline risk is essential. The same odds ratio corresponds to different risk ratios in populations with different baseline risks.

At a baseline risk of 1 percent, an odds ratio of 2.0 is almost the same as a risk ratio of 2.0. At 50 percent, the same odds ratio corresponds to a risk ratio of 1.33. The odds ratio is the same figure throughout, but the meaning for an individual changes a great deal. This is why a single odds ratio from a trial in a high-risk population should not be transferred to a low-risk population as though it were a risk ratio.

Converting an odds ratio to a risk ratio is an approximation when it uses a baseline risk that is not the one observed in the study, and the conversion of a confidence interval requires care. Treat the converted figure as an illustration of the likely size of the effect, and state the baseline risk used.

Getting to an absolute effect

Readers and decision makers usually need the absolute effect, meaning how many events would be avoided or caused per 1,000 people. Neither ratio gives this without a baseline risk. Take a pooled odds ratio of 0.70 for an adverse event, and suppose the baseline risk in the population of interest is 20 percent.

• Baseline odds: 0.20 / 0.80 = 0.25.
• Odds with treatment: 0.25 x 0.70 = 0.175.
• Risk with treatment: 0.175 / (1 + 0.175) = 0.149.
• Equivalent risk ratio: 0.74.
• Absolute risk reduction: 0.200 - 0.149 = 0.051, about 5.1 percentage points, or roughly 51 fewer events per 1,000 people.
• Number needed to treat: 1 / 0.051, about 20.

Choosing a different baseline changes the answer. At a 2 percent baseline, the same odds ratio of 0.70 gives about 6 fewer events per 1,000 people. This is the logic behind summary of findings tables in GRADE, which present absolute effects for a stated baseline risk and, where the population varies, for more than one.

Why the odds ratio is so widely used

Given that the risk ratio is easier to interpret, it is fair to ask why the odds ratio is everywhere. There are four reasons.

• Logistic regression. The coefficients of logistic regression, the standard model for a binary outcome with covariates, are log odds ratios. An observational study that adjusts for confounders usually reports an adjusted odds ratio.
• Case-control studies. Because case-control studies sample on the outcome, risks cannot be estimated, but the odds ratio can be, and it is valid. A risk ratio from a case-control study is not available without extra information.
• Symmetry. The odds ratio for the event is the reciprocal of the odds ratio for the non-event. A risk ratio does not have this property: the risk ratio for survival is not the reciprocal of the risk ratio for death. Analysts who code the outcome in the opposite direction obtain a different risk ratio and a consistent odds ratio.
• Range. The odds ratio can take any value above zero for any baseline risk, whereas a risk ratio is bounded by the baseline. If the baseline risk is 40 percent, a risk ratio above 2.5 is impossible.

These properties are real advantages for modelling. They do not remove the interpretation problem, which falls on whoever reads the result.

Pooling in a meta-analysis

Both ratios are analysed on the log scale, so that the sampling distribution is closer to normal and the intervals are symmetric around the estimate. The pooled log ratio is a weighted average across studies, and the result is exponentiated back. The choice between ratios therefore affects what the pooled value means, not how the pooling is done.

Simulated data show the divergence. Four studies compare an exposed with an unexposed group on a common outcome, with an event rate of about 26 percent in the unexposed groups.

Pooling with the Mantel-Haenszel method gives a risk ratio of 1.36 and an odds ratio of 1.57. The odds ratio is further from 1, as expected, and the difference is not trivial for decisions: one suggests a 36 percent higher risk and the other a 57 percent higher odds.

Which should be pooled? Many methodological sources suggest a pragmatic approach. Consider the types of study that are included. If all are randomized trials with complete risk data, the risk ratio is a reasonable default for common outcomes because it is easier to interpret, and the pooled estimate can be combined with a baseline risk to give absolute effects. If the studies include case-control designs or adjusted observational estimates, the odds ratio is the only measure that is available in all of them. The measure should be chosen in the protocol, before the results are seen, and a sensitivity analysis can show whether the conclusion depends on the choice.

Another consideration is heterogeneity. Some empirical work suggests that the odds ratio is more consistent across studies with different baseline risks than the risk ratio, which would favor it as a summary of the relative effect. Others show little difference in practice. The evidence is mixed, and it is best to avoid claiming that either is always more stable.

Rare events and zero cells

Binary outcomes in meta-analysis are often rare, and some studies have zero events in one or both groups. This affects both ratios. When a group has no events, the risk ratio and the odds ratio are zero or undefined, and the usual remedy of adding 0.5 to each cell distorts the estimate and is not neutral. The Mantel-Haenszel and Peto methods handle many sparse situations more gracefully than inverse-variance weighting, and some analysts use a generalized linear mixed model on the original counts. For the rare-event case the odds ratio and risk ratio nearly coincide, so the choice between them matters little, and the more important decision is how to handle the zero cells. Studies with no events in either group provide limited information about the relative effect and are excluded by default in some methods, which should be reported.

Communicating the result

Presenting a result well is as important as calculating it. A few habits help readers avoid the misreading described above.

• State clearly which measure was used and which group is the reference. A statement such as "the odds ratio was 0.70 for the event in the intervention group compared with control" leaves no doubt.
• Avoid describing an odds ratio as "times more likely" or as a percentage change in risk unless the outcome is rare. Describe it as a change in the odds.
• Give the baseline risk and the absolute effect, using natural frequencies such as "about 51 fewer events per 1,000 people" rather than relative terms alone.
• Show the confidence interval and, for a random-effects analysis, a prediction interval as well, since the range of true effects is often wider than the interval around the average.
• Where the baseline risk differs between settings, show the absolute effect for more than one baseline, and say that the figures are an application of the pooled relative effect and not a separate finding.

These steps do not make the odds ratio a risk ratio. They keep a reader from treating it as one, which is the practical source of most errors.

Common mistakes

• Reading an odds ratio as a risk ratio for an outcome that is not rare, which overstates the effect.
• Transferring a ratio between populations with different baseline risks without recalculating the absolute effect.
• Mixing ratios in one pooled estimate, for example combining odds ratios from some studies with risk ratios from others as if they were interchangeable.
• Converting to a risk ratio with an invented baseline instead of one drawn from the data or from a stated population.
• Changing the measure after seeing results, which is a form of analytic flexibility that should be avoided by prespecification.
• Ignoring the direction of coding. Reversing the outcome changes the risk ratio but not the odds ratio, so a risk ratio is not comparable across analyses unless the coding is the same.

How we can help

Choosing and reporting the effect measure is a protocol decision, and it affects the interpretation of everything that follows. We can help you select a measure that suits your included designs, extract data in a form that supports the conversion, run Mantel-Haenszel, inverse-variance and mixed-model analyses, calculate absolute effects for stated baselines and prepare summary of findings tables. [OWNER VERIFICATION REQUIRED] The relevant services are meta-analysis and statistical analysis.

Frequently asked questions