Surrogate-variable-based model-free feature screening for survival data under the general censoring mechanism

Abstract

Feature screening has been seen as the first step in analyzing the ultrahigh-dimensional data with the censored survival time. In this article, we develop a surrogate-variable-based model-free feature screening approach for the censored data under the general censoring mechanism, where the censoring variable may depend on the survival variable and the covariates. This approach is developed by finding some observable variables whose active covariates contain the active covariates of the survival variable as a subset, respectively. Then, any existing model-free feature screening method with the sure screening property for full data can be applied to estimating the sets of the active covariates of the observable variables and hence the set of the active covariates of the survival variable. The sure screening property of the proposed approach is established, and its finite sample performances are demonstrated through some simulations. Further, we illustrate the proposed approach by analyzing two real datasets.

Appendix

Proof of Lemma 1

To facilitate the presentation, we write $\mathbf{X} _{\mathcal {A}}=\{X_{k}:k\in \mathcal {A}\}$ for any non-negative integer set $\mathcal {A}$. First, we prove Lemma 1 (i).

Under the GC mechanism, for any $t\in [0,\tau )$, we have

$$\begin{aligned} pr(Y>t|\mathbf{X} )=pr(\text {min}(T,C)>t|\mathbf{X} )= pr(T>t|\mathbf{X} )\cdot pr(C>t|T>t, \mathbf{X} ). \end{aligned}$$

(1)

Recalling the definition of $\mathcal {A}(Y|\mathbf{X} )$, it is easy to see $\mathcal {A}(Y|\mathbf{X} )\subseteq \mathcal {A}(T|\mathbf{X} ) \cup \mathcal {A}^{*}(C|\mathbf{X} )$. On the other hand, for any $X_{j}\in \mathbf{X} _{\mathcal {A}(T|\mathbf{X} ) \cup \mathcal {A}^{*}(C|\mathbf{X} )}$, we have $X_{j}\in \mathbf{X} _{\mathcal {A}(T|\mathbf{X} )}$ or $X_{j}\in \mathbf{X} _{\mathcal {A}^{*}(C|\mathbf{X} )}$. That is, $pr(T>t|\mathbf{X} )$ or $pr(C>t|T>t, \mathbf{X} )$ depend functionally on $X_{j}$ for some $t\in [0,\tau )$, and hence $pr(Y>t|\mathbf{X} )$ depends functionally on $X_{j}$ by the conditions of Lemma 1. This proves $X_{j}\in \mathbf{X} _{\mathcal {A}(Y|\mathbf{X} )}$. Lemma 1 (i) is then proved.

Lemma 1 (ii) can be proved similar to Lemma 1(i) by noting

$$\begin{aligned} pr(\delta T>t|\mathbf{X} ) =&pr(\delta =1,T>t|\mathbf{X} ) \\ =&pr(T>t|\mathbf{X} )\cdot pr(\delta =1|T>t, \mathbf{X} ) \end{aligned}$$

for $t\in [0,\tau )$. $\square$

Proofs of Theorems 1 and 2

The proofs are direct based on Lemma 1, and hence we omit it. $\square$

Proof of Lemma 2

Under the CRC mechanism, namely, , we then have $\mathcal {A}^{*}(C|\mathbf{X} )$ is an empty set and $\mathcal {A}^{*}(\delta |\mathbf{X} )$ is a subset of $\mathcal {A}(T|\mathbf{X} )$. This proves Lemma 2. $\square$

Proof of Lemma 3

Under the RC mechanism, Lemma 3 is a direct result of Lemma 1 by noting $\mathcal {A}^{*}(C|\mathbf{X} )=\mathcal {A}(C|\mathbf{X} ).$ $\square$