{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:40:53Z","timestamp":1760222453158,"version":"build-2065373602"},"reference-count":7,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2008,12,22]],"date-time":"2008-12-22T00:00:00Z","timestamp":1229904000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>For 2 x 2 x K contingency tables, Tomizawa considered a Shannon entropy type measure to represent the degree of departure from a log-linear model of no three-factor interaction (the NOTFI model). This paper proposes a generalization of Tomizawa's measure for 2 x 2 x K tables. The measure proposed is expressed by using Patil-Taillie diversity index or Cressie-Read power-divergence. A special case of the proposed measure includes Tomizawa's measure. The proposed measure would be useful for comparing the degrees of departure from the NOTFI model in several tables.<\/jats:p>","DOI":"10.3390\/e10040776","type":"journal-article","created":{"date-parts":[[2008,12,21]],"date-time":"2008-12-21T14:29:26Z","timestamp":1229869766000},"page":"776-785","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Generalized Measure of Departure from No Three-Factor Interaction Model for 2 x 2 x K Contingency Tables"],"prefix":"10.3390","volume":"10","author":[{"given":"Kouji","family":"Yamamoto","sequence":"first","affiliation":[{"name":"Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Noda City, Chiba, 278-8510, Japan"}]},{"given":"Yohei","family":"Ban","sequence":"additional","affiliation":[{"name":"Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Noda City, Chiba, 278-8510, Japan"}]},{"given":"Sadao","family":"Tomizawa","sequence":"additional","affiliation":[{"name":"Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Noda City, Chiba, 278-8510, Japan"}]}],"member":"1968","published-online":{"date-parts":[[2008,12,22]]},"reference":[{"key":"ref_1","unstructured":"Bishop, Y.M.M., Fienberg, S.E., and Holland, P.W. (1975). Discrete Multivariate Analysis: Theory and Practice, The MIT Press."},{"key":"ref_2","unstructured":"Fienberg, S.E. (1980). The Analysis of Cross-Classified Categorical Data, The MIT Press. [2nd Ed.]."},{"key":"ref_3","first-page":"1","article-title":"A measure of departure from no three-factor interaction model in a 2 \u00d7 2 \u00d7 k contingency table","volume":"27","author":"Tomizawa","year":"1993","journal-title":"J. Stat. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1080\/01621459.1982.10477845","article-title":"Diversity as a concept and its measurement","volume":"77","author":"Patil","year":"1982","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1111\/j.2517-6161.1984.tb01318.x","article-title":"Multinomial goodness-of-fit tests","volume":"46","author":"Cressie","year":"1984","journal-title":"J. Royal Stat. Soc., Series B"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Read, T.R.C., and Cressie, N.A.C. (1988). Goodness-of-Fit Statistics for Discrete Multivariate Data, Springer.","DOI":"10.1007\/978-1-4612-4578-0"},{"key":"ref_7","unstructured":"Agresti, A. (1996). An Introduction to Categorical Data Analysis, Wiley."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/4\/776\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:21:17Z","timestamp":1760221277000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/4\/776"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,12,22]]},"references-count":7,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2008,12]]}},"alternative-id":["e10040776"],"URL":"https:\/\/doi.org\/10.3390\/e10040776","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2008,12,22]]}}}