Conditional screening for ultra-high dimensional covariates with survival outcomes

Abstract

Identifying important biomarkers that are predictive for cancer patients’ prognosis is key in gaining better insights into the biological influences on the disease and has become a critical component of precision medicine. The emergence of large-scale biomedical survival studies, which typically involve excessive number of biomarkers, has brought high demand in designing efficient screening tools for selecting predictive biomarkers. The vast amount of biomarkers defies any existing variable selection methods via regularization. The recently developed variable screening methods, though powerful in many practical setting, fail to incorporate prior information on the importance of each biomarker and are less powerful in detecting marginally weak while jointly important signals. We propose a new conditional screening method for survival outcome data by computing the marginal contribution of each biomarker given priorily known biological information. This is based on the premise that some biomarkers are known to be associated with disease outcomes a priori. Our method possesses sure screening properties and a vanishing false selection rate. The utility of the proposal is further confirmed with extensive simulation studies and analysis of a diffuse large B-cell lymphoma dataset. We are pleased to dedicate this work to Jack Kalbfleisch, who has made instrumental contributions to the development of modern methods of analyzing survival data.

Feature selection of ultrahigh-dimensional covariates with survival outcomes: a selective review

“…To fill the gap, we will review and compare several representative works in this field. Specifically, we first review modelmotivated screening methods, including the principled sure screening by [27], the feature aberration at survival times screening by [8] and the conditional screening by [11]. We then review several model-free methods, including the quantile adaptive sure independence screening by [9], the censored rank independence screening procedure by [19], the survival impact index screening by [17] and the integrated powered density screening by [10].…”

“…Let ℳ − = {j ∉ , β j ≠ 0}, q = | |, and X = (X j , j ∈ ) T . [11] proposed to fit the marginal Cox regression by including the known covariates in X . Specifically, for each X j ∉ X , they considered the following Cox regression model …”

“…If the conditioning set X is empty, the method reduces to PSIS. Using the magnitude of βĵ as a marginal utility to rank the importance of X j , the set of selected variables among j ∉ is given byfor a pre-defined λ n .To study the asymptotic behavior of the proposed procedure, [11] assumed the following "beta-min" condition: for constants c 1 > 0 and 0 < κ < 1/2,where cov * (ζ 1 , ζ 2 |ξ) indicates the conditional linear covariance between ζ 1 and ζ 2 given ξ.[11] introduced this new concept in order to approximate the ordinary conditional variance and facilitate the proof of the sure screening property.Under (7) and some other conditions, [11] proved that the estimated βĵ converges uniformly to β j in probability. That is, for any given ε 1 > 0 and ε 2 > 0, where c 2 and c 3 are some positive constants, q is the size of , and s is the size of ℳ − .…”

“…The censoring time was generated from a uniform distribution on (0, 10). This example was adopted from [11]. Following the literature, we used three metrics as the criteria for comparisons, the minimum model size (MMS; the minimum number of variables that need to be selected in order to include all active variables), the true positive rate (TPR; the proportion of active variables selected in the first [n/log n] variables), and the probability of inclusion of the true model (PIT; the probability of all active variables selected in the first [n/log n] variables).…”

Joint variable screening in the censored accelerated failure time model

“…To overcome the drawbacks of marginal screening methods, Hong, Kang and Li (2018) has recently proposed a conditional screening approach for survival data with ultrahigh dimensional covariates. However, the conditional screening approach requires pre-selection of a set of covariates, and the computational burden is heavy since it requires fitting a multivariate survival model for each covariate not pre-selected.…”