What is an Odds Ratio?
An odds ratio (OR) is a measure of association that represents the odds of an event occurring in one group divided by the odds of that event occurring in another group, typically comparing an exposed/treatment group to an unexposed/control group. 1, 2
Mathematical Definition
The odds ratio is calculated as the quotient between two odds, where odds represent an alternative way to express the possibility of occurrence of an outcome or presence of an exposure. 3
Mathematically, odds are defined as the probability of an event occurring divided by the probability of the event not occurring. 1, 4
Calculation Using a 2×2 Contingency Table
For a standard 2×2 table with the following structure:
| Disease Present (Cases) | Disease Absent (Controls) | |
|---|---|---|
| Exposed | A | B |
| Unexposed | C | D |
The odds ratio is calculated as: OR = (A × D) / (B × C) 1, 2
- Odds in the exposed group = A/B 2
- Odds in the unexposed group = C/D 2
- The OR divides the odds in the treatment/exposed group by the odds in the control/unexposed group 1, 2
Practical Example
Using the example from Clinical Microbiology Reviews: 1
- With first-void urine collection: 10 contaminated per 100 cultures (10% contamination)
- With midstream clean-catch: 3 contaminated per 100 cultures (3% contamination)
- The odds ratio = (3/10)/(97/90) = 0.28 1
Interpretation Framework
Basic Interpretation Rules
- OR = 1: No association exists; the two groups have equal odds of the outcome 2
- OR > 1: The exposure/treatment increases the odds of the outcome 2
- OR < 1: The exposure/treatment decreases the odds of the outcome 2
Important Caveats
The odds ratio always exaggerates the true relative risk to some degree. 4 This exaggeration becomes more pronounced as the outcome becomes more common:
- When the disease/outcome probability is low (<10%), the OR approximates the true relative risk closely 4, 5
- As the event becomes more common, the exaggeration grows substantially, and the OR no longer serves as a useful proxy for relative risk 4
- In small risk scenarios, odds ratios are very similar to risk ratios 1, 2
Context-Dependent Meaning
The meaning of the odds ratio obtained in a case-control study differs according to how controls are selected: 6
- Controls selected from person-time at risk → estimates the rate ratio
- Controls selected from persons at risk at baseline → estimates the risk ratio
- Controls selected from survivors at end of follow-up → estimates the odds ratio
None of these estimation procedures depends on any rare disease assumption. 6 The rare disease assumption is only relevant for determining whether the estimated effect measure approximates another effect measure of interest in the underlying study population. 6
Reporting Requirements
Confidence intervals (typically 95% CI) must always be provided with any report of an odds ratio. 1, 2
Both unadjusted and confounder-adjusted estimates should be reported with their precision measures. 1, 2 The STROBE guidelines emphasize that authors should make clear which confounders were adjusted for and why they were included. 1
Appropriate Use
Although the odds ratio is always a valid measure of association, it is not always a good substitute for the relative risk. 4 Because of the difficulty in understanding odds ratios and their potential to exaggerate effects, their use should probably be limited to case-control studies and logistic regression, for which odds ratios are the proper measures of association. 4
As long as the odds ratio is not used uncritically as an estimate of the relative risk, it remains an attractive statistic for epidemiologists to calculate. 5