The effect of kurtosis on the power of two test statistics in covariance structure analysis

We study the effect of excess kurtosis on the non-centrality parameters of the rescaled and the residual-based test statistics for covariance structure models. The analysis is based on population matrices and parameters, which eliminates the sampling variability inherent in simulation studies. We show that the non-centrality parameters, and consequently the asymptotic power, decrease as kurtosis in the data increases. Examples are provided to compare this decrease for the two test statistics, and to illustrate how substantial it is.     

A Simple Simulation Technique for Nonnormal Data with Prespecified Skewness,Kurtosis, and Covariance Matrix

We present and investigate a simple way to generate nonnormal data using linear combinations of independent generator (IG) variables. The simulated data have prespecified univariate skewness and kurtosis and a given covariance matrix. In contrast to the widely used Vale-Maurelli (VM) transform, the obtained data are shown to have a non-Gaussian copula. We analytically obtain asymptotic robustness conditions for the IG distribution. We show empirically that popular test statistics in covariance analysis tend to reject true models more often under the IG transform than under the VM transform. This implies that overly optimistic evaluations of estimators and fit statistics in covariance structure analysis may be tempered by including the IG transform for nonnormal data generation. We provide an implementation of the IG transform in the R environment.     

On Identification and Non-normal Simulation in Ordinal Covariance and Item Response Models

Psychometrika - A standard approach for handling ordinal data in covariance analysis such as structural equation modeling is to assume that the data were produced by discretizing a multivariate...     

A Problem with Discretizing Vale–Maurelli in Simulation Studies

Psychometrika - Previous influential simulation studies investigate the effect of underlying non-normality in ordinal data using the Vale–Maurelli (VM) simulation method. We show that...     

Correcting Too Much or Too Little? The Performance of Three Chi-Square Corrections

This simulation study investigates the performance of three test statistics, T₁, T₂, and T₃, used to evaluate structural equation model fit under non normal data conditions. T₁ is the well-known mean-adjusted statistic of Satorra and Bentler. T₂ is the mean-and-variance adjusted statistic of Sattertwaithe type where the degrees of freedom is manipulated. T₃ is a recently proposed version of T₂ that does not manipulate degrees of freedom. Discrepancies between these statistics and their nominal chi-square distribution in terms of errors of Type I and Type II are investigated. All statistics are shown to be sensitive to increasing kurtosis in the data, with Type I error rates often far off the nominal level. Under excess kurtosis true models are generally over-rejected by T₁ and under-rejected by T₂ and T₃, which have similar performance in all conditions. Under misspecification there is a loss of power with increasing kurtosis, especially for T₂ and T₃. The coefficient of variation of the nonzero eigenvalues of a certain matrix is shown to be a reliable indicator for the adequacy of these statistics.