# Symmetry of Square Contingency Tables Using Simplicial Geometry

## Authors

• Keita Nakamura Department of Information Sciences, Tokyo University of Science
• Tomoyuki Nakagawa School of Data Science, Meisei University
• Kouji Tahata Department of Information Sciences, Tokyo University of Science

## Abstract

Two-way contingency tables illustrate the relationship between two discrete variables. Their corresponding probability tables can be
regarded as an element in a simplex. Herein we discuss the symmetry of a square contingency table with the same row and column
classifications. Specifically, we identify symmetric probability tables as a linear subspace using the Aitchison geometry of the
simplex. Then given a probability table, an orthogonal projection onto the symmetric subspace yields the nearest symmetric table. The
(i, j) cell of the nearest symmetric table is characterized as the geometric mean of symmetric cells. This characterization does not
agree with the standard maximum likelihood estimators, except in the symmetric case. The original probability table is subsequently
decomposed into symmetric and skew-symmetric tables, which are orthogonal to each other. Finally, we develop a method to test the
symmetry of a contingency table based on a parametric bootstrap and provide an example.

2024-09-16

## How to Cite

Nakamura, K., Nakagawa, T., & Tahata, K. (2024). Symmetry of Square Contingency Tables Using Simplicial Geometry. Austrian Journal of Statistics, 53(4), 85–98. https://doi.org/10.17713/ajs.v53i4.1845

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