https://doi.org/10.1351/goldbook.15939
Quotient obtained by dividing one set of odds by another. The terms "odds" or "odds ratio" are defined differently according to the situation under discussion. Consider the following notation for the distribution of a binary exposure and a disease in a population or a sample. \[\begin{array}{l c c} & Exposed & Nonexposed \\ \rm{Disease} & a & b \\ \rm{No\ disease} & c & d \\ \end{array}\] The odds ratio (cross-product ratio) is \(ad/bc\).
Notes:
- The exposure-odds ratio for a set of case control data is the ratio of the odds in favor of exposure among the cases (\(a/b\)) to the odds in favor of exposure among non-cases (\(c/d\)), which is equal to \(ad/(bc)\). With incident cases, unbiased subject selection, and a "rare" disease (say, under \(\pu{2\%}\) cumulative incidence rate over the study period), \(ad/bc\) is an approximate estimate of the risk ratio. With incident cases, unbiased subject selection, and density sampling of controls, \(ad/bc\) is an estimate of the ratio of the person-time incidence rates (force of morbidity) in the exposed and unexposed. No rarity assumption is required for this.
- The disease-odds (rate-odds) ratio for a cohort or cross-section is the ratio of the odds in favor of disease among the exposed population (\(a/c\)) to the odds in favor of disease among the unexposed (\(b/d\)), which is equal to \(ad/bc\) and hence is equal to the exposure-odds ratio for the cohort or cross-section.
- The prevalence-odds ratio refers to an odds ratio derived cross-sectionally, as, for example, an odds ratio derived from studies of prevalent (rather than incident) cases.
- The risk-odds ratio is the ratio of the odds in favor of getting disease, if exposed, to the odds in favor of getting disease if not exposed. The odds ratio derived from a cohort study is an estimate of this.