Cohen's linearly weighted kappa is a weighted average

An agreement table with n ∈N≥3 ordered categories can be collapsed into n−1 distinct 2×2 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 2× 2 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 2 × 2 tables, where the weights are the denominators of the 2 × 2 kappas. In addition, it is shown that similar results and interpretations hold for linearly weighted kappas for multiple raters.