2025/06/03 by Keita Nakamura, Tomoyuki Nakagawa, Nakamura, Keita +3
Chemistry · Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Graph theory and applications #History and advancements in chemistry #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2506.02474
openalex publication_date 2025/06/03 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
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.