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-robust parameter estimation for non-normal error distributions. Compared to the conventional mean regression approach, quantile regression (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. This paper develops a likelihood-based approach to analyzing QR models for correlated continuous longitudinal data via the asymmetric Laplace (AL) distribution. Exploiting the nice hierarchical representation of the AL distribution, our classical approach follows the Stochastic Approximation of the EM (SAEM) algorithm for deriving exact maximum likelihood estimates of the fixed-effects and variance components in nonlinear mixed effects models (NLMEMs). 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. The proposed SAEM algorithm is implemented in the R package qrNLMM.