A Kraemer-type Rescaling That Transforms the Odds Ratio into the Weighted Kappa Coefficient

In a validity study a dichotomous variable Y is often compared to a ‘gold standard’ variable X. For example, in a medical test evaluation one has a ‘gold standard’ evaluation of the presence/absence or type of a disease against which a test is assessed. A 2× 2 study can be summarized in a table like Table 1 (Warrens, 2008a, 2008b, 2009). In Table 1, the four proportions a, b, c, and d characterize the joint distribution of the variables X and Y . The row and column totals are the marginal distributions that result from summing the joint proportions. We denote these by p1 and q1 for variable X and by p2 and q2 for variable Y (Warrens, 2008c, 2008d). The odds ratio is a widely used measure of 2 × 2 association, and probably the most widely used measure in epidemiology (Edwards, 1963; Fleiss, 2003; Kraemer, 2004). The formula of the odds ratio in terms of proportions a, b, c, and d is OR = ad/bc. The odds ratio is the ratio of the odds of an event occurring in one group to the odds of it occurring in another group. These groups might be any other dichotomous classification. 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 event is more likely in the first group. Another statistic of 2 × 2 association is the weighted kappa index (Spitzer, Cohen, Fleis, & Endicott, 1967; Vanbelle & Albert, 2009). It is the unique measure that is based on an acknowledgment that the clinical consequences of a false negative may be quite different from the clinical consequences of a false positive (Bloch & Kraemer, 1989; Kraemer, Periyakoil, & Noda, 2004). A real number r ∈ [0,1] must be specified a priori indicating the relative importance of false negatives to false positives. The sample estimator of the weighted kappa (Bloch & Kraemer, 1989) is κ(r)= ad − bc rp1q2 + (1 − r)p2q1 . (1)