Researchers are often interested in examining between-individual differences in within-individual change. 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 rate-of-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 rate-of-change 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 change-from-baseline, a common measure 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:研究人员往往有兴趣研究个人内部变化中的个人差异。如果对调查过程进行长期跟踪,那么其轨迹可能显示一定程度的非线性,从而使变化速度不固定。模型模拟这种非线性进程的一个基本目标是估计反映变化中有意义的各方面的模型参数,包括变化率和揭示实质性假设的其他参数。但是,如果测量时间不结构化,现有模型无法同时估计这两类参数。这一条有三个目标。首先,我们把时间的变化看成是变化率相对于时间(r-t)图的曲线(AUC)下的一个区域。第二,利用瞬时变化率中间隔时间间隔来估计这一间隔期间的平均变化率,我们提出新的规格来描述纵向过程。除了获得个别变化率和与具体研究问题有关的其他参数外,新的规格允许在每一波上进行不均匀的空间研究波和单个测量时间段。第三,我们用基于模型的变换速度分析从时标值到每个时标值,我们又用一个基于模型的模型的模型的模型和新标准来评估。