Sufficient variable screening with high-dimensional controls

Abstract

Variable screening for ultrahigh-dimensional data has attracted extensive attention in the past decade. In many applications, researchers learn from previous studies about certain important predictors or control variables related to the response of interest. Such knowledge should be taken into account in the screening procedure. The development of variable screening conditional on prior information, however, has been less fruitful, compared to the vast literature for generic unconditional screening. In this paper, we propose a model-free variable screening paradigm that allows for high-dimensional controls and applies to either continuous or categorical responses. The contribution of each individual predictor is quantified marginally and conditionally in the presence of the control variables as well as the other candidates by reproducing-kernel-based $R^2$ and partial $R^2$ statistics. As a result, the proposed method enjoys the sure screening property and the rank consistency property in the notion of sufficiency, with which its superiority over existing methods is well-established. The advantages of the proposed method are demonstrated by simulation studies encompassing a variety of regression and classification models, and an application to high-throughput gene expression data.