  Empirical researchers are usually interested in investigating the impacts of
baseline covariates have when uncovering sample heterogeneity and separating
samples into more homogeneous groups. However, a considerable number of studies
in the structural equation modeling (SEM) framework usually start with vague
hypotheses in terms of heterogeneity and possible reasons. It suggests that (1)
the determination and specification of a proper model with covariates is not
straightforward, and (2) the exploration process may be computational intensive
given that a model in the SEM framework is usually complicated and the pool of
candidate covariates is usually huge in the psychological and educational
domain where the SEM framework is widely employed. Following Bakk and Kuha
(2017), this article presents a two-step growth mixture model (GMM) that
examines the relationship between latent classes of nonlinear trajectories and
baseline characteristics. Our simulation studies demonstrate that the proposed
model is capable of clustering the nonlinear change patterns, and estimating
the parameters of interest unbiasedly, precisely, as well as exhibiting
appropriate confidence interval coverage. Considering the pool of candidate
covariates is usually huge and highly correlated, this study also proposes
implementing exploratory factor analysis (EFA) to reduce the dimension of
covariate space. We illustrate how to use the hybrid method, the two-step GMM
and EFA, to efficiently explore the heterogeneity of nonlinear trajectories of
longitudinal mathematics achievement data.
