Statistics and Its Interface

Volume 10 (2017)

Number 3

Quantile regression in linear mixed models: a stochastic approximation EM approach

Pages: 471 – 482



Christian E. Galarza (Departamento de Matemáticas, Escuela Superior Politécnica del Litoral, ESPOL, Guayaquil, Ecuador)

Victor H. Lachos (Departamento de Estatística, Universidade Estadual de Campinas, UNICAMP, Campinas, São Paulo, Brazil)

Dipankar Bandyopadhyay (Department of Biostatistics, Virginia Commonwealth University, Richmond, Va., U.S.A.)


This paper develops a likelihood-based approach to analyze quantile regression (QR) models for continuous longitudinal data via the asymmetric Laplace distribution (ALD). Compared to the conventional mean regression approach, QR can characterize the entire conditional distribution of the outcome variable and is more robust to the presence of outliers and misspecification of the error distribution. Exploiting the nice hierarchical representation of the ALD, our classical approach follows a Stochastic Approximation of the EM (SAEM) algorithm in deriving exact maximum likelihood estimates of the fixed-effects and variance components. We evaluate the finite sample performance of the algorithm and the asymptotic properties of the ML estimates through empirical experiments and applications to two real life datasets. Our empirical results clearly indicate that the SAEM estimates outperforms the estimates obtained via the combination of Gaussian quadrature and non-smooth optimization routines of the Geraci and Bottai (2014) approach in terms of standard errors and mean square error. The proposed SAEM algorithm is implemented in the $\mathrm{R}$ package $\texttt{qrLMM()}$.


quantile regression, linear mixed-effects models, asymmetric Laplace distribution, SAEM algorithm

Published 31 January 2017