Affiliations: Department of Medical Informatics, Shimane University,
School of Medicine, Enya-cho Izumo City, Shimane 693-8501 Japan.
tsumoto@computer.org, hirano@ieee.org
Note: [] Address for correspondence: Department of Medical Informatics,
Faculty of Medicine, Shimane University, Enya-cho Izumo City, Shimane 693-8501
Japan
Abstract: This paper analyzes pearson residuals, which is an important element
of chi-square test statistic, in a contingency table from the viewpoint of
matrix theory as follows. First, a given contingency table is viewed as a
matrix and the residual of each element in a matrix are obtained as the
difference bewteen observed values and expected values calculated by marginal
distributions. Then, each residual σ_{ij} is decomposed into the linear sum of the
2 × 2 subderminants of a original matrix, except for i-th column
and j-th row. Furthermore, the number of the determinants is equal to the
degree of freedom for the chi-square test statistic for a given contingency
table. Thus, 2 × 2 subdeterminants in a contingencymatrix
determine the degree of statistical independence of two attributes as
elementary granules.
Keywords: Contingency Table, Chi-square Distribution, Matrix Theory
