An Index for the Degree and Directionality of Asymmetry for Square Contingency Tables with Ordered Categories

Abstract

For square contingency tables with ordered categories, an index based on Cressie and Read's power divergence (or Patil and Taillie's diversity index) has been proposed in order to measure the degree of departure from symmetry. Although there are two types of maximum asymmetry (i.e., whether (1) all the observations concentrate in the top-right cell in the table, or (2) they concentrate in the bottom-left cell), the existing index cannot distinguish the two directions of maximum asymmetry. This paper proposes a directional index based on an arc-cosine function in order to simultaneously represent the degree and directionality of asymmetry. The proposed index would be useful for comparing degrees of asymmetry for several square contingency tables. Numerical examples show the utility of the proposed index using some datasets. We evaluate the usefulness of the proposed index by applying it to real data of the clinical study. The proposed index provides analysis results that are easier to interpret than the existing index.