Signature
ARR = p_C * (1 - RR); RRR = 1 - RR
| Inputs | Definition | Unit |
|---|---|---|
p_C | Baseline risk of the event without the intervention in the population of interest, over the same period as RR | probability from 0 to 1 |
RR | Ratio of the intervention risk to the comparator risk | ratio, no unit |
ARR | Events avoided per person treated at the assumed comparator risk | probability difference, over the period of the relative risk |
|---|---|---|
RRR | Proportional reduction in risk with the intervention, 1 minus RR | proportion, no unit |
Function
Absolute risk reduction function
Maps the risks of an adverse event with the comparator and with the intervention, over the same period, to their difference, oriented so that a benefit is positive. The difference converts a relative treatment effect into events avoided per person treated, the quantity that costs and QALYs in an economic evaluation are built from.
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Implementations
Excel
Absolute and relative risk reduction in two cells
With the assumed comparator risk in BaselineRisk and the relative risk in RelativeRisk, the two formulas return the ARR and the relative risk reduction.
=BaselineRisk*(1-RelativeRisk); =1-RelativeRisk
Assumptions
Relative risk constant across baseline risk
The relative risk estimated in the trials holds at the baseline risk of the target population. The assumption is stated and, where evidence allows, tested across risk groups.
Baseline risk and relative risk over the same period
p_C and RR refer to the same outcome and time horizon. The Cochrane Handbook recommends computing the ARR for a range of assumed comparator risks.
Worked examples
High-risk group with a relative risk of 0.75
A relative risk of 0.75 applied to a five-year baseline risk of 0.20 gives an ARR of 0.05 and a relative risk reduction of 0.25, as in the article's high-risk group. The figures are illustrative.
p_C = 0.20; RR = 0.75; ARR = 0.05; RRR = 0.25
Cochrane Handbook example with a relative risk of 0.92
A risk ratio of 0.92 and an assumed comparator risk of 0.3 give an ARR of 0.024, or 24 fewer events per 1,000, as in section 15.4.4.2 of the Cochrane Handbook.
p_C = 0.3; RR = 0.92; ARR = 0.024; RRR = 0.08
Common errors
Using the trial baseline for a different population
An ARR computed at the trial's comparator risk describes the trial population. Patients at a quarter of that risk gain a quarter of the absolute benefit under a constant relative risk, so the NNT and the cost per event avoided change accordingly.
Treating an odds ratio as a risk ratio
An odds ratio of 0.73 at a comparator risk of 0.3 implies an intervention risk of about 0.238 and an ARR of about 0.0617, as in section 15.4.4.3 of the Cochrane Handbook. Using 0.73 as if it were RR gives 0.081, about 30 per cent too high. The odds conversion is set out on the Network Meta-Analysis page.
Substituting a hazard ratio for the relative risk
A hazard ratio is not a ratio of risks. Using it in place of RR is only an approximation, and a poor one when event risks over the horizon are high; the treatment risk comes from applying the hazard ratio to a baseline survival curve, as set out on the Hazard Ratio page.
Sources
Absolute effects from a relative effect and an assumed comparator risk
Schünemann HJ, Vist GE, Higgins JPT, Santesso N, Deeks JJ, Glasziou P, Akl EA, Guyatt GH. Chapter 15: Interpreting results and drawing conclusions (last updated August 2023). In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. Section 15.4.1 (relative risk reduction from a risk ratio), section 15.4.4.2 (computing risk differences or NNT from a risk ratio, with the example RR 0.92 and assumed comparator risk 0.3) and section 15.4.4.3 (from an odds ratio, with the example OR 0.73).
Treated-group probability from a relative risk
Gidwani R, Russell LB. Estimating transition probabilities from published evidence: a tutorial for decision modelers. PharmacoEconomics. 2020;38(11):1153-1164. Section on using relative risks to derive transition probabilities for the treated group (the treated probability equals the relative risk times the untreated probability).
Canonical Identity
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