Statistics notes. The odds ratio.

In recent years odds ratios have become widely used in medical reports—almost certainly some will appear in today’s BMJ. There are three reasons for this. Firstly, they provide an estimate (with confidence interval) for the relationship between two binary (“yes or no”) variables. Secondly, they enable us to examine the effects of other variables on that relationship, using logistic regression. Thirdly, they have a special and very convenient interpretation in case-control studies (dealt with in a future note). The odds are a way of representing probability, especially familiar for betting. For example, the odds that a single throw of a die will produce a six are 1 to 5, or 1/5. The odds is the ratio of the probability that the event of interest occurs to the probability that it does not. This is often estimated by the ratio of the number of times that the event of interest occurs to the number of times that it does not. The table shows data from a cross sectional study showing the prevalence of hay fever and eczema in 11 year old children. The probability that a child with eczema will also have hay fever is estimated by the proportion 141/561 (25.1%). The odds is estimated by 141/420. Similarly, for children without eczema the probability of having hay fever is estimated by 928/14 453 (6.4%) and the odds is 928/13 525. We can compare the groups in several ways: by the difference between the proportions, 141/561 − 928/14 453 = 0.187 (or 18.7 percentage points); the ratio of the proportions, (141/ 561)/(928/14 453) = 3.91 (also called the relative risk); or the odds ratio, (141/420)/(928/ 13 525) = 4.89.