Directional Index for Measuring Global Symmetry in Square Tables

Abstract

An index based on the power divergence, which represents the degree of deviation from global symmetry, has been proposed for square tables with ordinal categories. However, this index cannot distinguish between the two directions of global asymmetry (i.e., whether (1) all observations concentrate in the lower triangular cells of the table, or whether (2) they concentrate in the upper triangular cells). This study proposes a directional index based on an arc–cosine function that can distinguish direction. The proposed index is valuable for comparing degrees of deviation from global symmetry for several types of tables. Numerical examples show the utility of the proposed index using two datasets, in which the existing index has the same value. We evaluate the utility of the proposed index by applying to real data of clinical study. We consider that the proposed index gives results easier to interpret than the existing index.