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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
$$ \begin{align}
OR & = \frac {a/b} {c/d} \\
& = \frac {a \times d } { b \times c}
\end{align}$$
$$ \begin{align}
OR & = \frac {a/b} {c/d} \\
& = \frac {a \times d } { b \times c}
\end{align}$$
with the standard error of the log odds ratio being
$$ \operatorname{SE} \left \{ \operatorname{ln}\left(OR\right) \right \} = \sqrt { \frac {1}{a} + \frac {1}{b} + \frac {1}{c} + \frac {1}{d} } $$
$$ \operatorname{SE} \left \{ \operatorname{ln}\left(OR\right) \right \} = \sqrt { \frac {1}{a} + \frac {1}{b} + \frac {1}{c} + \frac {1}{d} } $$
and 95% confidence interval
$$ \operatorname{95\%\text{ } CI} = \operatorname{exp} \Big( \text{ } \operatorname{ln}\left(OR\right) - 1.96 \times \operatorname{SE} \left \{ \operatorname{ln}\left(OR\right) \right \} \text{ }\Big) \quad \text{ to }\quad \operatorname{exp} \Big(\text{ } \operatorname{ln}\left(OR\right) + 1.96 \times \operatorname{SE} \left \{ \operatorname{ln}\left(OR\right) \right \} \text{ }\Big)$$
$$ \operatorname{95\%\text{ } CI} = \operatorname{exp} \Big( \text{ } \operatorname{ln}\left(OR\right) - 1.96 \times \operatorname{SE} \left \{ \operatorname{ln}\left(OR\right) \right \} \text{ }\Big) \quad \text{ to }\quad \operatorname{exp} \Big(\text{ } \operatorname{ln}\left(OR\right) + 1.96 \times \operatorname{SE} \left \{ \operatorname{ln}\left(OR\right) \right \} \text{ }\Big)$$
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 non-parametric statistical procedures. 3rd 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 23.0.5; accessed October 9, 2024)
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