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Measure of Proportional Reduction in Variation and Measure of Agreement for Contingency Tables with Ordered Categories

Kouji Yamamoto, Sadao Tomizawa


For a two-way table with a nominal explanatory and a nominal response variables, Goodman and Kruskal (1954), and Theil (1970) proposed the measures which describe the proportional reduction in variation (PRV) from the marginal distribution to the conditional distributions of the response. Tomizawa, Seo and Ebi (1997) proposed a generalization of those measures. Tomizawa, Miyamoto and Yajima (2002) proposed a PRV measure for a nominal-ordinal contingency table. Tomizawa and Yukawa (2003) proposed a PRV measure for a square contingency table with ordered categories. First, we propose a PRV measure for a ordinal-ordinal contingency table in which the explanatory and response variables are not defined clearly. The measure proposed is expressed by using Patil-Taillie diversity index including Shannon entropy. Secondly, we propose a measure of degree of agreement for a square contingency table with ordered categories. The proposed measure takes a minimum value zero if and only if the independence between the classifications holds, and a maximum value unity if and only if perfect agreement occurs. The measure is different from the Cohen’s Kappa measure. Examples are given.


agreement, independence, Kappa measure, Kullback-Leibler information, powerdivergence, Shannon entropy

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