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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.
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 [[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 for regression equations at the level of the individual. Thus, as a growth curve modeling technique, it allows for 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.|coauthors=Obeidat, Khawla, Losardo, Diane|title=Twelve Frequently Asked Questions About Growth Curve Modeling|journal=Journal of Cognition and Development|date=NaN undefined NaN|volume=11|issue=2|pages=121–136|doi=10.1080/15248371003699969}}</ref> In other words, it allows for 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==
The [[statistical assumptions|assumptions]] of MLM that hold for clustered data also apply to repeated measures:
:(1) Random components are assumed to have a normal distribution with a mean of zero
:(2) The dependent variable is assumed to be normally distributed. ''However,'' binary and discrete dependent variables may be examined in MLM using specialized procedures (i.e. employ different [[
One of the assumptions of using MLM for growth curve modeling is that all subjects show the same relationship over time (e.g. linear, quadratic etc.). Another assumption of MLM for growth curve modeling is that the observed changes are related to the passage of time.
==Statistics & Interpretation==
Mathematically, multilevel analysis with repeated measures is very similar to the analysis of data in which subjects are clustered in groups. However, one point to note is that time-related predictors must be explicitly entered into the model to evaluate trend analyses and to obtain an overall test of the repeated measure. Furthermore, interpretation of these analyses is dependent on the scale of the time variable (i.e. how it is coded).
*'''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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*'''Modeling Non-Linear Trends (Polynomial Models):'''
:*Non-linear trends (quadratic, cubic, etc.) may be evaluated in MLM by adding the products of Time (TimeXTime, TimeXTimeXTime etc.) as either random or fixed effects to the model.
*'''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 for 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 for 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 other statistical techniques for repeated measures==
===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:
::'''1. MLM has Less Stringent Assumptions:''' MLM can be used if the assumptions of constant variances (homogeneity of variance, or [[homoschedasticity]]), constant covariances (compound symmetry), or constant variances of differences scores ([[sphericity]]) are violated for RM-ANOVA. MLM allows for modeling of the variance-covariance matrix from the data; thus, unlike in RM-ANOVA, these assumptions are not necessary.
::'''2. MLM Allows for 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.
::'''3. MLM can Handle Missing Data:''' Missing data is permitted in MLM without causing additional complications. With RM-ANOVA, subject’s data must be excluded if they are missing a single data point. Missing data and attempts to resolve missing data (i.e. using the subject’s mean for non-missing data) can raise additional problems in RM-ANOVA.
::MLM can also handle data in which there is variation in the exact timing of data collection (i.e. variable timing versus fixed timing). For example, data for a longitudinal study may attempt to collect measurements at age 6 months, 9 months, 12 months, and 15 months. However, participant availability, bank holidays, and other scheduling issues may result in variation regarding when data is collected. This variation may be addressed in MLM by adding “age” into the regression equation. There is also no need for equal intervals between measurement points in MLM.
::''Note:'' Although [[missing data]] is permitted in MLM, it is assumed to be missing at random. Thus, systematically missing data can present problems.
===Multilevel Modeling versus Structural Equation Modeling (SEM; Latent Growth Model)===
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::*When there are many data points per subject
::*When the growth model is nested in additional levels of analysis (i.e. hierarchical structure)
::*Multilevel modeling programs have for more options in terms of handling non-continuous dependent variables ([[
:'''Structural equation modeling approach:'''
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==See
* [[Multilevel model]]
* [[Repeated measures design]]
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* [[Longitudinal study]]
==Further
*{{cite journal|last=Heo|first=Moonseong|coauthors=Faith, Myles S., Mott, John W., Gorman, Bernard S., Redden, David T., 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|
*{{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|
*{{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=0195152964}}
*{{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.|coauthors=Bosker, Roel J.}}
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==Notes==
{{reflist | 2}}
==References==
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{{cite book|last=Fidell|first=Barbara G. Tabachnick, Linda S.|title=Using multivariate statistics|year=2007|publisher=Pearson/A & B|___location=Boston ; Montreal|isbn=0205459382|edition=5th ed.}}
{{cite journal|last=Hoffman|first=Lesa|coauthors=Rovine, Michael J.|title=Multilevel models for the experimental psychologist: Foundations and illustrative examples|journal=Behavior Research Methods|
{{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 ed.}}
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{{cite book|last=Hox|first=Joop|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.|coauthors=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|
{{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|
{{cite journal|last=Quené|first=Hugo|coauthors=van den Bergh, Huub|title=On multi-level modeling of data from repeated measures designs: a tutorial|journal=Speech Communication|date=NaN undefined NaN|volume=43|issue=1-2|pages=103–121|doi=10.1016/j.specom.2004.02.004}}
{{Uncategorized|date=April 2012}}
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