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Charcteristics of Pearson Residuals in a Contingency Matrix

This paper shows a formal approach to the analysis of pearson residuals in a contingency matrix. Interestingly, the residual of each element of a matrix, which is defined as the difference between observed value and expected value is represented by linear combination of 2 times 2 submatrices. This fact shows that a 2 times 2 subdeterminant is an elementary granule for statistical independence in a contingency matrix. Furthermore, when the rank of a m times n contingency matrix is r(< min(m, n)), the subdeterminant of a contingency matrix is represented by linear combination of (r - 1)2 subdeterminants.