Relative Risk

What relative risk measures

It is also called the risk ratio and is commonly used to express treatment effects, exposure associations, and event probabilities in trials, cohort studies, meta-analysis, and health-economic models.

The measure is meaningful only when the event, population, groups, and follow-up period are clearly defined. A relative risk is a ratio of cumulative risks, not a ratio of odds, instantaneous hazards, or person-time incidence rates.

The basic formula

Let $p_1$ be the event risk in the treatment or exposed group and $p_0$ the event risk in the comparator or unexposed group. The relative risk is:

$$ RR=\frac{p_1}{p_0} $$

For a binary outcome observed in two groups, the data can be arranged as follows:

The sample relative risk is:

$$ \widehat{RR}=\frac{a/(a+b)}{c/(c+d)} =\frac{a/n_1}{c/n_0} $$

The numerator and denominator must refer to comparable populations and the same outcome window. Changing the reference group inverts the ratio, so the group order must always be reported.

How to interpret the value

Relative risk is centred at one, which represents equal risks in the two groups. Values below or above one indicate the direction of association relative to the stated event and reference group.

An $RR$ of $0.70$ means the event risk is 0.70 times the comparator risk, or 30% lower on the relative scale. It does not mean that 30 percentage points are removed, because the absolute change depends on the comparator risk.

A worked two-group calculation

Suppose 14 of 100 treated patients and 20 of 100 comparator patients experience an adverse event within one year. The group risks are $0.14$ and $0.20$.

$$ \widehat{RR}=\frac{14/100}{20/100}=\frac{0.14}{0.20}=0.70 $$

The estimated one-year event risk is 30% lower in the treatment group relative to the comparator. The absolute risk difference is:

$$ RD=p_1-p_0=0.14-0.20=-0.06 $$

This corresponds to 6 fewer events per 100 treated patients over one year. Under a causal interpretation, the number needed to treat to prevent one additional event is:

$$ NNT=\frac{1}{|RD|}=\frac{1}{0.06}=16.67 $$

The conventional whole-person presentation is 17 patients, with the direction, follow-up, and uncertainty reported. The NNT should be derived from unrounded absolute risks and interpreted only when the effect estimate supports a causal treatment comparison.

Relative and absolute effects answer different questions

Relative risk describes proportional change, while risk difference describes the change in percentage points. Decision makers usually need both because the same relative effect can produce very different numbers of events across baseline-risk groups.

If a treatment has $RR=0.70$:

The relative effect is identical in all three rows, but the absolute benefit differs markedly. Health-economic models therefore need an appropriate baseline risk as well as a relative treatment effect.

Relative risk reduction and relative risk increase

For an adverse event with $RR<1$, the relative risk reduction is one minus the relative risk. When $RR>1$, the relative risk increase is the relative risk minus one.

$$ RRR=1-RR $$

$$ RRI=RR-1 $$

With $RR=0.70$, the relative risk reduction is $1-0.70=0.30$, or 30%. These measures should be labelled explicitly because “30% reduction” can be mistaken for an absolute reduction of 30 percentage points.

Confidence intervals on the log scale

The sampling distribution of the log relative risk is often more nearly symmetric than that of the relative risk itself. A confidence interval is therefore commonly calculated on the log scale and exponentiated back to the ratio scale.

For the simple independent two-group table with non-zero event counts:

$$ SE{\log(\widehat{RR})}= \sqrt{\left(\frac{1}{a}-\frac{1}{n_1}\right) +\left(\frac{1}{c}-\frac{1}{n_0}\right)} $$

An approximate 95% confidence interval is:

$$ \exp\left[ \log(\widehat{RR})\pm1.96,SE{\log(\widehat{RR})} \right] $$

The formula assumes an appropriate large-sample setting and independent groups. Clustered trials, matched data, repeated observations, survey designs, adjusted models, sparse events, and zero cells require methods that reflect their actual design and estimator.

A confidence-interval example

Consider 40 events among 100 exposed people and 20 events among 100 unexposed people. The estimated relative risk is $0.40/0.20=2.00$.

The standard error is:

$$ SE{\log(\widehat{RR})}= \sqrt{\left(\frac{1}{40}-\frac{1}{100}\right) +\left(\frac{1}{20}-\frac{1}{100}\right)} =\sqrt{0.055}\approx0.235 $$

The 95% confidence interval is:

$$ \exp\left[\log(2.00)\pm1.96(0.235)\right] \approx(1.26,3.17) $$

The interval excludes one under this approximation, but it still describes sampling uncertainty rather than proving a causal exposure effect. Bias from confounding, selection, misclassification, missing data, or an inappropriate model is not captured by this interval.

Relative risk versus odds ratio

Risk is the probability of an event, while odds are the probability of an event divided by the probability of no event. The odds ratio and relative risk are therefore different effect measures.

$$ OR=\frac{p_1/(1-p_1)}{p_0/(1-p_0)} $$

When events are rare in both groups, the odds ratio may approximate the relative risk. For common events, the odds ratio is usually farther from one and should not be reported as though it were a relative risk.

If an odds ratio must be converted using a known comparator risk $p_0$, one relationship is:

$$ RR=\frac{OR}{(1-p_0)+p_0OR} $$

This conversion is conditional on the supplied baseline risk and compatible estimand. Applying one converted risk across populations with different baseline risks can be inappropriate.

