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# Odds ratio

The odds ratio is a measure of effect size particularly important in Bayesian statistics and logistic regression.

It is defined as the ratio of the odds of an event occurring in one group to the odds of it occurring in another group, or to a sample-based estimate of that ratio. These groups might be men and women, an experimental group and a control group, or any other dichotomous classification. If the probabilities of the event in each of the groups are p (first group) and q (second group), then the odds ratio is:

${ p/(1-p) \over q/(1-q)}=\frac{\;p(1-q)\;}{\;q(1-p)\;}.$

An odds ratio of 1 indicates that the condition or event under study is equally likely in both groups. An odds ratio greater than 1 indicates that the condition or event is more likely in the first group. And an odds ratio less than 1 indicates that the condition or event is less likely in the first group. The odds ratio must be greater than or equal to zero. As the odds of the first group approaches zero, the odds ratio approaches zero. As the odds of the second group approaches zero, the odds ratio approaches positive infinity.

For example, suppose that in a sample of 100 men, 90 have drunk wine in the previous week, while in a sample of 100 women only 20 have drunk wine in the same period. The odds of a man drinking wine are 90 to 10, or 9:1, while the odds of a woman drinking wine are only 20 to 80, or 1:4 = 0.25:1. Now, 9/0.25 = 36, so the odds ratio is 36, showing that men are much more likely to drink wine than women. Using the above formula for the calculation yields:

${ 0.9/0.1 \over 0.2/0.8}=\frac{\;0.9\times 0.8\;}{\;0.1\times 0.2\;} ={0.72 \over 0.02} = 36.$

This example also shows how odds ratios can sometimes seem to overstate relative positions: in this sample men are 4.5 times more likely to have drunk wine than women, but have 36 times the odds.

Taking the logarithm of the odds ratio ameliorates this effect, and also improves symmetry. For example, using natural logarithms, an odds ratio of 36 maps to 3.584, an odds ratio of one maps to zero, and an odds ratio of 1/36 maps to -3.584.

The logarithm of the odds ratio is the difference of the logits of the probabilities.

The increased use of logistic regression in medical and social science research means that the odds ratio is commonly used as a means of expressing the results in some forms of clinical trials, in survey research, and in epidemiology, such as in case-control studies. It is often abbreviated "OR" in reports. When data from multiple surveys is combined, it will often be expressed as "Pooled OR".