Hadamard matrices, quaternions, and the Pearson chi-square statistic

The symbolic partitioning of the Pearson chi-square statistic with unequal cell probabilities into asymptotically independent component tests is revisited. We introduce Hadamard-like matrices whose resulting component tests compares the full vector of cell counts. This contributes to making these component tests intuitively interpretable. We present a simple way to construct the Hadamard-like matrices when the number of cell counts is 2, 4 or 8 without assuming any relations between cell probabilities. For higher powers of 2, the theory of orthogonal designs is used to set a priori relations between cell probabilities, in order to establish the construction. 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.