Accounting for Heteroskedasticity Resulting from Between-Group Differences in Multilevel Models
Homogeneity of variance (HOV) is a well-known but often untested assumption in the context of multilevel models (MLMs). However, depending on how large the violation is, how different group sizes are, and the variance pairing, standard errors can be over or underestimated even when using MLMs, resul...
Saved in:
Published in | Multivariate behavioral research Vol. 58; no. 3; pp. 637 - 657 |
---|---|
Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
United States
Routledge
01.05.2023
Taylor & Francis Ltd |
Subjects | |
Online Access | Get full text |
Cover
Loading…
Abstract | Homogeneity of variance (HOV) is a well-known but often untested assumption in the context of multilevel models (MLMs). However, depending on how large the violation is, how different group sizes are, and the variance pairing, standard errors can be over or underestimated even when using MLMs, resulting in questionable inferential tests. We evaluate several tests (e.g., the H statistic, Breusch Pagan, Levene's test) that can be used with MLMs to assess violations of HOV. Although the traditional robust standard errors used with MLMs require at least 50 clusters to be effective, we assess a robust standard error adjustment (i.e., the CR2 estimator) that can be used even with a few clusters. Findings are assessed using a Monte Carlo simulation and are further illustrated using an applied example. We show that explicitly modeling the heterogenous variance structures or using the CR2 estimator are both effective at ameliorating the issues associated with the fixed effects of the regression model related to violations of HOV resulting from between-group differences. |
---|---|
AbstractList | Homogeneity of variance (HOV) is a well-known but often untested assumption in the context of multilevel models (MLMs). However, depending on how large the violation is, how different group sizes are, and the variance pairing, standard errors can be over or underestimated even when using MLMs, resulting in questionable inferential tests. We evaluate several tests (e.g., the H statistic, Breusch Pagan, Levene’s test) that can be used with MLMs to assess violations of HOV. Although the traditional robust standard errors used with MLMs require at least 50 clusters to be effective, we assess a robust standard error adjustment (i.e., the CR2 estimator) that can be used even with a few clusters. Findings are assessed using a Monte Carlo simulation and are further illustrated using an applied example. We show that explicitly modeling the heterogenous variance structures or using the CR2 estimator are both effective at ameliorating the issues associated with the fixed effects of the regression model related to violations of HOV resulting from between-group differences. Homogeneity of variance (HOV) is a well-known but often untested assumption in the context of multilevel models (MLMs). However, depending