Extended Second-Order Moment Symmetry Model and Decomposition of Symmetry Model for Ordinal Square Contingency Tables

This study proposes the extended second-order moment symmetry model in square contingency tables with same row and column ordinal classifications. The proposed model extends the previous second-order moment symmetry model by incorporating an asymmetry parameter to capture complex structures. The proposed and previous second-order moment symmetry models represent the asymmetry and symmetry structure of the covariance matrix, respectively. The previous study showed the decomposition of the symmetry model using the second-order moment symmetry model. This study shows the decomposition of the second-order moment symmetry model using the proposed model and the decomposition of the symmetry model applying its decomposition. The usefulness of proposed model is illustrated with examples of two real world datasets (cross-classified education of parents and cross-classified social group of fathers and their daughters).