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Relative risk compares the event proportion in an exposed cohort with the proportion in a non-exposed cohort.
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Formula
RR is dimensionless · a, b, c and d are person counts; a = event exposed, c = no event exposed, b = event unexposed, d = no event unexposed
| Symbol | Meaning | Unit |
|---|---|---|
| Risk in the exposed group divided by risk in the unexposed comparison group | dimensionless | |
| People with the event in the exposed group | people | |
| People without the event in the exposed group | people | |
| People with the event in the unexposed group | people | |
| People without the event in the unexposed group | people |
RR = \frac{a/(a+c)}{b/(b+d)}Risk ratio compares cumulative risks over the same defined follow-up period: RR = risk_exposed/risk_unexposed = [a/(a+c)]/[b/(b+d)]. RR=1 means equal observed risks; RR>1 means higher and RR<1 lower risk in the exposed group. It is not the odds ratio, which compares [a/c]/[b/d]; the two can differ substantially for common outcomes. RR requires observable risks, for example in a cohort or randomized study. It quantifies an association but does not establish causality by itself; confidence intervals, bias and study design matter.
Compares cumulative event risks over the same follow-up period in groups with known denominators, typically cohorts or randomized studies. It does not establish causality by itself.
First calculate the event proportion in each group, then divide exposed risk by unexposed risk.
Find exposed risk from RR
The relative measure does not replace absolute risks or their difference.
20/200 exposed and 10/200 unexposed people experience the event. Calculate RR.
Risks: 0.10 and 0.05.
RR=0.10/0.05=2; observed risk is twice as high in the exposed group.
What does RR=0.75 mean in an intervention study?
Event risk in the intervention arm is 75% of the comparison risk, a 25% relative reduction.
Absolute risks and a confidence interval are also needed for clinical meaning.
Confusing RR with the odds ratio.
RR compares a/(a+c) with b/(b+d); OR compares a/c with b/d. For common events, they are not similar.
Inferring causality automatically from RR>1.
Observational studies can be distorted by confounding and bias. Check design, adjustment, uncertainty and alternative explanations.
Formula cluster
A relative measure needs absolute risks, uncertainty and study design for sound interpretation.
In a cohort followed over the same period, exposed risk=20/200=0.10 and unexposed risk=10/200=0.05; RR=2. Risk was twice as high in the exposed group. This association alone does not establish causality.
Comparing cumulative event risks in cohort and randomized studies, clinical research and epidemiological analyses of exposures
Here is how to work through a typical Relative Risk (Risk Ratio) (RR = [a/(a+c)] / [b/(b+d)]) task step by step:
Task
20/200 exposed and 10/200 unexposed people experience the event. Calculate RR.
Solution path
Risks: 0.10 and 0.05.
RR=0.10/0.05=2; observed risk is twice as high in the exposed group.
Task
What does RR=0.75 mean in an intervention study?
Solution path
Event risk in the intervention arm is 75% of the comparison risk, a 25% relative reduction.
Absolute risks and a confidence interval are also needed for clinical meaning.
First label the cells clearly: a = event among exposed, c = no event among exposed, b = event among unexposed and d = no event among unexposed. Risk in the exposed group is a/(a+c); risk in the comparison group is b/(b+d). Then RR = [a/(a+c)]/[b/(b+d)]. Example: 20 events among 200 exposed and 10 among 200 unexposed give 0.10/0.05 = 2. Observed risk was twice as high in the exposed group. The follow-up period must be defined equally for both risks.
RR=2 means the observed cumulative event risk in the exposed group is twice that in the unexposed comparison group over the period studied. It does not mean that two out of three people experience the event, and by itself says nothing about absolute frequency. An increase from 1% to 2% has the same RR=2 as an increase from 30% to 60%, but very different absolute consequences. Therefore report both group risks and the absolute difference as well. A confidence interval also shows how uncertain the estimated ratio is because of sampling variation.
Relative risk compares probabilities: a/(a+c) divided by b/(b+d). The odds ratio instead compares event odds: (a/c)/(b/d) = ad/bc. For rare events, OR and RR are often close; for common events, the OR can lie farther from 1 and make the effect look larger. In a case-control study, participants are selected by outcome status, so risks cannot be estimated directly from sample proportions and OR is usually the appropriate measure. In a cohort with observed denominators, RR can be calculated directly.
You need event counts and the total number of people in each comparison group over a defined period. This is typical of a cohort study following exposed and unexposed people, or a randomized trial with treatment and control arms. Both cumulative risks can then be calculated as events divided by group size. A case-control study, however, fixes the number of cases and controls by design; its sample proportions are not population risks. An odds ratio is usually calculated there. If follow-up time differs substantially, an incidence rate per person-time may be more suitable than a cumulative RR.
No. RR>1 first describes only that the observed event risk was higher in the exposed group. In an observational study, age, pre-existing conditions, behavior or other confounders can influence both exposure and outcome. Selection and measurement errors can also create or alter an association. Randomization can balance known and unknown confounders on average, but it does not guarantee a flawless study. Therefore examine design, confounder control, temporal order, measurement quality and the confidence interval. Causality comes from the totality of evidence, not from one ratio.
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Curated exam set for "Relative Risk (Risk Ratio)":
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Which formula describes Relative Risk (Risk Ratio)?
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How do you rearrange Relative Risk (Risk Ratio) for Find exposed risk from RR?
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Which common mistake happens with Relative Risk (Risk Ratio)?
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