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.
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 . Thus, level-1 units consist of the repeated measures for each subject, and the 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. In other words, it allows the testing of individual differences in patterns of responses over time. This characteristic of multilevel modeling makes it preferable to other repeated measures statistical techniques such as repeated measures-analysis of variance for certain research questions.

Assumptions

The assumptions of MLM that hold for clustered data also apply to repeated measures:
One of the assumptions of using MLM for growth curve modeling is that all subjects show the same relationship over time. 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.

Multilevel Modeling versus RM-ANOVA

Repeated measures analysis of variance has been traditionally used for analysis of repeated measures designs. However, violation of the assumptions of RM-ANOVA can be problematic. Multilevel modeling 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:

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. 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:
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.

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. 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:
SubjectGroupTime0Time1Time2
111284
211176
32151210
4211109

Long form:
SubjectGroupTimeDepVar
11012
1118
1124
............
42011
42110
4229