Efficiency and Robustness of Alternative Estimators for Two- and Three-level Models: The Case of NAEP

This article investigates the efficiency and robustness of alternative estimators of regression coefficients for three-level data. To study student achievement, researchers might formulate a standard regression model or a hierarchical model with a two- or three-level structure. Having chosen the model, the researchers might employ either a model-based or a robust estimator of the standard errors. A simulation study showed that, as expected, the hierarchical model analyses produced more efficient point estimates than did analyses that ignored the covariance structure in the data, even when the normality assumption was violated. When samples were fairly large, the three-level analyses produced sound standard errors. In contrast, single-level analysis yielded seriously biased standard errors for coefficients defined at level 3 and level 2; and two-level analysis yielded biased standard errors for coefficients defined at level 2. These biases in standard error estimates were largely corrected by robust variance estimation. Implications of the results for analyzing NAEP and other large-scale surveys such as the Early Childhood Longitudinal Study (ECLS) and the Third International Mathematics and Science Study (TIMSS) are discussed.