Cohen’s linearly weighted kappa is a weighted average

An n × n agreement table F = {fij} with n ≥ 3 ordered categories can for fixed m (2 ≤ m ≤ n − 1) be collapsed into $${\binom{n-1}{m-1}}$$ distinct m × m tables by combining adjacent categories. It is shown that the components (observed and expected agreement) of Cohen’s weighted kappa with linear weights can be obtained from the m × m subtables. A consequence is that weighted kappa with linear weights can be interpreted as a weighted average of the linearly weighted kappas corresponding to the m × m tables, where the weights are the denominators of the kappas. Moreover, weighted kappa with linear weights can be interpreted as a weighted average of the linearly weighted kappas corresponding to all nontrivial subtables.