Using SAS to conduct nonparametric residual bootstrap multilevel modeling with a small number of groups.

In multilevel modeling, researchers often encounter data with a relatively small number of units at the higher levels. As a result, of this and/or non-normality of the residuals, model parameter estimates, particularly the variance components and standard errors of parameter estimates at the group level, may be biased, thus the corresponding statistical inferences may not be trustworthy. This problem can be addressed by using bootstrap methods to estimate the standard errors of the parameter estimates for significance testing. This study illustrates how to use statistical analysis system (SAS) to conduct nonparametric residual bootstrap multilevel modeling. Specific SAS programs for such modeling are provided

Parametric Bootstrap Interval Approach to Inference for Fixed Effects in the Mixed Linear Model

In mixed models, empirical best linear unbiased estimators of fixed effects generally have mean square errors (MSEs) that cannot be written in closed form. Standard methods of inference depend upon approximation of the estimator MSE, as well as upon approximation of the test statistic distribution by some known distribution, and may not perform well under small samples. The parametric bootstrap interval is presented as an alternative to standard methods of inference. Several parametric bootstrap intervals (Efron percentile, bias-corrected [BC], Hall percentile, and bootstrap-t) were compared using simulated data, along with analytic intervals based on the naïve MSE approximation and the Kenward-Roger method. Among the bootstrap methods, the bootstrap-t seems especially promising

Residual Normality Assumption and the Estimation of Multiple Membership Random Effects Models

Data collected in the human and biological sciences often have multilevel structures. While conventional hierarchical linear modeling is applicable to purely hierarchical data, multiple membership random effects modeling is appropriate for non-purely nested data wherein some lower-level units manifest mobility across higher-level units. Fitting a multiple membership random effects model (MMrem) to non-purely nested data may account for lower-level observation interdependencies and the contextual effects of higher-level units on the outcomes of lower-level units. One important assumption in multilevel modeling is normality of the residual distributions. Although a few recent studies have investigated the effect of cluster-level residual non-normality on hierarchical linear modeling estimation for purely hierarchical data, no research has examined MMrem robustness issues given residual non-normality. The purpose of the present research was to extend prior research on the influence of residual non-normality from purely nested data structures to multiple membership data structures. To investigate the statistical performance of an MMrem when the level-two residual distributional assumption was unmet, this research inquiry employed a Monte Carlo simulation study to examine two-level MMrem fixed effect and variance component parameter estimate biases and inferential errors under a fully crossed study design. Simulation factors included the level-two residual distribution, number of level-two clusters, number of level-one units per cluster, intra-cluster correlation coefficient, and mobility rate. The generating parameters for the Monte Carlo simulation study were based on an analysis of a subset of the newly-released publicly-available data of the Early Childhood Longitudinal Study, Kindergarten Class of 2010-11. By building upon previous MMrem methodological studies, this research inquiry sought answers to the following questions: When the level-two residual normality assumption was violated, (1) how accurate were MMrem fixed effect and variance component parameter estimates, and (2) what sample size was adequate with respect to MMrem estimation? The findings should be useful for research in education, public health, psychology, and other fields, and contribute to the literature on the importance of residual normality for the accuracy of MMrem estimates