Effect Size Measure
Odds Ratio Calculator
Quantify the association between exposure and outcome in a 2×2 table. Get the odds ratio plus a 95% confidence interval — the standard output for case-control and cohort studies.
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| Total | |||
|---|---|---|---|
| 100 | |||
| 100 | |||
| Total | 60 | 140 | 200 |
Odds ratio
2.67
95% CI 1.42 – 5.02 · Fisher's exact p = .003
χ²(1, N = 200) = 9.52, p = .002, V = .22
All statistics
| Pearson chi-square | χ² = 9.524, df = 1, p = .002 |
| G-test (likelihood ratio) | G = 9.663, df = 1, p = .002 |
| Chi-square with Yates' correction | χ² = 8.595, p = .003 |
| Fisher's exact test (two-sided) | p = .003 |
| Odds ratio | 2.667 (95% CI 1.417 – 5.020) |
| Cramér's V | 0.218 (small) |
| Phi coefficient (φ) | 0.218 |
| Contingency coefficient (C) | 0.213 |
| Lambda (symmetric / row|col / col|row) | 0.125 / 0.200 / 0.000 |
| Goodman–Kruskal gamma (γ) | 0.455 |
| Kendall's tau-b / tau-c | 0.218 / 0.200 |
| Somers' d (symmetric / row|col / col|row) | 0.217 / 0.200 / 0.238 |
| Theil's U (symmetric / row|col / col|row) | 0.037 / 0.035 / 0.040 |
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Reviewed by the crosstabs.com methods team · Last updated
What is an odds ratio?
The odds ratio (OR) measures the association between an exposure and an outcome by comparing the odds of the outcome in the exposed group to the odds in the unexposed group. An OR of 1 means the exposure has no effect; an OR greater than 1 indicates the exposure is associated with increased odds of the outcome; an OR less than 1 indicates decreased odds.
Unlike relative risk, the odds ratio can be computed from case-control studies where you cannot directly measure incidence. It is commonly used in medical and epidemiological research and is usually reported alongside a 95% confidence interval (CI) to convey uncertainty. If the CI excludes 1.0, the association is statistically significant at α = 0.05.
When the outcome is rare (<10% prevalence), the odds ratio closely approximates relative risk. In higher-prevalence settings the OR will be further from 1.0 than the true relative risk — a common source of over-interpretation.
When to use it
- Case-control studies where incidence data is unavailable.
- Logistic regression output (logistic regression coefficients exponentiated yield ORs).
- Any 2×2 contingency table where you want to express the strength of association as a ratio.
- Meta-analyses that pool multiple studies using the log odds ratio.
Formula
2×2 table layout
OR = (a × d) / (b × c)
95% CI = eln(OR) ± 1.96 × SE
SE = √(1/a + 1/b + 1/c + 1/d) — Woolf standard error of ln(OR)
If any cell is 0, a continuity correction of 0.5 is applied.
Interpreting the odds ratio
OR < 1
Protective
Exposure is associated with lower odds of the outcome. The further from 1, the stronger the protective effect.
OR = 1
No association
Exposure has no apparent effect on the odds of the outcome.
OR > 1
Risk factor
Exposure is associated with higher odds of the outcome. An OR of 2.0 means twice the odds.
The 95% confidence interval tells you the range of plausible values. A narrow CI indicates a precise estimate; a wide CI reflects uncertainty, usually due to small sample size. If the CI includes 1.0, the result is not statistically significant at α = 0.05.
Worked example
Worked example
Take the 2×2 table [[20, 10], [10, 40]], where rows are exposed/unexposed and columns are outcome yes/no:
a=20, b=10, c=10, d=40
OR = (a × d) / (b × c) = (20 × 40) / (10 × 10) = 800 / 100 = 8.0.
The odds of the outcome are 8 times higher in the exposed group. Note that an odds ratio is not the same as a risk ratio — see odds ratio vs risk ratio.
Frequently asked questions
- How do you interpret an odds ratio?
- An odds ratio above 1 means the outcome's odds are higher in the first group; below 1 means lower; exactly 1 means no association. For example, an odds ratio of 8 means the odds are eight times higher in the exposed group.
- What does an odds ratio of 1 mean?
- An odds ratio of 1 means the odds of the outcome are the same in both groups — there is no association between the exposure and the outcome.
- Is the odds ratio the same as the risk ratio?
- No. The risk ratio compares probabilities directly, while the odds ratio compares odds; the odds ratio only approximates the risk ratio when the outcome is rare.
References & further reading
- Cornfield, J. (1951). A method of estimating comparative rates from clinical data.
- Odds ratio — Wikipedia