We consider a flexible semiparametric quantile regression model for analyzing high dimensional heterogeneous data. This model has several appealing features: (1) By considering different conditional quantiles, we may obtain a more complete picture of the conditional distribution of a response variable given high dimensional covariates. (2) The sparsity level is allowed to be different at different quantile levels. (3) The partially linear additive structure accommodates nonlinearity and circumvents the curse of dimensionality. (4) It is naturally robust to heavy-tailed distributions. In this paper, we approximate the nonlinear components using B-spline basis functions. We first study estimation under this model when the nonzero components are known in advance and the number of covariates in the linear part diverges. We then investigate a nonconvex penalized estimator for simultaneous variable selection and estimation. We derive its oracle property for a general class of nonconvex penalty functions in the presence of ultra-high dimensional covariates under relaxed conditions. To tackle the challenges of nonsmooth loss function, nonconvex penalty function and the presence of nonlinear components, we combine a recently developed convex-differencing method with modern empirical process techniques. Monte Carlo simulations and an application to a microarray study demonstrate the effectiveness of the proposed method. We also discuss how the method for a single quantile of interest can be extended to simultaneous variable selection and estimation at multiple quantiles.Comment: Published at http://dx.doi.org/10.1214/15-AOS1367 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Summary We demonstrates that DNA methylation alterations at multiple genes, including those in the TH1-TH2 pathways and some novel genes, are associated with cow's milk allergy (CMA), which shed new light on the epigenetic underpinnings of CMA.

Finding the optimal treatment regime (or a series of sequential treatment regimes) based on individual characteristics has important applications in areas such as precision medicine, government policies and active labor market interventions. In the current literature, the optimal treatment regime is usually defined as the one that maximizes the average benefit in the potential population. This paper studies a general framework for estimating the quantile-optimal treatment regime, which is of importance in many real-world applications. Given a collection of treatment regimes, we consider robust estimation of the quantile-optimal treatment regime, which does not require the analyst to specify an outcome regression model. We propose an alternative formulation of the estimator as a solution of an optimization problem with an estimated nuisance parameter. This novel representation allows us to investigate the asymptotic theory of the estimated optimal treatment regime using empirical process techniques. We derive theory involving a nonstandard convergence rate and a non-normal limiting distribution. The same nonstandard convergence rate would also occur if the mean optimality criterion is applied, but this has not been studied. Thus, our results fill an important theoretical gap for a general class of policy search methods in the literature. The paper investigates both static and dynamic treatment regimes. In addition, doubly robust estimation and alternative optimality criterion such as that based on Gini’s mean difference or weighted quantiles are investigated. Numerical simulations demonstrate the performance of the proposed estimator. A data example from a trial in HIV+ patients is used to illustrate the application.

Preterm birth (PTB) affects one in six Black babies in the United States. Epigenetics is believed to play a role in PTB; however, only a limited number of epigenetic studies of PTB have been reported, most of which have focused on cord blood DNA methylation (DNAm) and/or were conducted in white populations. Here we conducted, by far, the largest epigenome-wide DNAm analysis in 300 Black women who delivered early spontaneous preterm (sPTB, n = 150) or full-term babies (n = 150) and replicated the findings in an independent set of Black mother-newborn pairs from the Boston Birth Cohort. DNAm in maternal blood and/or cord blood was measured using the Illumina HumanMethylation450 BeadChip. We identified 45 DNAm loci in maternal blood associated with early sPTB, with a false discovery rate (FDR) <5%. Replication analyses confirmed sPTB associations for cg03915055 and cg06804705, located in the promoter regions of the CYTIP and LINC00114 genes, respectively. Both loci had comparable associations with early sPTB and early medically-indicated PTB, but attenuated associations with late sPTB. These associations could not be explained by cell composition, gestational complications, and/or nearby maternal genetic variants. Analyses in the newborns of the 110 Black women showed that cord blood methylation levels at both loci had no associations with PTB. The findings from this study underscore the role of maternal DNAm in PTB risk, and provide a set of maternal loci that may serve as biomarkers for PTB. Longitudinal studies are needed to clarify temporal relationships between maternal DNAm and PTB risk.