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# Odds ratio

- Measure

Compares the odds of having been exposed in cases and in controls; it is the measure of case-control studies.

Updated Nov 5, 2026

## Key points

- OR = (a×d) / (b×c), the cross-product of the 2×2 table.
- Odds = probability it happens / probability it does not; a 20% probability is odds of 1 to 4 (0.25).
- It is the measure of case-control studies and of logistic regression.
- With a rare outcome, OR ≈ RR (the rare disease assumption).
- With a common outcome, the OR lies further from 1 than the RR.
- OR = 1: no association; above 1: more exposure among cases; below 1: less.

On this page

- The basics Key points
- What it is
- Formula and reading
- Worked example
- Relative risk versus odds ratio
- Markers
- How to read it
- Common pitfalls
- Memory aid

- Study Test yourself

- Connections Connections

## What it is

The odds ratio (OR) is the ratio of two odds: the odds of exposure among cases and the odds of exposure among controls. It is the natural measure of the case-control study, where risks cannot be calculated, and it is also what logistic regression produces. When the outcome is rare (roughly under 10%), the OR is very close to the relative risk; when the outcome is common, the OR exaggerates it.

## Formula and reading

- OR = (a/c) / (b/d) = (a×d) / (b×c).
- Odds of exposure in cases = a/c; in controls = b/d.
- The exposure OR equals the disease OR: (a/b)/(c/d) gives the same number.
- Read it like the RR: 1 neutral, above 1 linked to more outcome, below 1 to less.

## Worked example

- Data: 100 cases of a rare liver disease and 200 controls. 40 cases and 30 controls used a pesticide.
- Table: a = 40, c = 60 (unexposed cases), b = 30, d = 170 (unexposed controls).
- Odds of exposure in cases = 40/60 ≈ 0.67; in controls = 30/170 ≈ 0.18.
- OR = (40 × 170) / (30 × 60) = 6,800 / 1,800 ≈ 3.8 (rounded to one decimal).
- Because the disease is rare, the OR approximates the relative risk.
- Reading: cases have about 3.8 times the odds of having used the pesticide compared with controls.

## Relative risk versus odds ratio

| Feature | Relative risk (RR) | Odds ratio (OR) |
| --- | --- | --- |
| Formula | [a/(a+b)] / [c/(c+d)] | (a×d) / (b×c) |
| Design | Cohort, clinical trial | Case-control, logistic regression (also cohorts) |
| What it compares | Risks | Odds |
| Common outcome | Direct value | Exaggerates the effect relative to the RR |
| Rare outcome | Direct value | Approximates the RR |

## Markers

- OR
- odds
- 95% CI

## How to read it

- Many studies of risk factors for rare diseases (uncommon cancers, birth defects) report ORs because they are case-control studies.
- When reading an OR of 3 for an outcome that affects 40% of patients, remember that the true RR is smaller.

## Common pitfalls

- Calling an OR a risk when the outcome is common.
- Flipping the table and getting 1/OR.
- Confusing odds (0.25) with probability (20%).

## Memory aid

Cross your arms: the OR multiplies the diagonals, a times d over b times c, like arms folded across the table.

## Test yourself

Think of the answer, then tap the question to see it.

## Cases: 30 exposed and 70 unexposed. Controls: 10 exposed and 90 unexposed. What is the OR?

OR = (30 × 90) / (70 × 10) = 2,700 / 700 ≈ 3.9.

## When does the OR approximate the relative risk?

When the outcome is rare in the population studied (generally under 10%).

## What odds does a 50% probability represent?

Odds of 1 (0.5 / 0.5): one to one.

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## Connections

## Related 5

- [Relative risk](https://durgical.com/en/epidemiology/relative-risk/)
- [The 2×2 table](https://durgical.com/en/epidemiology/the-2-2-table/)
- [Case-control study](https://durgical.com/en/epidemiology/case-control-study/)
- [Logistic regression](https://durgical.com/en/epidemiology/logistic-regression/)
- [Confidence interval](https://durgical.com/en/epidemiology/confidence-interval/)

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Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.

Sources: [Principles of Epidemiology | Lesson 3 - Section 5](https://archive.cdc.gov/www_cdc_gov/csels/dsepd/ss1978/lesson3/section5.html) (Centers for Disease Control and Prevention, 2012); [Chapter 6: Choosing effect measures and computing estimates of effect](https://www.cochrane.org/authors/handbooks-and-manuals/handbook/current/chapter-06) (Cochrane, 2024).

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Representative model, for educational purposes only.

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License: CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Cite as: DurgiPedia (durgical.com), https://durgical.com/en/epidemiology/odds-ratio/