Testing Moderation in Business and Psychological Studies with Latent Moderated Structural Equations

Most organizational researchers understand the detrimental effects of measurement errors in testing relationships among latent variables and hence adopt structural equation modeling (SEM) to control for measurement errors. Nonetheless, many of them revert to regression-based approaches, such as moderated multiple regression (MMR), when testing for moderating and other nonlinear effects. The predominance of MMR is likely due to the limited evidence showing the superiority of latent interaction approaches over regression-based approaches combined with the previous complicated procedures for testing latent interactions. In this teaching note, we first briefly explain the latent moderated structural equations (LMS) approach, which estimates latent interaction effects while controlling for measurement errors. Then we explain the reliability-corrected single-indicator LMS (RCSLMS) approach to testing latent interactions with summated scales and correcting for measurement errors, yielding results which approximate those from LMS. Next, we report simulation results illustrating that LMS and RCSLMS outperform MMR in terms of accuracy of point estimates and confidence intervals for interaction effects under various conditions. Then, we show how LMS and RCSLMS can be implemented with Mplus, providing an example-based tutorial to demonstrate a 4-step procedure for testing a range of latent interactions, as well as the decisions at each step. Finally, we conclude with answers to some frequently asked questions when testing latent interactions. As supplementary files to support researchers, we provide a narrated PowerPoint presentation, all Mplus syntax and output files, data sets for numerical examples, and Excel files for conducting the loglikelihood values difference test and plotting the latent interaction effects.