{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T14:11:31Z","timestamp":1783692691589,"version":"3.55.0"},"reference-count":35,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,5,28]],"date-time":"2022-05-28T00:00:00Z","timestamp":1653696000000},"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>Testing the existence of association between a multivariate response and predictors is an important statistical problem. In this paper, we present nonparametric procedures that make no specific distributional, regression function, and covariance matrix assumptions. Our test is motivated by recent results in MANOVA tests for a large number of groups. Two types of tests are proposed. While it is natural to consider the classical approach for constructing the test by jointly considering all the variables together, we also investigate a composite test where variable-by-variable univariate tests are combined to form a multivariate test. The asymptotic distributions of the test statistics are derived in a unified manner by deriving the asymptotic matrix variate normal distribution of random matrices involved in the construction of the statistics. The tests have good numerical performance in finite samples. The application of the methods is illustrated with gene expression profiling of bronchial airway brushings.<\/jats:p>","DOI":"10.3390\/sym14061112","type":"journal-article","created":{"date-parts":[[2022,5,31]],"date-time":"2022-05-31T02:30:06Z","timestamp":1653964206000},"page":"1112","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Nonparametric Tests for Multivariate Association"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6802-340X","authenticated-orcid":false,"given":"Solomon W.","family":"Harrar","sequence":"first","affiliation":[{"name":"Dr. Bing Zhang Department of Statistics, College of Arts and Sciences, University of Kentucky, Lexington, KY 40506, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yan","family":"Xu","sequence":"additional","affiliation":[{"name":"Merck & Co., 351 North Sumneytown Pike, P.O. Box 1000, North Wales, PA 19454, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1098\/rspl.1895.0041","article-title":"Note on regression and inheritance in the case of two parents","volume":"58","author":"Pearson","year":"1895","journal-title":"Proc. R. Soc. Lond."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"441","DOI":"10.2307\/1422689","article-title":"The Proof and Measurement of Association between Two Things","volume":"100","author":"Spearman","year":"1987","journal-title":"Am. J. Psychol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1093\/biomet\/30.1-2.81","article-title":"A new measure of rank correlation","volume":"30","author":"Kendall","year":"1938","journal-title":"Biometrika"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"104767","DOI":"10.1016\/j.jmva.2021.104767","article-title":"Simultaneous inference for Kendall\u2019s tau","volume":"185","author":"Nowak","year":"2021","journal-title":"J. Multivar. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"542","DOI":"10.2307\/2533399","article-title":"Type I error robustness of ANOVA and ANOVA on ranks when the number of treatments is large","volume":"50","author":"Brownie","year":"1994","journal-title":"Biometrics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/0167-7152(94)00112-L","article-title":"ANOVA and rank tests when the number of treatments is large","volume":"23","author":"Boos","year":"1995","journal-title":"Stat. Probab. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1080\/01621459.2000.10473915","article-title":"Asymptotics for analysis of variance when the number of levels is large","volume":"95","author":"Akritas","year":"2000","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1007\/s10463-006-0055-7","article-title":"Asymptotic expansion for the null distribution of the F-statistic in one-way ANOVA under non-normality","volume":"59","author":"Harrar","year":"2007","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1198\/016214504000000412","article-title":"Heteroscedastic one-way ANOVA and lack-of-fit tests","volume":"99","author":"Akritas","year":"2004","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1080\/10485250802066112","article-title":"An ANOVA-type nonparametric diagnostic test for heteroscedastic regression models","volume":"20","author":"Wang","year":"2008","journal-title":"J. Nonparametric Stat."},{"key":"ref_11","first-page":"1837","article-title":"Nonparametric lack-of-fit testing and consistent variable selection","volume":"20","author":"Zambom","year":"2014","journal-title":"Stat. Sin."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1002\/cjs.5550330406","article-title":"A fully nonparametric diagnostic test for homogeneity of variances","volume":"33","author":"Wang","year":"2005","journal-title":"Can. J. Stat."},{"key":"ref_13","first-page":"143","article-title":"A study on the satisfaction with distance learning of university students with disabilities: Bivariate regression analysis using a multiple permutation test","volume":"33","author":"Bonnini","year":"2021","journal-title":"Stat. Appl. Ital. J. Appl. Stat."