Exploring the Test of Covariate Moderation Effects in Multilevel MIMIC Models

Abstract

In multilevel multiple-indicator multiple-cause (MIMIC) models, covariates can interact at the within level, at the between level, or across levels. This study examines the performance of multilevel MIMIC models in estimating and detecting the interaction effect of two covariates through a simulation and provides an empirical demonstration of modeling the interaction in multilevel MIMIC models. The design factors include the location of the interaction effect (i.e., between, within, or across levels), cluster number, cluster size, intraclass correlation (ICC) level, magnitude of the interaction effect, and cross-level measurement invariance status. Type I error, power, relative bias, and root mean square of error of the interaction effects are examined. The results showed that multilevel MIMIC models performed well in detecting the interaction effect at the within or across levels. However, when the interaction effect was at the between level, the performance of multilevel MIMIC models depended on the magnitude of the interaction effect, ICC, and sample size, especially cluster number. Overall, cross-level measurement noninvariance did not make a notable impact on the estimation of interaction in the structural part of multilevel MIMIC models when factor loadings were allowed to be different across levels.

Examining the Impact of and Sensitivity of Fit Indices to Omitting Covariates Interaction Effect in Multilevel Multiple-Indicator Multiple-Cause Models

“…In this study, for data generation, we adopted the ML MIMIC models that Cao et al (2019) used to detect covariates interaction effects in correctly specified ML MIMIC models. To be more specific, the population (true) models had a within-level covariates interaction effect, a between-level interaction effect, and a cross-level interaction effect, for the within-level interaction conditions, the between-level interaction conditions, and cross-level interaction conditions, respectively.…”

“…To be more specific, the population (true) models had a within-level covariates interaction effect, a between-level interaction effect, and a cross-level interaction effect, for the within-level interaction conditions, the between-level interaction conditions, and cross-level interaction conditions, respectively. In the study of Cao et al (2019), the focus was on the performance of ML MIMIC in detecting a significant covariates interaction effect using the true model. The current study used the true model proposed by them to generate the data and to examine the impact of excluding the interaction effect.…”

“…For the structural part of ML MIMIC models, three types of covariates interaction effect were described by Cao et al (2019). Between-level covariates interaction occurs when both covariates have an effect on the between-level latent factor, and the effect of one covariate on the latent effect is moderated by the other one as shown in Equation 8:…”

“…In ML MIMIC with cross-level interaction, the effect of the within-level covariate on the within-level factor is a random effect (Equation 10) that is predicted by the between-level covariate (Equation 11), which also predicts the between-level factor as shown in Equation 12. For more details about model specification of the three types of covariates interaction in ML MIMIC models, refer to Cao et al (2019).…”

“…Multiple-indicator multiple-cause (MIMIC) models under the framework of SEM have been gaining popularity in educational and psychological research because of their inclusion of causal indicators that predict the latent factor as well as effect indicators (Jöreskog & Goldberger, 1975;Kaplan, 2009). The multilevel MIMIC (ML MIMIC) model is used to account for the dependence of individuals clustered in the same unit, and it possesses the flexibility of simultaneously modeling covariates at the between level and at the within level, as well as interaction effects of the covariates (Cao et al, 2019).The results of this study showed adequate performance of ML MIMIC models in detecting the within-level and cross-level covariates interaction effect in most conditions, but the power of testing between-level interaction was lower, depending primarily on the number of clusters (CN) and intraclass correlation (ICC). However, when applied researchers employed ML MIMIC to examine the effects of Level 1 covariates and Level 2 covariates on the latent factor, they tended not to test covariates interaction effects in the models (e.g., Tsai et al, 2017).…”