Estimating nonlinear effects of random slopes: A comparison of multilevel structural equation modeling with a...

Multilevel structural equation modeling (MSEM) is a statistical framework of major relevance for research concerned with people's intrapersonal dynamics. An application domain that is rapidly gaining relevance is the study of individual differences in the within-person association (WPA) of variables that fluctuate over time. For instance, an individual's social reactivity - their emotional response to social situations - can be represented as the association between repeated measurements of the individual's social interaction quantity and momentary well-being. MSEM allows researchers to investigate the associations between WPAs and person-level outcome variables (e.g., life satisfaction) by specifying the WPAs as random slopes in the structural equation on level 1 and using the latent representations of the slopes to predict outcomes on level 2. Here, we are concerned with the case in which a researcher is interested in nonlinear effects of WPAs on person-level outcomes - a U-shaped effect of a WPA, a moderation effect of two WPAs, or an effect of congruence between two WPAs - such that the corresponding MSEM includes latent interactions between random slopes. We evaluate the nonlinear MSEM approach for the three classes of nonlinear effects (U-shaped, moderation, congruence) and compare it with three simpler approaches: a simple two-step approach, a single-indicator approach, and a plausible values approach. We use a simulation study to compare the approaches on accuracy of parameter estimates and inference. We derive recommendations for practice and provide code templates and an illustrative example to help researchers implement the approaches.