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Bonnini, S., Corain, L., Marozzi, M., and Salmaso, L. (2014). Nonparametric Hypothesis Testing: Rank and Permutation Methods with Applications in R, John Wiley & Sons.","DOI":"10.1002\/9781118763490"},{"key":"ref_15","unstructured":"Salmaso, L., and Pesarin, F. (2010). Permutation Tests for Complex Data: Theory, Applications and Software, John Wiley & Sons."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1007\/s11749-006-0026-6","article-title":"MANOVA for large hypothesis degrees of freedom under non-normality","volume":"17","author":"Gupta","year":"2008","journal-title":"Test"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1016\/j.jspi.2006.11.004","article-title":"Nonparametric methods in multivariate factorial designs for large number of factor levels","volume":"138","author":"Bathke","year":"2008","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1016\/j.jmva.2004.12.001","article-title":"Asymptotics for testing hypothesis in some multivariate variance components model under non-normality","volume":"97","author":"Gupta","year":"2006","journal-title":"J. Multivar. Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1635","DOI":"10.1016\/j.jmva.2008.01.005","article-title":"Nonparametric methods for unbalanced multivariate data and many factor levels","volume":"99","author":"Harrar","year":"2008","journal-title":"J. Multivar. Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1002\/cjs.11333","article-title":"A nonparametric hypothesis test for heteroscedasticity in multiple regression","volume":"45","author":"Zambom","year":"2017","journal-title":"Can. J. Stat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1214\/aoms\/1177699599","article-title":"Designs for regression problems with correlated errors","volume":"37","author":"Sacks","year":"1966","journal-title":"Ann. Math. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/s10463-010-0299-0","article-title":"A modified two-factor multivariate analysis of variance: Asymptotics and small sample approximations","volume":"64","author":"Harrar","year":"2012","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Bathke, A.C., and Harrar, S.W. (2016). Rank-based inference for multivariate data in factorial designs. Robust Rank-Based and Nonparametric Methods, Springer.","DOI":"10.1007\/978-3-319-39065-9_7"},{"key":"ref_24","unstructured":"Anderson, T. (2003). An Introduction to Multivariate Analysis, John Wiley & Sons."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"995","DOI":"10.1214\/aoms\/1177706437","article-title":"A high dimensional two sample significance test","volume":"29","author":"Dempster","year":"1958","journal-title":"Ann. Math. Stat."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1111\/1467-9868.00149","article-title":"Testing heteroscedasticity in nonparametric regression","volume":"60","author":"Dette","year":"1998","journal-title":"J. R. Stat. Soc. Ser. B (Stat. Methodol.)"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1214\/aos\/1176346788","article-title":"Bandwidth choice for nonparametric regression","volume":"12","author":"Rice","year":"1984","journal-title":"Ann. Stat."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Bretz, F., Hothorn, T., and Westfall, P. (2016). Multiple Comparisons Using R, CRC Press.","DOI":"10.1201\/9781420010909"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Genz, A., and Bretz, F. (2009). Computation of Multivariate Normal and T Probabilities, Springer Science & Business Media.","DOI":"10.1007\/978-3-642-01689-9"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"654","DOI":"10.3389\/fgene.2020.00654","article-title":"Gene set analysis: Challenges, opportunities, and future research","volume":"11","author":"Maleki","year":"2020","journal-title":"Front. Genet."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1783","DOI":"10.1128\/MCB.01003-09","article-title":"Establishment of extracellular signal-regulated kinase 1\/2 bistability and sustained activation through Sprouty 2 and its relevance for epithelial function","volume":"30","author":"Liu","year":"2010","journal-title":"Mol. Cell. Biol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1146\/annurev.immunol.21.120601.141142","article-title":"IL-13 effector functions","volume":"21","author":"Wynn","year":"2003","journal-title":"Annu. Rev. Immunol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1164\/rccm.201302-0264OC","article-title":"Early-life air pollution and asthma risk in minority children. The GALA II and SAGE II studies","volume":"188","author":"Nishimura","year":"2013","journal-title":"Am. J. Respir. Crit. Care Med."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.spl.2014.07.024","article-title":"A general central limit theorem for strong mixing sequences","volume":"94","year":"2014","journal-title":"Stat. Probab. Lett."},{"key":"ref_35","unstructured":"Serfling, R.J. (2009). Approximation Theorems of Mathematical Statistics, John Wiley & Sons."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1112\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:20:30Z","timestamp":1760138430000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1112"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,28]]},"references-count":35,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061112"],"URL":"https:\/\/doi.org\/10.3390\/sym14061112","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,28]]}}}