During the past two decades, nonlinear structural equation modeling (nonlinear SEM) has received much attention due to its increasing number of applications in several areas such as the prevention sciences. By adding latent moderator terms and quadratic effects, nonlinear SEM enables a more detailed specification of the structural part of a model. Goodness of fit of conventional SEM is usually evaluated by a likelihood ratio test (the chi-square test), which compares the target model to a saturated comparison model. Until now, there did not exist a comparable global model test for nonlinear SEM.
Based on the quasi-maximum likelihood method (QML, Klein & Muthén, 2006), we propose a quasi-likelihood ratio test (Q-LRT) equivalent to the chi-square test of linear SEM. This test is based on quasi-ML instead of normal ML and includes a proper comparison model especially tailored for nonlinear SEM.
Results from a Monte Carlo study show that the Q-LRT performs well with regard to Type I error rates and power rates, when sample size is sufficiently large. In a robustness study, nonnormally distributed predictors resulted in moderately inflated Type I error rates.
The application of the Q-LRT to empirical data is demonstrated by analyzing data from a study of aging in men.
We believe that this new method provides researchers with a tool for assessing the model fit of nonlinear SEM models, which has been lacking until now.