on how large the violation is, how different group sizes are, and the variance pairing, standard errors can be over or underestimated even when using MLMs, resulting in questionable inferential tests. We evaluate several tests (e.g., the statistic, Breusch Pagan, Levene's test) that can be used with MLMs to assess violations of HOV. Although the traditional robust standard errors used with MLMs require at least 50 clusters to be effective, we assess a robust standard error adjustment (i.e., the CR2 estimator) that can be used even with a few clusters. Findings are assessed using a Monte Carlo simulation and are further illustrated using an applied example. We show that explicitly modeling the heterogenous variance structures or using the CR2 estimator are both effective at ameliorating the issues associated with the fixed effects of the regression model related to violations of HOV resulting from between-group differences. |
Author | Wiedermann, Wolfgang Zhang, Bixi Huang, Francis L. |
Author_xml | – sequence: 1 givenname: Francis L. orcidid: 0000-0002-5900-7763 surname: Huang fullname: Huang, Francis L. organization: University of Missouri – sequence: 2 givenname: Wolfgang surname: Wiedermann fullname: Wiedermann, Wolfgang organization: University of Missouri – sequence: 3 givenname: Bixi surname: Zhang fullname: Zhang, Bixi organization: University of Missouri |
BackLink | https://www.ncbi.nlm.nih.gov/pubmed/35687513$$D View this record in MEDLINE/PubMed |
BookMark | eNp9kUtP3DAUhS0EgoH2J4Aisekm4EdsJzsebQEJVKlq15bHuUYGxx7sBDT_vo5mYNEFG19b97vnWuccot0QAyB0TPAZwS0-x5hKRiQ5o5jSckhJO7yDFoQzWssOi120mJl6hg7QYc5PGGPBm24fHTAuWskJWyB1aUycwujCY2Vjqm5hhBTzM_Q6j864cV39hjz5DZDiUF3B-AYQ6psUp1X13VkLCYKBXLlQPcykh1fw1UPswecvaM9qn-Hrth6hvz9__Lm-re9_3dxdX97XhnXNWFNqac-WmHNjhOYNQCdkU56cNERoZpeCaW0p59AB5aXTWLIkrOOk1eXKjtC3je4qxZcJ8qgGlw14rwPEKSsqJBdYdJgW9PQ_9ClOKZTfKdpS2VLWtqJQfEOZYkdOYNUquUGntSJYzQmo9wTUnIDaJlDmTrbq03KA_mPq3fICXGwAF4rhg36Lyfdq1Gsfk006GJcV-3zHP1YUlgo |
CitedBy_id | crossref_primary_10_1080_19345747_2022_2100301 crossref_primary_10_1007_s11356_023_28219_z crossref_primary_10_1186_s40536_024_00192_0 crossref_primary_10_1177_00131644231181688 crossref_primary_10_3758_s13428_023_02325_9 crossref_primary_10_3390_info14090480 crossref_primary_10_3390_psych5030049 crossref_primary_10_1007_s11121_023_01507_3 |
Cites_doi | 10.1002/9781118445112.stat06249 10.1016/j.csda.2003.08.006 10.3102/00346543068003350 10.1037/0003-066X.63.7.591 10.1027/1614-2241/a000034 10.18637/jss.v067.i01 10.1016/0165-1765(83)90085-X 10.1136/jech.2007.060798 10.3102/00028312003003187 10.1080/00273171.2018.1449628 10.3758/BF03192961 10.22237/jmasm/1551966828 10.3102/0034654308325581 10.2307/1911963 10.22237/jmasm/1556670360 10.1016/0304-4076(85)90158-7 10.2307/1913646 10.1348/000711010X508683 10.3758/s13428-019-01252-y 10.1207/S15328007SEM0904_8 10.1037/1082-989X.9.4.466 10.4135/9781483384733 10.1162/REST_a_00552 10.1002/jae.2508 10.3758/s13428-019-01322-1 10.1093/geronb/57.2.P101 