An index to simultaneously analyze the degree and directionality of departure from global marginal homogeneity in square contingency tables

For square contingency tables with ordered categories, an index based on Kullback–Leibler information (or Shannon entropy) has been proposed in order to measure the degree of departure from global marginal homogeneity. Although there are two types of maximum global marginal inhomogeneity [i.e., whether (1) all observations concentrate in the lower-left triangle cells in the table, or whether (2) they concentrate only in the upper-right triangle cells], the existing index cannot distinguish the two directions of global marginal inhomogeneity. This study proposes a directional index based on an arc-cosine function in order to simultaneously analyze the degree and directionality of departure from global marginal homogeneity. The proposed index would be useful for comparing degrees of departure from global marginal homogeneity 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 useful of the proposed index by applying it to real data of clinical study, and consider that the proposed index produces results that are easier to interpret than the existing index.