Degree of Dependence as Granularity in a Contingency Table

The degree of granularity of a contingency table is closely related with that of dependence of contingency tables. We investigate these relations from the viewpoints of determinantal devisors and determinants. From the results of determinantal divisors, it seems that the devisors provide information on the degree of dependencies between the matrix of the whole elements and its submatrices and the increase of the degree of granularity may lead to that of dependence. However, the other approach shows that 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.