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 A in the presence of B and the odds of A in the absence of B, or equivalently (due to symmetry), the ratio of the odds of B 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: 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. 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 A in the presence of B and the odds of A in the absence of B, or equivalently (due to symmetry), the ratio of the odds of B 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: 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. Note that the odds ratio is symmetric in the two events, and there is no causal direction implied (correlation does not imply causation): a positive OR does not establish that B causes A, or that A causes B. Two similar statistics that are often used to quantify associations are the risk ratio (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. Imagine there is a rare disease, afflicting, say, only one in many thousands of adults in a country. Imagine we suspect that being exposed to something (say, having had a particular sort of injury in childhood) makes one more likely to develop that disease in adulthood. The most informative thing to compute would be the risk ratio, RR. To do this in the ideal case, for all the adults in the population we would need to know whether they (a) had the exposure to the injury as children and (b) whether they developed the disease as adults. From this we would extract the following information: the total number of people exposed to the childhood injury, N E , {displaystyle N_{E},} out of which D E {displaystyle D_{E}} developed the disease and H E {displaystyle H_{E}} stayed healthy; and the total number of people not exposed, N N , {displaystyle N_{N},} out of which D N {displaystyle D_{N}} developed the disease and H N {displaystyle H_{N}} stayed healthy. Since N E = D E + H E {displaystyle N_{E}=D_{E}+H_{E}} and similarly for the N N {displaystyle N_{N}} numbers, we only have four independent numbers, which we can organize in a table: