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One application of [[multilevel modeling]] (MLM) is the analysis of repeated measures data. '''Multilevel modeling for repeated measures''' data is most often discussed in the context of modeling change over time (i.e. growth curve modeling for longitudinal designs); however, it may also be used for repeated measures data in which time is not a factor.<ref>{{cite journal|last=Hoffman|first=Lesa|author2=Rovine, Michael J.|title=Multilevel models for the experimental psychologist: Foundations and illustrative examples|journal=Behavior Research Methods|year=2007|volume=39|issue=1|pages=101–117|doi=10.3758/BF03192848|pmid=17552476|doi-access=free}}</ref>
In multilevel modeling, an overall change function (e.g. linear, quadratic, cubic etc.) is fitted to the whole sample and, just as in multilevel modeling for clustered data, the [[slope]] and [[Y-intercept|intercept]] may be allowed to vary. For example, in a study looking at income growth with age, individuals might be assumed to show linear improvement over time. However, the exact intercept and slope could be allowed to vary across individuals (i.e. defined as random coefficients).
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Repeated measures analysis of variance ([[RM-ANOVA]]) has been traditionally used for analysis of [[repeated measures]] designs. However, violation of the assumptions of RM-ANOVA can be problematic. Multilevel modeling (MLM) is commonly used for repeated measures designs because it presents an alternative approach to analyzing this type of data with three main advantages over RM-ANOVA:<ref name=quene>{{cite journal|last=Quené|first=Hugo|author2=van den Bergh, Huub|title=On multi-level modeling of data from repeated measures designs: a tutorial|journal=Speech Communication|year=2004|volume=43|issue=1–2|pages=103–121|doi=10.1016/j.specom.2004.02.004|citeseerx=10.1.1.2.8982}}</ref>
::'''1. MLM has Less Stringent Assumptions:''' MLM can be used if the assumptions of constant variances (homogeneity of variance, or [[homoscedasticity]]), constant covariances (compound symmetry), or constant variances of differences scores ([[sphericity]]) are violated for RM-ANOVA. MLM allows modeling of the variance-covariance matrix from the data; thus, unlike in RM-ANOVA, these assumptions are not necessary.<ref name=cohen>{{cite book|first1=Jacob|last1=Cohen|first2=Patricia|last2=Cohen|first3=Stephen G.|last3=West|first4=Leona S.|last4=Aiken|author4-link= Leona S. Aiken |title=Applied multiple regression/correlation analysis for the behavioral sciences|publisher=Erlbaum|___location=Mahwah, NJ [u.a.]|isbn=9780805822236|edition=3.|date=2003-10-03}}</ref>
::'''2. MLM Allows Hierarchical Structure:''' MLM can be used for higher-order sampling procedures, whereas RM-ANOVA is limited to examining two-level sampling procedures. In other words, MLM can look at repeated measures within subjects, within a third level of analysis etc., whereas RM-ANOVA is limited to repeated measures within subjects.
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Multilevel modeling with repeated measures data is computationally complex. Computer software capable of performing these analyses may require data to be represented in “long form” as opposed to “wide form” prior to analysis. In long form, each subject’s data is represented in several rows – one for every “time” point (observation of the dependent variable). This is opposed to wide form in which there is one row per subject, and the repeated measures are represented in separate columns. Also note that, in long form, time invariant variables are repeated across rows for each subject. See below for an example of wide form data transposed into long form:
'''Wide form:'''
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'''Long form:'''
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==References==
*{{cite book|first1=Jacob|last1=Cohen|first2=Patricia|last2=Cohen|first3=Stephen G.|last3= West|first4=Leona S.|last4= Aiken |author4-link= Leona S. Aiken |year=2002|title=Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences|publisher=Routledge Academic|isbn=9780805822236|edition=3.}}
*{{cite journal|last=Curran|first=Patrick J. |author2=Obeidat, Khawla |author3=Losardo, Diane|title=Twelve Frequently Asked Questions About Growth Curve Modeling|journal=Journal of Cognition and Development|year=2010|volume=11|issue=2|pages=121–136|doi=10.1080/15248371003699969|pmid=21743795 |pmc=3131138}}
*{{cite book|last1=Fidell|first1=Barbara G.|last2= Tabachnick|first2= Linda S.|title=Using Multivariate Statistics|year=2007|publisher=Pearson/A & B|___location=Boston
*{{cite journal|last=Hoffman|first=Lesa|author2=Rovine, Michael J.|title=Multilevel models for the experimental psychologist: Foundations and illustrative examples|journal=Behavior Research Methods|year=2007|volume=39|issue=1|pages=101–117|doi=10.3758/BF03192848|pmid=17552476|doi-access=free}}
*{{cite book|last=Howell|first=David C.|title=Statistical methods for psychology|year=2010|publisher=Thomson Wadsworth|___location=Belmont, CA|isbn=978-0-495-59784-1|edition=7th}}
*{{cite book|last=Hox|first=Joop|authorlink=Joop Hox|title=Multilevel and SEM Approached to Growth Curve Modeling|year=2005|publisher=Wiley|___location=Chichester|isbn=978-0-470-86080-9|url=http://joophox.net/publist/ebs05.pdf|edition=[Repr.].}}
*{{cite journal|last=Overall|first=John E.|author2=Tonidandel, Scott|title=Analysis of Data from a Controlled Repeated Measurements Design with Baseline-Dependent Dropouts|journal=Methodology: European Journal of Research Methods for the Behavioral and Social Sciences|year=2007|volume=3|issue=2|pages=58–66|doi=10.1027/1614-2241.3.2.58}}
*{{cite journal|last=Overall|first=John |author2=Ahn, Chul |author3=Shivakumar, C. |author4=Kalburgi, Yallapa
*{{cite journal|last=Quené|first=Hugo|author2=van den Bergh, Huub|title=On multi-level modeling of data from repeated measures designs: a tutorial|journal=Speech Communication|year=2004|volume=43|issue=1–2|pages=103–121|doi=10.1016/j.specom.2004.02.004|citeseerx=10.1.1.2.8982}}
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