Bayesian regularized quantile mixed models for longitudinal studies.

Abstract

In longitudinal studies, the same subjects are measured repeatedly over time, leading to correlations among the repeated measurements. Properly accounting for the intra-cluster correlations in the presence of data heterogeneity and long tailed distributions of the disease phenotype is challenging, especially in the context of high dimensional regressions. Here, we aim at developing novel Bayesian regularized quantile mixed effect models to tackle these challenges. In the first project, we have proposed a Bayesian variable selection method in the mixed effect models for longitudinal lipidomics studies. To dissect important lipid-environment interactions, our model can simultaneously identify important main and interaction effects on the individual and group level, which have been facilitated by imposing the spike-and- slab priors through Laplacian shrinkage in the Bayesian quantile hierarchical models. The within- subject dependence among data can be accommodated by incorporating the random effects. The Gibbs sampler has been developed along with the Markov Chain Monte Carlo (MCMC). We have established the advantage of the proposed method over multiple compet- ing methods in extensive simulation studies and a high-dimensional lipidomics study with repeated measurements. In the second project, we further extend the sparse Bayesian quantile mixed models to nonlinear longitudinal interactions. Specifically, the proposed Bayesian quantile semipara- metric model is robust not only to outliers and heavy-tailed distributions of the response variable, but also to the misspecification of interaction effect in the forms other than non- linear interactions. We have developed the Gibbs sampler with the spike-and-slab priors to promote sparse identification of appropriate forms of main and interaction effects. Simula- tion results reveal superior performance in identification, estimation and statistical inference. In particular, the proposed method that incorporates the spike-and-slab priors can enable exact statistical inference by yielding Bayesian credible intervals with nominal coverage prob- abilities on parametric and nonparametric fixed effects simultaneously. Application of the proposed method on high dimensional longitudinal biomedical studies shed novel insight on disease etiology. The Bayesian regularized quantile mixed models proposed in this dissertation aim to tackle challenges arising from longitudinal gene environment interaction studies under the linear and nonlinear interaction assumption in chapter 2 and chapter 3, respectively. In a regression analysis framework, gene-environment interactions can be divided into linear and non-linear types, based on whether the effect of genetic factors on disease traits can be represented by linear or non-linear functions of these genetic factors. The two models proposed in this dissertation fill a significant technical gap since robust identification and inference of the two types of interactions has rarely been explored in published longitudinal studies. We have also developed C++ based R packages mixedBayes on CRAN to facilitate re- producible and fast computations using all methods under comparison in this dissertation.