scholarly journals Ridit Score Type Quasi-Symmetry and Decomposition of Symmetry for Square Contingency Tables with Ordered Categories

2016 ◽  
Vol 38 (3) ◽  
Author(s):  
Kiyotaka Iki ◽  
Kouji Tahata ◽  
Sadao Tomizawa
Keyword(s):  
Contingency Tables ◽  
Ordered Categories ◽  
Marginal Distributions ◽  
Model Based ◽  
Symmetry Model ◽  
Proposed Model ◽  
Log Odds ◽  
The Difference ◽  
Mean Equality

For square contingency tables with the same row and column ordinal classifications, this paper proposes the quasi-symmetry model based on the marginal ridits. The model indicates that the log-odds that an observation will fall in the (i; j) cell instead of in the (j; i) cell, i < j, is proportional to the difference between the average ridit score of row and column marginal distributions for category j and that for category i. This paper also gives atheorem such that the symmetry model holds if and only if both the proposed model and the marginal mean equality model hold. Examples are given.

scholarly journals Asymmetry models based on ordered score and separations of symmetry model for square contingency tables

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shuji Ando
Keyword(s):  
Asymmetry Parameter ◽  
Contingency Tables ◽  
Real Data ◽  
Symmetry Model ◽  
Th Cell ◽  
Proposed Model ◽  
Log Odds ◽  
The Difference ◽  
Mean Equality ◽  
Column Variable

Summary This study proposes two original asymmetry models based on ordered scores for square contingency tables with the same row and column ordinal classifications. The proposed models can be applied to cases in which the scores of all categories are known or unknown. In the proposed models, the log odds for an observation falling in the (i, j)th cell instead of the (j, i)th cell are inversely proportional to the difference of the ordered scores corresponding to categories i and j. The asymmetry parameter of the proposed model can be useful for inferring whether the row variable is stochastically greater than the column variable or vice versa. The proposed models constantly hold when the symmetry model holds, but the converse is not necessarily true. This study also examines what is necessary for a model, in addition to the proposed models, to satisfy the symmetry model, and gives separations of the symmetry model using the proposed and marginal mean equality models. We apply real data to show the utility of the proposed models. The proposed models provide a better fit than that of the existing models.

scholarly journals Log-normal Distribution Type Symmetry Model for Square Contingency Tables with Ordered Categories

2018 ◽  
Vol 47 (3) ◽  
pp. 39-48
Author(s):  
Kiyotaka Iki
Keyword(s):  
Normal Distribution ◽  
Contingency Tables ◽  
Ordered Categories ◽  
Symmetry Model ◽  
Proposed Model ◽  
Type Symmetry ◽  
The Difference ◽  
Log Normal ◽  
Distribution Type ◽  
Log Normal Distribution

For the analysis of square contingency tables with the same row and column ordinal classications, this article proposes a new model which indicates that the log-ratios of symmetric cell probabilities are proportional to the difference between log-row category and log-column category. The proposed model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate log-normal distribution. Also, this article gives the decomposition of the symmetry model using the proposed model with the orthogonality of test statistics. Examples are given. The simulation studies based on bivariate log-normal distribution are given.

scholarly journals Diagonal Exponent Conditional Symmetry Model for Square Contingency Tables with Ordered Categories

2016 ◽  
Vol 5 (4) ◽  
pp. 38
Author(s):  
Kiyotaka Iki ◽  
Akira Shibuya ◽  
Sadao Tomizawa
Keyword(s):  
Contingency Tables ◽  
Main Diagonal ◽  
Test Statistics ◽  
Conditional Symmetry ◽  
Exponential Form ◽  
Ordered Categories ◽  
Expected Frequency ◽  
Symmetry Model ◽  
Proposed Model ◽  
New Models

For square contingency tables with ordered categories, this article proposes new models which indicate that in addition to the structure of asymmetry of the probabilities with respect to the main diagonal of the table, the expected frequency has an exponential form along every subdiagonal of the table. Also it gives the new three kinds of decompositions using the proposed model and proves the orthogonality of the test statistics.

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.

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