Role of Marginal Distribution in a Contingency Table

Marginal distributions play an central role in statistical analysis of a contingency table. However, when the number of partition becomes large, the contribution from marginal distributions decreases. This paper focuses on a formal analysis of marginal distributions in a contingency table. The main approach is to take the difference between two matrices with the same sample size and the same marginal distributions, which we call difference matrix. The important nature of the difference matrix is that the determinant is equal to 0: when the rank of a matrix is r, the difference between a original matrix and the expected matrix will become r - 1 at most. Since the sum of rows or columns of the will become zero, which means that the information of one rank corresponds to information on the frequency of a contingency matrix. Interestingly, if we take an expected matrix whose elements are the expected values based on marginal distributions, the difference between an original matrix and expected matrix can be represented by linear combination of determinants of 2 times 2 submatrices