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Symmetry
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Advances in Quasi-Symmetry Models
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Advances in Quasi-Symmetry Models

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (30 April 2022) | Viewed by 11852

Special Issue Editor

Dr. Kouji Tahata

E-Mail Website
Guest Editor
Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba-ken 278-8510, Japan
Interests: categorical data analysis

Special Issue Information

Dear Colleagues,

This Special Issue focuses on contingency table analysis in the statistical analysis of categorical data. The contingency table plays an important role in various fields. The issues of symmetry rather than independence (null association) are specifically considered for the analysis of square contingency tables because many observations tend to concentrate on or near the main diagonal. Caussinus proposed quasi-symmetry and used it as a bridge between symmetry and marginal homogeneity in square tables. The result had an impact on some methodological developments in the statistical analysis of categorical data. A special issue of Annales de la Faculté des Sciences de Toulouse, Mathématiques was published in 2002 which contained the papers written by internationally distinguished authors on topics related to quasi-symmetry. Almost twenty years have passed since then, and many more papers which treat quasi-symmetry from a variety of perspectives have been published. Therefore, we are soliciting contributions (research and review articles) covering a broad range of topics on symmetry and quasi-symmetry for the present Special Issue on “Advances in Quasi-Symmetry Models”.

Dr. Kouji Tahata
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Contingency table (multi-way contingency table)
  • Correspondence analysis (association model)
  • Cumulative probability (marginal distribution)
  • Decomposition (equivalence conditions for symmetry)
  • Information theory (KL divergence, f-divergence)
  • Log-linear model (multiplicative form)
  • Measure (index of degree of departure from symmetry)
  • Test (exact test, orthogonality)

Published Papers (8 papers)

