2.4286
0.8873
0.355
1.211
4.8704
2.4286
0.8873
0.355
1.211
4.8704
The Odds Ratio Calculator computes the odds ratio (OR) and its 95% confidence interval from a 2×2 contingency table. The odds ratio is a fundamental measure of association in epidemiology, clinical research, and case-control studies.
An odds ratio of 1 indicates no association between exposure and outcome. An OR greater than 1 suggests the exposure increases the odds of the outcome, while an OR less than 1 suggests a protective effect. The confidence interval tells you whether this association is statistically significant — if it does not contain 1, the association is significant at the 5% level.
The 2×2 contingency table is structured as:
| Event | No Event | |
|---|---|---|
| Exposed | a | b |
| Unexposed | c | d |
Odds Ratio:
$$OR = \frac{a \cdot d}{b \cdot c}$$
Log Odds Ratio:
$$\ln(OR) = \ln\left(\frac{a \cdot d}{b \cdot c}\right)$$
The log-odds ratio is approximately normally distributed, which allows construction of confidence intervals.
Standard Error of ln(OR):
$$SE(\ln OR) = \sqrt{\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d}}$$
95% Confidence Interval:
$$CI = \exp\left(\ln(OR) \pm 1.96 \cdot SE(\ln OR)\right)$$
This Woolf method (also called the log method) is the standard approach for large-sample odds ratio inference. It requires all four cells to be non-zero; adding 0.5 to each cell (Haldane correction) can handle zero cells if needed.
OR = 1: No association between exposure and outcome.
OR > 1: The exposure is associated with higher odds of the event. For example, OR = 2.43 means exposed individuals have 2.43 times the odds of the event compared to unexposed.
OR < 1: The exposure appears protective. OR = 0.5 means the exposed group has half the odds.
The 95% CI provides a range of plausible values. If the interval excludes 1, the result is statistically significant at α = 0.05.
Inputs
Results
OR = 8.5: smokers have 8.5 times the odds of lung cancer vs. non-smokers. CI excludes 1 — highly significant.
Inputs
Results
OR = 0.19: vaccinated group has 81% lower odds of infection. Strong protective effect.
The odds ratio compares odds (event / no-event ratios), while relative risk compares probabilities (event / total ratios). For rare events (< 10% prevalence), OR approximates RR. For common events, OR tends to exaggerate the association relative to RR.
Odds ratios are the natural measure for case-control studies (where you cannot directly compute incidence rates), logistic regression output, and meta-analyses. Relative risk is preferred in cohort studies and randomized trials when possible.
A zero cell makes the odds ratio 0 or infinity and the log-OR undefined. Apply the Haldane correction by adding 0.5 to each cell. This calculator requires all values ≥ 1.
The CI is symmetric on the log scale but asymmetric on the odds scale. This is appropriate because the odds ratio is bounded below by 0 and unbounded above, similar to a log-normal distribution.
A wide CI indicates low precision, typically due to small sample sizes. While the point estimate (OR) may suggest an association, the wide CI means you cannot rule out a range of effect sizes, including possibly no effect.
No. Matched studies require McNemar's test or conditional logistic regression. The cells in matched studies represent concordant/discordant pairs, not independent counts.
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