Odds ratio vs relative risk: the choice depends on study design and on how common the outcome is. Trials and cohorts can report a risk ratio (relative risk) plus a risk difference; case-control studies report an odds ratio. With a common outcome, an odds ratio sits further from 1 than the risk ratio, so never word it as a risk.
Which measure suits each outcome type is mapped in our guide to reporting p-values, confidence intervals and effect sizes. This guide covers binary outcomes and the number needed to treat (NNT). Hazard ratios are covered in reporting Kaplan-Meier curves and hazard ratios. Every study below is invented to illustrate the structure; its numbers are not real data.
Calculate risk, odds and each effect measure from one 2×2 table
Risk is the proportion of a group who have the event. Odds are the number with the event divided by the number without it. The two are close when events are rare and far apart when they are common: a risk of 50% equals odds of 1. A risk ratio (RR) divides one group's risk by the other's, and an odds ratio (OR) divides their odds. The risk difference, also called the absolute risk reduction, subtracts one risk from the other. The relative risk reduction is 1 minus the risk ratio, written as a percentage: a risk ratio of 0.60 is a 40% relative risk reduction.
In an invented trial, 36 of 160 people given the intervention had a wound complication within 30 days (22.5%; odds 36/124 = 0.29), against 60 of 160 with usual care (37.5%; odds 60/100 = 0.60).
| Group | Complication within 30 days | No complication | Total |
|---|---|---|---|
| Intervention | a = 36 | b = 124 | 160 |
| Usual care | c = 60 | d = 100 | 160 |
From these cells, the intervention risk is a ÷ (a + b), the usual-care risk is c ÷ (c + d) and the odds ratio is (a × d) ÷ (b × c). Confidence intervals (CIs) are 95%, from standard large-sample formulas: the log scale for the ratios and the Wald method for the difference.
| Measure | Calculation | Result (95% CI) | What it says |
|---|---|---|---|
| Risk ratio | 22.5% ÷ 37.5% | 0.60 (0.42 to 0.85) | Risk 40% lower in relative terms |
| Odds ratio | 0.29 ÷ 0.60 | 0.48 (0.30 to 0.79) | Odds about half; not a 52% fall in risk |
| Risk difference | 22.5% − 37.5% | −15.0 percentage points (−24.9 to −5.1) | 15 fewer complications per 100 people treated |
| Number needed to treat for benefit (NNTB) | 1 ÷ 0.15 = 6.7, rounded up | NNTB 7 (4.0 to 19.6) | On average, one complication avoided per 7 people given the intervention instead of usual care, within 30 days |
Odds ratio vs relative risk when the outcome is common
Odds ratios and risk ratios are both valid measures; the error is reading an odds ratio as a risk ratio. The Cochrane Handbook chapter on effect measures warns that this overstates the effect in both directions: above 1, the odds ratio is larger than the risk ratio; below 1, it is smaller. Davies and colleagues (1998) showed the gap grows with both baseline risk and effect size. The calculated rows below hold the risk ratio at 0.60.
| Comparator risk | Intervention risk | Risk ratio | Odds ratio | Risk difference | NNTB |
|---|---|---|---|---|---|
| 1% | 0.6% | 0.60 | 0.60 | −0.4 points | 250 |
| 10% | 6% | 0.60 | 0.57 | −4.0 points | 25 |
| 37.5% (the trial above) | 22.5% | 0.60 | 0.48 | −15.0 points | 7 |
| 60% | 36% | 0.60 | 0.38 | −24.0 points | 5 |
No single cut-off is agreed. Zhang and Yu (1998) used 10% as a rule of thumb, and the Cochrane Handbook flags risks above about 20%. Below such a level an odds ratio approximates the risk ratio, but still call it an odds ratio.
Study design limits which measure you can estimate
A randomised trial, or a cohort following everyone for the same fixed period, counts events among everyone at risk, so it can estimate risks, a risk ratio and a risk difference. When follow-up time varies, divide events by person-time (all participants' follow-up added together); comparing those rates gives a rate ratio. A cross-sectional study gives a prevalence ratio or a prevalence odds ratio. What else each design can claim is set out in choosing between a case report, case series and cohort study.
A case-control study is different: the investigator decides how many controls to recruit. In an invented example, 40 of 100 cases and 20% of controls were exposed. The odds ratio is 2.67 with 200 controls and still 2.67 with 400. A "risk ratio" calculated across the rows moves from 1.83 to 2.11, because those proportions reflect the sampling ratio, not anyone's risk.
A case-control study therefore cannot estimate risks, incidence, a risk difference or an NNT directly. It reports an odds ratio, and what that estimates depends on how controls were sampled (Pearce, 1993). Controls drawn from everyone at risk at the start estimate a risk ratio. Controls drawn from people still at risk when each case occurs (density sampling) estimate a rate ratio. Controls drawn from people still free of the outcome at the end estimate the odds ratio itself. State yours in the Methods. Sampling is why the conclusion wording ladder pairs case-control studies with odds ratios.
Logistic regression returns odds ratios, not risk ratios
Logistic regression, which the guide to choosing a statistical test names for adjusted binary outcomes, returns odds ratios. For an adjusted risk ratio in a trial or cohort with a common outcome, use log-binomial regression or Poisson regression with robust standard errors (modified Poisson; Zou, 2004). Log-binomial models can fail to converge; Knol and colleagues (2012) suggest robust Poisson as the fallback.
