{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,28]],"date-time":"2026-10-28T00:11:25Z","timestamp":1793146285319,"version":"4.5.0"},"reference-count":18,"publisher":"Oxford University Press (OUP)","issue":"1","funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,3,1]]},"abstract":"<jats:title>SUMMARY<\/jats:title>\n               <jats:p>Detecting dependence between two random variables is a fundamental problem. Although the Pearson correlation coefficient is effective for capturing linear dependence, it can be entirely powerless for detecting nonlinear and\/or heteroscedastic patterns. We introduce a new measure, G-squared, to test whether two univariate random variables are independent and to measure the strength of their relationship. The G-squared statistic is almost identical to the square of the Pearson correlation coefficient, R-squared, for linear relationships with constant error variance, and has the intuitive meaning of the piecewise R-squared between the variables. It is particularly effective in handling nonlinearity and heteroscedastic errors. We propose two estimators of G-squared and show their consistency. Simulations demonstrate that G-squared estimators are among the most powerful test statistics compared with several state-of-the-art methods.<\/jats:p>","DOI":"10.1093\/biomet\/asw071","type":"journal-article","created":{"date-parts":[[2016,12,16]],"date-time":"2016-12-16T13:02:03Z","timestamp":1481893323000},"page":"129-139","source":"Crossref","is-referenced-by-count":42,"title":["Generalized R-squared for detecting dependence"],"prefix":"10.1093","volume":"104","author":[{"given":"X.","family":"Wang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"B.","family":"Jiang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. S.","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2017,2,22]]},"reference":[{"key":"2019071815495818200_B1","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1093\/biomet\/81.3.579","article-title":"Local divergence and\n            association.","volume":"91","author":"Blyth","year":"1994","journal-title":"Biometrika"},{"key":"2019071815495818200_B2","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1080\/01621459.1985.10478157","article-title":"Estimating optimal transformations for multiple regression and\n            correlation.","volume":"80","author":"Breiman","year":"1985","journal-title":"J. Am. 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Learn.\n            Theory"},{"key":"2019071815495818200_B6","first-page":"723","article-title":"A kernel two-sample test.","volume":"13","author":"Gretton","year":"2012","journal-title":"J.\n            Mach. Learn.\n          Res."},{"key":"2019071815495818200_B7","first-page":"1","article-title":"Consistent distribution-free $K$-sample and\n            independence tests for univariate random variables.","volume":"17","author":"Heller","year":"2016","journal-title":"J. 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Math.\n            Sci."},{"key":"2019071815495818200_B11","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1080\/01621459.1995.10476572","article-title":"Bayes factors.","volume":"90","author":"Kass","year":"1995","journal-title":"J. Am. Statist.\n            Assoc."},{"key":"2019071815495818200_B12","doi-asserted-by":"crossref","first-page":"066138","DOI":"10.1103\/PhysRevE.69.066138","article-title":"Estimating mutual\n            information.","volume":"69.6","author":"Kraskov","year":"2004","journal-title":"Phys. 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Appl.\n            Statist."},{"key":"2019071815495818200_B18","doi-asserted-by":"crossref","first-page":"2769","DOI":"10.1214\/009053607000000505","article-title":"Measuring and testing dependence by correlation of\n            distances.","volume":"12","author":"Sz\u00e9kely","year":"2007","journal-title":"Ann.\n            Statist."}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/academic.oup.com\/biomet\/article-pdf\/104\/1\/129\/23590561\/asw071.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,9,20]],"date-time":"2019-09-20T00:41:37Z","timestamp":1568940097000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/104\/1\/129\/3045032"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,2,22]]},"references-count":18,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,2,22]]},"published-print":{"date-parts":[[2017,3,1]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asw071","relation":{},"ISSN":["0006-3444","1464-3510"],"issn-type":[{"value":"0006-3444","type":"print"},{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2017,3]]},"published":{"date-parts":[[2017,2,22]]}}}