Longitudinal processes are often associated with each other over time; therefore, it is important to investigate the associations among developmental processes and understand their joint development. The latent growth curve model (LGCM) with a time-varying covariate (TVC) provides a method to estimate the TVC's effect on a longitudinal outcome while simultaneously modeling the outcome's change. However, it does not allow the TVC to predict variations in the random growth coefficients. We propose decomposing the TVC's effect into initial trait and temporal states using three methods to address this limitation. In each method, the baseline of the TVC is viewed as an initial trait, and the corresponding effects are obtained by regressing random intercepts and slopes on the baseline value. Temporal states are characterized as (1) interval-specific slopes, (2) interval-specific changes, or (3) changes from the baseline at each measurement occasion, depending on the method. We demonstrate our methods through simulations and real-world data analyses, assuming a linear-linear functional form for the longitudinal outcome. The results demonstrate that LGCMs with a decomposed TVC can provide unbiased and precise estimates with target confidence intervals. We also provide OpenMx and Mplus 8 code for these methods with commonly used linear and nonlinear functions.

翻译：纵向过程通常随时间相互关联，因此研究发展过程之间的关联性并理解其联合发展具有重要意义。包含时变协变量的潜变量增长曲线模型提供了一种方法，可在估计结局变量变化的同时评估时变协变量对纵向结局的影响。然而，该模型无法使时变协变量预测随机增长系数的变异。为克服这一局限，我们提出三种将时变协变量效应分解为初始特质与时间状态的方法。在每种方法中，时变协变量的基线值被视为初始特质，通过将随机截距和斜率对基线值进行回归获得相应的效应。时间状态则根据方法不同被表征为：(1)区间特异性斜率，(2)区间特异性变化，或(3)每个测量时点相对于基线的变化。我们通过模拟研究和真实数据分析验证了所提方法，假设纵向结局服从线性-线性函数形式。结果表明，采用分解后时变协变量的潜变量增长曲线模型能在目标置信区间内提供无偏且精确的估计。此外，我们还提供了针对这些方法的OpenMx与Mplus 8代码，适用于常见的线性与非线性函数。