On the Performance of Likelihood-Based Difference Tests in Nonlinear Structural Equation Models

This article investigates likelihood-based difference statistics for testing nonlinear effects in structural equation modeling using the latent moderated structural equations (LMS) approach. In addition to the standard difference statistic T_D, 2 robust statistics have been developed in the literature to ensure valid results under the conditions of nonnormality or small sample sizes: the robust T_DR and the “strictly positive” T_DRP. These robust statistics have not been examined in combination with LMS yet. In 2 Monte Carlo studies we investigate the performance of these methods for testing quadratic or interaction effects subject to different sources of nonnormality, nonnormality due to the nonlinear terms, and nonnormality due to the distribution of the predictor variables. The results indicate that T_D is preferable to both T_DR and T_DRP. Under the condition of strong nonlinear effects and nonnormal predictors, T_DR often produced negative differences and T_DRP showed no desirable power.