A1323
Title: Orthogonal decomposition of probability tables with Aitchison geometry for symmetry assessment
Authors: Keita Nakamura - Tokyo University of Science (Japan) [presenting]
Tomoyuki Nakagawa - Meisei University (Japan)
Kouji Tahata - Tokyo University of Science (Japan)
Abstract: We propose a novel approach to analyzing symmetry in two-way contingency tables using Aitchison geometry of the simplex. We focus on square contingency tables with identical row and column classifications, introducing a method that identifies symmetric probability tables as a linear subspace within the Aitchison geometry. This enables orthogonal projection of any probability table onto the symmetric subspace, yielding the nearest symmetric table. A key feature of this projection is that each cell in the resulting symmetric table is represented by the normalized geometric mean of corresponding symmetric cells in the original table. This characterization differs from standard maximum likelihood estimators, except when the original table is already symmetric. The probability tables are decomposed into symmetric and skew-symmetric components, which are orthogonal to each other. We develop measures to quantify the degree of asymmetry and present a symmetry test based on these measures using a parametric bootstrap approach. The practical application of the method is demonstrated through an example using real-world data.
