Odds ratio vs relative risk vs hazard ratio: what is the difference and which should I report?

What are relative risk, odds ratio and hazard ratio?

All three are ratio measures of association: a value of 1 means no difference between groups, above 1 means the outcome is more frequent in the index group, below 1 means it is less frequent. They differ in what is being divided.

In the 2×2 table, a = exposed with the outcome, b = exposed without it, c = unexposed with the outcome, d = unexposed without it.

Risk is events divided by everyone in the group (20 events in 100 people = 0.20). Odds is events divided by non-events (20 / 80 = 0.25). That single difference in the denominator drives everything else in this guide. Bland and Altman's BMJ Statistics Note on the odds ratio is still the clearest short explanation of why the two diverge.

The hazard is the rate at which events occur at a given moment among people who have not had the event yet. A hazard ratio summarises that rate ratio across follow-up, usually assuming it stays constant over time (the proportional hazards assumption).

How do I calculate odds ratio and relative risk? A worked 2×2 example

Synthetic data, for illustration only. A cohort of 500 pregnant women: 200 exposed to a risk factor, 300 unexposed. The outcome (for example, preterm birth) occurs in 40 exposed and 30 unexposed women.

1. Risk in each group: 0.20 in exposed, 0.10 in unexposed.
2. Relative risk: 0.20 / 0.10 = 2.00 (95% CI, 1.29 to 3.10).
3. Odds in each group: 40/160 = 0.250 in exposed, 30/270 = 0.111 in unexposed.
4. Odds ratio: (40 × 270) / (160 × 30) = 10,800 / 4,800 = 2.25 (95% CI, 1.35 to 3.76).
5. Risk difference: 0.20 − 0.10 = 10.0 percentage points (95% CI, 3.5 to 16.5).
6. Number needed to harm: 1 / 0.10 = 10 (95% CI, 6.1 to 28.6).

Confidence intervals here use the standard log (Woolf) method for RR and OR and the Wald method for the risk difference; all numbers were recomputed with a script. The same data give an RR of 2.00 and an OR of 2.25. Neither is wrong, but they answer different questions, and writing "2.25 times the risk" would overstate the effect by 12.5%.

When does the odds ratio approximate the relative risk?

When the outcome is rare, non-events make up almost the whole group, so odds and risk are nearly the same number. With 1.0% vs 0.5% risk, the RR is 2.00 and the OR is 2.01. Norton and colleagues use exactly this kind of example in their JAMA Guide to Statistics and Methods, and note that the gap widens once baseline prevalence passes about 10%.

As the outcome becomes common, the OR moves further from 1 than the RR: it overstates harmful effects and makes protective effects look stronger than they are. The table below shows the OR you would observe for a given true RR at different baseline risks (computed exactly, not approximated).

At a 40% baseline risk, a doubling of risk appears as an OR of 6. This is why Altman, Deeks and Sackett argued that odds ratios should be avoided when events are common, and why reviewers push back when a cohort study or trial with a frequent outcome reports only ORs.

If you have an adjusted OR from logistic regression in a cohort, Zhang and Yu proposed an approximate conversion: RR ≈ OR / [(1 − P0) + (P0 × OR)], where P0 is the outcome risk in the unexposed group. In the example above, 2.25 / (0.90 + 0.225) = 2.00. The approximation is less reliable for adjusted estimates; a better route is to estimate the RR directly, for example with log-binomial regression or the modified Poisson regression with robust standard errors described by Zou.

Which study designs give an odds ratio, a relative risk or a hazard ratio?

For a design-by-design reporting checklist, see the STROBE checklist for observational studies and the CONSORT 2025 checklist for randomized trials.

What does an odds ratio of 1.5 mean? What does a relative risk of 1.2 mean?

Ratios describe relative change, so translate them into words carefully.

