Hadamard Matrices, Quaternions, and the Pearson Chi-square Statistic

TL;DR

This paper introduces a symbolic decomposition of the Pearson chi-square statistic using Hadamard-type matrices, revealing insights into the structure of cell probabilities and improving understanding of test sensitivity and power.

Contribution

It develops a novel decomposition method for the Pearson chi-square statistic based on Hadamard matrices and orthogonal designs, linking matrix theory with statistical testing.

Findings

Decomposition is feasible for small powers of 2 with distinct cell probabilities.

When cell probabilities are equal, the decomposition simplifies to Hadamard matrix-based methods.

Simulations show the method's sensitivity to distributional changes and potential power improvements.

Abstract

We present a symbolic decomposition of the Pearson chi-square statistic with unequal cell probabilities, by presenting Hadamard-type matrices whose columns are eigenvectors of the variance-covariance matrix of the cell counts. All of the eigenvectors have non-zero values so each component test uses all cell probabilities in a way that makes it intuitively interpretable. When all cell probabilities are distinct and unrelated we establish that such decomposition is only possible when the number of multinomial cells is a small power of 2. For higher powers of 2, we show, using the theory of orthogonal designs, that the targeted decomposition is possible when appropriate relations are imposed on the cell probabilities, the simplest of which is when the probabilities are equal and the decomposition is reduced to the one obtained by Hadamard matrices. Simulations are given to illustrate the sensitivity of various components to changes in location, scale skewness and tail probability, as well as to illustrate the potential improvement in power when the cell probabilities are changed.