Abstract
In high-dimensional regression models, the Bayesian lasso with the Gaussian spike and slab priors is widely adopted to select variables and estimate unknown parameters. However, it involves large matrix computations in a standard Gibbs sampler. To solve this issue, the Skinny Gibbs sampler is employed to draw observations required for Bayesian variable selection. However, when the sample size is much smaller than the number of variables, the computation is rather time-consuming. As an alternative to the Skinny Gibbs sampler, we develop a variational Bayesian approach to simultaneously select variables and estimate parameters in high-dimensional linear mixed models under the Gaussian spike and slab priors of population-specific fixed-effects regression coefficients, which are reformulated as a mixture of a normal distribution and an exponential distribution. The coordinate ascent algorithm, which can be implemented efficiently, is proposed to optimize the evidence lower bound. The Bayes factor, which can be computed with the path sampling technique, is presented to compare two competing models in the variational Bayesian framework. Simulation studies are conducted to assess the performance of the proposed variational Bayesian method. An empirical example is analyzed by the proposed methodologies.
Highlights
Linear mixed models are widely used to analyze longitudinal and correlated data by considering the between-subject and within-subject variations and incorporating the random effects to account for heterogeneity among the subjects in many fields, such as psychology, medicine, epidemiology and econometrics
Various methods have been developed to estimate fixed-effects parameters and variance–covariance matrices for unobservable random effects and noises or select fixed-effects and random-effects components, even if it is quite challenging for the problem of variable selection and parameter estimation in linear mixed models
Bondell, Krishna and Ghosh [7] and Ibrahim et al [8] proposed the penalized likelihood methods for joint selection of fixed and random effects; Schelldorfer, Buhlmann and van De Geer [9] proposed an 1–penalized estimation procedure; Fan and Li [10] investigated the problem of fixed and random effects selection when the cluster sizes are balanced; Li et al [11] presented a doubly regularized approach to simultaneously select fixed and random effects; Bradic, Claeskens and Gueuning [12] considered testing a single parameter of fixed effects in high-dimensional linear mixed models with fixed cluster sizes, fixed numbers of random effects and sub-Gaussian designs; Li, Cai and Li [13] proposed a penalized quasi-likelihood method for statistical inference on unknown parameters in high-dimensional linear mixed-effects models
Summary
Linear mixed models are widely used to analyze longitudinal and correlated data by considering the between-subject and within-subject variations and incorporating the random effects to account for heterogeneity among the subjects in many fields, such as psychology, medicine, epidemiology and econometrics. Motivated by the aforementioned variational Bayesian studies, we develop a novel variational Bayesian approach to estimate model parameters and select important variables under the Skinny Gibbs sampling framework in a linear mixed model with low-dimensional random effects and high-dimensional fixed effects. The merits of the proposed variational Bayesian method are (i) simultaneously estimating parameters and variance–covariance matrices and select fixed- and random-effects components with quite a low computation cost, (ii) efficiently analyzing high-dimensional data without requiring the non-convex optimization and avoiding the curse of dimensionality problem, (iii) automatically incorporating the shrinkage parameters and (iv) avoiding large matrix computations. Based on the above discussion, we can rewrite the considered linear mixed model together with the spike and slab lasso prior as the following hierarchical models: yij|bi ∼ N (μij, σj2), μij = xij β + zij bi, i = 1, .
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