Relative Risk Versus Odds Ratio

Two Ratios, Two Ways to Compare Groups

Relative risk and odds ratio both compare an outcome between two groups, but they compare different quantities. Relative risk compares probabilities of the outcome. Odds ratio compares the odds of the outcome. When an outcome is uncommon in both groups, these measures are often close; when it is common, they can differ substantially. Knowing which quantity a ratio describes is essential for interpreting a result accurately.

Throughout this tutorial, group 1 is the group of interest and group 0 is the reference group. The outcome is a specified event, such as a symptom occurring. Keeping the groups and event clear matters: swapping the group order or changing which outcome counts as the event changes the ratios.

A relative risk of 2 means the event probability in group 1 is twice the probability in group 0. An odds ratio of 2 means the event odds in group 1 are twice the event odds in group 0. These are different claims. In particular, odds are not probabilities: if the event probability is 0.20, its odds are \(0.20/0.80=0.25\), or 1 to 4.

Calculating Both Measures From the Same Data

For two groups with known event and non-event counts, organize the counts so each group forms a row. Let \(a\) and \(b\) be the event and non-event counts in group 1, and \(c\) and \(d\) the corresponding counts in group 0. The group sizes are \(a+b\) and \(c+d\). Thus the observed risks are \(a/(a+b)\) and \(c/(c+d)\).

The hats indicate that these are estimates calculated from observed counts. The cross-product form of the odds ratio follows by dividing the group 1 odds \(a/b\) by the group 0 odds \(c/d\). It is a convenient calculation, but the risk-ratio calculation must use each group’s full size as its denominator.

Why Rarity Makes the Measures Similar

The relationship between the two measures can be derived directly from their definitions. This result explains both when the measures are close and how they diverge.

If both event risks are small, then both \(1-p_0\) and \(1-p_1\) are close to 1. Their ratio is therefore close to 1, making \(OR\) close to \(RR\). This is an approximation, not an identity: rarity does not make odds and risks exactly equal.

The relationship also shows the direction of the difference. If \(p_1>p_0\), then \(1-p_0>1-p_1\), so the multiplying factor is greater than 1 and \(OR>RR\). If \(p_1<p_0\), the factor is less than 1 and \(OR<RR\). In either case, both measures point in the same direction: they are both above 1 when group 1 has the higher event risk and both below 1 when group 1 has the lower event risk.

When the Outcome Is Common

As event probabilities increase, the probability of no event is no longer close to 1. The factor \((1-p_0)/(1-p_1)\) can then pull the odds ratio noticeably farther from 1 than the relative risk. This is why an odds ratio can sound more dramatic if it is mistakenly described as a ratio of probabilities.

Interpreting a Ratio Below One

A ratio below 1 indicates a lower event risk or event odds in group 1 than in group 0. But the percentage reduction implied by each measure has a different meaning. For a relative risk \(RR\), \(1-RR\) describes the proportional reduction in event risk. For an odds ratio \(OR\), \(1-OR\) describes the proportional reduction in event odds, not in event risk.

Choosing and Reporting the Measure

The ratio to report depends on the question and how the data were collected. If the aim is to compare probabilities of an event and those probabilities can be estimated from the sampling design, relative risk gives that direct comparison. Odds ratios are useful when the scientific question concerns odds and arise naturally in odds-based models. In a case-control sample selected according to outcome status, the sample proportions with the outcome are set by the design; those proportions do not directly estimate population risks. An odds ratio can still be estimated from the group comparisons, but a relative risk generally cannot be obtained directly from those sampled proportions.

Whichever ratio is used, state the event, name group 1 and group 0, report the measure by its standard name, and include the quantities being compared. For example, say “the event risk was twice as high in group 1” for a relative risk of 2, but “the event odds were twice as high” for an odds ratio of 2. A ratio summarizes an association between groups; by itself, it does not establish that group membership caused the outcome.

Common Mistakes

• Calling an odds ratio a risk ratio. An odds ratio of 3.5 means odds are 3.5 times as large, not that the event is 3.5 times as probable.
• Assuming the measures are always nearly equal. They are often close when the event is rare in both groups, but can differ substantially for common outcomes.
• Using “rare” to describe the exposure instead of the outcome. The approximation depends on the event probabilities \(p_0\) and \(p_1\), not on how common group membership is.
• Reporting a ratio without its comparison order. State which group is in the numerator and which event is being counted. Reversing group order replaces a ratio by its reciprocal.
• Interpreting a reduction in odds as the same reduction in risk. For instance, \(OR=0.444\) represents about 55.6% lower odds, not necessarily 55.6% lower risk.