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

Keita Nakamura; Tomoyuki Nakagawa; Kouji Tahata

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arXiv · 2506.02474

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

Keita Nakamura·Tomoyuki Nakagawa·Kouji Tahata

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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.

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Keita Nakamura, Tomoyuki Nakagawa, Kouji Tahata. 2025-06-03. Quasi-symmetry and geometric marginal homogeneity: A simplicial approach to square contingency tables. https://arxiv.org/abs/2506.02474

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