Censored cumulative residual independent screening for ultrahigh-dimensional survival data

Abstract

For complete ultrahigh-dimensional data, sure independent screening methods can effectively reduce the dimensionality while retaining all the active variables with high probability. However, limited screening methods have been developed for ultrahigh-dimensional survival data subject to censoring. We propose a censored cumulative residual independent screening method that is model-free and enjoys the sure independent screening property. Active variables tend to be ranked above the inactive ones in terms of their association with the survival times. Compared with several existing methods, our model-free screening method works well with general survival models, and it is invariant to the monotone transformation of the responses, as well as requiring substantially weaker moment conditions. Numerical studies demonstrate the usefulness of the censored cumulative residual independent screening method, and the new approach is illustrated with a gene expression data set.

Appendix: Theoretic Proofs

Proof of Theorem 1

Let

$$\begin{aligned} \widetilde{d_k}(t,z)&=n^{-1}\sum _{i=1}^{n} \left[ \left\{ \frac{\Delta _{i}I(X_i>t)}{G(X_i)}-H(t)\right\} I(Z_{ik}<z)\right] \end{aligned}$$

and define

$$\begin{aligned} \left\| \widetilde{d_k}\right\| _n^2&=n^{-1}\sum _{j=1}^{n}\widetilde{d_k}\left( X_j,Z_{jk}\right) ^2. \end{aligned}$$

Straightforward calculations entail that

$$\begin{aligned} \left\| \widetilde{d_k}\right\| _n^2&=\frac{(n-1)(n-2)}{n^2}\left( \frac{1}{n-2} \widetilde{D}_{k1}+\widetilde{D}_{k2}\right) , \end{aligned}$$

(A.1)

where

$$\begin{aligned} \widetilde{D}_{k1}= & {} \frac{2}{n(n-1)}\sum _{i<j}\frac{1}{2} \left[ \left\{ \frac{\Delta _{i}I(X_i>X_j)}{G(X_i)}-H(X_j)\right\} ^2 I(Z_{ik}<Z_{jk})\right. \\&+\left. \left\{ \frac{\Delta _{j}I(X_j>X_i)}{G(X_j)}-H(X_i)\right\} ^2I \left( Z_{jk}<Z_{ik}\right) \right] \\\equiv & {} \frac{2}{n(n-1)} \sum _{i<j}h_{1}\left( {\mathcal {O}}_{ik};{\mathcal {O}}_{jk};G,H\right) ,\\ \end{aligned}$$

${{\mathcal {O}}}_{ik}=(X_i,\Delta _i,Z_{ik})$, and the definitions of kernels $h_{1}({{\mathcal {O}}}_{ik};{{\mathcal {O}}}_{jk};G,H)$ and $h_{2}({{\mathcal {O}}}_{ik};{{\mathcal {O}}}_{jk};{{\mathcal {O}}}_{lk};G,H)$ in the U-statistics are clear from the context. Likewise, we have

$$\begin{aligned} \left\| \widehat{d_k}\right\| _n^2&=\frac{(n-1)(n-2)}{n^2} \left( \frac{1}{n-2}\widehat{D}_{k1}+\widehat{D}_{k2}\right) , \end{aligned}$$

(A.2)

where $\widehat{D}_{ks}, s=1,2,$ are obtained by replacing G and H in $\widetilde{D}_{ks}$ with $\widehat{G}_n$ and $\widehat{H}_n$ respectively.

First, we derive the exponential tail probability bound of $P\big (\big |\Vert \widehat{d_k}\Vert _n^2-\Vert \widetilde{d_k}\Vert _n^2\big |\ge \upsilon n^{-\alpha }\big )$ for any positive constants $\upsilon$ and $\alpha \in [0, 1/2)$. Consider $P(|\widehat{D}_{k1}-\widetilde{D}_{k1}|\ge \upsilon n^{-\alpha }/2)$ and note that

By condition C1 and the boundness of the indicator function, there exists a constant $c_1$ such that

$$\begin{aligned}&\left| \left\{ \frac{\Delta _{i}I(X_i>X_j)}{\widehat{G}_n(X_i)}-\widehat{H}_n(X_j)\right\} ^2 -\left\{ \frac{\Delta _{i}I(X_i>X_j)}{G(X_i)}-H(X_j)\right\} ^2\right| \\&\quad \le c_1\left\{ \left| \widehat{G}_n(X_i)-G(X_i)\right| +\left| \widehat{H}_n(X_j)-H(X_j)\right| \right\} . \end{aligned}$$

Denoting $c_2=\min \{G(\tau ),H(\tau )\}$, we immediately have

$$\begin{aligned} \left| \widehat{D}_{k1}-\widetilde{D}_{k1}\right|\le & {} \frac{c_1}{n}\sum _{i=1}^{n} \Big [\Big |\widehat{G}_n(X_i)-G(X_i)\Big | +\Big |\widehat{H}_n(X_i)-H(X_i)\Big |\Big ]\nonumber \\\le & {} \frac{c_1}{c_2 n} \sum _{i=1}^{n}\Big [\Big |H(X_i)\Big \{\widehat{G}_n(X_i)-G(X_i)\Big \}\Big |+ \Big |G(X_i)\Big \{\widehat{H}_n(X_i)-H(X_i)\Big \}\Big |\Big ]\nonumber \\\le & {} c_3\sup _{0\le t \le \tau }\Big |H(t)\Big \{\widehat{G}_n(t)-G(t)\Big \}\Big |+ c_3\sup _{0\le t\le \tau }\Big |G(t)\Big \{\widehat{H}_n(t)-H(t)\Big \}\Big |,\nonumber \\ \end{aligned}$$

(A.3)

where $c_3=c_1/c_2$.

