Risk Ratios and Odds Ratios: Computation and Interpretation

The risk ratio and the odds ratio are the two ratio measures most often used to express the association between a binary exposure and a binary outcome from a 2×2 table. The risk ratio compares the probability (risk) of the outcome between exposed and unexposed groups; the odds ratio compares the odds. They coincide when the outcome is rare but diverge as it becomes common, and choosing and reading them correctly is a recurring source of confusion in health research.

Definition

The risk ratio is the risk of the outcome in the exposed group divided by the risk in the unexposed group; the odds ratio is the odds of the outcome in the exposed group divided by the odds in the unexposed, equal to the cross-product of the 2×2 table cells. Both equal one under no association.

Scope

This entry covers how each measure is computed from the four cells of a 2×2 table, the difference between odds and risk, why the design of a study dictates which measure is estimable, the conditions under which the odds ratio approximates the risk ratio, the ways an odds ratio can be misread as a risk ratio for common outcomes, and the regression approaches used to estimate risk and prevalence ratios directly. It presents these as effect measures for interpreting evidence, not as clinical guidance.

Core questions

• How are risk and odds defined for a binary outcome, and how do their ratios differ?
• Which cells of the 2×2 table enter the computation of each measure?
• Why can a case-control study estimate the odds ratio but not the risk ratio directly?
• When does the odds ratio approximate the risk ratio, and how is it misleading when the outcome is common?
• How can risk ratios or prevalence ratios be estimated directly in regression?

Key concepts

• Risk versus odds of an outcome
• Risk ratio (relative risk)
• Odds ratio as the 2×2 cross-product
• Reference (null) value of one
• Rare-outcome approximation of OR to RR
• Inflation of the OR for common outcomes
• Study design determines the estimable measure
• Log-binomial and modified-Poisson regression for risk/prevalence ratios

Mechanisms

From a 2×2 table with cells a (exposed cases), b (exposed non-cases), c (unexposed cases) and d (unexposed non-cases), the risk in the exposed is a/(a+b) and in the unexposed c/(c+d), so the risk ratio is [a/(a+b)] ÷ [c/(c+d)]. The odds of being a case are a/b in the exposed and c/d in the unexposed, so the odds ratio is (a/b) ÷ (c/d) = ad/bc, the cross-product. Because a case-control study fixes the numbers of cases and non-cases by sampling, it cannot estimate the underlying risks and therefore reports the odds ratio, which by its symmetry still estimates the disease odds ratio; cohort and cross-sectional studies can estimate risks (or prevalences) directly and so can report risk or prevalence ratios. When the outcome is rare the odds and the risk are close, so the odds ratio approximates the risk ratio; when the outcome is common the odds ratio lies further from one than the risk ratio, so reading it as a relative risk overstates the effect. To obtain a risk or prevalence ratio directly from adjusted analyses, log-binomial regression and the modified-Poisson (robust-variance) approach are used in place of logistic regression.

Clinical relevance

Risk ratios and odds ratios are among the most frequently reported numbers in the health-sciences literature, and confusing one for the other can materially distort how a result is understood, so interpreting them in light of how common the outcome is and how the study was designed is essential to appraising evidence. These measures quantify associations for interpreting research and are not a basis for individual diagnostic or treatment decisions.

Epidemiology

The choice of measure follows the design: case-control studies yield odds ratios, cohort studies yield risk or rate ratios, and cross-sectional studies yield prevalence ratios or odds. Because logistic regression returns odds ratios even when outcomes are common, the methodological literature has emphasised direct estimation of risk and prevalence ratios via log-binomial and modified-Poisson models to avoid overstating effects.

History

Cornfield's 1951 argument established that case-control odds ratios estimate the disease odds ratio and approximate the relative risk for rare outcomes, anchoring the use of the odds ratio. As logistic regression spread, the late-1990s literature (Davies and colleagues; Zhang and Yu) returned to the problem of odds ratios being misread as relative risks for common outcomes, and subsequent work (Barros and Hirakata; Zou) developed regression methods that estimate risk and prevalence ratios directly, with later guidance on communicating odds ratios as plausible relative risks.

Debates

Reporting odds ratios for common outcomes

For common outcomes the odds ratio exceeds the risk ratio in magnitude, so reporting logistic-regression odds ratios as if they were relative risks exaggerates effects; commentators recommend either direct estimation of risk/prevalence ratios or explicit conversion, while others defend the odds ratio's mathematical properties.

Key figures

• Jerome Cornfield
• Kenneth Rothman
• Sander Greenland
• Jun Zhang
• Guangyong Zou

Seminal works

• davies-1998
• zhang-yu-1998
• zou-2004

Frequently asked questions

What is the difference between a risk ratio and an odds ratio?

A risk ratio compares the probability of the outcome between groups, while an odds ratio compares the odds; they are close when the outcome is rare but the odds ratio is further from one than the risk ratio when the outcome is common.

Why do case-control studies report odds ratios rather than risk ratios?

Because a case-control study fixes how many cases and non-cases are sampled, it cannot estimate the underlying risks, so it reports the odds ratio, which can still be computed from the table and estimates the association of interest.

How can I estimate a risk ratio directly in an adjusted analysis?

Log-binomial regression and the modified-Poisson approach with robust variance estimate risk or prevalence ratios directly, avoiding the odds-ratio inflation that logistic regression produces when the outcome is common.