{"id":"https://openalex.org/W7203937652","doi":"https://doi.org/10.3390/math14163020","title":"A Dimension-Reduction Method for Detecting Non-Multinormality in Two-Level Structural Equation Models","display_name":"A Dimension-Reduction Method for Detecting Non-Multinormality in Two-Level Structural Equation Models","publication_year":2026,"publication_date":"2026-08-21","ids":{"openalex":"https://openalex.org/W7203937652","doi":"https://doi.org/10.3390/math14163020"},"language":"en","primary_location":{"id":"doi:10.3390/math14163020","doi":"https://doi.org/10.3390/math14163020","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14163020","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://doi.org/10.3390/math14163020","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5086473668","display_name":"Yiwen Cao","orcid":"https://orcid.org/0009-0005-8242-189X","observed_orcids":["https://orcid.org/0009-0005-8242-189X"]},"institutions":[{"id":"https://openalex.org/I49835588","display_name":"Macao Polytechnic University","ror":"https://ror.org/02sf5td35","country_code":"MO","type":"education","lineage":["https://openalex.org/I49835588"]}],"countries":["MO"],"is_corresponding":false,"raw_author_name":"Yiwen Cao","raw_affiliation_strings":["Faculty of Applied Sciences, Macao Polytechnic University, Rua de Luis Gonzaga Gomes, Macao, China"],"raw_orcid":"https://orcid.org/0009-0005-8242-189X","affiliations":[{"raw_affiliation_string":"Faculty of Applied Sciences, Macao Polytechnic University, Rua de Luis Gonzaga Gomes, Macao, China","institution_ids":["https://openalex.org/I49835588"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110683819","display_name":"Jiajuan Liang","orcid":"https://orcid.org/0000-0003-2993-7692","observed_orcids":["https://orcid.org/0000-0003-2993-7692"]},"institutions":[{"id":"https://openalex.org/I12615008","display_name":"Beijing Normal-Hong Kong Baptist University","ror":"https://ror.org/04snvc712","country_code":"CN","type":"education","lineage":["https://openalex.org/I12615008"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Jiajuan Liang","raw_affiliation_strings":["Department of Statistics and Data Science, Beijing Normal-Hong Kong Baptist University, Zhuhai 519087, China","Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, Beijing Normal-Hong Kong Baptist University, Zhuhai 519087, China"],"raw_orcid":"https://orcid.org/0000-0003-2993-7692","affiliations":[{"raw_affiliation_string":"Department of Statistics and Data Science, Beijing Normal-Hong Kong Baptist University, Zhuhai 519087, China","institution_ids":["https://openalex.org/I12615008"]},{"raw_affiliation_string":"Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science, Beijing Normal-Hong Kong Baptist University, Zhuhai 519087, China","institution_ids":["https://openalex.org/I12615008"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5108221944","display_name":"Charlene Lam","orcid":"https://orcid.org/0000-0002-1916-1184","observed_orcids":["https://orcid.org/0000-0002-1916-1184"]},"institutions":[{"id":"https://openalex.org/I49835588","display_name":"Macao Polytechnic University","ror":"https://ror.org/02sf5td35","country_code":"MO","type":"education","lineage":["https://openalex.org/I49835588"]}],"countries":["MO"],"is_corresponding":true,"raw_author_name":"Chi-Kin Lam","raw_affiliation_strings":["Faculty of Applied Sciences, Macao Polytechnic University, Rua de Luis Gonzaga Gomes, Macao, China"],"raw_orcid":"https://orcid.org/0000-0002-1916-1184","affiliations":[{"raw_affiliation_string":"Faculty of Applied Sciences, Macao Polytechnic University, Rua de Luis Gonzaga Gomes, Macao, China","institution_ids":["https://openalex.org/I49835588"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5108221944"],"corresponding_institution_ids":["https://openalex.org/I49835588"],"apc_list":{"value":2600,"currency":"CHF","value_usd":3137},"apc_paid":{"value":2600,"currency":"CHF","value_usd":3137},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":"16","first_page":"3020","last_page":"3020"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.9487000107765198,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.9487000107765198,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11871","display_name":"Advanced Statistical Methods and Models","score":0.033799998462200165,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13141","display_name":"Statistical Methods and