Modeling Individual Differences in the Timing of Change Onset and Offset

Individual differences in the timing of developmental processes are often of interest in longitudinal studies, yet common statistical approaches to modeling change cannot directly estimate the timing of when change occurs. The time-to-criterion framework was recently developed to incorporate the timing of a prespecified criterion value; however, this framework has difficulty accommodating contexts where the criterion value differs across people or when the criterion value is not known a priori, such as when the interest is in individual differences in when change starts or stops. This article combines aspects of reparameterized quadratic models and multiphase models to provide information on the timing of change. We first consider the more common situation of modeling decelerating change to an offset point, defined as the point in time at which change ceases. For increasing trajectories, the offset occurs when the criterion attains its maximum (“inverted J-shaped” trajectories). For decreasing trajectories, offset instead occurs at the minimum. Our model allows for individual differences in both the timing of offset and ultimate level of the outcome. The same model, reparameterized slightly, captures accelerating change from a point of onset (“J-shaped” trajectories). We then extend the framework to accommodate “S-shaped” curves where both the onset and offset of change are within the observation window. We provide demonstrations that span neuroscience, educational psychology, developmental psychology, and cognitive science, illustrating the applicability of the modeling framework to a variety of research questions about individual differences in the timing of change. Translational Abstract Developmental processes are of interest in many fields including education, neuroscience, developmental psychology, and cognitive science. One common interest in these processes is individual differences in timing of developmental milestones and variables that predict earlier or later timing of these milestones. What differentiates children who learn to talk earlier than other children? What variables are associated with faster brain maturation? Do educational interventions change how quickly students master content? Many conventional statistical models focus on growth rates but do not explicitly capture information about timing of important milestones in developmental processes and such information can only be obtained indirectly. In this article, we demonstrate how to reparameterize longitudinal models such that timing is explicitly captured by parameters in the model. Parameters associated with timing can then be directly predicted by other variables to better accommodate hypotheses concerning how covariates relate to timing of developmental milestones. We focus on two types of developmental milestones. First is “onset,” which captures when a developmental process begins such as when children being learning to talk or when cognitive decline associated with dementia begins. Second is “offset,” which captures when a developmental process is completed such as when a student has mastered a particular concept or when a brain fully matures. We present four examples from four different subfields of psychology to demonstrate how our proposed model can address research questions related to timing more easily and more directly than conventional longitudinal models.