  Researchers continue to be interested in exploring the effects that
covariates have on the heterogeneity in trajectories. The inclusion of
covariates associated with latent classes allows for a more clear understanding
of individual differences and a more meaningful interpretation of latent class
membership. Many theoretical and empirical studies have focused on
investigating heterogeneity in change patterns of a univariate repeated outcome
and examining the effects on baseline covariates that inform the cluster
formation. However, developmental processes rarely unfold in isolation;
therefore, empirical researchers often desire to examine two or more outcomes
over time, hoping to understand their joint development where these outcomes
and their change patterns are correlated. This study examines the heterogeneity
in parallel nonlinear trajectories and identifies baseline characteristics as
predictors of latent classes. Our simulation studies show that the proposed
model can tell the clusters of parallel trajectories apart and provide unbiased
and accurate point estimates with target coverage probabilities for the
parameters of interest in general. We illustrate how to apply the model to
investigate the heterogeneity in the joint development of reading and
mathematics ability from Grade K to 5. In this real-world example, we also
demonstrate how to select covariates that contribute the most to the latent
classes and transform candidate covariates from a large set into a more
manageable set with retaining the meaningful properties of the original set in
the structural equation modeling framework.