Applications","score":0.005100000184029341,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/dimensionality-reduction","display_name":"dimensionality reduction","score":0.9129999876022339},{"id":"https://openalex.org/keywords/principal-component-analysis","display_name":"principal component analysis","score":0.8379999995231628},{"id":"https://openalex.org/keywords/sphericity-test","display_name":"sphericity test","score":0.7070000171661377},{"id":"https://openalex.org/keywords/heavy-tailed-distributions","display_name":"heavy-tailed distributions","score":0.6930000185966492},{"id":"https://openalex.org/keywords/skewed-distributions","display_name":"skewed distributions","score":0.9359999895095825},{"id":"https://openalex.org/keywords/correlated-observations","display_name":"correlated observations","score":0.34299999475479126}],"concepts":[{"id":"https://openalex.org/C169857963","wikidata":"https://www.wikidata.org/wiki/Q1461038","display_name":"Test statistic","level":3,"score":0.5945000052452087},{"id":"https://openalex.org/C35651441","wikidata":"https://www.wikidata.org/wiki/Q625303","display_name":"Independence (probability theory)","level":2,"score":0.574999988079071},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5622000098228455},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5400999784469604},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.515500009059906},{"id":"https://openalex.org/C27438332","wikidata":"https://www.wikidata.org/wiki/Q2873","display_name":"Principal component analysis","level":2,"score":0.46790000796318054},{"id":"https://openalex.org/C120639","wikidata":"https://www.wikidata.org/wiki/Q7068958","display_name":"Null distribution","level":4,"score":0.45320001244544983},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.42879998683929443},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.40869998931884766},{"id":"https://openalex.org/C203763787","wikidata":"https://www.wikidata.org/wiki/Q371029","display_name":"Null (SQL)","level":2,"score":0.3905999958515167},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.38659998774528503},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.38359999656677246},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.38019999861717224},{"id":"https://openalex.org/C87007009","wikidata":"https://www.wikidata.org/wiki/Q210832","display_name":"Statistical hypothesis testing","level":2,"score":0.3630000054836273},{"id":"https://openalex.org/C2776157432","wikidata":"https://www.wikidata.org/wiki/Q1375683","display_name":"Normality","level":2,"score":0.3582000136375427},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3544999957084656},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.33640000224113464},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.3296000063419342},{"id":"https://openalex.org/C143380155","wikidata":"https://www.wikidata.org/wiki/Q5160088","display_name":"Confidence region","level":3,"score":0.326200008392334},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.3181000053882599},{"id":"https://openalex.org/C40696583","wikidata":"https://www.wikidata.org/wiki/Q989120","display_name":"Type I and type II errors","level":2,"score":0.3174999952316284},{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.310699999332428},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.3001999855041504},{"id":"https://openalex.org/C89128539","wikidata":"https://www.wikidata.org/wiki/Q1949963","display_name":"Statistic","level":2,"score":0.2955000102519989},{"id":"https://openalex.org/C191988596","wikidata":"https://www.wikidata.org/wiki/Q628374","display_name":"Null hypothesis","level":2,"score":0.2818000018596649},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.2694999873638153},{"id":"https://openalex.org/C96608239","wikidata":"https://www.wikidata.org/wiki/Q1199823","display_name":"Statistical power","level":2,"score":0.26899999380111694},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.26109999418258667},{"id":"https://openalex.org/C65778772","wikidata":"https://www.wikidata.org/wiki/Q12345341","display_name":"Asymptotic distribution","level":3,"score":0.2558000087738037},{"id":"https://openalex.org/C102094743","wikidata":"https://www.wikidata.org/wiki/Q133871","display_name":"Normal