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Research

25 pages, 1328 KiB  
Article
Visualising Departures from Symmetry and Bowker’s X2 Statistic
by Eric J. Beh and Rosaria Lombardo
Symmetry 2022, 14(6), 1103; https://doi.org/10.3390/sym14061103 - 27 May 2022
Cited by 1 | Viewed by 1501
Abstract
Sometimes, the same categorical variable is studied over different time periods or across different cohorts at the same time. One may consider, for example, a study of voting behaviour of different age groups across different elections, or the study of the same variable [...] Read more.
Sometimes, the same categorical variable is studied over different time periods or across different cohorts at the same time. One may consider, for example, a study of voting behaviour of different age groups across different elections, or the study of the same variable exposed to a child and a parent. For such studies, it is interesting to investigate how similar, or different, the variable is between the two time points or cohorts and so a study of the departure from symmetry of the variable is important. In this paper, we present a method of visualising any departures from symmetry using correspondence analysis. Typically, correspondence analysis uses Pearson’s chi-squared statistic as the foundation for all of its numerical and visual features. In the case of studying the symmetry of a variable, Bowker’s chi-squared statistic, presented in 1948, provides a simple numerical means of assessing symmetry. Therefore, this paper shall discuss how a correspondence analysis can be performed to study the symmetry (or lack thereof) of a categorical variable when Bowker’s statistic is considered. The technique presented here provides an extension to the approach developed by Michael Greenacre in 2000. Full article
(This article belongs to the Special Issue Advances in Quasi-Symmetry Models)
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15 pages, 316 KiB  
Article
Measures of Departure from Local Marginal Homogeneity for Square Contingency Tables
by Ken Saito, Nozomi Takakubo, Aki Ishii, Tomoyuki Nakagawa and Sadao Tomizawa
Symmetry 2022, 14(6), 1075; https://doi.org/10.3390/sym14061075 - 24 May 2022
Viewed by 1133
Abstract
When focusing on changes in political party support, it is crucial to determine whether or not there has been a change in the aggregate. From this perspective, various types of marginal homogeneity models have been proposed. We propose local marginal homogeneity models, which [...] Read more.
When focusing on changes in political party support, it is crucial to determine whether or not there has been a change in the aggregate. From this perspective, various types of marginal homogeneity models have been proposed. We propose local marginal homogeneity models, which indicate that there are symmetric structures of probabilities for only one pair of symmetric marginal probabilities or cumulative probabilities. In addition, we propose two measures, one for nominal categories and one for ordered categories, to express the degree of departure from local marginal homogeneity models. We also apply the measures to data and confirm that the measures help compare the degree of departure from the model in several tables. Full article
(This article belongs to the Special Issue Advances in Quasi-Symmetry Models)
9 pages, 262 KiB  
Article
Asymmetry Model Based on Quasi Local Odds Symmetry for Square Contingency Tables
by Shuji Ando
Symmetry 2022, 14(5), 1031; https://doi.org/10.3390/sym14051031 - 18 May 2022
Cited by 1 | Viewed by 1151
Abstract
For the analysis of square contingency tables, the primary objective is to estimate an unknown distribution from presented data. To achieve this objective, we generally use a statistical model that fits the presented data well and has a parsimony. The recently proposed quasi [...] Read more.
For the analysis of square contingency tables, the primary objective is to estimate an unknown distribution from presented data. To achieve this objective, we generally use a statistical model that fits the presented data well and has a parsimony. The recently proposed quasi local odds symmetry (QLOS) model was compared to various models that represent the structure of symmetry or asymmetry, and it provided the best fit performance compared with other models for real data. However, the QLOS model has many parameters, that is, the QLOS model is not the parsimonious model. To address this issue, this study proposes a new model that is more parsimonious than the QLOS model. The proposed model is identical to the QLOS model under the specified condition; it is the asymmetry model based on the QLOS model.Moreover, we compare the proposed model with the existing models, including the QLOS model, and show that the proposed model provides better fit performance than the existing models for real datasets. Full article
(This article belongs to the Special Issue Advances in Quasi-Symmetry Models)
10 pages, 284 KiB  
Article
Quasi Association Models for Square Contingency Tables with Ordinal Categories
by Kengo Fujisawa and Kouji Tahata
Symmetry 2022, 14(4), 805; https://doi.org/10.3390/sym14040805 - 12 Apr 2022
Cited by 3 | Viewed by 1244
Abstract
The analysis of contingency tables focuses on a statistical model instead of independence when the independence between row and column variables does not hold. Many association models have been proposed to indicate the structure of odds ratios. Additionally, symmetry and asymmetry models have [...] Read more.
The analysis of contingency tables focuses on a statistical model instead of independence when the independence between row and column variables does not hold. Many association models have been proposed to indicate the structure of odds ratios. Additionally, symmetry and asymmetry models have been proposed to analyze the cell probabilities of square contingency tables with symmetric or asymmetric structures. This paper proposes an asymmetry plus association model for square contingency tables with ordinal categories and partitioning of the test statistic for goodness-of-fit using our proposed model. Full article
(This article belongs to the Special Issue Advances in Quasi-Symmetry Models)
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11 pages, 296 KiB  
Article
Families of Generalized Quasisymmetry Models: A ϕ-Divergence Approach
by Maria Kateri
Symmetry 2021, 13(12), 2297; https://doi.org/10.3390/sym13122297 - 02 Dec 2021
Cited by 2 | Viewed by 1151
Abstract
The quasisymmetry (QS) model for square contingency tables is revisited, highlighting properties and features on the basis of its alternative definitions. More parsimonious QS-type models, such as the ordinal QS model for ordinal classification variables and models based [...] Read more.
