Relative Risk Reduction

Relative Risk Reduction in Trials and Economic Models

Relative risk reduction (RRR) states a treatment effect as the share of the comparator group's events that the treatment prevents, so a risk ratio of 0.75 is reported as a 25% reduction. Health economists meet it in trial reports and vaccine efficacy figures, and a decision model turns it into events avoided by applying it to the baseline risk of the population the decision covers. The same RRR can mean many events avoided in a high-risk group and very few in a low-risk group, so cost-effectiveness depends on that multiplication as much as on the trial result. The quoted value also depends on whether the source effect was a relative risk, an odds ratio or a hazard ratio.

Re-expressing a risk ratio as a percentage reduction

The Cochrane Handbook (chapter 6) explains that a risk ratio of 0.25 can be restated as the intervention decreasing the risk of events by 75%, and calls this the relative risk reduction. The RRR therefore carries exactly the information in the risk ratio and nothing more.

$$\text{RRR} = 1 - \text{RR} = 1 - \frac{\text{EER}}{\text{CER}} = \frac{\text{CER} - \text{EER}}{\text{CER}}$$

where RR is the risk ratio, CER is the control event rate (the absolute risk of the event in the comparator group over follow-up) and EER is the experimental event rate in the intervention group. The numerator is the absolute risk reduction, so the RRR is that reduction as a share of the comparator risk.

When a treatment raises the risk of an event, RR exceeds 1 and the RRR is negative. The Handbook expresses that case as an increase in risk of $(\text{RR} - 1) \times 100\%$. The RRR cannot exceed 1, reached when treatment removes the event entirely.

Why decision models carry the effect in relative form

Economic models commonly take the treatment effect from trials in relative form and the event risk from elsewhere. Chapter 10 of the Cochrane Handbook reports empirical evidence that relative effect measures are, on average, more consistent across studies than absolute measures, and advises against meta-analysing risk differences unless there is a clear reason to expect them to be consistent. Deeks found, in 114 meta-analyses of interventions to prevent an undesirable event, that the largest absolute benefits came in trials with the highest baseline event rates, the pattern expected if the risk ratio for the harmful outcome is constant.

The NICE manual (PMG36, section 4.6.16) states that relative treatment effects seen in randomised trials may be applied to data on the baseline risk of health outcomes for the populations or subgroups of interest. With an RRR the step is a multiplication.

$$p_1 = p_0 \times (1 - \text{RRR}), \qquad \text{ARR} = p_0 - p_1 = p_0 \times \text{RRR}$$

where $p_0$ is the baseline risk of the event under the comparator over the period the model uses, $p_1$ is the risk under the new treatment and ARR is the absolute risk reduction, the events avoided per person treated, whose reciprocal is the number needed to treat. For a fixed RRR, events avoided rise in direct proportion to baseline risk.

The Handbook (chapter 15) cites a review of oral anticoagulants in atrial fibrillation. In high-risk patients with a stroke rate of about 120 per 1000 per year, warfarin prevents about 70 strokes per 1000 patients yearly, while in low-risk patients with about 20 strokes per 1000 per year it prevents only 12. The implied relative reductions are similar, 70/120 = 0.58 and 12/20 = 0.60, yet strokes avoided differ almost sixfold.

Vaccine efficacy is an RRR under another name. The CDC's Principles of Epidemiology in Public Health Practice gives vaccine efficacy as one minus the risk ratio of vaccinated to unvaccinated people, and in its varicella example, worked as vaccine effectiveness, 42.9 − 11.8 = 31.1 and 31.1 / 42.9 = 0.72, or 72%. A vaccination model still needs a baseline attack rate to turn that figure into cases avoided.

Choices that change the reported figure

Several choices made before an RRR reaches a model alter its value. They change how the effect is defined, not how precisely it is estimated.

