# Relative risk, Risk difference and Odds ratio

When the data to be analyzed consist of counts in a cross-classification of two groups (or conditions) and two outcomes, the data can be represented in a fourfold table as follows:

Group 1 | Group 2 | Total | |
---|---|---|---|

Number with positive outcome | a | c | a+c |

Number with negative outcome | b | d | b+d |

Total | a+b | c+d | a+b+c+d |

Several statistics can be calculated such as relative risk and risk difference, relevant in prospective studies, and odds ratio, relevant in retrospective case controls studies.

## How to calculate Relative Risk

The relative risk (RR), its standard error and 95% confidence interval are calculated as follows (Altman, 1991).

The relative risk or risk ratio is given by

with the standard error of the log relative risk being

and 95% confidence interval

## Risk difference

The risk difference (RD) and its 95% confidence interval are calculated according to Newcombe & Altman (2000)

The recommended method for the calculation of the risk difference, which is a difference between proportions, requires the calculation of the confidence intervals of the two proportions separately. MedCalc calculates exact binomial confidence intervals for proportions (Armitage et al., 2002). With l_{1} to u_{1} being the 95% CI of the first proportion p_{1} and l_{2} to u_{2} being the 95% CI of the second proportion p_{2}, the 95% confidence interval for the difference is given by

In the context of meta-analysis, the standard error and 95% confidence interval are calculated according to Deeks & Higgins (2010), where the standard error is defined as

and 95% confidence interval

## How to calculate Odds Ratio

The odds ratio (OR), its standard error and 95% confidence interval are calculated as follows (Altman, 1991).

The formula for odds ratio is:

with the standard error of the log odds ratio being

and 95% confidence interval

## Notes

Where zeros cause problems with computation of effects or standard errors, 0.5 is added to all cells (a, b, c, d) (Pagano & Gauvreau, 2000; Deeks & Higgins, 2010).

In meta-analysis for relative risk and odds ratio, studies where a=c=0 or b=d=0 are excluded from the analysis (Higgins & Thomas, 2021).

## Literature

- Altman DG (1991) Practical statistics for medical research. London: Chapman and Hall.
- Armitage P, Berry G, Matthews JNS (2002) Statistical methods in medical research. 4
^{th}ed. Blackwell Science. - Deeks JJ, Higgins JPT (2010) Statistical algorithms in Review Manager 5. Retrieved from https://training.cochrane.org/
- Higgins JPT, Thomas J (editors) (2021) Cochrane Handbook for Systematic Reviews of Interventions Version 6.2. The Cochrane Collaboration, 2021. Available from https://training.cochrane.org
- Newcombe RG, Altman DG (2000) Proportions and their differences. In: Altman DG, Machin D, Bryant TN, Gardner MJ (Eds) Statistics with confidence, 2
^{nd}ed. BMJ Books, 2000. - Pagano M, Gauvreau K (2000) Principles of biostatistics. 2
^{nd}ed. Belmont, CA: Brooks/Cole.