Conditional inequalities between Cohen’s kappa and weighted kappas

Cohen’s kappa and weighted kappa are two standard tools for describing the degree of agreement between two observers on a categorical scale. For agreement tables with three or more categories, popular weights for weighted kappa are the so-called linear and quadratic weights. It has been frequently observed in the literature that, when Cohen’s kappa and the two weighted kappas are applied to the same agreement table, the value of the quadratically weighted kappa is higher than the value of the linearly weighted kappa, which in turn is higher than the value of Cohen’s kappa. This paper considers a sufficient condition for this double inequality.