The Asymptotic Covariance Matrix and its Use in Simulation Studies

The asymptotic performance of structural equation modeling tests and standard errors are influenced by two factors: the model and the asymptotic covariance matrix of the sample covariances. Although most simulation studies clearly specify model conditions, specification of is usually limited to values of univariate skewness and kurtosis. We illustrate that marginal skewness and kurtosis are not sufficient to adequately specify a nonnormal simulation condition by showing that asymptotic standard errors and test statistics vary substantially among distributions with skewness and kurtosis that are identical. We argue therefore that should be reported when presenting the design of simulation studies. We show how can be exactly calculated under the widely used Vale-Maurelli transform. We suggest plotting the elements of and reporting the eigenvalues associated with the test statistic. R code is provided.