Multilevel modeling for repeated measures: Difference between revisions

Revision as of 22:10, 5 January 2013

Revision as of 12:36, 20 March 2013

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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==

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:*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:'''

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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|coauthors=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}}</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 for modeling of the variance-covariance matrix from the data; thus, unlike in RM-ANOVA, these assumptions are not necessary.<ref name=cohen>{{cite book|first=Jacob Cohen|title=Applied multiple regression/correlation analysis for the behavioral sciences|publisher=Erlbaum|___location=Mahwah, NJ [u.a.]|isbn=9780805822236|edition=3. ed.}}</ref>

::'''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.

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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 ([[link function]]s) and allowing for different error structures

:'''Structural equation modeling approach:'''

::*Better suited for extended models in which the model is embedded into a larger path model, or the intercept and slope are used as predictors for other variables. In this way, SEM allows for greater flexibility.

The distinction between multilevel modeling and latent growth curve analysis is become less defined. Some statistical programs incorporate multilevel features within their structural equation modeling software, and some multilevel modeling software is beginning to add latent growth curve features.