Symmetry of Square Contingency Tables Using Simplicial Geometry

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.