Relative risk versus hazard ratio and rate ratio

A hazard ratio compares instantaneous event rates among those still event-free, while a rate ratio compares events per unit of person-time. Neither is generally equal to a cumulative-risk ratio.

Converting a hazard ratio or rate ratio directly into an RR without a survival model and time horizon can produce incorrect event probabilities. The source measure, underlying time process, and competing risks must be retained.

Estimating adjusted relative risks

Crude relative risks can be distorted when treatment or exposure groups differ in prognostic characteristics. Adjustment aims to estimate a defined conditional or marginal effect under the chosen study design and causal assumptions.

Log-binomial regression models risk with a log link but can encounter convergence or boundary problems. Modified Poisson regression with robust standard errors is often used for adjusted relative risks in cohort data, while standardisation or inverse-probability weighting can estimate marginal risks and their ratio when correctly specified.

An adjusted association is not automatically causal. A causal RR requires a defined estimand, exchangeability, positivity, consistency, correct time alignment, and appropriate handling of missingness, selection, measurement error, and interference.

Baseline risk and transportability

Relative effects may be more stable across risk groups than absolute effects in some settings, but constancy should not be assumed. Biological effect modification, adherence, competing risks, follow-up, case mix, outcome definition, and healthcare context can all change the RR.

Transporting an RR from one study to another population requires assessing whether the treatment contrast and relative effect are applicable. The new absolute risk is not established until the target population's comparator risk is also estimated coherently.

Applying relative risk in an economic model

Health-economic models often combine a baseline event risk with a relative treatment effect. For a common binary event over the same interval, a simple proportional-risk application is:

$$ p_{\text{treat}}=RR\times p_{\text{control}} $$

If $p_{\text{control}}=0.20$ and $RR=0.70$, the modelled treatment risk is $0.14$. This calculation is valid only when the RR and baseline risk refer to compatible populations, outcomes, follow-up periods, and competing-event structures.

Applying an annual RR independently in every cycle may imply a treatment-effect duration not supported by evidence. Models should specify whether the effect is constant, wanes, stops, depends on history, or operates on a hazard or rate scale instead.

Correlation and uncertainty in model inputs

The baseline risk and log relative risk may be statistically dependent, particularly when estimated from the same dataset or model. Sampling them independently in probabilistic sensitivity analysis can misrepresent outcome uncertainty.

Relative risks are commonly sampled on the log scale so that simulated values remain positive. The parameter distribution, covariance, treatment-effect duration, and transformation to cycle probabilities should be documented and tested for clinically plausible values.

Meta-analysis of relative risks

Meta-analysis commonly combines study-specific log relative risks because the log scale is unbounded and has convenient statistical properties. The pooled result depends on study design, heterogeneity model, weighting, zero-event handling, and whether the studies estimate a sufficiently common effect.

$$ \log(RR_{\text{pooled}})= \frac{\sum_{k=1}^{K}w_k\log(RR_k)} {\sum_{k=1}^{K}w_k} $$

The pooled RR is obtained by exponentiation. A numerical pooled estimate should not conceal important differences in outcome definition, follow-up, bias, adherence, or population that make a single common interpretation questionable.

Sparse events and zero cells

If one group has no events, the ordinary log-RR and standard-error formulas are undefined. Adding a continuity correction is simple but can materially affect results when groups are small or unbalanced.

Exact, likelihood-based, Bayesian, or other sparse-data methods may be preferable depending on the design and question. Studies with zero events in both groups contain information about absolute event rarity but do not identify a study-specific relative effect through the ordinary ratio.

A practical workflow

A defensible relative-risk analysis starts with the estimand and event window rather than the formula alone. The following workflow keeps the calculation, interpretation, and downstream use aligned.

1. Define the population, treatment or exposure, comparator, event, time horizon, and analysis estimand.
2. Verify that numerator and denominator groups use compatible eligibility, follow-up, and outcome ascertainment.
3. Calculate each group risk using the correct number at risk and event count.
4. Estimate the RR with a method appropriate to the study design, clustering, adjustment, and event frequency.
5. Quantify uncertainty on a suitable scale and address sparse or zero-event data explicitly.
6. Report absolute risks and risk differences alongside the RR whenever they are decision-relevant.
7. Assess confounding, effect modification, missingness, selection, misclassification, and competing risks.
8. Translate the effect into target-population risk only with a compatible baseline risk and time horizon.
9. Propagate parameter uncertainty, dependence, and treatment-effect duration through any economic model.
10. Document the group order, event direction, formula, model, assumptions, diagnostics, and limitations.

Interpreting and reporting the result

A relative risk is a dimensionless comparison of two event probabilities over a defined interval. It should be reported with both group risks, an uncertainty interval, the reference group, event definition, follow-up, and study design.

Common errors include calling an odds ratio or hazard ratio a relative risk, interpreting a 30% relative reduction as 30 percentage points, omitting the baseline risk, reversing the reference group, applying a fixed RR indefinitely, and claiming causation from an unadjusted observational association. Clear reporting makes both the proportional effect and its absolute consequences visible.