Using Monte Carlo Normal Distributions to Evaluate Structural Models with Nonnormal Data

One of the main problems of statistical inference in Structural Equation Modeling (SEM) is the overall goodness of fit test. Many statistical theories have been developed based on asymptotic distributions of test statistics. When the model includes a large number of variables or the population is not from the multivariate normal distribution, the rates of convergence of these asymptotic distributions are very slow, and thus in these situations the asymptotic distributions do not approximate the distribution of the test statistics very well. Modifications to theoretical models and also bootstrap methods have been developed by researchers to improve the accuracy of hypothesis testing, mainly accuracy of Type I error, but when the sample size is small or the number of variables is large those methods have their limitations. Here we propose a Monte Carlo test that is able to control Type I error with more accuracy and it overcomes some of the limitations in the bootstrapping and theoretical approaches. Our simulation study shows that the suggested Monte Carlo test has more accurate observed significance level, as compared to other tests. Problems that occur in the bootstrapping are highlighted and it is shown that the new Monte Carlo test can overcome those problems. A power analysis shows that the new test has a reasonable power