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:

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

$$ RR = \frac {a/(a+b) } { c/(c+d) } $$

with the standard error of the log relative risk being

$$ \operatorname{SE} \left \{ \operatorname{ln}\left(RR\right) \right \} = \sqrt { \frac {1}{a} + \frac {1}{c} - \frac {1}{a+b} - \frac {1}{c+d} } $$

and 95% confidence interval

$$ \operatorname{95\%\text{ } CI} = \operatorname{exp} \Big( \text{ } \operatorname{ln}\left(RR\right) - 1.96 \times \operatorname{SE} \left \{ \operatorname{ln}\left(RR\right) \right \} \text{ } \Big) \quad \text{ to }\quad \operatorname{exp} \Big(\text{ } \operatorname{ln}\left(RR\right) + 1.96 \times \operatorname{SE} \left \{ \operatorname{ln}\left(RR\right) \right \} \text{ }\Big)$$

Risk difference

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

$$ RD = \frac {a} {a+b} - \frac {c} {c+d} $$

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 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

$$ \operatorname{95\%\text{ } CI} = RD - \sqrt { (p_1-l_1)^2 + (u_2-p_2)^2 } \quad \text{ to }\quad RD + \sqrt { (p_2-l_2)^2 + (u_1-p_1)^2} $$

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

$$ \operatorname{SE} \left \{ RD \right \} = \sqrt { \frac {a \times b}{ \left ( a+b \right )^3} + \frac {c\times d}{\left (c+d\right )^3} } $$

and 95% confidence interval

$$ \operatorname{95\%\text{ } CI} = RD - 1.96 \times \operatorname{SE} \left \{ RD \right \} \quad \text{ to }\quad RD + 1.96 \times \operatorname{SE} \left \{ RD \right \} $$

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:

$$ \begin{align} OR & = \frac {a/b} {c/d} \\ & = \frac {a \times d } { b \times c} \end{align}$$

with the standard error of the log odds ratio being

$$ \operatorname{SE} \left \{ \operatorname{ln}\left(OR\right) \right \} = \sqrt { \frac {1}{a} + \frac {1}{b} + \frac {1}{c} + \frac {1}{d} } $$

and 95% confidence interval

$$ \operatorname{95\%\text{ } CI} = \operatorname{exp} \Big( \text{ } \operatorname{ln}\left(OR\right) - 1.96 \times \operatorname{SE} \left \{ \operatorname{ln}\left(OR\right) \right \} \text{ }\Big) \quad \text{ to }\quad \operatorname{exp} \Big(\text{ } \operatorname{ln}\left(OR\right) + 1.96 \times \operatorname{SE} \left \{ \operatorname{ln}\left(OR\right) \right \} \text{ }\Big)$$

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.