ⓘ Multilevel modeling for repeated measures
One application of multilevel modeling 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 ; however, it may also be used for repeated measures data in which time is not a factor.
In multilevel modeling, an overall change function 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, level1 units consist of the repeated measures for each subject, and the level2 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 interindividual differences in intraindividual change over time by modeling the variances and covariances. 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 measuresanalysis of variance RMANOVA for certain research questions.
1. Assumptions
The 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 link functions.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.
2. 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 timerelated 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.
 Estimation Procedures & Comparing Models: These procedures are identical to those used in multilevel analysis where subjects are clustered in groups.
 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.
 Fixed Effects: Fixed regression coefficients may be obtained for an overall equation that represents how, averaging across subjects, the subjects change over time.
2.1. Statistics & Interpretation Extensions
 Modeling NonLinear Trends Polynomial Models
 Nonlinear trends may be evaluated in MLM by adding the products of Time TimeXTime, TimeXTimeXTime etc. as either random or fixed effects to the model.
 Alternative Specifications
 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 RMANOVA, 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 timevarying covariates.
 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.
 Dependent Variable: Dichotomous dependent variables may be analyzed with multilevel analysis by using more specialized analysis i.e. using the logit or probit link functions.
3.1. Multilevel modeling versus other statistical techniques for repeated measures Multilevel Modeling versus RMANOVA
Repeated measures analysis of variance RMANOVA has been traditionally used for analysis of repeated measures designs. However, violation of the assumptions of RMANOVA 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 RMANOVA:
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 RMANOVA. MLM allows modeling of the variancecovariance matrix from the data; thus, unlike in RMANOVA, these assumptions are not necessary. 2. MLM Allows Hierarchical Structure: MLM can be used for higherorder sampling procedures, whereas RMANOVA is limited to examining twolevel sampling procedures. In other words, MLM can look at repeated measures within subjects, within a third level of analysis etc., whereas RMANOVA is limited to repeated measures within subjects. 3. MLM can Handle Missing Data: Missing data is permitted in MLM without causing additional complications. With RMANOVA, 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 nonmissing data can raise additional problems in RMANOVA. 4. 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. 5. MLM is relatively easily extended to discrete data. Note: Although missing data is permitted in MLM, it is assumed to be missing at random. Thus, systematically missing data can present problems.3.2. Multilevel modeling versus other statistical techniques for repeated measures Multilevel Modeling versus Structural Equation Modeling SEM; Latent Growth Model
An alternative method of growth curve analysis is latent growth curve modeling using structural equation modeling SEM. This approach will provide the same estimates as the multilevel modeling approach, provided that the model is specified identically in SEM. However, there are circumstances in which either MLM or SEM are preferable:
Multilevel modeling approach: For designs with a large number of unequal intervals between time points SEM cannot manage data with a lot of variation in time points
 When the growth model is nested in additional levels of analysis i.e. hierarchical structure
 When there are many data points per subject
 Multilevel modeling programs have for more options in terms of handling noncontinuous dependent variables link functions and allowing different error structures
 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 greater flexibility.
The distinction between multilevel modeling and latent growth curve analysis has 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.
4. Data Structure
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:
Long form:
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