MedCalc

# Free statistical calculators

a
c

b
d

## Computational notes

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

The odds ratio is given by

\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

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

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

Test of significance: the P-value is calculated according to Sheskin, 2004 (p. 542). A standard normal deviate (z-value) is calculated as ln(OR)/SE{ln(OR)}, and the P-value is the area of the normal distribution that falls outside ±z (see Values of the Normal distribution table).

## Literature

• Altman DG (1991) Practical statistics for medical research. London: Chapman and Hall.
• Altman DG, Deeks JJ, Sackett DL. Odds ratios should be avoided when events are common [letter]. BMJ 1998;317:1318.
• Deeks JJ, Higgins JPT (2010) Statistical algorithms in Review Manager 5. Retrieved from https://training.cochrane.org/
• Kirkwood BR, Sterne JAC (2003) Essential medical statistics, 2nd ed. Oxford: Blackwell Science.
• Pagano M, Gauvreau K (2000) Principles of biostatistics. 2nd ed. Belmont, CA: Brooks/Cole.
• Parshall MB (2013) Unpacking the 2 x 2 table. Heart & Lung 42:221-226.
• Sheskin DJ (2004) Handbook of parametric and nonparametric statistical procedures. 3rd ed. Boca Raton: Chapman & Hall /CRC.