Quantile Regression for Nonlinear Mixed Effects Models: A Likelihood Based Perspective

Christian E. Galarza
Luis M. Castro
Francisco Louzada
Víctor H. Lachos

Longitudinal data are frequently analyzed using normal mixed effects models. Moreover,the traditional estimation methods are based on mean regression, which leads to non-robustparameter estimation for non-normal error distributions. Compared to the conventional meanregression approach, quantile regression (QR) can characterize the entire conditional distribu-tion of the outcome variable and is more robust to the presence of outliers and misspecificationof the error distribution. This paper develops a likelihood-based approach to analyzing QRmodels for correlated continuous longitudinal data via the asymmetric Laplace (AL) distri-bution. Exploiting the nice hierarchical representation of the AL distribution, our classicalapproach follows the Stochastic Approximation of the EM (SAEM) algorithm for deriving ex-act maximum likelihood estimates of the fixed-effects and variance components in nonlinearmixed effects models (NLMEMs). We evaluate the finite sample performance of the algorithmand the asymptotic properties of the ML estimates through empirical experiments and applica-tions to two real life datasets. The proposed SAEM algorithm is implemented in the R packageqrNLMM.

Asymmetric Laplace distribution
Nonlinear mixed effects models
Quantile re- gression
SAEM algorithm
Stochastic Approximations