Odds ratio or relative risk: which to report?

When the design lets you estimate risks (a cohort study, a trial, a cross-sectional survey), report the relative risk together with the risk difference. When it does not, as in a case-control study, or when the estimate comes out of a logistic model, report the odds ratio and call it an odds ratio.

The odds ratio and the relative risk (risk ratio) both compare how often an outcome happens in two groups, and they are routinely confused. They are different quantities, and whenever there is an effect the odds ratio is further from 1 than the relative risk. Reporting one while describing it as the other overstates or understates an effect, most often by a lot when the outcome is common. Whichever you choose, say which one it is.

The reasons odds differ from probabilities are covered in the logistic regression guide; this guide is about choosing between the measures and converting between them.

Odds ratio vs relative risk: what each one measures

Arrange the data as a 2 x 2 table. Exposed participants have a events and b non-events; unexposed participants have c events and d non-events.

• Risk is the proportion with the event: a / (a + b) among the exposed, c / (c + d) among the unexposed.
• Relative risk (RR) is the ratio of those two risks. An RR of 2 means the event is twice as likely in the exposed group.
• Risk difference (RD) is the difference between the two risks, an absolute quantity.
• Odds ratio (OR) is (a / b) / (c / d), which simplifies to ad / bc. It compares odds, not risks.

When the event is rare, a is small next to b and c is small next to d, so a / b is close to a / (a + b) and the OR lands close to the RR. As the event becomes common the two drift apart, and the OR always sits on the far side of 1. An OR of 2.4 can sit alongside an RR of 2.0, or alongside an RR of 1.4, depending on how common the event is in the unexposed group. The OR on its own does not say which.

Why a case-control study can only give you an odds ratio

A case-control study samples by outcome: the investigators choose how many cases and how many controls to recruit. The proportion of cases in the sample is therefore a number the investigators set, not a property of the population. Any “risk” computed from the sample, such as cases among the exposed, simply reflects the chosen case-to-control ratio. Change the ratio and the apparent risks change, with the population unchanged.

The odds ratio is not affected by that choice. Exposure odds among cases compared with exposure odds among controls give the same cross-product ratio ad / bc as disease odds among the exposed compared with the unexposed. Cornfield (1951) used this to show that a case-control study can estimate the odds ratio of disease, and that when the disease is rare that odds ratio approximates the relative risk. That rare-disease assumption is the only bridge from a case-control OR to an RR, and it is a real assumption: if the outcome is common in your source population, the bridge fails.

Choosing the measure by study design

Design	What you can estimate	What to report
Cohort, trial	Risks, so RR, RD and OR	RR and RD; add the OR if a model needs it
Cross-sectional survey	Prevalences, so prevalence ratio, difference and OR	Prevalence ratio and difference
Case-control	Only the OR	The OR, with the rare-disease caveat if you interpret it as an RR
Any design, adjusted by logistic regression	Adjusted OR	The adjusted OR; use a different model for an adjusted RR

A cross-sectional survey produces prevalences, and a ratio of prevalences is the natural analogue of the RR. The logic of the table is the same: ask whether the group sizes were fixed by exposure, by the outcome, or by neither.

Worked example

Take a synthetic survey of 400 workers, 200 in an exposed occupational group and 200 in an unexposed group. A symptom is reported by 60 of the exposed workers (30%) and 30 of the unexposed (15%). The numbers come from a base R script, not from real participants.

Measure	Estimate	95% CI
Relative risk	2.00	1.35 to 2.96
Risk difference	0.15	0.07 to 0.23
Odds ratio	2.43	1.49 to 3.97

The OR of 2.43 is noticeably larger than the RR of 2.00, because the symptom is common (15% in the baseline group). Reading “2.4 times as likely” off the OR would overstate the effect by about a fifth.

Now suppose the same population had been studied case-control: 90 cases and 90 controls. In expectation, 60 of the cases and 41 of the controls are exposed. The odds ratio from that table is 2.39 (95% CI 1.31 to 4.37), close to the cohort value, which is what the invariance above predicts. But the “risk” in the sample is meaningless: 59% of exposed participants are cases and 38% of the unexposed, a ratio of 1.56. That figure comes from the 1-to-1 sampling ratio and says nothing about the population. Only the OR from this design carries over.

Converting an odds ratio to a relative risk

If you have only an OR, you can approximate the RR when you also know the baseline risk P0, the event risk in the unexposed group (Zhang and Yu, 1998):

RR = OR / (1 - P0 + P0 x OR)

With the OR of 2.43 and P0 = 0.15, this gives 2.43 / (0.85 + 0.15 x 2.43) = 2.00, recovering the crude RR exactly, as it should. The conversion has limits worth knowing before you use it:

• It needs the baseline risk. A case-control study cannot supply it from its own sample. Grant (2014) suggests reporting a range of plausible RRs over a range of plausible baseline risks. In the example above, baseline risks of 5%, 15%, 30% and 50% correspond to RRs of 2.27, 2.00, 1.70 and 1.42 for the same OR.
• It is biased for an adjusted OR. The formula was derived for a crude OR with a crude baseline risk. Feeding it a covariate-adjusted OR does not produce an adjusted RR (McNutt et al., 2003).
• Do not convert confidence limits. Putting the two OR limits into the formula gives an interval, 1.38 to 2.75 here, that does not match the correct RR interval of 1.35 to 2.96. Intervals do not transform that way.

If the question is an adjusted RR in a cohort or trial, a model that estimates it directly is the better route: log-binomial regression, or Poisson regression with a robust variance estimator (McNutt et al., 2003).

Risk difference and number needed to treat

Relative measures hide the absolute size of an effect. An RR of 2 can mean going from 15% to 30% or from 0.15% to 0.30%, which matter very differently at population level. The risk difference reports the absolute change directly: 0.15 in the example, an extra 15 events per 100 exposed.

The number needed to treat (or, for a harmful exposure, number needed to harm) is 1 / RD, here 1 / 0.15, about 6.7. Read it as a population summary: on average, about seven exposed people per additional event, a statement about groups and not about any individual. Its confidence interval needs care. Here the RD interval of 0.07 to 0.23 inverts to about 4.3 to 14.4. If the RD interval crosses zero, the NNT interval is not a single range but extends through infinity, and Altman (1998) explains how to present it.

Which one to report

• Cohort, trial or survey, one grouping variable: report the RR and the RD with confidence intervals, and give both groups’ absolute risks.
• Case-control study: report the OR and say it is an OR. Call it an approximation of the RR only if you can defend the rare-outcome assumption.
• Adjusted analysis with a binary outcome: the logistic regression OR is the standard output. If readers need an adjusted RR, use log-binomial or modified Poisson regression instead of converting.
• Reporting to a general audience: lead with the absolute risks, then the RD, then the ratio.

Avoid describing an OR as “times as likely” unless the outcome is rare. Say “times the odds”.

How Celeus handles this

For a 2 x 2 table, Celeus reports the risk ratio, risk difference and odds ratio, each with a 95% confidence interval, and names the reference group, the comparison group and the event level. It uses Wald intervals when every cell count is at least 5 and score or exact intervals when any cell is smaller, and attaches a note that only the odds ratio is interpretable if participants were sampled by outcome. Logistic regression reports adjusted odds ratios with profile-likelihood confidence intervals. Celeus does not compute a number needed to treat, and logistic regression does not report an adjusted risk ratio.