Which outcome is counted as the event

The Handbook (chapter 6) notes that switching events and non-events only changes the sign of a risk difference and inverts an odds ratio, but can make a substantial difference for risk ratios. As an illustration, a trial with death risks of 0.15 under control and 0.10 under treatment has a risk ratio of 0.10/0.15 = 0.667, an RRR for death of 33%. Stated as survival, 0.90 against 0.85, the same result is a risk ratio of 1.059, a relative increase in survival of only 5.9%. The Handbook suggests treating the less common state as the event.

An odds ratio read as a risk ratio

The Handbook warns that, for interventions that reduce events, misinterpreting an odds ratio as a risk ratio overestimates the effect, especially when events are common, and that the error is quite common in published reports. Converting requires an assumed comparator risk.

$$\text{RR} = \frac{\text{OR}}{1 - \text{ACR} \times (1 - \text{OR})}$$

where OR is the odds ratio and ACR is the assumed comparator risk. In the Handbook's example an odds ratio of 0.73 with an assumed comparator risk of 0.3 means 62 fewer events per 1000. Since 0.3 × 0.27 = 0.081, RR = 0.73 / 0.919 = 0.794, an RRR of 20.6%, and 0.3 × 0.794 = 0.238 reproduces the 62 fewer per 1000. Reading the odds ratio directly as a 27% reduction overstates the effect by about 30%.

A hazard ratio treated as a risk ratio

For time-to-event outcomes the trial effect is usually a hazard ratio, and one minus the hazard ratio is a relative reduction in the event rate at each moment, not in the cumulative risk over the horizon. PMG36 (section 4.6.22) says the proportional hazards assumption should always be assessed. Under that assumption, treated survival equals baseline survival raised to the power of the hazard ratio, so with an illustrative hazard ratio of 0.7 and five-year baseline survival of 0.5, treated survival is 0.616 and the cumulative RRR is 1 − 0.3844/0.5 = 0.231, not 0.30.

Uncertainty in an RRR and its use in probabilistic analysis

The Handbook (chapter 6) notes that ratio measures are analysed on the natural log scale. A confidence interval for an RRR is therefore built for the log risk ratio, transformed back and subtracted from 1. The standard error of the log risk ratio is set out on the risk ratio page. The Handbook (chapter 15) adds that absolute effects computed from these limits for an assumed comparator risk ignore uncertainty in that risk.

The ISPOR-SMDM task force report by Briggs and colleagues names the log-normal as a standard choice for relative risks in probabilistic sensitivity analysis. In its example the baseline progression probability and the relative risk reduction each get a distribution and the on-treatment probability is the baseline probability multiplied by one minus the RRR, with odds ratios suggested where this gives out-of-range probabilities. PMG36 (section 4.7.22) describes a switching value as the value a parameter would need to take for the decision to change at a threshold such as GBP 25,000 or GBP 35,000 per QALY gained.

Worked example with one trial result and three risk groups

All numbers in this example are illustrative. A trial randomises 800 people to usual care and 800 to a preventive treatment, and over five years 80 people in the usual care arm and 60 in the treatment arm have a single serious event. The treatment costs GBP 800 per person, each event costs GBP 6,000 to treat and causes a loss of 1.2 QALYs, and discounting is ignored.

1. Risk ratio and RRR. The control event rate is 80/800 = 0.10 and the experimental event rate is 60/800 = 0.075. The risk ratio is 0.075/0.10 = 0.75 and the RRR is 1 − 0.75 = 0.25, or 25%.

2. Confidence interval. The standard error of the log risk ratio is $\sqrt{1/60 - 1/800 + 1/80 - 1/800} = \sqrt{0.02667} = 0.1633$. With ln 0.75 = −0.2877 and 1.96 × 0.1633 = 0.3201, the log limits are −0.6077 and 0.0324, giving a risk ratio interval of 0.545 to 1.033 and an RRR interval of minus 3.3% to 45.5%, which includes no effect.

