Cohen's Linearly Weighted Kappa is a Weighted Average of 2x2 Kappas

An agreement table with naacenter dot(a parts per thousand yen3) ordered categories can be collapsed into n-1 distinct 2x2 tables by combining adjacent categories. Vanbelle and Albert (Stat. Methodol. 6:157-163, 2009c) showed that the components of Cohen's weighted kappa with linear weights can be obtained from these n-1 collapsed 2x2 tables. In this paper we consider several consequences of this result. One is that the weighted kappa with linear weights can be interpreted as a weighted arithmetic mean of the kappas corresponding to the 2x2 tables, where the weights are the denominators of the 2x2 kappas. In addition, it is shown that similar results and interpretations hold for linearly weighted kappas for multiple raters.