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

Growth mixture modeling (GMM) is an analytical tool for identifying multiple unobserved sub-populations of longitudinal processes. In particular, it describes change patterns within each latent sub-population and examines between-individual differences in within-individual change for each sub-group. One research interest in utilizing GMMs is to explore how covariates affect such heterogeneity in change patterns. Liu and Perera (2022c) extended mixture-of-experts (MoE) models, which mainly focus on time-invariant covariates, for allowing the covariates to account for within-group and between-group differences simultaneously and examining the heterogeneity in nonlinear trajectories. The present study further extends Liu and Perera (2022c) and examines the effects on trajectory heterogeneity of time-varying covariates (TVCs). Specifically, we propose methods to decompose a TVC into a trait 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 to account for within-group differences in growth factors of trajectories (i.e., trait effect), and the state features are allowed to impact observed values of a longitudinal process (i.e., state effect). We examine the proposed models using a simulation study and a real-world data analysis. The simulation study demonstrated that the proposed models are capable of separating trajectories into several clusters and generally generating unbiased and accurate estimates with target coverage probabilities. With the proposed models, we showed the heterogeneity in the trait and state features of reading ability across latent classes of students' mathematics performance. Meanwhile, the trait and state effects on mathematics development of reading ability are also heterogeneous across the clusters of students.