Anti-Linear-Diagonals-Parameter Symmetry Model and Orthogonal Decomposition of Anti-Symmetry for Square Contingency Tables with Ordinal Classifications

Abstract

For $R \times R$ square contingency tables with the same ordinal classifications for rows and columns, this study investigates models in which the relationship between the row and column variables is symmetric or asymmetric with respect to the anti-diagonal, rather than the main diagonal. The recently proposed anti-diagonals-parameter symmetry model includes $R - 1$ asymmetric parameters and is capable of representing complex asymmetric structures. However, this model is saturated in the following cells: the cell in the first row and first column, the cell in the $R$th row and $R$th column, and all anti-diagonal cells. Since observed frequencies in square contingency tables tend to concentrate along the main diagonal, a model saturated only in the anti-diagonal cells may be more appropriate. We propose the anti-linear diagonals-parameter symmetry model, which captures asymmetry with respect to the anti-diagonal. The proposed model is saturated solely in the anti-diagonal cells and, like the anti-diagonals-parameter symmetry model, expresses how the degree of asymmetry varies according to the distance from the anti-diagonal. Furthermore, we demonstrate a decomposition of the anti-symmetry model using the proposed model and derive an orthogonal decomposition of the test statistic for the anti-symmetry model.