How an Odds Ratio is Calculated
An odds ratio is calculated as the ratio of the odds of an event occurring in one group divided by the odds of that same event occurring in another group, typically comparing a treatment/intervention group to a control group. 1
Basic Calculation Method
The odds ratio (OR) represents the ratio of the odds of an event in one group to the odds of an event in another group 1. For a standard 2×2 contingency table:
- The odds in the treatment group are divided by the odds in the control or usual-practice group 1
- The calculation uses the cross-product ratio from the 2×2 table cells 2
Interpretation Framework
The interpretation follows a straightforward pattern 1:
- OR = 1: The two practices are equally successful (no difference in risk with respect to the outcome evaluated)
- OR > 1: The treatment or intervention practice is more successful
- OR < 1: The treatment or intervention practice is less successful
Important Relationship to Risk Ratios
When the outcome is rare (occurring in less than 10% of the unexposed population), the OR provides a reasonable approximation of the relative risk (RR) 3. However, a critical caveat exists:
- When an outcome is common (≥10% in the unexposed group), the OR will exaggerate the RR 3
- In small risk scenarios, odds ratios are very similar to risk ratios 1
Calculation Considerations
For Standard 2×2 Tables
- Multiple estimation methods exist, including unconditioned maximum likelihood estimators (UMLEs) and conditional MLEs (CMLEs) 2
- For the log(OR), the UMLE usually has the lowest bias 2
- Estimation on the log scale using the UMLE is generally recommended 2
For Complex Tables
- The odds ratio can be generalized beyond simple 2×2 tables to r×k tables when there are multiple degrees of severity of both exposure and disease outcome 4
- A general odds ratio (OR_G) has been proposed for such situations 4
- Standard errors and 95% confidence intervals can be calculated for these generalized measures 4
Adjustment for Correlation
- The standard variance expression for log-odds ratio should be modified in the presence of correlation 5
- This adjustment is particularly important for matched-pair data or multiple 2×2 tables 5
Reporting Requirements
Confidence intervals should always be provided with any report of an OR 3. The STROBE guidelines emphasize that both unadjusted and confounder-adjusted estimates should be reported with their precision measures (e.g., 95% confidence intervals) 1.