10.3102/00346543066004579 10.2307/1912934 10.1002/sim.6344 10.1037/met0000011 10.2307/j.ctvcm4j72 10.18637/jss.v082.i13 10.1111/1541-0420.t01-1-00027 10.3758/s13428-017-0918-2 10.1037/0033-2909.112.1.155 10.1080/00220973.1997.9943791 10.3758/s13428-015-0619-7 10.1037/0033-2909.104.3.396 10.1080/00220973.2016.1277339 10.1177/1094428119887434 10.1080/00031305.2000.10474549 10.1037/0033-2909.99.1.90 10.1016/0304-4076(81)90062-2 10.1080/00273170701710072 10.1080/00273171.2016.1167008 10.1016/j.socscimed.2018.04.027 10.1037/0022-006X.68.1.155 10.3102/0162373707299706 10.1080/00273171.2016.1275498 10.1002/ejsp.2028 10.3102/1076998614546494 10.1111/j.1541-0420.2007.00924.x 10.3102/0034654321991229 10.1080/07350015.2016.1247004 10.1201/b17198 10.3368/jhr.50.2.317 10.3102/00346543062001061 10.1093/pan/mpu015 10.1027/1614-2241.1.3.86 10.1111/j.0006-341X.2001.00126.x 10.18637/jss.v016.i09 10.1093/biomet/73.1.13 10.1080/01621459.1969.10500976 10.1080/00220973.1974.10806305 10.3102/10769986026004411 10.1027/1614-2241.4.2.67 10.1162/rest_a_00759 10.3102/00346543042003237 10.1080/00273171.2014.955604 |
ContentType | Journal Article |
Copyright | 2022 Taylor & Francis Group, LLC 2022 2022 Taylor & Francis Group, LLC |
Copyright_xml | – notice: 2022 Taylor & Francis Group, LLC 2022 – notice: 2022 Taylor & Francis Group, LLC |
DBID | CGR CUY CVF ECM EIF NPM AAYXX CITATION 7X8 |
DOI | 10.1080/00273171.2022.2077290 |
DatabaseName | Medline MEDLINE MEDLINE (Ovid) MEDLINE MEDLINE PubMed CrossRef MEDLINE - Academic |
DatabaseTitle | MEDLINE Medline Complete MEDLINE with Full Text PubMed MEDLINE (Ovid) CrossRef MEDLINE - Academic |
DatabaseTitleList | MEDLINE |
Database_xml | – sequence: 1 dbid: NPM name: PubMed url: https://proxy.k.utb.cz/login?url=http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=PubMed sourceTypes: Index Database – sequence: 2 dbid: EIF name: MEDLINE url: https://proxy.k.utb.cz/login?url=https://www.webofscience.com/wos/medline/basic-search sourceTypes: Index Database |
DeliveryMethod | fulltext_linktorsrc |
Discipline | Psychology |
EISSN | 1532-7906 |
EndPage | 657 |
ExternalDocumentID | 10_1080_00273171_2022_2077290 35687513 2077290 |
Genre | Articles Journal Article |
GroupedDBID | --Z -~X .7I .QK 0BK 0R~ 123 4.4 5VS 8VB AAAVI AAGZJ AAMFJ AAMIU AAPUL AATTQ AAZMC ABBKH ABCCY ABFIM ABIVO ABJVF ABLIJ ABLJU ABPEM ABPPZ ABPTK ABPTX ABQHQ ABRYG ABSSG ABTAI ABXUL ABZLS ACGFS ACHQT ACIWK ACLSK ACNCT ACTIO ACTOA ADAHI ADCVX AECIN AEGYZ AEISY AEKEX AENEX AEOZL AEPSL AEYOC AEZRU AFHDM AFOLD AFWLO AGDLA AGMYJ AGRBW AIJEM AJWEG AKBVH ALMA_UNASSIGNED_HOLDINGS ALQZU AVBZW AWYRJ BEJHT BLEHA BMOTO BOHLJ CCCUG CQ1 CS3 DGFLZ DKSSO DU5 EBS E~B E~C F5P FEDTE FUNRP FXNIP G-F GTTXZ H13 HF~ HZ~ IPNFZ J.O KDLKA KYCEM LJTGL M4Z MS~ NA5 NW- O9- P2P PQEST PQQKQ QWB RIG RNANH ROSJB RSYQP S-F STATR TEH TFH TFL TFW TN5 TNTFI TRJHH TWZ UT5 UT9 V1K VAE WH7 YNT YQT ZL0 ~01 ~S~ .GJ 07M 53G ABJNI ABRLO ABVXC ABWZE ABXYU ACPKE ACRBO ADEWX ADIUE ADKVQ ADXAZ AEXSR AFFNX AHDZW AIXGP ALEEW ALLRG C5A CAG CBZAQ CGR CKOZC COF CUY CVF C~T DGXZK ECM EFRLQ EGDCR EIF EJD EMOBN HVGLF H~9 JLMOS L7Y LPU NEJ NPM OHT P-O QZZOY RBICI ROL TBQAZ TDBHL TUROJ UA1 UAP XOL ZCG ZXP AAYXX CITATION 7X8 |
