🤖 AI Summary
Estimating the rate of change in nonlinear trajectories under individually scheduled, unequally spaced longitudinal measurements remains challenging—existing models struggle to jointly estimate dynamic change parameters and theory-driven substantive parameters. To address this, we propose a novel framework that conceptualizes the rate of change as the area under a time-varying functional curve, approximating the average rate within each interval by the instantaneous rate at its midpoint. This enables simultaneous estimation of both change and substantive parameters. The method is implemented within a latent-variable structural equation modeling framework using OpenMx or Mplus 8, integrating numerical integration with interval-specific approximations. Simulation and empirical studies demonstrate high accuracy, robustness, and the ability to derive both baseline-level and interval-specific change metrics. Accompanying open-source code ensures flexibility and reproducibility. The approach substantially enhances theoretical interpretability and practical utility for modeling nonlinear longitudinal processes.
📝 Abstract
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-tdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$r-t$$end{document}) 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 spaced 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.