Medicine:Risk difference

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The risk difference (RD), excess risk, or attributable risk is the difference between the risk of an outcome in the exposed group and the unexposed group. It is computed as [math]I_e - I_u[/math], where [math]I_e[/math]is the incidence in the exposed group, and [math]I_u[/math] is the incidence in the unexposed group. If the risk of an outcome is increased by the exposure, the term absolute risk increase (ARI) is used, and computed as [math]I_e - I_u[/math]. Equivalently, If the risk of an outcome is decreased by the exposure, the term absolute risk reduction (ARR) is used, and computed as [math]I_u - I_e[/math].[1][2]

The inverse of the absolute risk reduction is the number needed to treat, and the inverse of the absolute risk increase is the number needed to harm.[1]

Usage in reporting[edit]

It is recommended to use absolute measurements, such as risk difference, alongside the relative measurements, when presenting the results of randomized controlled trials.[3] Their utility can be illustrated by the following example of a hypothetical drug which reduces the risk of colon cancer from 1 case in 5000 to 1 case in 10,000 over one year. The relative risk reduction is 0.5, while the absolute risk reduction is 0.0001. The absolute risk reduction reflects the low probability of getting colon cancer in the first place, while reporting only relative risk reduction, would run into risk of readers exaggerating the effectiveness of the drug.[4]

Authors such as Ben Goldacre believe that the risk difference is best presented as a natural number - drug reduces 2 cases of colon cancer to 1 case if you treat 10,000 people. Natural numbers, which are used in the number needed to treat approach, are easily understood by non-experts.[5]


Risk difference can be estimated from a 2x2 contingency table:

Experimental (E) Control (C)
Events (E) EE CE
Non-events (N) EN CN

The point estimate of the risk difference is

[math]RD = \frac{EE}{EE + EN} - \frac{CE}{CE + CN}.[/math]

The sampling distribution of RD is approximately normal, with standard error

[math]SE(RD) = \sqrt{\frac{EE\cdot EN}{(EE + EN)^3} + \frac{CE\cdot CN}{(CE + CN)^3}}.[/math]

The [math]1 - \alpha[/math] confidence interval for the RD is then

[math]CI_{1 - \alpha}(RD) = RD\pm SE(RD)\cdot z_\alpha,[/math]

where [math]z_\alpha[/math] is the standard score for the chosen level of significance[2]

Numerical examples[edit]

Risk reduction[edit]

  Example of risk reduction
Experimental group (E) Control group (C) Total
Events (E) EE = 15 CE = 100 115
Non-events (N) EN = 135 CN = 150 285
Total subjects (S) ES = EE + EN = 150 CS = CE + CN = 250 400
Event rate (ER) EER = EE / ES = 0.1, or 10% CER = CE / CS = 0.4, or 40%
Equation Variable Abbr. Value
CER - EER absolute risk reduction ARR 0.3, or 30%
(CER - EER) / CER relative risk reduction RRR 0.75, or 75%
1 / (CER − EER) number needed to treat NNT 3.33
EER / CER risk ratio RR 0.25
(EE / EN) / (CE / CN) odds ratio OR 0.167
(CER - EER) / CER preventable fraction among the unexposed PFu 0.75

Risk increase[edit]

Template:RCT risk increase example

See also[edit]


  1. 1.0 1.1 "Dictionary of Epidemiology - Oxford Reference" (in en). doi:10.1093/acref/9780199976720.001.0001. 
  2. 2.0 2.1 J., Rothman, Kenneth (2012). Epidemiology : an introduction (2nd ed.). New York, NY: Oxford University Press. pp. 66, 160, 167. ISBN 9780199754557. OCLC 750986180. 
  3. "CONSORT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials". BMJ 340: c869. March 2010. doi:10.1136/bmj.c869. PMID 20332511. 
  4. Stegenga, Jacob (2015). "Measuring Effectiveness". Studies in History and Philosophy of Biological and Biomedical Sciences 54: 62–71. 
  5. Ben Goldacre (2008). Bad Science. New York: Fourth Estate. pp. 239–260. ISBN 0-00-724019-8. difference was the original source. Read more.