A note on Cohen’s weighted kappa coefficient of agreement with linear weights

Vanbelle and Albert [S. Vanbelle, A. Albert, A note on the linearly weighted kappa coefficient for ordinal scales, Statistical Methodology 6 (2008) 157–163] showed that the observed and expected proportions of agreement of a linearly weighted kappa for an r×rr×r classification table are simply the mean values of the corresponding proportions of r−1r−1 unweighted kappa values for embedded 2×2 classification tables. Utilizing linear agreement weights, this note shows that the extension of linearly weighted kappa to m≥2m≥2 raters and r≥3r≥3 ordered categories for an rmrm classification table is the mean value of the corresponding proportions of r−1r−1 unweighted kappa values for embedded 2m2m classification tables. The procedure is illustrated with m=3m=3 raters and r=4r=4 ordered categories.