For a crude 2×2 table, RR = OR ÷ (1 − p + p × OR) converts an odds ratio into a risk ratio, with p the comparator risk. In the invented wound-complication trial, p = 0.375 turns an odds ratio of 0.48 into a risk ratio of 0.60. Do not apply it to an adjusted odds ratio: the result is biased, and more so as confounding increases.
Word an odds ratio as odds, never as a risk
In an invented cohort, 45 of 150 exposed and 18 of 150 unexposed people had the event (30% vs 12%). Copy the middle column, replace the example numbers with your own and fill the bracketed slots.
| Measure (example) | Correct wording | Wording that misleads |
|---|---|---|
| Risk ratio (cohort) | Exposed participants had 2.5 times the risk of [event] compared with unexposed participants (risk ratio 2.50, 95% CI 1.52 to 4.11; a 150% relative increase) | "A 250% increase in risk" |
| Odds ratio (cohort) | Exposed participants had 3.1 times the odds of [event] compared with unexposed participants (odds ratio 3.14, 95% CI 1.72 to 5.75) | "3.1 times more likely to have [event]"; "3.1 times the risk" |
| Odds ratio (wound-complication trial) | The odds of [event] were about half as high with [intervention] as with [comparator] (odds ratio 0.48, 95% CI 0.30 to 0.79) | "Reduced the risk by 52%" (the relative risk reduction was 40%) |
| Risk difference (cohort) | The risk of [event] was 18 percentage points higher in exposed than in unexposed participants (30% vs 12%; 95% CI 9.0 to 27.0 points) | "An 18% increase" (the relative increase was 150%) |
| NNTB (wound-complication trial) | On average, for every 7 people given [intervention] rather than [comparator], one fewer has [event] within 30 days (NNTB 7, 95% CI 4.0 to 19.6) | "NNT 7", with no direction, comparator or time period |
| Hazard ratio | A [higher or lower] hazard of [event] than in [reference group], never a risk; full sentence in the hazard ratio reporting template | "[value] times the risk of [event]" |
Always name the event and the reference group: swapping event and non-event turns an odds ratio into its reciprocal, but can change a risk ratio substantially, by an amount that depends on both risks. In a trial abstract, the CONSORT 2025 explanation paper asks for each group's primary-outcome result and the effect size with its precision; the same wording rules apply.
Put the absolute effect and an NNT beside the ratio
For binary outcomes, item 26 of the CONSORT 2025 checklist asks for "presentation of both absolute and relative effect size"; the 2010 version numbered it 17b. Its explanation notes that readers tend to overestimate effects presented only in relative terms. For observational studies, item 16(c) of the STROBE checklist says "if relevant, consider translating estimates of relative risk into absolute risk for a meaningful time period". Holding the risk ratio at 0.60 shows why: on average, treating 250 people avoids one event at a 1% comparator risk, but treating 5 does so at a 60% comparator risk.
The Results section guide gives a sentence reporting both. For the NNT, which the CONSORT explanation calls helpful but does not require:
- Calculate 1 ÷ the absolute risk difference (0.15, not 15) and round up to a whole number.
- Label the direction: number needed to treat for an additional beneficial outcome (NNTB) or harmful outcome (NNTH). The Cochrane Handbook chapter on interpreting results prefers these to "number needed to harm".
- State the comparator, the time period and the exact outcome, including any cut-off on a scale.
- Give a CI from the reciprocals of the risk difference's limits, here to one decimal place.
- If the risk difference's CI crosses zero, the NNT interval runs through infinity. In an invented trial, 27 of 140 versus 33 of 140 events give a risk difference of −4.3 points (95% CI −13.9 to 5.3), written in Altman's notation as NNTB 24 (95% CI NNTH 18.8 to ∞ to NNTB 7.2).
- Never derive an NNT from a ratio alone; you need a baseline risk.
Decision table: design and outcome frequency pick the measure
"Rare" means a risk below about 10% in every group compared, a rule of thumb rather than a threshold.
| Design | Outcome frequency | Relative measure | Absolute companion | Adjusted model |
|---|---|---|---|---|
| Trial, or cohort with equal fixed follow-up | Rare | Risk ratio (an odds ratio will be close) | Risk difference; NNTB or NNTH for interventions | Logistic; log-binomial or robust Poisson for a risk ratio |
| Trial, or cohort with equal fixed follow-up | Common | Risk ratio | Risk difference; NNTB or NNTH for interventions | Log-binomial; robust Poisson if it fails to converge |
| Cohort with varying follow-up | Any | Rate ratio | Rate difference, such as per 1,000 person-years | Poisson regression |
| Time-to-event with censoring | Any | Hazard ratio | Survival in each group at a fixed time | Cox regression |
| Case-control | Any | Odds ratio, stating how controls were sampled | None without an external baseline risk | Logistic; conditional logistic if matched |
| Cross-sectional | Any | Prevalence ratio or prevalence odds ratio | Prevalence in each group | Logistic gives the prevalence odds ratio |
At Directive Publications, the author guidelines ask you to match the manuscript to the relevant EQUATOR checklist, such as CONSORT or STROBE; they set no separate rule on effect measures. Review is double-blind, with at least two independent expert reviewers sought per research manuscript, and reviewers are asked whether the statistics are appropriate and the claims free of overstatement. If you find an odds ratio worded as a risk in an article we published, report the problem to us.