Estimating Rate of Change for nonlinear Trajectories in the Framework of Individual Measurement Occasions: A New Perspective of Growth Curves
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Researchers are interested in employing longitudinal data analysis to examine 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 this nonlinear process is to estimate model parameters that reflect the meaningful aspects of individual changes over time, including the rate-of-change and other parameters that shed light on substantive hypotheses. Theoretical and empirical researchers have developed and applied nonparametric and various parametric functions to describe nonlinear trajectories in the latent growth curve and latent change score modeling frameworks. 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 and propose a new specification for describing a longitudinal process. In addition to simultaneously obtaining the individual rate-of-change and other parameters related to a specific research question, the new specification allows for (1) unequal-spaced study waves and (2) individual measurement occasions around each wave. Second, we derive the model-based change-from-baseline, a common measure to evaluate change over time in an observational study or a treatment effect in a clinical trial. Third, we use a simulation study and a real-world data analysis to illustrate the extended models. We also provide OpenMx code for each model with the novel specification.
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