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Growth mixture modeling (GMM) is an analytical tool for identifying multipleunobserved sub-populations of longitudinal processes. In particular, itdescribes change patterns within each latent sub-population and examinesbetween-individual differences in within-individual change for each sub-group.One research interest in utilizing GMMs is to explore how covariates affectsuch heterogeneity in change patterns. Liu and Perera (2022c) extendedmixture-of-experts (MoE) models, which mainly focus on time-invariantcovariates, for allowing the covariates to account for within-group andbetween-group differences simultaneously and examining the heterogeneity innonlinear trajectories. The present study further extends Liu and Perera(2022c) and examines the effects on trajectory heterogeneity of time-varyingcovariates (TVCs). Specifically, we propose methods to decompose a TVC into atrait feature (e.g., the baseline value of the TVC) and a set of state features(e.g., interval-specific slopes or changes). The trait features are allowed toaccount for within-group differences in growth factors of trajectories (i.e.,trait effect), and the state features are allowed to impact observed values ofa longitudinal process (i.e., state effect). We examine the proposed modelsusing a simulation study and a real-world data analysis. The simulation studydemonstrated that the proposed models are capable of separating trajectoriesinto several clusters and generally generating unbiased and accurate estimateswith target coverage probabilities. With the proposed models, we showed theheterogeneity in the trait and state features of reading ability across latentclasses of students' mathematics performance. Meanwhile, the trait and stateeffects on mathematics development of reading ability are also heterogeneousacross the clusters of students.

Further Exploration of the Effects of Time-varying Covariate in Growth Mixture Models with Nonlinear Trajectories