Relative Risk Versus Odds Ratio

Relative risk compares two probabilities. An odds ratio compares two odds. They approach one another when an outcome is uncommon, but they can differ substantially when it is common, and a reported odds ratio of 2 does not generally mean the outcome is twice as likely. Both measures are valid when used and labeled correctly. The interpretation problem begins when the familiar language of risk is attached to an odds ratio without a baseline probability.

Key points#

• Risk is events divided by everyone in the group.
• Odds are events divided by nonevents in the group.
• Relative risk is the ratio of two risks; odds ratio is the ratio of two odds.
• Odds ratios preserve useful mathematical properties in logistic regression and case-control studies.
• Always pair a relative measure with absolute risks or a clear baseline risk when possible.

Work through one neutral example#

Imagine 100 people in group A and 100 in group B. In A, 40 experience the outcome. In B, 20 experience it.

The risk in A is:

40 / 100 = 0.40

The risk in B is:

20 / 100 = 0.20

The relative risk is:

0.40 / 0.20 = 2.0

So the observed probability is twice as high in A as in B.

The odds use nonevents in the denominator. A has 40 events and 60 nonevents, so its odds are 40/60, or 0.667. B has 20 events and 80 nonevents, so its odds are 20/80, or 0.25.

The odds ratio is:

0.667 / 0.25 = 2.67

The same table therefore gives a relative risk of 2.0 and an odds ratio of 2.67. Saying “2.67 times the risk” would overstate the actual risk ratio.

Odds are not probabilities#

A probability of 0.50 means one event among every two people on average. Its odds are 0.50 divided by 0.50, or 1 to 1.

A probability of 0.80 has odds of 0.80 divided by 0.20, or 4 to 1. Doubling odds from 1 to 2 does not double probability from 0.50 to 1.00. Odds of 2 correspond to probability 2/(1+2), or about 0.667.

Conversions are:

odds = probability / (1 - probability)

probability = odds / (1 + odds)

These formulas explain why odds rise sharply as probability approaches 1. Risk stays between 0 and 1, while odds range from 0 without an upper bound.

Rare outcomes make the measures look similar#

If risks are 2% and 1%, the relative risk is 2.0. The odds are approximately 0.0204 and 0.0101, giving an odds ratio near 2.02. The difference is tiny because the number of nonevents is almost the same as the total in each group.

If risks are 60% and 30%, the relative risk remains 2.0. The odds are 1.5 and about 0.429, giving an odds ratio of 3.5. The odds ratio moves farther from 1.

The “rare disease assumption” is about outcome frequency in the populations represented, not simply the number of cases in a dataset: a study can have thousands of events and still study a rare outcome in a much larger population. No universal prevalence threshold makes substitution safe, and even at moderate frequencies the difference can matter when you are explaining benefit or harm to somebody.

Why logistic regression reports odds ratios#

Logistic regression models the log odds of a binary outcome as a function of predictors. Exponentiating a coefficient produces an odds ratio, conditional on the other model variables.

This approach keeps predicted probabilities between 0 and 1 and has useful computational properties. It is widely available and can handle continuous and categorical predictors, interactions, and adjustment.

The adjusted odds ratio is not necessarily the same as the unadjusted odds ratio from a two-by-two table; it compares odds across values of a predictor conditional on included covariates and model form. Odds ratios are also non-collapsible: an adjusted value can differ from an unadjusted value even without confounding. That does not automatically mean one is biased; they may be different statistical summaries.

When you need a risk ratio or a risk difference, the alternatives include log-binomial models, modified Poisson regression with robust variance, standardization, or model-based marginal risks, and each has assumptions and implementation details. Authors should choose a measure that matches the scientific question and communicate it directly.

Why case-control studies use odds ratios#

In a case-control study, investigators deliberately select people with the outcome and a sample without it, so the proportion of cases in the dataset is set by design and usually does not estimate population risk. A risk ratio cannot be calculated directly from those sampled proportions.

The factor odds among cases compared with controls yields an odds ratio. Under appropriate sampling, it estimates the same odds ratio that would arise in the source population. With incidence-density sampling, the odds ratio can estimate an incidence rate ratio without requiring the outcome to be rare.

The common statement that case-control studies use odds ratios only because disease is rare is incomplete. The sampling design is the central reason. You need to know how the controls were sampled and which source population they represent.

Baseline risk changes the translation#

An odds ratio alone does not determine an absolute risk. If the baseline risk is p0 and the odds ratio is OR, a corresponding treated or factor-group risk under a simple setting can be calculated as:

p1 = (OR x p0) / (1 - p0 + OR x p0)

For an odds ratio of 2:

• baseline risk 1% translates to about 2.0%;
• baseline risk 10% translates to about 18.2%;
• baseline risk 40% translates to about 57.1%.

The same odds ratio creates different risk ratios and risk differences, which is why translating one for a patient requires a credible baseline risk for the population and time horizon you actually mean. In adjusted analyses, simply inserting the average baseline risk can be an approximation. Marginal standardization across the actual covariate distribution gives you a clearer population-level adjusted risk estimate.

Direction and outcome coding matter#

If the outcome is recovery rather than non-recovery, an odds ratio changes form. Reversing the group comparison takes the reciprocal. An odds ratio of 2 for recovery becomes 0.5 when the group order is reversed.

Switching from the odds of an event to the odds of its complement also takes the reciprocal for the same group comparison. This can catch you out when a paper alternates between benefit and harm framing.

Relative risk has related framing issues. A treatment that reduces risk from 20% to 10% has a risk ratio of 0.5 for the adverse outcome, but the probability of avoiding it rises from 80% to 90%, a risk ratio of 1.125 for the favorable outcome. Absolute change is 10 percentage points either way. So a report should name the outcome, the reference group, the comparison group, the follow-up time, and whether the estimate is adjusted.

Limits: confidence intervals and causality remain separate#

A confidence interval describes sampling uncertainty under the design and model. It does not repair confounding, selection bias, missing data, outcome misclassification, or model misspecification.

In a randomized trial, an odds ratio can estimate an assignment effect for the prespecified outcome. In an observational study, an adjusted odds ratio remains an association unless the design and assumptions support a causal interpretation.

Statistical interaction also depends on the scale. Two factors can interact on an odds-ratio scale but not on a risk-difference scale. The clinically relevant scale should be justified rather than chosen for the most favorable p value.

A fast reading checklist#

When a paper reports a binary outcome, ask:

1. Is the measure a risk ratio, odds ratio, rate ratio, or hazard ratio?
2. What event and follow-up period does it describe?
3. What are the raw or standardized absolute risks?
4. Is the outcome common enough that OR and RR differ materially?
5. Was the estimate unadjusted, conditional, or marginal?
6. If case-control, how were controls sampled?
7. Which group and outcome are the references?
8. Does the confidence interval show adequate precision?
9. Does the study design support a causal claim?
10. Would a risk difference or number needed to treat make the decision clearer?

CONSORT encourages randomized trial reports to provide outcomes for each group and an effect size with precision. STROBE asks observational reports to distinguish relative and absolute estimates where relevant. Clear denominators are the simplest protection against misinterpretation.