The odds ratio and the risk ratio both compare how often a binary outcome occurs in a treatment group versus a control group, but they answer subtly different questions: the risk ratio (also called relative risk) compares the probability of the event between groups, while the odds ratio compares the odds, which are the probability of the event divided by the probability of no event. When the outcome is uncommon the two measures are almost identical, but as the outcome becomes common the odds ratio drifts further from 1 than the risk ratio and exaggerates the apparent strength of the effect. Knowing which one a number represents, and choosing the right one for your design, is the difference between a defensible result and an overstated one.
How each is calculated from a two-by-two table
Every binary comparison reduces to a two-by-two table with four counts: the number of events and non-events in the treatment arm, and the number of events and non-events in the control arm. Call the treatment events a, the treatment non-events b, the control events c, and the control non-events d. The two measures are built from these four numbers in two different ways, which is exactly why they diverge.
- Risk in a group is events divided by the total in that group: the treatment risk is a divided by (a plus b), and the control risk is c divided by (c plus d).
- Odds in a group is events divided by non-events: the treatment odds are a divided by b, and the control odds are c divided by d.
- The risk ratio is the treatment risk divided by the control risk.
- The odds ratio is the treatment odds divided by the control odds, which reduces to the familiar cross-product (a times d) divided by (b times c).
The contrast is structural. The risk ratio puts the whole group size in the denominator, so it is bounded by how common the event can be; the odds ratio puts only the non-events in the denominator, so it can keep growing as events crowd out non-events. That single difference drives every divergence discussed below.
A worked numeric example
Suppose a trial of a preventive drug enrols 100 people on treatment and 100 on control. In the treatment arm 20 people have the event and 80 do not; in the control arm 40 people have the event and 60 do not. Read straight off this two-by-two table, the treatment risk is 20 divided by 100, or 0.20, and the control risk is 40 divided by 100, or 0.40. The risk ratio is 0.20 divided by 0.40, which equals 0.50: treatment halves the probability of the event, an intuitive and honest statement.
Now compute the odds. The treatment odds are 20 divided by 80, or 0.25; the control odds are 40 divided by 60, or 0.667. The odds ratio is 0.25 divided by 0.667, which equals 0.375. Both numbers point the same direction, but the odds ratio of 0.375 looks like a larger protective effect than the risk ratio of 0.50, even though they describe the identical data. This is the common-outcome divergence in action: with a baseline event rate of 40 percent, the odds ratio overstates the effect relative to what most readers picture when they hear "the risk was cut in half." If you want to see how the same counts behave once they are pooled across studies, you can enter event data directly into our event-count calculator and switch between measures.
Odds ratio versus relative risk when the outcome is common
The mathematical link is that odds and risk are nearly equal only when the event is rare. When a probability p is small, the odds p divided by (1 minus p) are close to p itself, because the denominator is close to 1. So for a rare outcome, the treatment and control odds each approximate their respective risks, and the odds ratio approximates the risk ratio. This is why epidemiologists treat the odds ratio as a usable estimate of relative risk only when the disease is uncommon, the so-called rare-disease assumption.
As the event rate climbs, the (1 minus p) denominator shrinks and the odds inflate faster than the risk. The odds ratio then moves away from 1 more aggressively than the risk ratio, so an odds ratio of 2.0 for a common outcome can correspond to a risk ratio closer to 1.3 or 1.4. The numbers are not wrong; they answer the odds question rather than the risk question. The trap is reading an odds ratio aloud as if it were a relative risk, which inflates the perceived effect size and misleads clinicians and patients alike.
Why study design forces the choice
The measure is not always a free decision; sometimes the design dictates it. In a case-control study, investigators start by selecting people who already have the outcome (cases) and people who do not (controls), then look backward at exposure. Because the ratio of cases to controls is fixed by the sampling, the data contain no valid estimate of the underlying event probability, so a risk ratio cannot be computed honestly. The odds ratio, however, is invariant to that sampling scheme, which is precisely why case-control studies must report odds ratios. The odds ratio is the only association measure that survives outcome-based sampling.
Why logistic regression returns odds ratios
The same logic explains the output of one of the most widely used models. A logistic regression models the log-odds of the outcome as a linear function of the predictors, so when you exponentiate a coefficient you get an odds ratio, not a risk ratio. This is convenient because odds ratios behave well in regression and adjust cleanly for covariates, but it means that adjusted analyses in cohort and cross-sectional studies routinely report odds ratios even when the outcome is common. Whenever you read an adjusted odds ratio for a frequent event, remember that it will look larger than the corresponding adjusted relative risk.
The risk difference as a third option
Ratios are not the only way to summarise a binary contrast. The risk difference, also called the absolute risk reduction, is simply the control risk minus the treatment risk. In the worked example it is 0.40 minus 0.20, or 0.20, meaning twenty fewer events per hundred people treated. The risk difference is the most clinically tangible measure because its reciprocal gives the number needed to treat, but it does not travel well across populations with different baseline risks, which is why ratio measures are usually preferred for pooling.
What to report in a meta-analysis
When you combine studies, two practical principles guide the choice. First, both the odds ratio and the risk ratio are pooled on the log scale: the software takes the natural logarithm of each study's ratio, weights and averages those log values, then exponentiates the result, because ratios are multiplicative and their sampling distribution is far more symmetric in log space. This log-scale handling is the same regardless of which ratio you choose, and it is built into any tool that will build a forest plot from your study counts.
- Prefer the risk ratio for prospective designs with a common outcome, because it is interpretable as a true relative risk and will not exaggerate the effect.
- Use the odds ratio when the included studies are case-control, when results come from logistic regression, or when sparse cells make the odds ratio more stable to pool.
- Report the risk difference alongside a ratio when clinical decision-making needs an absolute number such as the number needed to treat.
- Never mix measures within one pooled estimate; convert every study to a single common metric before combining, and state the chosen measure plainly in the methods.
Above all, label the measure correctly in the text and on the figure, because the most common reviewer complaint is an odds ratio narrated as if it were a relative risk. If you are still deciding how to frame the underlying review, our overview of how a meta-analysis pools evidence sets the context, the choice of fixed-effect or random-effects pooling determines how the studies are combined, and the guide on turning extracted counts into a forest plot shows where the measure you pick ends up on the published figure.