Dynamic Modeling of Conditional Quantile Trajectories, with Application to Longitudinal Snippet Data
Longitudinal data are often plagued with sparsity of time points where measurements are available. The functional data analysis perspective has been shown to provide an effective and flexible approach to address this problem for the case where measurements are sparse but their times are randomly distributed over an interval. Here we focus on a different scenario where available data can be characterized as snippets, which are very short stretches of longitudinal measurements. For each subject the stretch of available data is much shorter than the time frame of interest, a common occurrence in accelerated longitudinal studies. An added challenge is introduced if a time proxy that is basic for usual longitudinal modeling is not available. This situation arises in the case of Alzheimer's disease and comparable scenarios, where one is interested in time dynamics of declining performance, but the time of disease onset is unknown and chronological age does not provide a meaningful time reference for longitudinal modeling. Our main methodological contribution to address these challenges is to introduce conditional quantile trajectories for monotonic processes that emerge as solutions of a dynamic system. Our proposed estimates for these trajectories are shown to be uniformly consistent. Conditional quantile trajectories are useful descriptors of processes that quantify deterioration over time, such as hippocampal volumes in Alzheimer's patients. We demonstrate how the proposed approach can be applied to longitudinal snippets data sampled from such processes.