  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.