3. Applying the RRR to three baseline risks. The model assumes the RRR of 0.25 holds at every baseline risk. Events avoided are $p_0 \times 0.25$, QALYs gained are 1.2 × ARR and incremental cost is 800 − 6,000 × ARR.

For the 0.30 group the event costs avoided are 6,000 × 0.075 = GBP 450, so the incremental cost is 800 − 450 = GBP 350 and the ICER is 350 / 0.09 = GBP 3,889.

4. Switching values for the RRR. The ICER equals a threshold when the value of the QALYs gained at that threshold matches the incremental cost.

$$\lambda \times 1.2\, p_0 r = 800 - 6{,}000\, p_0 r \quad \Rightarrow \quad r = \frac{800}{p_0 (1.2 \lambda + 6{,}000)}$$

where $\lambda$ is the threshold in GBP per QALY, $r$ is the RRR and $p_0$ is the baseline risk. At GBP 25,000 per QALY the term in brackets is 1.2 × 25,000 + 6,000 = 36,000, so the switching RRR is 800 / (0.30 × 36,000) = 0.074 for the high-risk group, 800 / (0.10 × 36,000) = 0.222 for the trial-like group and 800 / (0.04 × 36,000) = 0.556 for the low-risk group. At GBP 35,000 the values are 0.056, 0.167 and 0.417.

5. Interpretation. One RRR supports treatment comfortably in the high-risk group, lies just above the switching value in the trial-like group and fails in the low-risk group, where even the upper confidence limit of 45.5% falls short of the 55.6% needed at GBP 25,000. The confidence interval spans at least one switching value in every group, but the trial-like group's point estimate sits closest to its switching values, so that decision is the most sensitive to uncertainty in the relative effect. A headline of "25% fewer events" fits all three groups and cannot say which of them to fund.

Relative risk reduction beside related measures

The absolute risk reduction and the number needed to treat are the absolute counterparts of the RRR, with the linking arithmetic set out on their own pages. The complements of the odds ratio and the hazard ratio approximate the RRR only when events are rare.

Misreadings of relative risk reduction

A Cochrane review by Akl and colleagues, covering 35 studies, found the RRR perceived as larger and more persuasive than the same result given as an ARR or an NNT, and judged it uncertain whether presenting the RRR helps people make decisions consistent with their own values. A meta-analysis by Covey of 31 experiments found treatments evaluated more favourably when benefits were given in relative format. Item 26 of the CONSORT 2025 statement asks trial reports to present both absolute and relative effect sizes for binary outcomes.

1. Quoting an RRR without the baseline risk. The same 25% means 1 or 7.5 events avoided per 100 people in the example.
2. Assuming the RRR is constant without saying so. Deeks' evidence supports a constant relative effect on average, not for every treatment, and PMG36 (section 4.9.7) asks subgroup analyses to specify the scale on which any effect modification is defined.
3. Setting negative values to zero. A spreadsheet rule that replaces a negative RRR with zero hides a harmful effect and truncates the uncertainty the model should carry.

Sources

• Akl EA, Oxman AD, Herrin J, et al. Using alternative statistical formats for presenting risks and risk reductions. Cochrane Database of Systematic Reviews. 2011;(3):CD006776. https://doi.org/10.1002/14651858.CD006776.pub2
• Briggs AH, Weinstein MC, Fenwick EAL, Karnon J, Sculpher MJ, Paltiel AD. Model parameter estimation and uncertainty analysis: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force Working Group-6. Medical Decision Making. 2012;32(5):722-732. https://doi.org/10.1177/0272989X12458348
• Centers for Disease Control and Prevention. Principles of Epidemiology in Public Health Practice, 3rd ed. Lesson 3, Measures of Risk, Section 6, Measures of Public Health Impact. Last reviewed 18 May 2012. https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson3/section6.html
• Covey J. A meta-analysis of the effects of presenting treatment benefits in different formats. Medical Decision Making. 2007;27(5):638-654. https://doi.org/10.1177/0272989X07306783
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