Statistical Independence and Determinants in a Contingency Table: Interpretation of Pearson Residuals based on Linear Algebra

Shusaku Tsumoto; Shoji Hirano

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Statistical Independence and Determinants in a Contingency Table: Interpretation of Pearson Residuals based on Linear Algebra

机译：列联表中的统计独立性和行列式：基于线性代数的皮尔逊残差解释

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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 subde-terminants in a contingency matrix determine the degree of statistical independence of two attributes as elementary granules.

机译：本文从矩阵理论的角度，在列联表中分析了作为卡方检验统计量重要组成部分的皮尔逊残差。首先，将给定的列联表视为矩阵，并将矩阵中每个元素的残差作为观察值与通过边际分布计算的期望值之间的差来获得。然后，每个残差σ_（ij）分解为原始矩阵（第I列和第j行除外）的2×2子线性的线性和。此外，行列式的数量等于给定列联表的卡方检验统计量的自由度。因此，列联矩阵中的2×2个子行列式确定了作为基本粒子的两个属性的统计独立性程度。

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来源
《Fundamenta Informaticae》 |2009年第3期|251-267|共17页
作者
Shusaku Tsumoto; Shoji Hirano;
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作者单位

Department of Medical Informatics Shimane University, School of Medicine Enya-cho Izumo City, Shimane 693-8501 Japan;

Department of Medical Informatics Shimane University, School of Medicine Enya-cho Izumo City, Shimane 693-8501 Japan;

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收录信息美国《科学引文索引》(SCI);美国《工程索引》(EI);
原文格式 PDF
正文语种 eng
中图分类
关键词
contingency table; chi-square distribution; matrix theory;

机译：列联表;卡方分布;矩阵论;

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Statistical Independence and Determinants in a Contingency Table: Interpretation of Pearson Residuals based on Linear Algebra

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