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Multilevel Factor Analysis by Model Segregation

Measures of classroom environments have become central to policy efforts that assess school and teacher quality. This has sparked a wide interest in using multilevel factor analysis to test measurement hypotheses about classroom-level variables. One approach partitions the total covariance matrix and tests models separately on the between-classroom and within-classroom levels. This article shows that when using this approach, robust test statistics, including rescaled and residual-based test statistics provide better inferences about the classroom-level measurement structure than the widely used likelihood ratio test statistic even when the number of classrooms is large, and there is no excess kurtosis in the observed variables. This article then presents an empirical example and a simulation study to demonstrate how item intraclass correlations and within-group sample sizes influence test statistic performance. The results have implications for the study of classroom environments.