Absolute Risk Reduction

Absolute Risk Reduction Formula, Number Needed to Treat and Economic Use

Absolute risk reduction (ARR) turns a treatment effect into the number of events actually prevented. A relative effect says a treatment cuts risk by a third; the absolute reduction says how many people out of a hundred avoid the event, and only an absolute figure can be carried forward into events, costs and QALYs. The formula is simplest in event-rate notation, where the same two rates also give the relative risk reduction and the number needed to treat, while a decision model usually reaches ARR another way, from a pooled relative effect and a baseline risk, and that route rests on assumptions that change the answer.

The ARR formula in event-rate notation

Evidence-based medicine texts write ARR in terms of two event rates. The control event rate is the proportion of people in the comparator group who have the event during follow-up, and the experimental event rate is the same proportion in the intervention group. Each is an absolute risk over a stated period.

$$\text{ARR} = \text{CER} - \text{EER}$$

where CER is the control event rate and EER is the experimental event rate, both measured over the same follow-up period. For an adverse outcome a positive ARR means the intervention lowers risk, and a negative value means it raises risk. With counts from a trial the two rates come straight from the arms.

$$\text{CER} = \frac{a}{n_C}, \qquad \text{EER} = \frac{b}{n_I}$$

where $a$ is the number of events among the $n_C$ people in the control arm and $b$ is the number of events among the $n_I$ people in the intervention arm. The Cochrane Handbook computes the risk difference the other way round, as the intervention risk minus the comparator risk, so for an adverse outcome ARR equals minus the Cochrane risk difference.

Relative risk reduction and number needed to treat from the same two rates

The same two event rates give the relative measures and the number needed to treat. Laupacis, Sackett and Roberts set these measures side by side in 1988 to show how differently one trial result can look.

$$\text{RR} = \frac{\text{EER}}{\text{CER}}, \qquad \text{RRR} = \frac{\text{CER} - \text{EER}}{\text{CER}} = 1 - \text{RR}, \qquad \text{NNT} = \frac{1}{\text{ARR}}$$

where RR is the relative risk, RRR is the relative risk reduction and NNT is the number needed to treat, the average number of people who must receive the intervention for one more person to avoid the event over the study period. Altman gives NNT as 1/ARR, or 100/ARR when ARR is expressed as a percentage, and NNT is usually rounded up to the next whole number.

Worked example from a two-arm trial

The figures in this example are illustrative. A trial randomises 1,000 people to usual care and 1,000 to a new treatment, and over three years 150 people in the usual care arm and 100 in the treatment arm have a stroke.

$$\text{CER} = \frac{150}{1{,}000} = 0.15, \qquad \text{EER} = \frac{100}{1{,}000} = 0.10, \qquad \text{ARR} = 0.15 - 0.10 = 0.05$$

where 0.05 means five strokes avoided per hundred people treated for three years. The relative measures follow from the same rates.

$$\text{RR} = \frac{0.10}{0.15} = 0.667, \qquad \text{RRR} = 1 - 0.667 = 0.333, \qquad \text{NNT} = \frac{1}{0.05} = 20$$

where the RRR of 0.333 is the one-third reduction a press release would quote, and the NNT of 20 says that 20 people need three years of treatment for one stroke to be avoided. A large-sample 95% confidence interval for ARR uses the standard error of a difference in proportions.

$$SE = \sqrt{\frac{0.15 \times 0.85}{1{,}000} + \frac{0.10 \times 0.90}{1{,}000}} = 0.0147, \qquad 0.05 \pm 1.96 \times 0.0147 = 0.021 \text{ to } 0.079$$

where 1.96 is the normal multiplier for 95% coverage. Taking reciprocals of the limits, as Altman describes, gives a 95% interval for NNT of about 13 to 47. If the treatment costs £500 more per person over the three years, before any savings from strokes avoided, the extra cost per stroke avoided is the extra cost divided by ARR.

$$\frac{500}{0.05} = 10{,}000$$

where the result is £10,000 per stroke avoided, which equals the NNT of 20 multiplied by £500. A relative reduction of one third gives no figure of this kind on its own, because the cost per event avoided depends on the absolute number of events the treatment prevents.

