Databases Reference
In-Depth Information
Role of Sample Size and Determinants
in Granularity of Contingency Matrix
Shusaku Tsumoto
Department of Medical Informatics,
Shimane University, School of Medicine
89-1 Enya-cho, Izumo 693-8501, Japan
tsumoto@med.shimane-u.ac.jp
Summary.
This paper gives a empirical analysis of determinant, which empirically
validates the trade-off between sample size and size of matrix. 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.
1 Introduction
Although independence is a very important concept, it has not been fully and
formally investigated as a relation between two attributes. Tsumoto introduces
linear algebra into formal analysis of a contigency table [1]. The results give
the following interesting results. First, a contingency table can be viewed as
comparison between two attributes with respect to information granularity.
Second, algebra is a key point of analysis of this table. A contingency table can
be viewed as a matrix and several operations and ideas of matrix theory are
introduced into the analysis of the contingency table. Especially, The degree
of independence, rank plays a very important role in extracting a probabilistic
model from a given contingency table.
Then, thirdly, the results of determinantal divisors show that 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 [2]. This gives a
