Odds ratio

An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of event A taking place in the presence of B, and the odds of A in the absence of B. Due to symmetry, odds ratio reciprocally calculates the ratio of the odds of B occurring in the presence of A, and the odds of B in the absence of A. Two events are independent if and only if the OR equals 1, i.e., the odds of one event are the same in either the presence or absence of the other event. If the OR is greater than 1, then A and B are associated (correlated) in the sense that, compared to the absence of B, the presence of B raises the odds of A, and symmetrically the presence of A raises the odds of B. Conversely, if the OR is less than 1, then A and B are negatively correlated, and the presence of one event reduces the odds of the other event occurring.

Note that the odds ratio is symmetric in the two events, and no causal direction is implied (correlation does not imply causation): an OR greater than 1 does not establish that B causes A, or that A causes B.

Two similar statistics that are often used to quantify associations are the relative risk (RR) and the absolute risk reduction (ARR). Often, the parameter of greatest interest is actually the RR, which is the ratio of the probabilities analogous to the odds used in the OR. However, available data frequently do not allow for the computation of the RR or the ARR, but do allow for the computation of the OR, as in case-control studies, as explained below. On the other hand, if one of the properties (A or B) is sufficiently rare (in epidemiology this is called the rare disease assumption), then the OR is approximately equal to the corresponding RR.

The OR plays an important role in the logistic model.

If we flip an unbiased coin, the probability of getting heads and the probability of getting tails are equal — both are 50%. Imagine we get a biased coin such that, if one flips it, one is twice as likely to get heads than tails (i.e., the odds double: from 1:1 to 2:1). The new probabilities would be 66.666...% for heads and 33.333...% for tails.

Suppose a radiation leak in a village of 1,000 people increased the incidence of a rare disease. The total number of people exposed to the radiation was ${\displaystyle V_{E}=400,}$ out of which ${\displaystyle D_{E}=20}$ developed the disease and ${\displaystyle H_{E}=380}$ stayed healthy. The total number of people not exposed was ${\displaystyle V_{N}=600,}$ out of which ${\displaystyle D_{N}=6}$ developed the disease and ${\displaystyle H_{N}=594}$ stayed healthy. We can organize this in a contingency table:

The risk of developing the disease given exposure is ${\displaystyle D_{E}/V_{E}=20/400=.05}$ and of developing the disease given non-exposure is ${\displaystyle D_{N}/V_{N}=6/600=.01}$. One obvious way to compare the risks is to use the ratio of the two, the relative risk.

The odds ratio is different. The odds of getting the disease if exposed is ${\displaystyle D_{E}/H_{E}=20/380\approx .0526,}$ and the odds if not exposed is ${\displaystyle D_{N}/H_{N}=6/594\approx .0101\,.}$ The odds ratio is the ratio of the two,