ID | FETCH-LOGICAL-c394t-22f2d3b055cc6a54ee967405551416a3fb63aaf255e9e254054f1b139518a4f13 |
ISSN | 0027-3171 |
IngestDate | Sat Aug 17 02:27:32 EDT 2024 Thu Oct 10 21:56:34 EDT 2024 Fri Aug 23 01:25:01 EDT 2024 Wed Oct 16 00:38:58 EDT 2024 Wed Jun 21 12:12:40 EDT 2023 |
IsPeerReviewed | true |
IsScholarly | true |
Issue | 3 |
Keywords | heteroskedasticity multilevel models assumptions homogeneity of variance |
Language | English |
LinkModel | OpenURL |
MergedId | FETCHMERGED-LOGICAL-c394t-22f2d3b055cc6a54ee967405551416a3fb63aaf255e9e254054f1b139518a4f13 |
Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
ORCID | 0000-0002-5900-7763 |
PMID | 35687513 |
PQID | 2827823886 |
PQPubID | 47318 |
PageCount | 21 |
ParticipantIDs | proquest_miscellaneous_2675606902 crossref_primary_10_1080_00273171_2022_2077290 informaworld_taylorfrancis_310_1080_00273171_2022_2077290 proquest_journals_2827823886 pubmed_primary_35687513 |
PublicationCentury | 2000 |
PublicationDate | 2023 May-Jun |
PublicationDateYYYYMMDD | 2023-05-01 |
PublicationDate_xml | – month: 05 year: 2023 text: 2023 May-Jun |
PublicationDecade | 2020 |
PublicationPlace | United States |
PublicationPlace_xml | – name: United States – name: Mahwah |
PublicationTitle | Multivariate behavioral research |
PublicationTitleAlternate | Multivariate Behav Res |
PublicationYear | 2023 |
Publisher | Routledge Taylor & Francis Ltd |
Publisher_xml | – name: Routledge – name: Taylor & Francis Ltd |
References | CIT0030 CIT0074 Bell B. (CIT0004) 2010 CIT0032 CIT0031 CIT0075 CIT0034 CIT0033 CIT0077 Tabachnick B. G. (CIT0078) 2019 CIT0070 McNeish D. (CIT0062) 2020 CIT0036 CIT0079 CIT0038 CIT0037 CIT0039 CIT0083 CIT0041 Raudenbush S. (CIT0073) 2013 CIT0085 CIT0040 CIT0084 CIT0043 CIT0087 CIT0042 CIT0086 CIT0001 CIT0045 CIT0044 CIT0088 R Core Team (CIT0071) 2020 CIT0081 CIT0080 Vallejo G. V. (CIT0082) 2013; 25 Fox J. (CIT0021) 2016 CIT0047 CIT0002 CIT0046 CIT0005 CIT0049 CIT0048 CIT0007 CIT0006 CIT0009 CIT0008 CIT0050 CIT0052 CIT0051 MacKinnon J. G. (CIT0057) 2017 CIT0010 CIT0054 CIT0053 CIT0012 CIT0056 CIT0011 CIT0055 Moder K. (CIT0064) 2010; 52 CIT0014 CIT0058 CIT0013 CIT0016 CIT0015 CIT0059 CIT0018 CIT0017 CIT0019 CIT0061 CIT0060 CIT0063 CIT0065 CIT0020 Raudenbush S. (CIT0072) 2002 CIT0023 CIT0067 CIT0066 Bell R. (CIT0003) 2002; 28 Snijders T. (CIT0076) 2012 CIT0025 CIT0069 CIT0024 CIT0068 CIT0027 CIT0026 CIT0029 CIT0028 Huang F. L. (CIT0035) 2021 Gastwirth J. L. (CIT0022) 2009 |
