The Dependence of Chance-Corrected Weighted Agreement Coefficients on the Power Parameter of the Weighting Scheme: Analysis and Measurement

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We consider the dependence of a broad class of chance-corrected weighted agreement coefficients on the weighting scheme that penalizes rater disagreements. The considered class encompasses many existing coefficients with any number of raters, and one real-valued power parameter defines the weighting scheme that includes linear, quadratic, identity, and radical weights. We obtain the first-order and second-order derivatives of the coefficients with respect to the power parameter and decompose them into components corresponding to all pairs of different category distances. Each component compares its two distances in terms of the ratio of observed to expected-by-chance frequency. A larger ratio for the smaller distance than the larger distance contributes to a positive relationship between the power parameter and the coefficient value; the opposite contributes to a negative relationship. We provide necessary and sufficient conditions for the coefficient value to increase or decrease and the relationship to intensify or weaken as the power parameter increases. We use the first-order and second-order derivatives for corresponding measurement. Furthermore, we show how these two derivatives allow other researchers to obtain quite accurate estimates of the coefficient value for unreported values of the power parameter, even without access to the original data.