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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).
Multilevel modeling with repeated measures employs the same statistical techniques as MLM with clustered data. In multilevel modeling for repeated measures data, the measurement occasions are nested within cases (e.g. individual or subject). Thus, [[Multilevel model#Level 1 regression equation|level-1]] units consist of the repeated measures for each subject, and the [[Multilevel model#Level 1 regression equation|level-2]] unit is the individual or subject. In addition to estimating overall parameter estimates, MLM allows regression equations at the level of the individual. Thus, as a growth curve modeling technique, it allows the estimation of inter-individual differences in intra-individual change over time by modeling the variances and covariances.<ref>{{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|volume=11|issue=2|pages=121–136|doi=10.1080/15248371003699969|pmid=21743795 |pmc=3131138|year=2010 }}</ref> In other words, it allows the testing of individual differences in patterns of responses over time (i.e. growth curves). This characteristic of multilevel modeling makes it preferable to other repeated measures statistical techniques such as repeated measures-analysis of variance ([[RM-ANOVA]]) for certain research questions.
==Assumptions==
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*'''Fixed Effects:''' Fixed regression coefficients may be obtained for an overall equation that represents how, averaging across subjects, the subjects change over time.
*'''Random Effects:''' Random effects are the variance components that arise from measuring the relationship of the predictors to Y for each subject separately. These variance components include: (1) differences in the intercepts of these equations at the level of the subject; (2) differences across subjects in the slopes of these equations; and (3) covariance between subject slopes and intercepts across all subjects. When random coefficients are specified, each subject has its own regression equation, making it possible to evaluate whether subjects differ in their means and/or response patterns over time.
*'''Estimation Procedures & Comparing Models:''' These procedures are identical to those used in multilevel analysis where subjects are clustered in groups.
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*'''Adding Predictors to the Model:''' It is possible that some of the random variance (i.e. variance associated with individual differences) may be attributed to fixed predictors other than time. Unlike RM-ANOVA, multilevel analysis allows the use of continuous predictors (rather than only categorical), and these predictors may or may not account for individual differences in the intercepts as well as for differences in slopes. Furthermore, multilevel modeling also allows time-varying covariates.
*'''Alternative Specifications:'''
:*''Covariance Structure:'' Multilevel software provides several different covariance or error structures to choose from for the analysis of multilevel data (e.g. autoregressive). These may be applied to the growth model as appropriate.
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===Multilevel Modeling versus RM-ANOVA===
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. 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|
::'''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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::'''5. MLM is relatively easily extended to discrete data.''' <ref>{{cite book | last = Molenberghs | first = Geert | title = Models for discrete longitudinal data | publisher = Springer Science+Business Media, Inc | ___location = New York | year = 2005 | isbn = 978-0387251448 }}</ref>
::''Note:'' Although [[missing data]] is permitted in MLM, it is assumed to be missing at random. Thus, systematically missing data can present problems.<ref name=quene /><ref>{{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}}</ref><ref>{{cite journal|
===Multilevel Modeling versus Structural Equation Modeling (SEM; Latent Growth Model)===
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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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==Further reading==
*{{cite journal|last=Heo|first=Moonseong|author2=Faith, Myles S. |author3=Mott, John W. |author4=Gorman, Bernard S. |author5=Redden, David T. |author6=Allison, David B. |title=Hierarchical linear models for the development of growth curves: an example with body mass index in overweight/obese adults|journal=Statistics in Medicine|year=2003|volume=22|issue=11|pages=1911–1942|doi=10.1002/sim.1218|pmid=12754724}}
*{{cite journal|last=Singer|first=J. D.|title=Using SAS PROC MIXED to Fit Multilevel Models, Hierarchical Models, and Individual Growth Models|journal=Journal of Educational and Behavioral Statistics|year=1998|volume=23|issue=4|pages=323–355|doi=10.3102/10769986023004323}}
*{{cite book|last=Willett|first=Judith D. Singer, John B.|title=Applied longitudinal data analysis : modeling change and event occurrence|year=2003|publisher=Oxford University Press|___location=Oxford|isbn=
*{{cite book|last=Snijders|first=Tom A.B.|title=Multilevel analysis : an introduction to basic and advanced multilevel modeling|year=2002|publisher=Sage Publications|___location=London|isbn=978-0761958901|edition=Reprint.|author2=Bosker, Roel J.}}
*{{cite book | last = Hedeker | first = Donald | title = Longitudinal data analysis | publisher = Wiley-Interscience | ___location = Hoboken, N.J | year = 2006 | isbn = 978-0471420279 }} Covers many models and shows the advantages of MLM over other approaches
*{{cite book | last = Verbeke | first = Geert | title = Linear mixed models for longitudinal data | publisher = Springer-Verlag New York | ___location = S.l | year = 2013 | isbn = 978-1475773842 }} Has extensive SAS code.
*{{cite book | last = Molenberghs | first = Geert | title = Models for discrete longitudinal data | publisher = Springer Science+Business Media, Inc | ___location = New York | year = 2005 | isbn = 978-0387251448 }} Covers non-linear models. Has SAS code.
*{{cite book | last = Pinheiro | first = Jose | author2 = Bates, Douglas M. | title = Mixed-effects models in S and S-PLUS | publisher = Springer | ___location = New York, NY u.a | year = 2000 | isbn = 978-1441903174 }} Uses S and S-plus but will be useful for R users as well.
==Notes==
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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=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}}
*{{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|last1=Fidell|first1=Barbara G.|last2= Tabachnick|first2= Linda S.|title=Using Multivariate Statistics|year=2007|publisher=Pearson/A & B|___location=Boston ; Montreal|isbn=0205459382|edition=5th}}
▲*{{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}}
*{{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 |
*{{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=
▲*{{cite journal|last=Overall|first=John|coauthors=Ahn, Chul, Shivakumar, C., Kalburgi, Yallapa|title=PROBLEMATIC FORMULATIONS OF SAS PROC.MIXED MODELS FOR REPEATED MEASUREMENTS|journal=Journal of Biopharmaceutical Statistics|year=2007|volume=9|issue=1|pages=189–216|doi=10.1081/BIP-100101008}}
▲*{{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}}
▲[[Category:Statistical models]]
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