Parametric Modeling of Quantile Regression Coefficient Functions With Longitudinal Data

In ordinary quantile regression, quantiles of different order are estimated one at a time. An alternative approach, which is referred to as <i>quantile regression coefficients modeling</i> (qrcm), is to model quantile regression coefficients as parametric functions of the order of the quantile. In this article, we describe how the qrcm paradigm can be applied to longitudinal data. We introduce a two-level quantile function, in which two different quantile regression models are used to describe the (conditional) distribution of the within-subject response and that of the individual effects. We propose a novel type of penalized fixed-effects estimator, and discuss its advantages over standard methods based on l1 and l2 penalization. We provide model identifiability conditions, derive asymptotic properties, describe goodness-of-fit measures and model selection criteria, present simulation results, and discuss an application. The proposed method has been implemented in the R package qrcm.