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<p>In the analysis of square contingency tables, wherein the row and column variables are composed of the same ordered categories, researchers typically seek to ascertain whether the row and column variables are symmetrical. The degree of asymmetry (i.e., degree of departure from symmetry) between the row and column variables is measured using the index based on Kullback–Leibler divergence. This index value of 1 indicates a perfect upper asymmetry (namely, the cell probabilities in the lower triangle of the table, excluding the diagonal elements, are all zero) or a perfect lower asymmetry (namely, the cell probabilities in the upper triangle of the table, excluding the diagonal elements, are all zero). Consequently, this index cannot discriminate between these two types of asymmetries. The objective of the present paper is to propose a new directional index to discriminate between perfect lower and upper asymmetries. An asymptotically unbiased estimator and an approximate confidence interval for the proposed index are given. We evaluate the performances of those in a finite sample through numerical experiments. The usefulness of the proposed index is demonstrated though real data analysis.</p>

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A directional index to measure the degree of asymmetry for square contingency tables

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  • 发表日期：

    2025年7月21日

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  • Shuji Ando

摘要

In the analysis of square contingency tables, wherein the row and column variables are composed of the same ordered categories, researchers typically seek to ascertain whether the row and column variables are symmetrical. The degree of asymmetry (i.e., degree of departure from symmetry) between the row and column variables is measured using the index based on Kullback–Leibler divergence. This index value of 1 indicates a perfect upper asymmetry (namely, the cell probabilities in the lower triangle of the table, excluding the diagonal elements, are all zero) or a perfect lower asymmetry (namely, the cell probabilities in the upper triangle of the table, excluding the diagonal elements, are all zero). Consequently, this index cannot discriminate between these two types of asymmetries. The objective of the present paper is to propose a new directional index to discriminate between perfect lower and upper asymmetries. An asymptotically unbiased estimator and an approximate confidence interval for the proposed index are given. We evaluate the performances of those in a finite sample through numerical experiments. The usefulness of the proposed index is demonstrated though real data analysis.