Decomposing Impact on Longitudinal Outcome of Time-varying Covariate into Trait Effect and State Effect

Developmental processes are often associated with each other over time; therefore, examining such associations and understanding the joint development of multiple processes is of interest. One statistical method is the latent growth curve model (LGCM) with a time-varying covariate (TVC), which estimates the effect on a longitudinal outcome of a TVC while simultaneously modeling change in the longitudinal outcome. However, this existing model does not allow the TVC to predict variation in the random growth coefficients. Our study proposes decomposing the effect of a TVC into trait and state effects to address this limitation. Specifically, we proposed three methods to decompose the impact of a TVC. In all three methods, we consider the baseline value of a TVC as the trait feature, and by regressing random intercepts and slopes on the baseline value, we obtain trait effects. Meanwhile, we characterize (1) the interval-specific slopes, (2) the interval-specific changes, or (3) the change from baseline at each measurement occasion of the TVC as the state feature in three methods, respectively. We obtain state effects by regressing the longitudinal outcome on such state features. We demonstrate the proposed methods using simulation studies and real-world data analyses, assuming the longitudinal outcome takes a linear-linear functional form. Based on the simulation results, the LGCM with a TVC breaking into the baseline value and interval-specific slopes or changes can produce unbiased and precise estimates with target confidence intervals. We provide \textit{OpenMx} and \textit{Mplus 8} code for three methods with commonly used linear and nonlinear functions.