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

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.