Risk difference

The adverse outcome (black) risk difference between the group exposed to the treatment (left) and the group unexposed to the treatment (right) is −0.25 (RD = −0.25, ARR = 0.25).

The risk difference (RD), excess risk, or attributable risk[1] is the difference between the risk of an outcome in the exposed group and the unexposed group. It is computed as [math]\displaystyle{ I_e - I_u }[/math], where [math]\displaystyle{ I_e }[/math]is the incidence in the exposed group, and [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ I_u - I_e }[/math].[2][3]

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.[2]

## Contents

* 1 Usage in reporting
* 2 Inference
* 3 Bayesian interpretation
* 4 Numerical examples
* 4.1 Risk reduction
* 4.2 Risk increase
* 5 See also
* 6 References

## 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.[4] 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 (50%), while the absolute risk reduction is 0.0001 (0.01%). 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.[5]

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.[6]

## Inference

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

| Group

|

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

The point estimate of the risk difference is

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

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

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

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

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

where [math]\displaystyle{ z_\alpha }[/math] is the standard score for the chosen level of significance[3]

## Bayesian interpretation

We could assume a disease noted by [math]\displaystyle{ D }[/math], and no disease noted by [math]\displaystyle{ \neg D }[/math], exposure noted by [math]\displaystyle{ E }[/math], and no exposure noted by [math]\displaystyle{ \neg E }[/math]. The risk difference can be written as

[math]\displaystyle{ RD = P(D\mid E)-P(D\mid \neg E). }[/math]

## 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%  |
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

| Example of risk increase

|

Experimental group (E)  | Control group (C)  | Total
Events (E)  | EE = 75  | CE = 100  | 175
Non-events (N)  | EN = 75  | CN = 150  | 225
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%  |
Equation | Variable | Abbr. | Value

|
|
|

EER − CER  | absolute risk increase | ARI | 0.1, or 10%
(EER − CER) / CER  | relative risk increase | RRI | 0.25, or 25%
1 / (EER − CER)  | number needed to harm | NNH | 10
EER / CER | risk ratio | RR | 1.25
(EE / EN) / (CE / CN) | odds ratio | OR | 1.5
(EER − CER) / EER | attributable fraction among the exposed | AFe | 0.2

## See also

* Population Impact Measures
* Relative risk reduction

## References

1. ↑ Dictionary of Epidemiology (6th ed.). Oxford University Press. 2014. pp. 14. doi:10.1093/acref/9780199976720.001.0001. ISBN 978-0-19-939006-9. http://www.oxfordreference.com/view/10.1093/acref/9780199976720.001.0001/acref-9780199976720.
2. ↑ 2.0 2.1 Porta, Miquel, ed (2014). "Dictionary of Epidemiology - Oxford Reference" (in en). Oxford University Press. doi:10.1093/acref/9780199976720.001.0001. http://www.oxfordreference.com/view/10.1093/acref/9780199976720.001.0001/acref-9780199976720.
3. ↑ 3.0 3.1 J., Rothman, Kenneth (2012). Epidemiology : an introduction (2nd ed.). New York, NY: Oxford University Press. pp. 66, 160, 167. ISBN 9780199754557. OCLC 750986180.
4. ↑ "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.
5. ↑ Stegenga, Jacob (2015). "Measuring Effectiveness". Studies in History and Philosophy of Biological and Biomedical Sciences 54: 62–71. doi:10.1016/j.shpsc.2015.06.003. PMID 26199055. https://www.academia.edu/16420844.
6. ↑ Ben Goldacre (2008). Bad Science. New York: Fourth Estate. pp. 239–260. ISBN 978-0-00-724019-7.

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Clinical research and experimental design

Overview|

* Clinical trial
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Controlled study
(EBM I to II-1)|

* Randomized controlled trial
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Measures|

| Occurrence|

Incidence, Cumulative incidence, Prevalence, Point prevalence, Period prevalence

|

Association|

Risk difference, Number needed to treat, Number needed to harm, Risk ratio, Relative risk reduction, Odds ratio, Hazard ratio

Population impact|

Attributable fraction among the exposed, Attributable fraction for the population, Preventable fraction among the unexposed, Preventable fraction for the population

Other|

Clinical endpoint, Virulence, Infectivity, Mortality rate, Morbidity, Case fatality rate, Specificity and sensitivity, Likelihood-ratios, Pre- and post-test probability

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Interpretation of results|

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* Null result

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