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

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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Measure of Departure From Local Symmetry for Square Contingency Tables

International Journal of Statistics and Probability ◽

10.5539/ijsp.v8n2p140 ◽

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.

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

Austrian Journal of Statistics ◽

10.17713/ajs.v47i3.701 ◽

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.

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

International Journal of Statistics and Probability ◽

10.5539/ijsp.v5n4p38 ◽

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.

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

Austrian Journal of Statistics ◽

10.17713/ajs.v38i3.271 ◽

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.

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