Contingency Table and Granularity

This paper gives a matrix-theory based approach to a contingency table and shows that sample size gives a strong constraints on its granularity. In the former studies, relations between degree of granularity and dependence of contingency tables are given from the viewpoint of determinantal divisors and sample size. The nature of determinantal divisors shows that the increase of the degree of granularity may lead to that of dependence. However, a constraint on the sample size of a contingency table is very strong, which leads to the evaluation formula where the increase of degree of granularity gives the decrease of dependency. This paper gives a further study of the nature of sample size effect on the degree of dependency in a contingency matrix. The results show that sample size will restrict the nature of matrix in a combinatorial way, which suggests that the dependency is closely related with integer programming.