# Absolute risk reduction

In epidemiology, the absolute risk reduction, risk difference or excess risk is the change in risk of a given activity or treatment in relation to a control activity or treatment.[1] It is the inverse of the number needed to treat.[2]

In general, absolute risk reduction is the difference between the control group’s event rate (CER) and the experimental group’s event rate (EER). The difference is usually calculated with respect to two treatments A and B, with A typically a drug and B a placebo. For example, A could be a 5-year treatment with a hypothetical drug, and B is treatment with placebo, i.e. no treatment. A defined endpoint has to be specified, such as a survival or a response rate. For example: the appearance of lung cancer in a 5 year period. If the probabilities pA and pB of this endpoint under treatments A and B, respectively, are known, then the absolute risk reduction is computed as (pBpA).

The inverse of the absolute risk reduction, NNT, is an important measure in pharmacoeconomics. If a clinical endpoint is devastating enough (e.g. death, heart attack), drugs with a low absolute risk reduction may still be indicated in particular situations. If the endpoint is minor, health insurers may decline to reimburse drugs with a low absolute risk reduction.

## Presenting results

Consider a hypothetical drug which reduces the relative risk of colon cancer by 50% over five years. Even without the drug, colon cancer is fairly rare, maybe 1 in 3,000 in every five-year period. The rate of colon cancer for a five-year treatment with the drug is therefore 1/6,000, as by treating 6,000 people with the drug, one can expect to reduce the number of colon cancer cases from 2 to 1.

The raw calculation of absolute risk reduction is a probability (0.003 fewer cases per person, using the colon cancer example above). Authors such as Ben Goldacre believe that this information is best presented as a natural number in the context of the baseline risk ("reduces 2 cases of colon cancer to 1 case if you treat 6,000 people for five years").[3] Natural numbers, which are used in the number needed to treat approach, are easily understood by non-experts.

## Worked example

Example 1: risk reductionExample 2: risk increase
Experimental group (E)Control group (C)Total(E)(C)Total
Events (E)EE = 15CE = 100115EE = 75CE = 100175
Non-events (N)EN = 135CN = 150285EN = 75CN = 150225
Total subjects (S)ES = EE + EN = 150CS = CE + CN = 250400ES = 150CS = 250400
Event rate (ER)EER = EE / ES = 0.1, or 10%CER = CE / CS = 0.4, or 40%EER = 0.5 (50%)CER = 0.4 (40%)
EquationVariableAbbr.Example 1Example 2
EER − CER< 0: absolute risk reductionARR(−)0.3, or (−)30%N/A
> 0: absolute risk increaseARIN/A0.1, or 10%
(EER − CER) / CER< 0: relative risk reductionRRR(−)0.75, or (−)75%N/A
> 0: relative risk increaseRRIN/A0.25, or 25%
1 / (EER − CER)< 0: number needed to treatNNT(−)3.33N/A
> 0: number needed to harmNNHN/A10
EER / CERrelative riskRR0.251.25
(EE / EN) / (CE / CN)odds ratioOR0.1671.5
EER − CERattributable riskAR(−)0.30, or (−)30%0.1, or 10%
(RR − 1) / RRattributable risk percentARPN/A20%
1 − RR (or 1 − OR)preventive fractionPF0.75, or 75%N/A