Calculating ARR from a relative effect for a decision model

Decision models commonly start from a pooled relative effect rather than one trial's two arms. The Cochrane Handbook notes that relative effect measures are, on average, more consistent across studies than absolute ones, and advises against meta-analysing risk differences unless there is a clear reason to expect them to be consistent. It notes that an absolute effect obtained from a pooled relative effect needs an assumed comparator risk, and recommends doing this for a range of comparator risks.

$$\text{EER} = \text{CER} \times \text{RR} \quad \Rightarrow \quad \text{ARR} = \text{CER} \times (1 - \text{RR})$$

where CER is now the baseline risk of the population the decision covers and RR is the pooled relative risk. For a fixed relative risk, ARR rises in direct proportion to baseline risk. With an illustrative RR of 0.75, a high-risk group with a five-year baseline risk of 0.20 has an ARR of 0.20 × 0.25 = 0.05 and an NNT of 20, while a low-risk group with a baseline risk of 0.04 has an ARR of 0.04 × 0.25 = 0.01 and an NNT of 100. At the same extra cost of £500 per person, the cost per event avoided is £10,000 in the high-risk group and £50,000 in the low-risk group, a fivefold gap from one relative effect.

What the calculation assumes

Deriving ARR from a relative effect assumes that the relative effect holds at the baseline risk of the target population. McAlister notes that the relative efficacy of drug therapies is often similar across patient subgroups at different risk, which supports the assumption in many settings, but it remains an assumption that should be stated and, where evidence allows, tested.

The choice of relative measure matters too. The Cochrane Handbook notes that where the assumed comparator risk differs from the typical comparator risk in the trials, predictions of absolute benefit differ according to which summary statistic is used, so an ARR derived from an odds ratio will not match one derived from a risk ratio. The source of the baseline risk, whether trial control arms or population and registry data, and the time horizon over which risks are measured are further choices that change the answer.

From events avoided to cost-effectiveness

In a simple decision tree with a single, non-recurring event over a fixed horizon, the events avoided per patient equal the ARR, and the downstream costs saved and QALYs gained follow from it. In a Markov model or other multi-cycle model, events avoided accumulate over cycles and can be affected by competing and recurrent events, but the same logic applies, and uncertainty in baseline risk passes directly into the incremental results of a cost-effectiveness analysis.

Reporting standards reflect the need for both views. Item 26 of the CONSORT 2025 statement, like item 17b of CONSORT 2010 before it, recommends that for binary outcomes both absolute and relative effect sizes are presented, because neither alone gives a complete picture of the effect and its implications. The CONSORT 2010 explanation adds that the risk difference is less generalisable to other populations than the relative risk, since it depends on baseline risk, which tends to vary across populations.

Risk difference, harm and hazard ratios

ARR is the absolute counterpart of the relative risk reduction, which divides the same difference by the control event rate. The risk difference has the same magnitude but a fixed order of subtraction, intervention minus comparator in the Cochrane Handbook, so a beneficial effect on an adverse outcome gives a negative risk difference and a positive ARR. When the intervention increases the risk of an adverse event, the corresponding measure is an absolute risk increase, and its reciprocal is the number needed to harm. Altman shows that when the confidence interval for ARR includes zero, the interval for NNT passes through infinity, and proposes labelling its two parts as the number needed to treat for benefit (NNTB) and for harm (NNTH).

ARR applies to binary outcomes measured as risks over a fixed period. When an effect is reported as a hazard ratio, an ARR refers to a chosen time point and is obtained either from both arms' survival estimates at that time or by applying the hazard ratio to a baseline survival curve, which assumes the hazard ratio does not vary over time. Substituting a hazard ratio for a relative risk in the formula above is only an approximation, and a poor one when event risks over the horizon are high.

Common misreadings of absolute risk reduction

ARR depends on time. McAlister notes that even when the relative risk reduction from a long-term therapy is constant over time, the number needed to treat falls as follow-up lengthens and events accrue, so an ARR or NNT quoted without its time horizon cannot be compared with one from a study of different length.

ARR also depends on the population. The Cochrane Handbook observes that the clinical importance of a risk difference may depend on the underlying risk, and an ARR from a trial describes the trial's baseline risk rather than the risk of the patients to whom a decision applies. Presentation affects perception as well. A Cochrane review by Akl and colleagues found that relative risk reductions tend to be perceived as larger, and to be more persuasive, than the equivalent absolute reductions or numbers needed to treat. Reading a percentage-point ARR as a percentage change, or comparing an ARR in one population with an RRR in another, are frequent errors of the same kind.

Sources

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• Laupacis A, Sackett DL, Roberts RS. An assessment of clinically useful measures of the consequences of treatment. New England Journal of Medicine. 1988;318(26):1728-1733.
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• Altman DG. Confidence intervals for the number needed to treat. BMJ. 1998;317(7168):1309-1312.
• McAlister FA. The "number needed to treat" turns 20, and continues to be used and misused. CMAJ. 2008;179(6):549-553.
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