In two-way contingency tables under an asymmetric situation, where the row and column variables are defined as explanatory and response variables, respectively, quantifying the extent to which the explanatory variable contributes to predicting the response variable is important. One quantification method is the association measure, which indicates the degree of association in a range from $0$ to $1$. Among various measures that have been proposed, those based on proportional reduction in error (PRE) are particularly notable for their simplicity and intuitive interpretation. These measures, including Goodman-Kruskal's lambda proposed in 1954, are widely implemented in statistical software such as R and SAS and remain extensively used. However, a well-known limitation of PRE measures is their potential to return a value of $0$ despite no independence. This issue arises because the measures are constructed based solely on the maximum joint and marginal probabilities, failing to make full use of the information available in the contingency table. To address this problem, we propose an extension of PRE measures designed for the proportional reduction in error with multiple categories. The properties of the proposed measures are examined, and their utility is demonstrated through numerical experiments. The results suggest their potential as practical tools in applied statistics.

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