  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.
