{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,1]],"date-time":"2025-12-01T06:17:11Z","timestamp":1764569831795,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T00:00:00Z","timestamp":1635811200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/sym13112067","type":"journal-article","created":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T22:14:52Z","timestamp":1635891292000},"page":"2067","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Generalized Two-Dimensional Index to Measure the Degree of Deviation from Double Symmetry in Square Contingency Tables"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1663-1897","authenticated-orcid":false,"given":"Shuji","family":"Ando","sequence":"first","affiliation":[{"name":"Department of Information and Computer Technology, Faculty of Engineering, Tokyo University of Science, Tokyo 125-8585, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hikaru","family":"Hoshi","sequence":"additional","affiliation":[{"name":"Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba 278-8510, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aki","family":"Ishii","sequence":"additional","affiliation":[{"name":"Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba 278-8510, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sadao","family":"Tomizawa","sequence":"additional","affiliation":[{"name":"Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Chiba 278-8510, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"572","DOI":"10.1080\/01621459.1948.10483284","article-title":"A test for symmetry in contingency tables","volume":"43","author":"Bowker","year":"1948","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_2","unstructured":"Bishop, Y.M.M., Fienberg, S.E., and Holland, P.W. (1975). Discrete Multivariate Analysis: Theory and Practice, The MIT Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Tan, T.K. (2017). Doubly Classified Model with R, Springer.","DOI":"10.1007\/978-981-10-6995-6"},{"key":"ref_4","unstructured":"Agresti, A. (2018). An Introduction to Categorical Data Analysis, Wiley. [3rd ed.]."},{"key":"ref_5","first-page":"259","article-title":"A test for point-symmetry in J-dimensional contingency-cubes","volume":"18","author":"Wall","year":"1976","journal-title":"Biom. J."},{"key":"ref_6","first-page":"17","article-title":"Double symmetry model and its decomposition in a square contingency table","volume":"15","author":"Tomizawa","year":"1985","journal-title":"J. Jpn. Stat. 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Introduction to the Statistical Analysis of Categorical Data, Springer.","DOI":"10.1007\/978-3-642-59123-5"},{"key":"ref_15","first-page":"91","article-title":"Linear column-parameter symmetry model for square contingency tables: Application to decayed teeth data","volume":"43","author":"Tomizawa","year":"2006","journal-title":"Biom. Lett."},{"key":"ref_16","first-page":"47","article-title":"Improved approximate unbiased estimators of measures of asymmetry for square contingency tables","volume":"7","author":"Tomizawa","year":"2007","journal-title":"Adv. Appl. 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