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 $$n\in \mathbb N _{\ge 3}$$ ordered categories can be collapsed into $$n-1$$ distinct $$2\times 2$$ tables by combining adjacent categories. Weighted kappa with linear weights is a weighted average of the kappas corresponding to the $$2\times 2$$ tables, where the weights are the denominators of the $$2\times 2$$ 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 $$2\times 2$$ 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.