Orthogonal Decomposition of the Anti-Symmetry Model via the Anti-Two-Ratios-Parameter Symmetry Model for Square Contingency Tables

Abstract

This paper proposes the anti-two-ratios-parameter symmetry (ATRPS) model for square contingency tables with ordered categories. The proposed model provides a parsimonious two-parameter extension of the anti-symmetry model, separates a global ratio effect from an anti-
diagonal-distance effect, contains the anti-conditional symmetry and anti-linear-diagonals-parameter symmetry models as special cases, and is nested within the anti-diagonals-parameter symmetry model. By applying a column-reversal transformation, we show that the anti-symmetry model holds if and only if the ATRPS, anti-global symmetry, and anti-marginal mean equality models hold simultaneously; equivalently, it holds if and only if the ATRPS and AGSME models hold simultaneously. We also establish an asymptotic orthogonal decomposition of the likelihood ratio statistic. An analysis of grip strength data illustrates the proposed methodology and shows that the departure from anti-symmetry is almost entirely attributable to the AGSME component.