Is an odds ratio of 1.5 high? It depends on the outcome, the baseline risk and the confidence interval. An OR of 1.5 for a rare, serious outcome may matter clinically; an OR of 1.5 with a 95% CI of 0.9 to 2.5 is compatible with no effect. Always read the ratio with its interval and the absolute risks.

What does a hazard ratio of 2.0 mean, and how do I interpret its 95% CI?

A hazard ratio is a rate ratio, not a ratio of cumulative risks and not a measure of time saved. Spruance and colleagues showed that an HR says nothing about how much sooner events happen: two trials with the same HR can have very different differences in median time to event. They also show one intuitive translation: HR / (1 + HR) estimates the probability that a patient in the index group has the event before a comparable patient in the other group. For HR 2.0 that is 2 / 3, about 67%.

Interpret the 95% CI the same way as for any ratio: a CI of 1.4 to 2.9 means the data are compatible with hazards from 40% to 190% higher; a CI that crosses 1 (for example 0.8 to 1.6) means the data are compatible with no difference. Hernán has shown two further traps: an average HR depends on how long follow-up lasted, and HRs for later time periods carry built-in selection bias because the most susceptible patients have already had the event. Report Kaplan–Meier or adjusted survival curves and absolute risks at fixed time points alongside the HR, and state how you checked proportional hazards.

How do I report OR, RR and HR in a manuscript?

1. Name the measure and the comparison: "risk ratio for preterm birth, exposed vs unexposed".
2. Give the counts behind it: events and denominators for each group (JAMA's author instructions ask for absolute numbers or rates alongside ratio measures).
3. Give the point estimate with its 95% CI, two decimals for ratios: "RR, 2.00; 95% CI, 1.29 to 3.10" (or follow your journal's house style for separators).
4. Say whether it is crude or adjusted, and list the covariates in the methods.
5. Add an absolute measure: risk difference with 95% CI, and NNT or NNH where clinically useful.
6. Match the verbs to the design: "was associated with" for observational data; causal language only for randomized comparisons.

Example sentence (synthetic): "Preterm birth occurred in 40 of 200 exposed women (20.0%) and 30 of 300 unexposed women (10.0%) (risk ratio, 2.00; 95% CI, 1.29 to 3.10; risk difference, 10.0 percentage points; 95% CI, 3.5 to 16.5)."

For P values alongside these estimates, see our guide to interpreting and reporting P values and confidence intervals.

Why do reviewers ask for absolute risk and NNT?

Relative measures hide the baseline. An RR of 2.0 can mean 0.5% vs 1.0% (risk difference 0.5 percentage points, NNH 200) or 10% vs 20% (risk difference 10 points, NNH 10). Laupacis, Sackett and Roberts introduced the number needed to treat to make that clinical consequence visible: NNT = 1 / absolute risk difference, rounded up to a whole patient. Altman showed how to give it a confidence interval (invert the limits of the CI for the risk difference) and how to report it when that CI crosses zero, labelling one side as number needed to treat for benefit and the other as number needed to treat for harm, instead of printing a meaningless negative interval.

Common mistakes reviewers flag

• Calling an OR a risk. "Twice the risk" from an OR of 2.0 when the outcome affects 30% of patients.
• Reporting RR from a case-control study. The design cannot estimate risk; only the OR is valid.
• Logistic regression in a cohort or trial with a common outcome, ORs only. Expect a request for RRs or risk differences.
• Comparing ORs across models or studies. Norton and colleagues note that ORs from logistic models with different covariates are not on the same scale and should not be compared directly.
• "Twice as fast" for HR 2.0, or no check of proportional hazards, or no survival curve.
• Ratios without counts, or no absolute effect anywhere in the paper.
• Inconsistent numbers: a ratio in the abstract that differs from the table or cannot be recomputed from the reported counts. CliniPaper recomputes OR, RR and risk differences from your tables and flags mismatches between abstract, results and tables.

Whatever tool helps you check your numbers, you remain responsible for the analysis, and if you use generative AI in preparing the manuscript, disclose it as your target journal requires.