{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:09:48Z","timestamp":1760231388074,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,9,17]],"date-time":"2022-09-17T00:00:00Z","timestamp":1663372800000},"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>In square contingency table analysis, we consider a partial measure that represents the degree of departure from symmetry for each of several pairs. It may be useful to pool the values of the measure into a single summary measure of partial asymmetry. We show that the estimator of partial measures is asymptotically mutually independent for a large sample size. The present paper proposes a symmetry measure in the class of weighted averages that is different from previous studies. The proposed measure is an approximation of the measure in the class of weighted averages that has the smallest variance.<\/jats:p>","DOI":"10.3390\/sym14091936","type":"journal-article","created":{"date-parts":[[2022,9,20]],"date-time":"2022-09-20T04:28:55Z","timestamp":1663648135000},"page":"1936","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Partial Asymmetry Measures for Square Contingency Tables"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1979-8310","authenticated-orcid":false,"given":"Takuma","family":"Ishihara","sequence":"first","affiliation":[{"name":"Innovative and Clinical Research Promotion Center, Gifu University Hospital, Gifu City 501-1194, Gifu, Japan"}]},{"given":"Kouji","family":"Yamamoto","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, Yokohama City University School of Medicine, Yokohama City 236-0004, Kanagawa, Japan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6583-1986","authenticated-orcid":false,"given":"Kouji","family":"Tahata","sequence":"additional","affiliation":[{"name":"Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Noda City 278-8510, Chiba, Japan"}]},{"given":"Sadao","family":"Tomizawa","sequence":"additional","affiliation":[{"name":"Department of Information Sciences, Faculty of Information Sciences, Meisei University, Hino City 191-0042, Tokyo, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2022,9,17]]},"reference":[{"key":"ref_1","unstructured":"Agresti, A. (2013). Categorical Data Analysis, Wiley. [3rd ed.]."},{"key":"ref_2","unstructured":"Bishop, Y.M.M., Fienberg, S.E., and Holl, P.W. (1975). Discrete Multivariate Analysis: Theory and Practice, The MIT Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.1016\/j.jmva.2007.01.011","article-title":"Partitioning a non-symmetric measure of association for three-way contingency tables","volume":"98","author":"Beh","year":"2007","journal-title":"J. Multivar. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1789","DOI":"10.1016\/j.jspi.2010.11.029","article-title":"Three-way association measure decompositions: The Delta index","volume":"141","author":"Lombardo","year":"2011","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"3875","DOI":"10.1002\/sim.7399","article-title":"Subcopula-based measure of asymmetric association for contingency tables","volume":"36","author":"Wei","year":"2017","journal-title":"Stat. Med."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"109183","DOI":"10.1016\/j.spl.2021.109183","article-title":"Measure of asymmetric association for ordinal contingency tables via the bilinear extension copula","volume":"178","author":"Wei","year":"2021","journal-title":"Stat. Probab. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5836","DOI":"10.1080\/03610926.2020.1737128","article-title":"The essential dependence for a group of random vectors","volume":"50","author":"Zhang","year":"2021","journal-title":"Commun. 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Sin."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"109909","DOI":"10.1016\/j.chaos.2020.109909","article-title":"Taxonomy of commodities assets via the complexity-entropy causality plane","volume":"137","author":"Fernandes","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","unstructured":"Agresti, A. (1984). Analysis of Ordinal Categorical Data, Wiley."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v028.i11","article-title":"Exact tests for two-way contingency tables with structural zeros","volume":"28","author":"West","year":"2008","journal-title":"J. Stat. 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