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## Odds ratio calculator

## 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, 2
^{nd}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. 3
^{rd}ed. Boca Raton: Chapman & Hall /CRC.

## How to cite this page

- MedCalc Software Ltd. Odds ratio calculator. https://www.medcalc.org/calc/odds_ratio.php (Version 22.021; accessed February 28, 2024)

## See also

## Recommended book

## An Introduction to Medical Statistics

Martin Bland

Buy from Amazon

This textbook is intended for medical researchers and includes the design of clinical trials and epidemiological studies, data collection, summarizing and presenting data, probability, standard error, confidence intervals and significance tests, techniques of data analysis including multifactorial methods and the choice of statistical method, problems of medical measurement and diagnosis, vital statistics, and calculation of sample size.