Quasi-symmetry and geometric marginal homogeneity: A simplicial approach to square contingency tables

TL;DR

This paper introduces a geometric framework for analyzing square contingency tables using Aitchison geometry, defining new concepts like geometric marginal homogeneity and decomposing skew-symmetry.

Contribution

It presents a novel geometric approach to symmetry analysis in contingency tables, including new subspace characterizations and a decomposition of skew-symmetry.

Findings

Quasi-symmetry and geometric marginal homogeneity form subspaces in the simplex.

Skew-symmetry can be orthogonally decomposed into measures of departure.

Application to vision data demonstrates the methodology's effectiveness.

Abstract

Square contingency tables are traditionally analyzed with a focus on the symmetric structure of the corresponding probability tables. We view probability tables as elements of a simplex equipped with the Aitchison geometry. This perspective allows us to present a novel approach to analyzing symmetric structure using a compositionally coherent framework. We present a geometric interpretation of quasi-symmetry as an e-flat subspace and introduce a new concept called geometric marginal homogeneity, which is also characterized as an e-flat structure. We prove that both quasi-symmetric tables and geometric marginal homogeneous tables form subspaces in the simplex, and demonstrate that the measure of skew-symmetry in Aitchison geometry can be orthogonally decomposed into measures of departure from quasi-symmetry and geometric marginal homogeneity. We illustrate the application and effectiveness of our proposed methodology using data on unaided distance vision from a sample of women.