Abstract
This paper presents a Bayesian analysis of linear mixed models for quantile regression based on a Cholesky decomposition for the covariance matrix of random effects. We develop a Bayesian shrinkage approach to quantile mixed regression models using a Bayesian adaptive lasso and an extended Bayesian adaptive group lasso. We also consider variable selection procedures for both fixed and random effects in a linear quantile mixed model via the Bayesian adaptive lasso and extended Bayesian adaptive group lasso with spike and slab priors. To improve mixing of the Markov chains, a simple and efficient partially collapsed Gibbs sampling algorithm is developed for posterior inference. Simulation experiments and an application to the Age-Related Macular Degeneration Trial data to demonstrate the proposed methods.
Highlights
Quantile regression(QR) in longitudinal or panel data models have received increasing attention in recent years
Later [2] generalized the work of [1] to be more flexible model with endogenous variables. [3] offered a estimation of individual effects for panel data quantile regression estimator (QR) models, which is widely applied by practitioners because of computationally simple. [4] studied quantile panel models with correlated random effects. [5] proposed a Stochastic Approximation of the EM (SAEM) algorithm to analyze linear quantile mixed regressions(LQMMs) via the asymmetric Laplace distribution
Like all regression issues, when there are many covariates in longitudinal or panel QR models, variable selection becomes necessary to avoid overfitting and multicollinearity. [10] presented a penalized quantile regression model for random intercept using the Bayesian lasso priors and Bayesian adaptive lasso priors, respectivly. [11] considered a Bayesian Lasso approach to jointly estimate a vector of covariate effects and a vector of random effects by introducing an l1 penalty
Summary
Quantile regression(QR) in longitudinal or panel data models have received increasing attention in recent years. We assume the following hierarchical Bayesian lasso with independent spike and slab type priors for β: bsjts; p00s p00sd0ðbsÞ þ ð1 À p00sÞNð0; tsÞ; l21s l21s Gammaðal ; bl1 Þ; where δ0( ) denotes a point mass at 0, p00s is the prior probability of excluding the sth fixed effect in the model, which is assigned a beta prior with parameters ap00 1⁄4 1 and bp00 1⁄4 1, resulting in a noninformative uniform prior on (0, 1). We assume the following extended hierarchical adaptive Bayesian group lasso with independent spike and slab type priors for γ: gkjZk; p0 p0d0ðgkÞ þ ð1 À p0ÞNkð0; ZkIkÞIðgkk > 0Þ; Gamma k þ l22k Gammaðal ; bl2 Þ; where π0 is the prior probability of excluding the kth random effect in the model, which is assumed to be a noninformative uniform prior on (0, 1).
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