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. 2017;10(3):471-482.
doi: 10.4310/SII.2017.v10.n3.a10.

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

Christian E Galarza  1 , Victor H Lachos  2 , Dipankar Bandyopadhyay  3
Affiliations

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

Christian E Galarza et al. Stat Interface. 2017.

Abstract

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 R package qrLMM().

Keywords: Asymmetric laplace distribution; Linear mixed-effects models; Quantile regression; SAEM algorithm.

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Figures

Figure 1
Figure 1
Standard asymmetric Laplace density.
Figure 2
Figure 2
Bias, Standard Deviation and RMSE for β1 (upper panel) and β2 (lower panel) for varying sample sizes over the quantiles p = 0.05, 0.10, 0.50, 0.90, 0.95.
Figure 3
Figure 3
Standard errors for the fixed effects β0, β1 and β2 and AIC over the quantiles p = {0.05, 0.10, …, 0.95} from fitting the proposed QR-LMM model and Geraci’s method to the Cholesterol data.
Figure 4
Figure 4
Point estimates (center solid line) and 95% confidence intervals across various quantiles for model parameters after fitting the proposed QR-LMM model to the Cholesterol data using the qrLMM package. The interpolated curves are spline-smoothed.
Figure 5
Figure 5
Point estimates (center solid line) and 95% confidence intervals for model parameters across various quantiles from fitting the QR-LMM using the qrLMM package to the orthodontic growth distance data. The interpolated curves are spline-smoothed.
See this image and copyright information in PMC

Comment in

  • Letter to the Editor.
    Geraci M. Geraci M. Stat Interface. 2019;12(1):71-75. doi: 10.4310/SII.2019.v12.n1.a7. Epub 2018 Oct 26. Stat Interface. 2019. PMID: 31423293 Free PMC article.

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