Cohen's weighted kappa with additive weights

Cohen's weighted kappa is a popular descriptive statistic for summarizing interrater agreement on an ordinal scale. An agreement table with ordered categories can be collapsed into distinct tables by combining adjacent categories. Weighted kappa with linear weights is a weighted average of the kappas corresponding to the tables, where the weights are the denominators of the kappas. It is shown that the linearly weighted kappa is a special case of a more general weighted kappa that is a weighted average of the kappas. This weighted kappa has additive weights, that is, given initial weights for pairs of adjacent categories the weight for two non-adjacent categories is obtained by adding the weights of all pairs of adjacent categories between the two.