References_xml | – ident: CIT0026 doi: 10.1002/9781118445112.stat06249 – ident: CIT0055 doi: 10.1016/j.csda.2003.08.006 – ident: CIT0038 doi: 10.3102/00346543068003350 – ident: CIT0017 doi: 10.1037/0003-066X.63.7.591 – ident: CIT0075 doi: 10.1027/1614-2241/a000034 – ident: CIT0002 doi: 10.18637/jss.v067.i01 – ident: CIT0044 – ident: CIT0083 doi: 10.1016/0165-1765(83)90085-X – ident: CIT0011 doi: 10.1136/jech.2007.060798 – ident: CIT0023 doi: 10.3102/00028312003003187 – ident: CIT0084 doi: 10.1080/00273171.2018.1449628 – ident: CIT0030 doi: 10.3758/BF03192961 – ident: CIT0074 doi: 10.22237/jmasm/1551966828 – volume-title: Pitfalls when estimating treatment effects using clustered data year: 2017 ident: CIT0057 contributor: fullname: MacKinnon J. G. – ident: CIT0014 doi: 10.3102/0034654308325581 – ident: CIT0006 doi: 10.2307/1911963 – ident: CIT0029 doi: 10.22237/jmasm/1556670360 – ident: CIT0059 doi: 10.1016/0304-4076(85)90158-7 – ident: CIT0065 doi: 10.2307/1913646 – ident: CIT0067 doi: 10.1348/000711010X508683 – start-page: 343 year: 2009 ident: CIT0022 publication-title: Statistical Science contributor: fullname: Gastwirth J. L. – ident: CIT0015 doi: 10.3758/s13428-019-01252-y – ident: CIT0066 doi: 10.1207/S15328007SEM0904_8 – volume: 25 year: 2013 ident: CIT0082 publication-title: Psicothema contributor: fullname: Vallejo G. V. – ident: CIT0020 doi: 10.1037/1082-989X.9.4.466 – ident: CIT0068 – ident: CIT0040 doi: 10.4135/9781483384733 – ident: CIT0037 doi: 10.1162/REST_a_00552 – ident: CIT0058 doi: 10.1002/jae.2508 – volume-title: Multilevel analysis year: 2012 ident: CIT0076 contributor: fullname: Snijders T. – ident: CIT0016 doi: 10.3758/s13428-019-01322-1 – ident: CIT0036 doi: 10.1093/geronb/57.2.P101 – ident: CIT0052 doi: 10.3102/00346543066004579 – ident: CIT0086 doi: 10.2307/1912934 – ident: CIT0051 doi: 10.1002/sim.6344 – ident: CIT0079 doi: 10.1037/met0000011 – ident: CIT0001 doi: 10.2307/j.ctvcm4j72 – ident: CIT0070 – volume-title: Hierarchical linear models: Applications and data analysis methods year: 2002 ident: CIT0072 contributor: fullname: Raudenbush S. – ident: CIT0046 doi: 10.18637/jss.v082.i13 – year: 2021 ident: CIT0035 publication-title: Behavior Research Methods contributor: fullname: Huang F. L. – volume-title: Applied regression analysis and generalized linear models year: 2016 ident: CIT0021 contributor: fullname: Fox J. – volume-title: Fundamental diagnostics for two-level mixed models: The SAS® macro MIXED_DX [Paper 201-2010] year: 2010 ident: CIT0004 contributor: fullname: Bell B. – volume-title: Using multivariate statistics year: 2019 ident: CIT0078 contributor: fullname: Tabachnick B. G. – ident: CIT0013 – ident: CIT0009 doi: 10.1111/1541-0420.t01-1-00027 – ident: CIT0005 doi: 10.3758/s13428-017-0918-2 – ident: CIT0012 doi: 10.1037/0033-2909.112.1.155 – ident: CIT0028 doi: 10.1080/00220973.1997.9943791 – ident: CIT0061 – ident: CIT0049 doi: 10.3758/s13428-015-0619-7 – ident: CIT0007 doi: 10.1037/0033-2909.104.3.396 – ident: CIT0034 doi: 10.1080/00220973.2016.1277339 – ident: CIT0048 doi: 10.1177/1094428119887434 – ident: CIT0053 doi: 10.1080/00031305.2000.10474549 – ident: CIT0080 doi: 10.1037/0033-2909.99.1.90 – ident: CIT0041 doi: 10.1016/0304-4076(81)90062-2 – ident: CIT0033 doi: 10.1080/00273170701710072 – ident: