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

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.