Eight test statistics for multilevel structural equation models

Abstract

Data in social and behavioral sciences are often hierarchically organized though seldom normal. 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 desirable. 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 finite sample sizes. However, the six statistics perform quite differently 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.