distribution","level":2,"score":0.2529999911785126}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/math14163020","doi":"https://doi.org/10.3390/math14163020","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14163020","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:9117371cf967480eb56b44e97197d250","doi":null,"is_oa":false,"landing_page_url":"https://doaj.org/article/9117371cf967480eb56b44e97197d250","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics, Vol 14, Iss 16, p 3020 (2026)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/math14163020","doi":"https://doi.org/10.3390/math14163020","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14163020","pdf_url":null,"source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":23,"referenced_works":["https://openalex.org/W1543298386","https://openalex.org/W1970957555","https://openalex.org/W1973272350","https://openalex.org/W1994149163","https://openalex.org/W1994884074","https://openalex.org/W2000031724","https://openalex.org/W2001674862","https://openalex.org/W2030630447","https://openalex.org/W2042350321","https://openalex.org/W2044236880","https://openalex.org/W2047317044","https://openalex.org/W2072155771","https://openalex.org/W2095968500","https://openalex.org/W2131070718","https://openalex.org/W2156661350","https://openalex.org/W2168745915","https://openalex.org/W2331897191","https://openalex.org/W2909748942","https://openalex.org/W4246177722","https://openalex.org/W4250407599","https://openalex.org/W4292023222","https://openalex.org/W4300246981","https://openalex.org/W4416777005"],"related_works":["https://openalex.org/W4416777005","https://openalex.org/W2079558941","https://openalex.org/W2028818971","https://openalex.org/W2413594434","https://openalex.org/W2610915164","https://openalex.org/W7037977982","https://openalex.org/W2270866878","https://openalex.org/W2001905677","https://openalex.org/W872926196","https://openalex.org/W4210437528"],"abstract_inverted_index":{"Testing":[0],"multinormality":[1,202],"in":[2,204],"two-level":[3,205],"structural":[4],"equation":[5],"models":[6],"(SEMs)":[7],"presents":[8],"a":[9,35,55,69,142,180,196],"fundamental":[10],"challenge":[11],"because":[12],"observations":[13,53],"from":[14,162],"the":[15,22,51,82,93,99,113,117,127,173],"same":[16],"level-2":[17],"unit":[18],"are":[19],"correlated,":[20],"violating":[21],"independence":[23],"assumption":[24],"required":[25],"by":[26],"classical":[27],"normality":[28],"tests.":[29],"In":[30],"this":[31,45,66],"paper,":[32],"we":[33],"develop":[34],"novel":[36],"generalized":[37],"Shapiro\u2013Wilk":[38,84],"(GW)":[39],"test":[40,89,119,130,175],"that":[41,95,126,172],"explicitly":[42],"accounts":[43],"for":[44,90,200],"dependence.":[46],"The":[47,105,152,184],"proposed":[48,128,182],"method":[49,153],"rearranges":[50],"dependent":[52],"into":[54],"random":[56],"matrix":[57,67],"and":[58,137,149,192],"employs":[59],"principal":[60,72],"component":[61],"analysis":[62],"(PCA)":[63],"to":[64,88,111,179,188],"project":[65],"onto":[68],"set":[70],"of":[71,102,116,144],"directions,":[73],"achieving":[74],"effective":[75],"dimension":[76],"reduction.":[77],"On":[78],"each":[79],"projected":[80],"direction,":[81],"scale-invariant":[83],"statistic":[85],"is":[86,108,154,186],"applied":[87],"sphericity,":[91],"leveraging":[92],"property":[94],"spherical":[96],"distributions":[97],"preserve":[98],"null":[100,114],"distribution":[101,115],"such":[103],"statistics.":[104],"Johnson":[106],"SB-transform":[107],"then":[109],"used":[110],"approximate":[112],"combined":[118],"statistic.":[120],"A":[121],"Monte":[122],"Carlo":[123],"study":[124],"demonstrates":[125],"GW":[129,174],"controls":[131],"type":[132],"I":[133],"error":[134],"rates":[135],"satisfactorily":[136],"exhibits":[138],"strong":[139],"power":[140],"against":[141],"range":[143],"non-normal":[145],"alternatives,":[146],"including":[147],"heavy-tailed":[148],"asymmetric":[150],"distributions.":[151],"further":[155],"illustrated":[156],"using":[157],"real":[158],"alcohol":[159],"use":[160],"data":[161],"nested":[163],"families,":[164],"highlighting":[165],"its":[166],"practical":[167],"utility.":[168],"Comparative":[169],"evaluation":[170],"indicates":[171],"performs":[176],"favorably":[177],"relative":[178],"recently":[181],"approach.":[183],"procedure":[185],"applicable":[187],"balanced":[189],"level-1":[190],"designs":[191],"provides":[193],"researchers":[194],"with":[195],"necessary":[197],"diagnostic":[198],"tool":[199],"assessing":[201],"assumptions":[203],"SEMs.":[206]},"counts_by_year":[],"updated_date":"2026-11-06T12:05:46.655746","created_date":"2026-08-22T00:00:00"}