Odds ratio vs risk ratio: read a 2 × 2 table and adjust for confounding

Guide · Updated 2026-11-10

How odds ratios and risk ratios are worked out from a 2 × 2 table, which one your study design allows, what the intervals and tests mean, and how Mantel-Haenszel pooling removes a confounder.

Open the Odds and risk ratio Free, in your browser, nothing uploaded.

Did exposed people fall ill more often than unexposed ones? The odds and risk ratio calculator answers that from a 2 × 2 table of counts: odds ratio, risk ratio when the design allows it, intervals, tests, and a pooled answer across groups such as age bands. It is for public health students and anyone checking a published table; general education, not medical advice. Plan how many people a study needs with the sample size tool before you collect data, and turn a risk difference into a number needed to treat with the NNT calculator.

What it is

Every study of this kind can be squeezed into four counts:

• a: exposed people who got the outcome
• b: exposed people who did not
• c: unexposed people who got the outcome
• d: unexposed people who did not

Risk is the share of a group that gets the outcome. The risk if exposed is a ÷ (a + b), and the risk if unexposed is c ÷ (c + d). The risk ratio (RR) divides one by the other. An RR of 2 means the exposed group had twice the risk.

Odds compare those who got the outcome with those who did not: a ÷ b for the exposed. The odds ratio (OR) is (a × d) ÷ (b × c).

Which one you may use depends on how people entered the study:

• In a cohort study or trial, groups are chosen by exposure and followed up. Risks are real, so the RR and the risk difference (risk if exposed minus risk if unexposed) make sense.
• In a case-control study, people are chosen because they have the outcome (cases) or not (controls). The researcher set the number of cases, so only the OR is valid.

When the outcome is rare, the OR and RR are close.

A confidence interval is the range of values the data agree with. If a 95 % interval for a ratio includes 1, the data fit “no difference” too.

A confounder is a third factor linked to both the exposure and the outcome. Splitting the data into strata (layers, one per level of the confounder) and pooling them by the Mantel-Haenszel method gives a ratio adjusted for it.

How to use it

1. Choose One table or Strata (Mantel-Haenszel) at the top.
2. Set Study design to Case-control or Cohort or trial. The column headings change to match (Cases and Controls, or Ill and Well).
3. Pick the Confidence level: 90 %, 95 % or 99 %.
4. For one table, type the four counts into the grid. For strata, paste one row per stratum (a name, then exposed cases, exposed non-cases, unexposed cases, unexposed non-cases) or open a CSV or Excel file. Tick the continuity correction box if you want a more cautious Mantel-Haenszel test.
5. Read the ratios and tests. With strata you also get the crude value, the Breslow-Day test and a forest plot, which you can save with Forest plot (PNG) beside Results (CSV).

A worked example

A made-up cohort follows 300 adults for one year. 150 live beside a busy road (exposed) and 150 do not. The outcome is a new wheeze.

Risks: 30 ÷ 150 = 20.0 % if exposed, 15 ÷ 150 = 10.0 % if not.

Risk ratio: 20.0 % ÷ 10.0 % = 2, with a 95 % interval of 1.12 to 3.56. The risk difference is 10.0 % (2.0 % to 18.0 %).

Odds ratio: (30 × 135) ÷ (120 × 15) = 4,050 ÷ 1,800 = 2.25 (1.16 to 4.38), further from 1 than the RR because wheeze is common here.

Tests: Pearson’s chi-square is 5.882 (p = 0.015). Fisher’s exact test gives p = 0.023. The smallest expected count is 22.5 people, so the plain chi-square is fine.

Now suppose 80 of the 150 exposed people smoke, against 40 of the 150 unexposed, and smoking also raises the risk of wheeze. Switch to Strata and paste:

Smokers	20	60	6	34
Non-smokers	10	60	9	101

They add up to the table above. Each stratum alone gives an RR of about 1.67 (smokers) and 1.75 (non-smokers). The Mantel-Haenszel RR is 1.70 (0.941 to 3.08), against the crude 2. The crude value is 17.4 % higher than the adjusted one, a clear sign of confounding by smoking. The Breslow-Day test (p = 0.989) finds no difference between the strata, so one pooled value is fair. After adjustment the interval includes 1, so the link is much less certain than it first looked.

Where it is used

• Outbreak investigations. Compare people who ate each food with those who did not.
• Reading papers. Re-enter a published 2 × 2 table to check a reported ratio and interval.
• Clinical trials. The RR and risk difference describe how much a treatment changed an outcome. For pooling many trials, use meta-analysis.
• Workplace safety and product tests. Compare injuries between sites, or sign-ups between two versions of a page.

Common mistakes

• Reporting a risk ratio from a case-control study. The design fixes the number of cases, so only the OR is valid. Set the design first.
• Reading an OR as a risk ratio. “OR 2.25” is not “2.25 times the risk”.
• Swapping rows. Exposed goes on the top row; swapping turns the ratio upside down.
• Trusting chi-square with tiny counts. If an expected count is below 5, use Fisher’s exact or the mid-p test and the exact interval. The page warns you.
• Pooling strata that disagree. If Breslow-Day is significant, report each stratum instead.

Questions people ask

What does a ratio below 1 mean? The exposure goes with less of the outcome. An RR of 0.5 means half the risk.

Does a significant odds ratio prove cause? No. Bias, chance and unmeasured confounders can all produce a link.

Why is the exact odds ratio slightly different from ad/bc? It is estimated in a different way that holds the table’s totals fixed. In large studies the two come close together.