  Quantile regression has demonstrated promising utility in longitudinal data
analysis. Existing work is primarily focused on modeling cross-sectional
outcomes, while outcome trajectories often carry more substantive information
in practice. In this work, we develop a trajectory quantile regression
framework that is designed to robustly and flexibly investigate how latent
individual trajectory features are related to observed subject characteristics.
The proposed models are built under modeling with usual parametric assumptions
lifted or relaxed. We derive our estimation procedure by novelly transforming
the problem at hand to quantile regression with perturbed responses and
adapting the bias correction technique for handling covariate measurement
errors. We establish desirable asymptotic properties of the proposed estimator,
including uniform consistency and weak convergence. Extensive simulation
studies confirm the validity of the proposed method as well as its robustness.
An application to the DURABLE trial uncovers sensible scientific findings and
illustrates the practical value of our proposals.
