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 in longitudinal processes. In particular, it describes change patterns within each latent sub-population and investigates between-individual differences in within-individual change for each sub-group. A key research interest in using GMMs is examining how covariates influence the heterogeneity in change patterns. Liu & Perera (2022b) extended mixture-of-experts (MoE) models, which primarily focus on time-invariant covariates, to allow covariates to account for both within-group and between-group differences and investigate the heterogeneity in nonlinear trajectories. The present study further extends Liu & Perera, 2022b by examining the effects of time-varying covariates (TVCs) on trajectory heterogeneity. Specifically, we propose methods to decompose a TVC into an initial trait (the baseline value of the TVC) and a set of temporal states (interval-specific slopes or changes of the TVC). The initial trait is allowed to account for within-group differences in growth factors of trajectories (i.e., baseline effect), while the temporal states are allowed to impact observed values of a longitudinal process (i.e., temporal effects). We evaluate the proposed models using a simulation study and real-world data analysis. The simulation study demonstrates that the proposed models are capable of separating trajectories into several clusters and generally producing unbiased and accurate estimates with target coverage probabilities. The proposed models reveal the heterogeneity in initial trait and temporal states of reading ability across latent classes of students' mathematics performance. Additionally, the baseline and temporal effects on mathematics development of reading ability are also heterogeneous across the clusters of students.