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：研究者通常关注个体内过程中个体间差异的检验。若被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基本目标是估计反映变化有意义层面的模型参数，包括与变化相关的参数及揭示实质性假设的其他参数。然而，当测量时机非结构化时，现有模型无法同时估计这两类参数。本文具有三个目标：第一，将随时间的变化视为变化率对时间（r-t）曲线下的面积；第二，利用时间区间中点处的瞬时变化率近似该区间内平均变化率，提出描述纵向过程的新规范。该新规范不仅能获取个体相关变化参数及与特定研究问题相关的其他参数，还允许研究波次间隔不等以及各波次周围个体测量时机存在差异；第三，推导基于模型的区间特定变化和基线变化量——两种评估随时间变化的常用指标。我们通过模拟研究和真实数据分析评估所提规范，并为采用新规范的每个模型提供OpenMx和Mplus 8代码。