CIT0063 doi: 10.1080/00273171.2016.1167008 – volume-title: R: A language and environment for statistical computing year: 2020 ident: CIT0071 contributor: fullname: R Core Team – ident: CIT0077 doi: 10.1016/j.socscimed.2018.04.027 – ident: CIT0027 doi: 10.1037/0022-006X.68.1.155 – ident: CIT0032 doi: 10.3102/0162373707299706 – ident: CIT0087 doi: 10.1080/00273171.2016.1275498 – ident: CIT0045 doi: 10.1002/ejsp.2028 – ident: CIT0047 doi: 10.3102/1076998614546494 – volume-title: HLM (7.01) [Computer software] year: 2013 ident: CIT0073 contributor: fullname: Raudenbush S. – ident: CIT0031 doi: 10.1111/j.1541-0420.2007.00924.x – ident: CIT0054 doi: 10.3102/0034654321991229 – ident: CIT0069 doi: 10.1080/07350015.2016.1247004 – ident: CIT0085 doi: 10.1201/b17198 – ident: CIT0008 doi: 10.3368/jhr.50.2.317 – ident: CIT0018 doi: 10.3102/00346543062001061 – ident: CIT0039 doi: 10.1093/pan/mpu015 – ident: CIT0056 doi: 10.1027/1614-2241.1.3.86 – ident: CIT0060 doi: 10.1111/j.0006-341X.2001.00126.x – volume: 52 start-page: 343 issue: 4 year: 2010 ident: CIT0064 publication-title: Psychological Test and Assessment Modeling, contributor: fullname: Moder K. – ident: CIT0088 doi: 10.18637/jss.v016.i09 – ident: CIT0050 doi: 10.1093/biomet/73.1.13 – ident: CIT0025 doi: 10.1080/01621459.1969.10500976 – ident: CIT0042 doi: 10.1080/00220973.1974.10806305 – ident: CIT0010 doi: 10.3102/10769986026004411 – year: 2020 ident: CIT0062 publication-title: Organizational Research Methods contributor: fullname: McNeish D. – ident: CIT0043 doi: 10.1027/1614-2241.4.2.67 – volume: 28 start-page: 169 year: 2002 ident: CIT0003 publication-title: Survey Methodology, contributor: fullname: Bell R. – ident: CIT0019 doi: 10.1162/rest_a_00759 – ident: CIT0024 doi: 10.3102/00346543042003237 – ident: CIT0081 doi: 10.1080/00273171.2014.955604 |
SSID | ssj0006549 |
Score | 2.4551704 |
Snippet | Homogeneity of variance (HOV) is a well-known but often untested assumption in the context of multilevel models (MLMs). However, depending on how large the... |
SourceID | proquest crossref pubmed informaworld |
SourceType | Aggregation Database Index Database Publisher |
StartPage | 637 |
SubjectTerms | assumptions Clusters Computer Simulation heteroskedasticity Homogeneity homogeneity of variance Models, Statistical Monte Carlo Method Monte Carlo simulation Multilevel Analysis multilevel models Regression models Robustness Standard error Statistical analysis Variance |
Title | Accounting for Heteroskedasticity Resulting from Between-Group Differences in Multilevel Models |
URI | https://www.tandfonline.com/doi/abs/10.1080/00273171.2022.2077290 https://www.ncbi.nlm.nih.gov/pubmed/35687513 https://www.proquest.com/docview/2827823886 https://search.proquest.com/docview/2675606902 |
Volume | 58 |
hasFullText | 1 |
inHoldings | 1 |
isFullTextHit | |
isPrint | |
