# Odds Ratio

## ODDS RATIO

The odds ratio (OR) provides a measure of the strength of relationship between two variables,

Table 1

 Frequencies in a 2 × 2 Table. OUTCOME +ve OUTCOME –ve SOURCE: Courtesy of author. Exposure (outcome positive) a b Exposure (outcome negative) c d

most commonly an exposure and a dichotomous outcome. It is most commonly used in a case-control study where it is defined as "the ratio of the odds of being exposed in the group with the outcome to the odds of being exposed in the group without the outcome." In the standard 2×2 epidemiological table, this ratio can be expressed as the "cross-product" (ad/bc), as seen in Table 1.

This concept can be extended to a situation with multiple levels of exposure (e.g., low, moderate, or high exposure to an environmental containment). One exposure level is assigned as the "reference" level. For each of the remaining exposure levels, one divides the odds of that exposure level in the outcome positive group (compared with the reference level) by the odds of that exposure level in the outcome negative group.

The OR ranges in value from 0 to infinity. Values close to 1.0 indicate no relationship between the exposure and the outcome. Values less than 1.0 suggest a protective effect, while values greater than 1.0 suggest a causative or adverse effect of exposure.

The OR is closely connected to logistic regression. This analytic method models the natural logarithm of the OR as a linear function of the predictor variables. It is a powerful and very common method for the analysis of epidemiological studies.

The OR is one of the most common measures encountered in observational epidemiology. The value of the OR for case-control research was first

Table 2

 Frequencies of Erysipelas by Obesity erysipelas No erysipelas SOURCE: Courtesy of author. Obese 68 97 Non-obese 61 197

recognized by Jerome Cornfield in 1951. His work provided the theoretical base for the application of the case-control approach to studying disease etiology. The OR estimates the incidence-density ratio or the cumulative incidence ratio that would have been observed if it had been feasible to perform a cohort study rather than a case-control study. Depending on the method used to obtain control subjects, the OR either is identical to one of the incidence ratios or is close to them if the disease is rare. Some epidemiologists modify the term to reflect the type of study being done (e.g., prevalence odds ratio, exposure odds ratio, or disease odds ratio).

Although mainly used for the analysis of case-control studies, the odds ratio can also be applied in cross-sectional and cohort studies. It also plays a major role in certain approaches to the metaanalysis of randomized clinical trials (e.g., the Peto method).

An example of the use of the odds ratio can be found in a paper published by A. Dupuy et al. This paper studied 129 patients with erysipelas of the leg and a control group of 294 people without erysipelas of the leg. Obesity was considered as a risk factor. Analysis of the data produced the 2×2 table shown in Table 2.

This gives an OR of (68×197)/(61×97) or 2.3. That is, people with erysipelas are 2.3 times more likely to be obese than people without erysipelas. This supports the suggestion that obesity increases the risk of developing erysipelas.

GEORGE WELLS

(SEE ALSO: Case-Control Study; Epidemiology; Statistics for Public Health)