Odds and Odds Ratio, Risk Ratio

The main goal of many medical studies is to evaluate the effect of a treatment or the risk of disease under given conditions. This entry introduces and discusses measures of effect and/or risk when the factors or variables of interest are categorical (i.e., nominal) in nature. The discussion focuses on the case of dichotomous variables (factors), that is, those that take only two values that often indicate the presence or absence of a characteristic (disease, treatment, exposure, etc.). The data consist of information on n individuals who have been categorized according to the presence or absence of two factors. The information is presented in a 2 × 2 contingency table like the one in Table 1.

The numbers in the table denote the frequency of each cell. For example, a is the number of individuals for whom both factors were present, and b those for whom A was present and B was absent. Thus, the total number of patients is n = a + b + c + d. Common types of factors are exposure, treatment, and disease. The statistical significance of the association between the two factors is tested using the chisquare test (or the likelihood test). The result of the test is a p value, which measures the chance of the observed relationship under the assumption that there is none. Accordingly, a small p value (e.g., < .05) leads to a “significant” result and the rejection of the assumption of no association. The size of the p value is determined, in a critical way, by the sample size and can't be used to assess the strength of the association. When the association (or effect, when one factor is the “cause” of the other) is significant, the measure of its strength is critical. Below are the most common measures of association.

Definition 1. The relative risk (RR) is defined as

where Pr[ ] indicates probability.

Definition 2. The risk difference (RD), also called attributable risk, is defined as

Definition 3. The odds ratio (OR) is defined as

Table 1 The 2 × 2 contingency table