Estimating Rate of Change for nonlinear Trajectories in the Framework of Individual Measurement Occasions: A New Perspective of Growth Curves

Researchers are interested in examining between-individual differences in within-individual changes. If the process under investigation is tracked for a long time, its trajectory may show a certain degree of nonlinearity, so 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, with 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 \textit{OpenMx} and \textit{Mplus 8} code for each model with the novel specification.