{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:28:15Z","timestamp":1760236095812,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,10,27]],"date-time":"2021-10-27T00:00:00Z","timestamp":1635292800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["JP20K03756","JP18K03425"],"award-info":[{"award-number":["JP20K03756","JP18K03425"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u2019s coefficients of association and of colligation, Cram\u00e9r\u2019s coefficient, and Goodman and Kruskal\u2019s 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.<\/jats:p>","DOI":"10.3390\/sym13112031","type":"journal-article","created":{"date-parts":[[2021,10,27]],"date-time":"2021-10-27T23:24:42Z","timestamp":1635377082000},"page":"2031","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Two-Dimensional Index of Departure from the Symmetry Model for Square Contingency Tables with Nominal Categories"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6735-0804","authenticated-orcid":false,"given":"Tomotaka","family":"Momozaki","sequence":"first","affiliation":[{"name":"Department of Information Sciences, Graduate School of Science and Technology, Tokyo University Science, Noda 278-8510, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5057-5927","authenticated-orcid":false,"given":"Tomoyuki","family":"Nakagawa","sequence":"additional","affiliation":[{"name":"Department of Information Sciences, Faculty of Science and Technology, Tokyo University Science, Noda 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 Science, Noda 278-8510, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4785-7653","authenticated-orcid":false,"given":"Yusuke","family":"Saigusa","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Yokohama City University School of Medicine, Yokohama 236-0004, 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 Science, Noda 278-8510, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,27]]},"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. 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