The quasisymmetry (QS) model for square contingency tables is revisited, highlighting properties and features on the basis of its alternative definitions. More parsimonious QS-type models, such as the ordinal QS model for ordinal classification variables and models based on association models (AMs) with homogeneous row and column scores, are discussed. All these models are linked to the local odds ratios (LOR). QS-type models and AMs were extended in the literature for generalized odds ratios other than LOR. Furthermore, in an information-theoretic context, they are expressed as distance models from a parsimonious reference model (the complete symmetry for QS and the independence for AMs), while they satisfy closeness properties with respect to Kullback–Leibler (KL) divergence. Replacing the KL by ϕ divergence, flexible classes of QS-type models for LOR, AMs for LOR, and AMs for generalized odds ratios were generated. However, special QS-type models that are based on homogeneous AMs for LOR have not been extended to ϕ-divergence-based classes so far, or the QS-type models for generalized odds ratios. In this work, we develop these missing extensions, and discuss QS-type models and their generalizations in depth. These flexible families enrich the modeling options, leading to models of better fit and sound interpretation, as illustrated by representative examples. Full article
(This article belongs to the Special Issue Advances in Quasi-Symmetry Models)
10 pages, 346 KiB  
Article
A Generalized Two-Dimensional Index to Measure the Degree of Deviation from Double Symmetry in Square Contingency Tables
by Shuji Ando, Hikaru Hoshi, Aki Ishii and Sadao Tomizawa
Symmetry 2021, 13(11), 2067; https://doi.org/10.3390/sym13112067 - 02 Nov 2021
Viewed by 1085
Abstract
The double symmetry model satisfies both the symmetry and point symmetry models simultaneously. To measure the degree of deviation from the double symmetry model, a two-dimensional index that can concurrently measure the degree of deviation from symmetry and point symmetry is considered. This [...] Read more.
The double symmetry model satisfies both the symmetry and point symmetry models simultaneously. To measure the degree of deviation from the double symmetry model, a two-dimensional index that can concurrently measure the degree of deviation from symmetry and point symmetry is considered. This two-dimensional index is constructed by combining two existing indexes. Although the existing indexes are constructed using power divergence, the existing two-dimensional index that can concurrently measure both symmetries is constructed using only Kullback-Leibler information, which is a special case of power divergence. Previous studies note the importance of using several indexes of divergence to compare the degrees of deviation from a model for several square contingency tables. This study, therefore, proposes a two-dimensional index based on power divergence in order to measure deviation from double symmetry for square contingency tables. Numerical examples show the utility of the proposed two-dimensional index using two datasets. Full article
(This article belongs to the Special Issue Advances in Quasi-Symmetry Models)
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9 pages, 315 KiB  
Article
Two-Dimensional Index of Departure from the Symmetry Model for Square Contingency Tables with Nominal Categories
by Tomotaka Momozaki, Tomoyuki Nakagawa, Aki Ishii, Yusuke Saigusa and Sadao Tomizawa
Symmetry 2021, 13(11), 2031; https://doi.org/10.3390/sym13112031 - 27 Oct 2021
Cited by 1 | Viewed by 1686
Abstract
In the analysis of two-way contingency tables, the degree of departure from independence is measured using measures of association between row and column variables (e.g., Yule’s coefficients of association and of colligation, Cramér’s coefficient, and Goodman and Kruskal’s coefficient). On the other hand, [...] Read more.
In the analysis of two-way contingency tables, the degree of departure from independence is measured using measures of association between row and column variables (e.g., Yule’s coefficients of association and of colligation, Cramér’s coefficient, and Goodman and Kruskal’s coefficient). On the other hand, in the analysis of square contingency tables with the same row and column classifications, we are interested in measuring the degree of departure from symmetry rather than independence. Over past years, many studies have proposed various types of indexes based on their power divergence (or diversity index) to represent the degree of departure from symmetry. This study proposes a two-dimensional index to measure the degree of departure from symmetry in terms of the log odds of each symmetric cell with respect to the main diagonal of the table. By measuring the degree of departure from symmetry in terms of the log odds of each symmetric cell, the analysis results are easier to interpret than existing indexes. Numerical experiments show the utility of the proposed two-dimensional index. We show the usefulness of the proposed two-dimensional index by using real data. Full article
(This article belongs to the Special Issue Advances in Quasi-Symmetry Models)
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10 pages, 283 KiB  
Article
Measure of Departure from Conditional Symmetry Based on Cumulative Probabilities for Square Contingency Tables
by Yusuke Saigusa, Yuta Teramoto and Sadao Tomizawa
Symmetry 2021, 13(10), 1897; https://doi.org/10.3390/sym13101897 - 08 Oct 2021
Viewed by 1267
Abstract
For the analysis of square contingency tables with ordered categories, a measure was developed to represent the degree of departure from the conditional symmetry model in which there is an asymmetric structure of the cell probabilities with respect to the main diagonal of [...] Read more.
For the analysis of square contingency tables with ordered categories, a measure was developed to represent the degree of departure from the conditional symmetry model in which there is an asymmetric structure of the cell probabilities with respect to the main diagonal of the table. The present paper proposes a novel measure for the departure from conditional symmetry based on the cumulative probabilities from the corners of the square table. In a given example, the proposed measure is applied to Japanese occupational status data, and the interpretation of the proposed measure is illustrated as the departure from a proportional structure of social mobility. Full article
(This article belongs to the Special Issue Advances in Quasi-Symmetry Models)
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