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 1Group 2Total
Number with positive outcomeaca+c
Number with negative outcomebdb+d
Totala+bc+da+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.

Relative risk

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

The relative risk or risk ratio is given by

Formula for relative risk

with the standard error of the log relative risk being

Formula for standard error of relative risk

and 95% confidence interval

Formula for confidence interval of relative risk

Risk difference

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

Formula for risk difference

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 seperately. MedCalc calculates exact binomial confidence intervals for proportions (Armitage et al., 2002). With l1 to u1 being the 95% CI of the first proportion p1 and l2 to u2 being the 95% CI of the second proportion p2, the 95% confidence interval for the difference is given by

Formula for confidence interval of risk difference

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

Formula for standard error of risk difference (meta-analysis context)

and 95% confidence interval

Formula for confidence interval of risk difference (meta-analysis context)

Odds ratio

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

The odds ratio is given by

Formula for odds ratio

with the standard error of the log odds ratio being

Formula for standard error of odds ratio

and 95% confidence interval

Formula for confidence interval of odds ratio

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 & Green, 2011).

Literature

  • Altman DG (1991) Practical statistics for medical research. London: Chapman and Hall. Book info
  • Armitage P, Berry G, Matthews JNS (2002) Statistical methods in medical research. 4th ed. Blackwell Science. Book info
  • Deeks JJ, Higgins JPT (2010) Statistical algorithms in Review Manager 5. Retrieved from http://ims.cochrane.org/
  • Higgins JPT, Green S (editors). Cochrane Handbook for Systematic Reviews of Interventions Version 5.1.0 [updated March 2011]. The Cochrane Collaboration, 2011. Available from www.cochrane-handbook.org
  • Newcombe RG, Altman DG (2000) Proportions and their differences. In: Altman DG, Machin D, Bryant TN, Gardner MJ (Eds) Statistics with confidence, 2nd ed. BMJ Books, 2000.
  • Pagano M, Gauvreau K (2000) Principles of biostatistics. 2nd ed. Belmont, CA: Brooks/Cole.