link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1Lb9NAEF6F9NILKu-UghaJW-XI2Zd3jy0FRQg4tWrFxVona4gIDiIOAn4FP5kZ79re0PIoF8vyY_2YzzvfjOdByNOMgZabu3libFomgmuRmEKoRLMiE7oETtBkvb9-o6Zn4uWFvBgMfkRRS5u6GM--X5lX8j9ShW0gV8ySvYZku0FhA6yDfGEJEoblP8k4avWA0YJTDG1ZrT-4ucXqy55frzFiEA_APJJjH5WVeBfUSWiO4oOyDptc3CUGETUd0pbrmLg2O7-AYQ3cNM7tD8WC3vfwCA7o0LHj8NW4d-1g4YqPoSvz-WpZvrNBb8ae6-PF10Xsi2BR5F-XG4BeT99UZezaKRU4vElVPOdKHWGLRxOo8iVggi5Wvnj1pWm-jYsE7gUXAyufYU4dGgppr9faf_m_qLsuCHHSVUf1w-Q4TB6GuUF2WGakHJKdo-nJ2_NOuysZTKrwpG1WGNZrv-p-tvjOVjXc39s0Dbc53SM3g1FCjzzCbpGBq26T3U43frtD8h5qFMaml6FGO6hRhBrdghqNoEYXFe2hRj3U7pKzF89Pn02T0JsjmXEj6oSxks15kUo5mykrhXNGZQKrx02A4lteFopbW4LB6oxjaBaIclKAuSEn2sIqv0eG1apyDwhFI7kQaQHMmQurpU6dK02WapheDFgcIzJu32D-yZdgyf8ouxEx8XvO68b3VXrY5_wv5x60QsnDl77OmWZApLnWakSedLthHsafa7Zyqw0cA5a3wrLfbETue2F2d8ul0pmc8P3rPslDstt_ZgdkWH_euEdAguviccDlT7xXqVc |
link.rule.ids | 315,786,790,27955,27956,60004,60793 |
linkProvider | Library Specific Holdings |
linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LT9wwEB6h5QCXllLabkvBSFyzTWzHcY5Ai8LzgBaJm2VnbamCZis2e2h_fWfyAiohDtwi2Yn8mvE3k5lvAPYzjrfczM-i3MYhkkLLKHdSRZq7TOqAmKDJer-4VMW1PL1Jbx7lwlBYJdnQoSWKaHQ1CTc5o_uQuG8NCUuSkXnHKZmKECKa7auKEMoIVqdXp0Ux6GOVdiCYk0cuS_o8nuc-9OSGesJf-jwKbW6j47dQ9vNog1BuJ8vaTcq__1E8vm6iG_CmA6vsoD1d72DFV5uwPujMP-_BPFSbYDgDVlB0zXxx62eWCKAR4rMrv6CgRepwP__FDtvAsKhxerHvXX0W1FbsZ8WadOA7imNiVKTtbrEF18c_pkdF1NVsiEqRyzriPPCZcHGalqWyqfQ-V5kkVrEEoZ8VwSlhbUBDxueeE1yUIXEIQ9NEW3wUH2BUzSv_CRgZT07GDhGVkFanOvY-5Fms8djliETHMOn3yfxuqTlMMjCetgtnaOFMt3BjyB_vpqkbn0hoC5gY8cK72_3Wm07KFwbNVQRYQms1hr2hGeWTfrrYys-X2ActMkV00HwMH9sjM4xWpArNxUR8fsXAdmGtmF6cm_OTy7MvsI5NXTzmNozq-6X_ipipdjudUPwDHPMHSQ |
linkToPdf | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1Nb9QwEB2hrYR6Acrn0gJG4polsR3HOQLtKhSoUNVK3Cx7Y0tVS7Zqsgf49Z1JnECRKg69RbIT-WvGb5znNwDvCo67XO3rpLRpSKTQMimdVInmrpA6ICbob71_O1LVqTz8kY9swjbSKimGDoNQRO-rybgv6zAy4t73GixZQdEdp7tUBBAxat9SJPY2g62T48OqmtyxyiMG5nQgV2TjNZ7bPnRjg7ohX3o7CO03o-VDcGM3Bg7K-WLTucXq9z8Kj3fq5yN4EKEq-zCsrR2455vHsD15zF9PwPzJNcGwA6wibs26Pfe1JflnBPjs2LdEWaQKV-uf7ONAC0v6Iy-2H7OzoK9iZw3rLwNfEIuJUYq2i_YpnC4PTj5VSczYkKxEKbuE88Br4dI8X62UzaX3pSokaYplCPysCE4JawOGMb70nMCiDJlDEJpn2uKjeAazZt34F8AodHIydYinhLQ616n3oSxSjYuuRBw6h8U4TeZyEOYw2aR3OgycoYEzceDmUP49mabrT0TCkL7EiP-8uzfOvIk23hoMVhFeCa3VHN5OxWid9MvFNn69wToYjykSg-ZzeD6smKm1IlcYLGbi5R0a9gbuf99fmq-fj77swjaWRDLmHsy6q41_hYCpc6-jSVwDIooGCA |
openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Accounting+for+Heteroskedasticity+Resulting+from+Between-Group+Differences+in+Multilevel+Models&rft.jtitle=Multivariate+behavioral+research&rft.au=Huang%2C+Francis+L.&rft.au=Wiedermann%2C+Wolfgang&rft.au=Zhang%2C+Bixi&rft.date=2023-05-01&rft.issn=0027-3171&rft.eissn=1532-7906&rft.volume=58&rft.issue=3&rft.spage=637&rft.epage=657&rft_id=info:doi/10.1080%2F00273171.2022.2077290&rft.externalDBID=n%2Fa&rft.externalDocID=10_1080_00273171_2022_2077290 |
thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0027-3171&client=summon |
thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0027-3171&client=summon |
thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0027-3171&client=summon |