Evaluating the performance of existing and novel equivalence tests for fit indices in structural equation modelling

It has been suggested that equivalence testing (otherwise known as negligible effect testing) should be used to evaluate model fit within structural equation modelling (SEM). In this study, we propose novel variations of equivalence tests based on the popular root mean squared error of approximation and comparative fit index fit indices. Using Monte Carlo simulations, we compare the performance of these novel tests to other existing equivalence testing-based fit indices in SEM, as well as to other methods commonly used to evaluate model fit. Results indicate that equivalence tests in SEM have good Type I error control and display considerable power for detecting well-fitting models in medium to large sample sizes. At small sample sizes, relative to traditional fit indices, equivalence tests limit the chance of supporting a poorly fitting model. We also present an illustrative example to demonstrate how equivalence tests may be incorporated in model fit reporting. Equivalence tests in SEM also have unique interpretational advantages compared to other methods of model fit evaluation. We recommend that equivalence tests be utilized in conjunction with descriptive fit indices to provide more evidence when evaluating model fit.