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.

对于具有有序类别的正方形列联表，已提出了基于Kullback-Leibler信息（或Shannon熵）的索引，以衡量与整体边际同质性的偏离程度。尽管存在两种最大的全局边际不均匀性[即，（1）所有观测值集中在表中的左下三角形单元格中，还是（2）它们仅集中在右上角三角形单元格中]，指数不能区分全局边际不均匀性的两个方向。这项研究提出了一个基于反余弦函数的方向索引，以便同时分析偏离全局边缘同质性的程度和方向。提议的索引对于比较几种类型的表与全局边际同质性的偏离程度将很有用。数值示例显示了使用两个数据集所提出的索引的效用，其中现有索引具有相同的值。我们通过将其应用于临床研究的真实数据来评估所提出的索引的有用性，并认为所提出的索引所产生的结果比现有索引更容易解释。