Researchers are often interested in examining between-individual differences in within-individual processes. If the process under investigation is tracked for a long time, its trajectory may show a certain degree of nonlinearity, so that the rate-of-change is not constant. A fundamental goal of modeling such nonlinear processes is to estimate model parameters that reflect meaningful aspects of change, including the parameters related to change and other parameters that shed light on substantive hypotheses. However, if the measurement occasion is unstructured, existing models cannot simultaneously estimate these two types of parameters. This article has three goals. First, we view the change over time as the area under the curve (AUC) of the rate-of-change versus time (r-t) graph. Second, using the instantaneous rate-of-change midway through a time interval to approximate the average rate-of-change during that interval, we propose a new specification to describe longitudinal processes. In addition to obtaining the individual change-related parameters and other parameters related to specific research questions, the new specification allows for unequally-space study waves and individual measurement occasions around each wave. Third, we derive the model-based interval-specific change and change-from-baseline, two common measures to evaluate change over time. We evaluate the proposed specification through a simulation study and a real-world data analysis. We also provide OpenMx and Mplus 8 code for each model with the novel specification.

翻译：研究者常关注个体内过程的个体间差异。若所研究过程被长期追踪，其轨迹可能呈现一定程度的非线性，导致变化率并非恒定。对此类非线性过程建模的核心目标在于估计反映变化关键特征的模型参数，包括与变化相关的参数以及揭示实质性假设的其他参数。然而，若测量时点缺乏结构化，现有模型无法同时估计这两类参数。本文旨在实现三个目标：首先，将随时间变化视为变化率-时间（r-t）曲线下面积（AUC）；其次，利用时间区间中点的瞬时变化率近似该区间的平均变化率，提出一种描述纵向过程的新模型设定。该设定不仅能获取个体变化相关参数及特定研究问题相关的其他参数，还允许研究波次间不等距及各波次内个体测量时点存在差异；第三，推导出基于模型的区间特异性变化及相对于基线的变化这两种评估时间变化性的常用指标。通过模拟研究和实际数据分析验证所提模型设定的有效性，并提供采用新设定的OpenMx与Mplus 8模型代码。