Quasi-symmetry and geometric marginal homogeneity: A simplicial approach to square contingency tables
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908391430946816 |
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| author | Nakamura, Keita Nakagawa, Tomoyuki Tahata, Kouji |
| author_facet | Nakamura, Keita Nakagawa, Tomoyuki Tahata, Kouji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02474 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quasi-symmetry and geometric marginal homogeneity: A simplicial approach to square contingency tables Nakamura, Keita Nakagawa, Tomoyuki Tahata, Kouji Statistics Theory 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. |
| title | Quasi-symmetry and geometric marginal homogeneity: A simplicial approach to square contingency tables |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2506.02474 |