Statistical Independence and Determinants in a Contingency Table – Interpretation of Pearson Residuals based on Linear Algebra –

Abstract

This paper analyzes pearson residuals, which is an important element of chi-square test statistic, in a contingency table from the viewpoint of matrix theory as follows. First, a given contingency table is viewed as a matrix and the residual of each element in a matrix are obtained as the difference bewteen observed values and expected values calculated by marginal distributions. Then, each residual σ _{ij} is decomposed into the linear sum of the 2 × 2 subderminants of a original matrix, except for i-th column and j-th row. Furthermore, the number of the determinants is equal to the degree of freedom for the chi-square test statistic for a given contingency table. Thus, 2 × 2 subdeterminants in a contingencymatrix determine the degree of statistical independence of two attributes as elementary granules.