scholarly journals Measure of Departure From Local Symmetry for Square Contingency Tables

2019 ◽  
Vol 8 (2) ◽  
pp. 140
Author(s):  
Yusuke Saigusa ◽  
Mitsuhiro Takami ◽  
Aki Ishii ◽  
Sadao Tomizawa
Keyword(s):  
Shannon Entropy ◽  
Diversity Index ◽  
Contingency Tables ◽  
Local Symmetry ◽  
Harmonic Mean ◽  
Symmetric Structure ◽  
Symmetry Model

For square contingency tables, this paper considers the local symmetry model which indicates that there is a symmetric structure of probabilities for only one of pairs of symmetric cells. Also it proposes the measure to express the degree of departure from the local symmetry model. The measure is expressed as the weighted harmonic mean of the diversity index including the Shannon entropy. Examples are given.

scholarly journals Measure of departure from cumulative local symmetry for square contingency tables having ordered categories

2020 ◽  
Vol 57 (1) ◽  
pp. 23-35
Author(s):  
Yusuke Saigusa ◽  
Tomomasa Takada ◽  
Aki Ishii ◽  
Tomoyuki Nakagawa ◽  
Sadao Tomizawa
Keyword(s):  
Diversity Index ◽  
Contingency Tables ◽  
Local Symmetry ◽  
Women Patients ◽  
Ordered Categories ◽  
Case Examples ◽  
Symmetry Model ◽  
Proposed Model ◽  
Nominal Categories ◽  
Table Data

SummaryFor square contingency tables with nominal categories, a local symmetry model which indicates the symmetric structure of probabilities for only one pair of symmetric cells is proposed. For ordinal square tables, the present paper proposes (1) another local symmetry model for cumulative probabilities from the upper-right and lower-left corners of the table, and (2) a measure to represent the degree of departure from the proposed model. The measure has the form of a weighted harmonic mean of the diversity index, which includes the Shannon entropy as a special case. Examples are given in which the proposed method is applied to square table data on decayed teeth in Japanese women patients.

Decomposition of Bivariate Symmetric Density Function

1996 ◽  
Vol 46 (1-2) ◽  
pp. 129-134
Author(s):  
Takashi Sadao Tomizawa Seo ◽  
Jun-Ichiro Minaguchi
Keyword(s):  
Density Function ◽  
Contingency Tables ◽  
Marginal Homogeneity ◽  
Symmetric Density ◽  
Symmetry Model ◽  
Bivariate Density

For the analysis of square contingency tables, it is known that the symmetry model holds if and only if both the quasi-symmetry and the marginal homogeneity models hold (Caussinus, 1965; Bishop et al., 1975, p. 287). This paper shows that a similar decomposition for bivariate density function (instead of cell probabilities) holds.

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