What is the Odds Ratio?
The odds ratio (OR) is a measure of association between an exposure and an outcome. It represents the odds that an outcome will occur given a particular exposure, compared to the odds of the outcome occurring in the absence of that exposure.
2×2 Table Layout
| Disease+ | Disease- |
|---|
| Exposed | a | b |
| Unexposed | c | d |
Formula
OR = (a × d) / (b × c)
Interpreting Odds Ratio
- OR = 1: No association between exposure and outcome
- OR > 1: Exposure associated with higher odds of outcome
- OR < 1: Exposure associated with lower odds of outcome
95% Confidence Interval
The confidence interval tells you the precision of your estimate:
- If CI includes 1: association not statistically significant
- If CI excludes 1: association is statistically significant
- Narrower CI = more precise estimate
Example: Smoking and Lung Cancer
| Lung Cancer | No Cancer |
|---|
| Smoker | 70 | 30 |
| Non-smoker | 20 | 80 |
OR = (70×80)/(30×20) = 9.33
Interpretation: Smokers have 9.33 times the odds of lung cancer compared to non-smokers.
Odds Ratio vs Relative Risk
- Odds Ratio: Used in case-control studies
- Relative Risk: Used in cohort studies and RCTs
- OR approximates RR when outcome is rare (<10%)
Try the Calculator
CrossTabs.com calculates odds ratio with:
- 95% confidence interval (Woolf method)
- Relative risk for comparison
- Fisher's exact test p-value
- Chi-square with Yates' correction
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Step-by-Step Example
A study examines whether smoking is associated with lung disease:
| Lung Disease | No Disease |
|---|
| Smokers | 30 (a) | 70 (b) |
| Non-smokers | 10 (c) | 90 (d) |
Odds ratio = (a × d) / (b × c) = (30 × 90) / (70 × 10) = 2700 / 700 = 3.86
Interpretation: Smokers have 3.86 times the odds of lung disease compared to non-smokers.
95% Confidence interval: Using the log method: ln(OR) = 1.35, SE = √(1/30 + 1/70 + 1/10 + 1/90) = 0.41. CI = exp(1.35 ± 1.96 × 0.41) = (1.73, 8.61). Since the CI doesn't include 1, the association is significant.
Common Mistakes to Avoid
- Interpreting OR as relative risk. OR = 3.86 does NOT mean smokers are 3.86 times more likely to get disease. OR approximates RR only when the outcome is rare (< 10%).
- Ignoring the confidence interval. An OR of 5.0 with CI (0.8, 31.2) is NOT significant because the interval crosses 1.0.
- Reversing the direction. Make sure your "exposure" is in the rows and "outcome" is in the columns, or the OR will be inverted.
- Using OR with prospective cohort data. In cohort studies, use relative risk (RR) instead. OR is most appropriate for case-control studies.
Interpreting Odds Ratios
| Odds Ratio | Interpretation |
|---|
| OR = 1.0 | No association between exposure and outcome |
| OR > 1.0 | Exposure increases odds of outcome |
| OR < 1.0 | Exposure decreases odds of outcome (protective) |
| 95% CI includes 1 | Association is not statistically significant |
| 95% CI does not include 1 | Association is statistically significant |