The composite quantile regression for longitudinal data using the mixture of asymmetric laplace distributions

Abstract

We propose a linear mixture quantile regression approach, with composite quantile regression (CQR) as a special case, to analyze continuous longitudinal data via a finite mixture of asymmetric Laplace distributions (ALD). Compared with the conventional mean regression approach, the proposed quantile regression model can characterize the entire conditional distribution of the response variable and is more robust to heavy tails and misspecification in the error distribution. To implement the model, we develop a two-layer MCEM-EM algorithm to approximate random effects through a Monte Carlo simulation and derive the exact maximum likelihood estimates of the parameters in each step with the nice hierarchical representation of the ALD. The proposed algorithm performs similarly to the traditional linear mixed model when the error term follows a Gaussian distribution, but provides more efficient estimates for heavy-tailed data or with few low leverage outliers. We evaluate the finite sample performance of the algorithm and asymptotic properties of the maximum likelihood estimates through simulation studies and illustrate its usefulness through an application to a real life data set.