Odds Ratio

The odds ratio is the ratio of the odds of an outcome in one group to the odds of that outcome in another. It is the natural measure of association for case-control studies and the effect estimate produced by logistic regression, and under certain conditions it approximates the relative risk.

Definition

The odds ratio is the ratio of the odds of an outcome (or exposure) in one group to the odds of that outcome (or exposure) in a comparison group, equal to the cross-product of the cells of a two-by-two table.

Scope

This entry covers the definition of odds and of the odds ratio, how it is read relative to its null value of one, why it is the measure case-control studies and logistic regression produce, the conditions under which it approximates the relative risk, and the ways it can mislead when an outcome is common. It treats the odds ratio as a methodological measure, not as clinical guidance.

Key concepts

• Odds versus probability
• Cross-product of a 2x2 table
• Null value of one
• Symmetry of exposure and disease odds ratios
• Rare-disease approximation to relative risk
• Logistic regression coefficient

Mechanisms

Odds express the probability of an event divided by the probability of its complement, and the odds ratio divides the odds in one group by the odds in another, which for a two-by-two table equals the cross-product of the diagonal cells. A value of 1 means no association. A defining property, set out by Cornfield, is that the odds ratio is symmetric: the ratio of exposure odds (cases versus controls) equals the ratio of disease odds (exposed versus unexposed), which is why case-control studies that sample on outcome can still estimate the disease odds ratio. When the outcome is rare, the odds ratio is close to the relative risk; when the outcome is common, the odds ratio is further from 1 than the corresponding risk ratio, so treating it as a relative risk overstates the effect. It is also the exponentiated coefficient of a logistic regression, which makes it ubiquitous in adjusted analyses.

Clinical relevance

The odds ratio is one of the most widely reported effect measures in the health sciences, especially from case-control studies and logistic-regression analyses, so interpreting it correctly is essential to appraising that evidence. It quantifies the strength of an association and approximates relative risk only when the outcome is rare; it is a measure for reading research, not a directive for individual care.

Epidemiology

The odds ratio is the principal measure of case-control studies, where risks cannot be estimated directly, and the standard output of logistic regression used for confounder adjustment across study designs. Because it is symmetric and mathematically convenient, it appears throughout meta-analysis, though its divergence from the relative risk for common outcomes means it must be interpreted with care.

History

Cornfield's 1951 paper gave the formal argument that the exposure odds ratio observed in a case-control study estimates the disease odds ratio and, when disease is rare, approximates the relative risk, supplying the quantitative justification for the case-control design. As logistic regression became the dominant tool for adjusted analysis in the later twentieth century, the odds ratio became correspondingly pervasive, and the late-1990s literature returned repeatedly to the problem of odds ratios being misread as relative risks for common outcomes.

Debates

Odds ratios misinterpreted as relative risks

When the outcome is common the odds ratio is further from the null than the risk ratio, so reporting or reading it as if it were a relative risk exaggerates the apparent effect; commentators recommend correction or explicit conversion to plausible relative risks.

Key figures

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

Seminal works

• cornfield-1951
• davies-1998
• grant-2014

Frequently asked questions

When does the odds ratio approximate the relative risk?

When the outcome is rare in both groups, the odds and the risks are close, so the odds ratio approximates the relative risk; as the outcome becomes more common, the odds ratio moves further from the null than the corresponding risk ratio.

Why do case-control studies report odds ratios?

Because case-control studies sample by outcome and cannot observe underlying risks, they can estimate odds but not risks; the odds ratio's symmetry means the exposure odds ratio they can measure equals the disease odds ratio of interest.