Detecting Transition Points in the Slope-Intercept Relation in Linear Latent Growth Models

In a linear latent growth model parameterized by intercept (α) and slope (β) factors, those factors' relation is often of interest. The model typically captures this through their covariance parameter, which inherently assumes linearity in their relation. However, this assumption may not always hold. For instance, α and β might be unrelated below a certain threshold along the α-axis but show a meaningful relation above it. That is, even though individual growth trajectories may follow a linear pattern over time, the relation between α and β can be nonlinear, potentially featuring distinct segments separated by a transition point. To address such relations, we propose a semiparametric approach that combines Bayesian P-splines for flexible nonlinear modeling of the α-β relation along with a segmented regression-based transition point detection method. This two-stage analytic approach provides for a more nuanced understanding of the α-β relation, including estimation of a potential transition point where the α-β relation structure fundamentally changes. Simulation results and an empirical data illustration support this approach's effectiveness with single transition point scenarios, offering deeper insights into aspects of the growth process.