Decompositions for Extended Double Symmetry Models in Square Contingency Tables with Ordered Categories

For square contingency tables with ordered categories, Tomizawa (1992) proposed three kinds of double symmetry models, whose each has a structure of both symmetry about the main diagonal and asymmetry about the reverse diagonal of the table. This paper proposes the extensions of those models and gives the decompositions for three kinds of double symmetry models into the extended quasi double symmetry models, the weighted marginal double symmetry models, and the balance models. Those decompositions are applied to two kinds of data on unaided distance vision.