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Researchers are usually interested in examining the impact of covariates whenseparating heterogeneous samples into latent classes that are more homogeneous.The majority of theoretical and empirical studies with such aims have focusedon identifying covariates as predictors of class membership in the structuralequation modeling framework. In other words, the covariates only indirectlyaffect the sample heterogeneity. However, the covariates' influence onbetween-individual differences can also be direct. This article presents amixture model that investigates covariates to explain within-cluster andbetween-cluster heterogeneity simultaneously, known as a mixture-of-experts(MoE) model. This study aims to extend the MoE framework to investigateheterogeneity in nonlinear trajectories: to identify latent classes, covariatesas predictors to clusters, and covariates that explain within-clusterdifferences in change patterns over time. Our simulation studies demonstratethat the proposed model generally estimates the parameters unbiasedly,precisely and exhibits appropriate empirical coverage for a nominal 95%confidence interval. This study also proposes implementing structural equationmodel forests to shrink the covariate space of the proposed mixture model. Weillustrate how to select covariates and construct the proposed model withlongitudinal mathematics achievement data. Additionally, we demonstrate thatthe proposed mixture model can be further extended in the structural equationmodeling framework by allowing the covariates that have direct effects to betime-varying.

Extending Mixture of Experts Model to Investigate Heterogeneity of Trajectories: When, Where and How to Add Which Covariates