odds ratio

The US Centers for Disease Control and Prevention, in its Principles of Epidemiology in Public Health Practice, describes an odds ratio as a measure of association that quantifies the relationship between an exposure with two categories and a health outcome. It is calculated from a two by two table as ad divided by bc, which is why it is sometimes called the cross-product ratio. As with any ratio measure, 1.0 means the two groups are alike, a figure above 1.0 means more of the outcome in the first group and a figure below 1.0 means less of it.

Odds and risk are different arithmetic. Risk divides the people who had the outcome by everyone in the group. Odds divides the people who had it by the people who did not. The CDC course notes that an odds ratio provides a reasonable approximation of the risk ratio when the health outcome is uncommon, and its own worked example shows the gap opening: a population with a 5.0 percent risk among the exposed and 1.0 percent among the unexposed yields a risk ratio of 5.0 and an odds ratio of 5.2. The commoner the outcome, the wider that gap grows, and always in the direction that puts the odds ratio further from 1 than the risk ratio, which makes the effect read as larger.

Odds ratios appear at all, rather than the plainer risk ratio, because some designs cannot produce anything else. The CDC course points out that an odds ratio can be calculated from a case-control study, where neither a risk ratio nor a rate ratio can be. They also fall out of logistic regression, the usual way of adjusting a yes or no outcome for other factors, which is why they turn up in trials and in pooled analyses too. Read one as a ratio of odds, not as a statement that one group's chance of the outcome was that many times the other's.