An index for discriminating the direction of anti-sum-asymmetry for square contingency tables

A square contingency table is a special case of a two-way contingency table wherein the row and column variables consist of the same classification. In square contingency tables, observations tend to be concentrated in the main diagonal cells of the table or nearby, and independence between row and column variables often does not hold. The anti-sum-symmetry model has a symmetrical structure with respect to the anti-diagonal cells of the table. The anti-sum-symmetry model should be applied to square contingency tables with ordered categories. The existing index, which can measure the degree of departure from the anti-sum-symmetry model, cannot discriminate between perfect upper and lower anti-sum-asymmetries. To address this issue, we propose here a directional index which can discriminate between the two types of anti-sum-asymmetries. The proposed index can be interpreted as an index for measuring the degree of perfect anti-sum-asymmetry. We derive an approximately unbiased estimator and an approximate confidence interval for the proposed index. The utility of the proposed index is demonstrated through a real data analysis.