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We evaluate the validity of a projection‐based test checking linear models when the number of covariates tends to infinity, and analyze two gene expression datasets. We show that the test is still consistent and derive the asymptotic distributions under the null and alternative hypotheses. The asymptotic properties are almost the same as those when the number of covariates is fixed as long asp/n → 0with additional mild assumptions. The test dramatically gains dimension reduction, and its numerical performance is remarkable.

In this article, we study the estimation of high‐dimensional single index models when the response variable is censored. We hybrid the estimation methods for high‐dimensional single‐index models (but without censorship) and univariate nonparametric models with randomly censored responses to estimate the index parameters and the link function and apply the proposed methods to analyze a genomic dataset from a study of diffuse large B‐cell lymphoma. We evaluate the finite sample performance of the proposed procedures via simulation studies and establish large sample theories for the proposed estimators of the index parameter and the nonparametric link function under certain regularity conditions.

Feature selection is an important initial step of exploratory analysis in biomedical studies. Its main objective is to eliminate the covariates that are uncorrelated with the outcome. For highly correlated covariates, traditional feature selection methods, such as the Lasso, tend to select one of them and eliminate the others, although some of the eliminated ones are still scientifically valuable. To alleviate this drawback, we propose a feature selection method based on covariate space decomposition, referred herein as the “Decomposition Feature Selection” (DFS), and show that this method can lead to scientifically meaningful results in studies with correlated high dimensional data. The DFS consists of two steps: (i) decomposing the covariate space into disjoint subsets such that each of the subsets contains only uncorrelated covariates and (ii) identifying significant predictors by traditional feature selection within each covariate subset. We demonstrate through simulation studies that the DFS has superior practical performance over the Lasso type methods when multiple highly correlated covariates need to be retained. Application of the DFS is demonstrated through a study of bipolar disorders with correlated biomarkers.