Normality assumption in latent interaction models

Latent moderated structural equation (LMS) is one of the most common techniques for estimating interaction effects involving latent variables (i.e., XWITH command in Mplus). However, empirical applications of LMS often overlook that this estimation technique assumes normally distributed variables and that violations of this assumption may lead to seriously biased parameter estimates. Against this backdrop, we study the robustness of LMS to different shapes and sources of nonnormality and examine whether various statistical tests can help researchers detect such distributional misspecifications. In four simulations, we show that LMS can be severely biased when the latent predictors or the structural disturbances are nonnormal. On the contrary, LMS is unaffected by nonnormality originating from measurement errors. As a result, testing for the multivariate normality of observed indicators of the latent predictors can lead to erroneous conclusions, flagging distributional misspecifications in perfectly unbiased LMS results and failing to reject seriously biased results. To solve this issue, we introduce a novel Hausman-type specification test to assess the distributional assumptions of LMS and demonstrate its performance.