Mixed-Effects Models for Conditional Quantiles with Longitudinal Data

Abstract

We propose a regression method for the estimation of conditional quantiles of a continuous response variable given a set of covariates when the data are dependent. Along with fixed regression coefficients, we introduce random coefficients which we assume to follow a form of multivariate Laplace distribution. In a simulation study, the proposed quantile mixed-effects regression is shown to model the dependence among longitudinal data correctly and estimate the fixed effects efficiently. It performs similarly to the linear mixed model at the central location when the regression errors are symmetrically distributed, but provides more efficient estimates when the errors are over-dispersed. At the same time, it allows the estimation at different locations of conditional distribution, which conveys a comprehensive understanding of data. We illustrate an application to clinical data where the outcome variable of interest is bounded within a closed interval.

Boosting Structured Additive Quantile Regression for Longitudinal Childhood Obesity Data

“…[7]. Since boosting does not directly provide standard errors for the estimated effects, we additionally conducted a block-wise bootstrap analysis in accordance with Liu and Bottai [10]. We obtained one single bootstrap sample by randomly choosing 2,226 children with replacement at the first stage.…”

“…The majority of existing approaches considers predictors with conventional linear population effects and individual-specific intercepts. They either rely on a penalized loss criterion where shrinkage of individual-specific effects to zero is imposed [8], on quasi-likelihood approaches based on the asymmetric Laplace error distribution as a working likelihood [9][10][11], or on full Bayesian inference with a nonparametric Dirichlet process error distribution [12]. On the other hand, additive quantile regression models without individual-specific effects have been recently proposed by Fenske et al [13], using boosting as computationally effective method for smoothing, and by Koenker [14,15], using a total variation penalty for enforcing smoothness of the functional effects.…”

Using Linear Mixed-Effects Models with Quantile Regression to Simulate the Crown Profile of Planted Pinus sylvestris var. Mongolica Trees

“…The parameters of quantile regression for the linear mixed-effects model (qrLMM) achieve an unbiased estimation due to the fact that the correlation of the measurements from the same subjects are adequately accounted for [36]. Liu and Bottai [38] studied the parameter estimation of the qrLMM based on Monte Carlo simulations for the fixed parameters of the model given the assumption that no correlation existed between the random effects parameters. Chen et al [39] proposed a marginally-conditional quantile regression approach to deal with the longitudinal dataset by including the random observing times, and the consistency and asymptotic normality for the estimators were developed.…”

Quantile Regression Models and Their Applications: A Review

“…Koenker [81] generalized his previous work on QR to longitudinal data via penalized least squares method. Other methods or algorithms used to QR includes Barrodale-Roberts algorithm [82], Expectation-Maximization (EM) algorithm [83], Monte Carlo Expectation-Maximization (MCEM) algorithm [13,84,85], and Bayesian approach by Markov chain Monte Carlo (MCMC) procedure [86][87][88][89][90][91][92][93]. Longitudinal QR has been rapidly expanded in many areas, including investment and finance [94,95], economics [96], environmental science [97,98], geography [99], public health [100,101] and biomedical research [102][103][104][105].…”