Dynamic modeling with conditional quantile trajectories for longitudinal snippet data, with application to cognitive decline of Alzheimer's patients

Abstract

Longitudinal data are often plagued with sparsity of time points where measurements are available and the functional data analysis perspective provides an effective and flexible approach to address this problem. The commonly studied case is 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. 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 the chronological age does not provide a meaningful time proxy for longitudinal modeling. Our main methodological contribution is to address this problem with a novel approach to obtain uniformly consistent estimates of conditional quantile trajectories for monotonic processes as solutions of a dynamic system. These trajectories are useful to describe processes that quantify deterioration over time, such as hippocampal volumes in Alzheimer's patients.