# Risk difference

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 ${\displaystyle I_{e}-I_{u}}$, where ${\displaystyle I_{e}}$is the incidence in the exposed group, and ${\displaystyle I_{u}}$ 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 ${\displaystyle I_{e}-I_{u}}$. Equivalently, If the risk of an outcome is decreased by the exposure, the term absolute risk reduction (ARR) is used, and computed as ${\displaystyle I_{u}-I_{e}}$.[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

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]

## Inference

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

Group
Experimental (E)Control (C)
Events (E)EECE
Non-events (N)ENCN

The point estimate of the risk difference is

${\displaystyle RD={\frac {EE}{EE+EN}}-{\frac {CE}{CE+CN}}.}$

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

${\displaystyle SE(RD)={\sqrt {{\frac {EE\cdot EN}{(EE+EN)^{3}}}+{\frac {CE\cdot CN}{(CE+CN)^{3}}}}}.}$

The ${\displaystyle 1-\alpha }$ confidence interval for the RD is then

${\displaystyle CI_{1-\alpha }(RD)=RD\pm SE(RD)\cdot z_{\alpha },}$

where ${\displaystyle z_{\alpha }}$ is the standard score for the chosen level of significance[2].

## Numerical examples

### Risk reduction

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%
EquationVariableAbbr.Value
CER - EERabsolute risk reduction ARR0.3, or 30%
(CER - EER) / CER relative risk reduction RRR 0.75, or 75%
1 / (CER EER)number needed to treatNNT3.33
EER / CERrisk ratioRR0.25
(EE / EN) / (CE / CN)odds ratioOR0.167
(CER - EER) / CERpreventable fraction among the unexposedPFu0.75

### Risk increase

Example of risk increase
Experimental group (E) Control group (C) Total
Events (E) EE = 75 CE = 100 115
Non-events (N) EN = 75 CN = 150 285
Total subjects (S) ES = EE + EN = 150 CS = CE + CN = 250 400
Event rate (ER) EER = EE / ES = 0.5, or 50% CER = CE / CS = 0.4, or 40%
EquationVariableAbbr.Value
EER CER absolute risk increaseARI0.1, or 10%
(EER CER) / CER relative risk increaseRRI0.25, or 25%
1 / (EER CER) number needed to harmNNH10
EER / CERrisk ratioRR1.25
(EE / EN) / (CE / CN)odds ratioOR1.5
(EER CER) / EERattributable fraction among the exposedAFe0.2

## References

1. "Dictionary of Epidemiology - Oxford Reference". doi:10.1093/acref/9780199976720.001.0001. Retrieved 2018-05-09.
2. J., Rothman, Kenneth (2012). Epidemiology : an introduction (2nd ed.). New York, NY: Oxford University Press. pp. 66, 160, 167. ISBN 9780199754557. OCLC 750986180.
3. Moher D, Hopewell S, Schulz KF, Montori V, Gøtzsche PC, Devereaux PJ, Elbourne D, Egger M, Altman DG (March 2010). "CONSORT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials". BMJ. 340: c869. doi:10.1136/bmj.c869. PMC 2844943. 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.