Doctor of Philosophy (Ph.D.)
Degree Granting Department
Educational & Psychological Studies
John Ferron, Ph.D.
Jeffrey Kromrey, Ph.D.
Eun Sook Kim, Ph.D.
Danielle Dennis, Ph.D.
Bayesian estimation, Hierarchical, Level-1 error structure, Multilevel, Single-Case study, Single-subject study
The Multilevel modeling (MLM) approach has a great flexibility in that can handle various methodological issues that may arise with single-case studies, such as the need to model possible dependency in the errors, linear or nonlinear trends, and count outcomes (e.g.,Van den Noortgate & Onghena, 2003a). By using the MLM framework, researchers can not only model dependency in the errors but also model a variety of level-1error structures.
The effect of misspecification in the level-1 error structure has been well studied for MLM analyses. Generally, it was found that the estimates of the fixed effects were unbiased but the estimates of variance parameters were substantially biased when level-1 error structure was misspecified. However, in previous misspecification studies as well as applied studies of multilevel models with single-case data, a critical assumption has been made. Researchers generally assumed that the level-1 error structure is constant across all participants.
It is possible that the level-1 error structure may not be same across participants. Previous studies show that there is a possibility that the level-1 error structure may not be same across participants (Baek & Ferron, 2011; Baek & Ferron, 2013; Maggin et al., 2011). If much variation in level-1 error structure exists, this can possibly impact estimation of the fixed effects and random effects. Despite the importance of this issue, the effects of modeling between-case variation in the level-1 error structure had not yet been systematically studied. The purpose of this simulation study was to extend the MLM modeling in growth curve models to allow the level-1 error structure to vary across cases, and to identify the consequences of modeling and not modeling between-case variation in the level-1 error structure for single-case studies.
A Monte Carlo simulation was conducted that examined conditions that varied in series length per case (10 or 20), the number of cases (4 or 8), the true level-1 errors structure (homogenous, moderately heterogeneous, severely heterogeneous), the level-2 error variance in baseline slope and shift in slope (.05 or .2 times the level-1 variance), and the method to analyze the data (allow level-1 error variance and autocorrelation to vary across cases (Model 2) or not allow level-1 error variance and autocorrelation to vary across cases (Model 1)). All simulated data sets were analyzed using Bayesian estimation. For each condition, 1000 data were simulated, and bias, RMSE and credible interval (CI) coverage and width were examined for the fixed treatment effects and the variance components.
The results of this study found that the different modeling methods in level-1 error structure had little to no impact on the estimates of the fixed treatment effects, but substantial impacts on the estimates of the variance components, especially the level-1 error standard deviation and the autocorrelation parameters. Modeling between case variation in the level-1 error structure (Model 2) performs relatively better than not modeling between case variation in the level-1 error structure (Model 1) for the estimates of the level-1 error standard deviation and the autocorrelation parameters. It was found that as degree of the heterogeneity in the data (i.e., homogeneous, moderately heterogeneous, severely heterogeneous) increased, the effectiveness of Model 2 increased.
The results also indicated that whether the level-1 error structure was under-specified, over-specified, or correctly-specified had little to no impact on the estimates of the fixed treatment effects, but a substantial impact on the level-1 error standard deviation and the autocorrelation. While the correctly-specified and the over-specified models perform fairly well, the under-specified model performs poorly.
Moreover, it was revealed that the form of heterogeneity in the data (i.e., one extreme case versus a more even spread of the level-1 variances) might have some impact on relative effectiveness of the two models, but the degree of the autocorrelation had little to no impact on the relative performance of the two models.
Scholar Commons Citation
Baek, Eun Kyeng, "Consequences of Non-Modeled and Modeled Between Case Variation in the Level-1 Error Structure in Multilevel Models for Single-Case Data: A Monte Carlo Study" (2015). Graduate Theses and Dissertations.