This JAMA Guide to Statistics and Methods explains the correct usage of odds ratios in the clinical literature to report the strength of the association between binary outcomes.

Odds ratios frequently are used to present strength of association between risk factors and outcomes in the clinical literature. Odds and odds ratios are related to the probability of a binary outcome (an outcome that is either present or absent, such as mortality). The *odds* are the ratio of the probability that an outcome occurs to the probability that the outcome does not occur. For example, suppose that the probability of mortality is 0.3 in a group of patients. This can be expressed as the odds of dying: 0.3/(1 − 0.3) = 0.43. When the probability is small, odds are virtually identical to the probability. For example, for a probability of 0.05, the odds are 0.05/(1 − 0.05) = 0.052. This similarity does not exist when the value of a probability is large.

Probability and odds are different ways of expressing similar concepts. For example, when randomly selecting a card from a deck, the probability of selecting a spade is 13/52 = 25%. The odds of selecting a card with a spade are 25%/75% = 1:3. Clinicians usually are interested in knowing probabilities, whereas gamblers think in terms of odds. Odds are useful when wagering because they represent fair payouts. If one were to bet $1 on selecting a spade from a deck of cards, a payout of $3 is necessary to have an even chance of winning your money back. From the gambler’s perspective, a payout smaller than $3 is unfavorable and greater than $3 is favorable.

Differences between 2 different groups having a binary outcome such as mortality can be compared using odds ratios, the ratio of 2 odds. Differences also can be compared using probabilities by calculating the *relative risk ratio*, which is the ratio of 2 probabilities. Odds ratios commonly are used to express strength of associations from logistic regression to predict a binary outcome.^{1}

Researchers often analyze a binary outcome using multivariable logistic regression. One potential limitation of logistic regression is that the results are not directly interpretable as either probabilities or relative risk ratios. However, the results from a logistic regression are converted easily into odds ratios because logistic regression estimates a parameter, known as the log odds, which is the natural logarithm of the odds ratio. For example, if a log odds estimated by logistic regression is 0.4 then the odds ratio can be derived by exponentiating the log odds (exp(0.4) = 1.5). It is the odds ratio that is usually reported in the medical literature. The odds ratio is always positive, although the estimated log odds can be positive or negative (log odds of −0.2 equals odds ratio of 0.82 = ...