Computational Statistics & Data Analysis 44 (2003) 89–107 www.elsevier.com/locate/csda Eight test statistics for multilevel structural equation models  Ke-Hai Yuan a , Peter M. Bentler b; ∗ a Department of Psychology, University of Notre Dame, Notre Dame, IN 46556, USA b Departments of Psychology and Statistics, Institute of Psychology and Statistics, University of California, Box 951563, UCLA, Los Angeles, CA 90095-1563, USA Received 7 November 2002 Abstract Data in social and behavioral sciences are often hierarchically organized though seldom nor- mal. They typically contain heterogeneous marginal skewnesses and kurtoses. With such data, the normal theory based likelihood ratio statistic is not reliable when evaluating a multilevel structural equation model. Statistics that are not sensitive to sampling distributions are desir- able. Six statistics for evaluating a structural equation model are extended from the conventional context to the multilevel context. These statistics are asymptotically distribution free, that is, their distributions do not depend on the sampling distribution when sample size at the highest level is large enough. The performance of these statistics in practical data analysis is evaluated with a Monte Carlo study simulating conditions encountered with real data. Results indicate that each of the statistics is very insensitive to the underlying sampling distributions even with nite sample sizes. However, the six statistics perform quite dierently at smaller sample sizes; some over-reject the correct model and some under-reject the correct model. Comparing the six statistics with two existing ones in the multilevel context, two of the six new statistics are recommended for model evaluation in practice. c  2002 Elsevier B.V. All rights reserved. Keywords: Nonnormal data; Asymptotically distribution free statistics; Generalized estimating equation; Monte Carlo 1. Introduction Data in social and behavioral sciences often exhibit a hierarchical structure. For ex- ample, households are nested within neighborhoods, neighborhood are nested within  This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and by a University of Notre Dame faculty research grant. ∗ Corresponding author. E-mail addresses: kyuan@nd.edu (K.-H. Yuan), bentler@ucla.edu (P.M. Bentler). 0167-9473/03/$-see front matter c  2002 Elsevier B.V. All rights reserved. doi:10.1016/S0167-9473(02)00349-3