Multilevel modeling for repeated measures

Updated on Nov 21, 2023

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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, 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 (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 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.

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.

Extensions

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 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.

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).

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:

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: Structural equation modeling approach:

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 (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:

References

Multilevel modeling for repeated measures Wikipedia

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