Using the similar argument, along with some tedious calculation, we also have

$$\begin{aligned} \Big |\widehat{D}_{k2}-\widetilde{D}_{k2}\Big | \le c_{4}\sup _{0\le t \le \tau }\Big |H(t)\Big \{\widehat{G}_n(t)-G(t)\Big \}\Big |+ c_{4}\sup _{0\le t\le \tau }\Big |G(t)\Big \{\widehat{H}_n(t)-H(t)\Big \}\Big |, \end{aligned}$$

where $c_{4}$ is a constant. It follows from (A.3) and Theorem 1 of Bitouzé et al. (1999) that

$$\begin{aligned} P\Big (\Big |\widehat{D}_{k1}-\widetilde{D}_{k1}\Big |\ge 2\upsilon n^{-\alpha }\Big )\le & {} P\left( c_3\sup _{0\le t \le \tau }\left| H(t)\left\{ \widehat{G}_n(t)-G(t)\right\} \right| \ge \upsilon n^{-\alpha }\right) \nonumber \\&+P\left( c_3\sup _{0\le t\le \tau }\left| G(t)\left\{ \widehat{H}_n(t)-H(t)\right\} \right| \ge \upsilon n^{-\alpha }\right) \nonumber \\\le & {} 5\exp \left( -2c_{3}^{-2}\upsilon ^{2}n^{1-2\alpha }+\mu _1 c_{3}^{-1}\upsilon n^{1/2-\alpha }\right) , \end{aligned}$$

(A.4)

where $\mu _1$ is a constant. Similarly, we also have

$$\begin{aligned} P\left( \left| \widehat{D}_{k2}-\widetilde{D}_{k2}\right| \ge 2\upsilon n^{-\alpha }\right) \le 5\exp \left( -2c_{4}^{-2}\upsilon ^{2}n^{1-2\alpha }+\mu _2 c_{4}^{-1}\upsilon n^{1/2-\alpha }\right) , \end{aligned}$$

(A.5)

where $\mu _2$ is a constant. Combining (A.1), (A.2), (A.4) and (A.5), we have

$$\begin{aligned}&P\left( \left| \left\| \widehat{d_k}\right\| _n^2-\left\| \widetilde{d_k}\right\| _n^2\right| \ge 4\upsilon n^{-\alpha }\right) \nonumber \\&\quad = P\left\{ \left| \frac{n-1}{n^2}\left( \widehat{D}_{k1}-\widetilde{D}_{k1}\right) + \frac{(n-1)(n-2)}{n^2}\left( \widehat{D}_{k2}-\widetilde{D}_{k2}\right) \right| \ge 4\upsilon n^{-\alpha }\right\} \nonumber \\&\quad \le P\left\{ \left| \widehat{D}_{k1} -\widetilde{D}_{k1}\right| \ge 2\upsilon n^{1-\alpha }\right\} + P\left\{ \left| \widehat{D}_{k2}-\widetilde{D}_{k2}\right| \ge 2\upsilon n^{-\alpha }\right\} \nonumber \\&\quad \le 5\exp \left( -2c_3^{-2} \upsilon ^{2}n^{3-2\alpha }+\mu _1 c_3^{-1}\upsilon n^{3/2-\alpha }\right) \nonumber \\&\qquad + 5\exp \left( -2c_{4}^{-2}\upsilon ^{2}n^{1-2\alpha }+\mu _2 c_{4}^{-1} \upsilon n^{1/2-\alpha }\right) . \end{aligned}$$

(A.6)

Second, we derive the exponential tail probability bound of $P\big (\big |\Vert \widetilde{d_k}\Vert _n^2-\Vert d_k\Vert _n^2\big |\ge \upsilon n^{-\alpha }\big )$ for any positive constants $\upsilon$ and $0\le \alpha <1/2$.

Note that $\Vert d_k\Vert _n^2=E\{h_{2}({{\mathcal {O}}}_{ik};{{\mathcal {O}}}_{jk};{{\mathcal {O}}}_{lk};G;H)\}=E(\widetilde{D}_{k2})$. Employing the Markov inequality, we obtain that, for any $\epsilon >0$ and $\xi >0$,

$$\begin{aligned} P\left( \widetilde{D}_{k2}-\Vert d_k\Vert _n^2\ge \epsilon \right) \le \exp (-\xi \epsilon )\exp \left( -\xi \left\| d_k\right\| _n^2\right) E\left\{ \exp (\xi \widetilde{D}_{k2})\right\} . \end{aligned}$$

Serfling (1980, Section 5.1.6) showed that any U-statistic can be represented as an average of averages of i.i.d. random variables. We can rewrite

$$\begin{aligned} \widetilde{D}_{k2}=(n!)^{-1}\Sigma _{n!}D_{2}\left( {{\mathcal {O}}}_{1k}; \cdots ;{{\mathcal {O}}}_{nk};G,H\right) , \end{aligned}$$

where $\Sigma _{n!}$ denotes the summation over all possible permutations of $(1,\ldots ,n)$, and each $D_{2}({{\mathcal {O}}}_{1k};\cdots ;{{\mathcal {O}}}_{nk};G,H)$ is an average of $m\equiv [n/3]$ i.i.d. random variables. Denote $\psi (\xi )=E[\exp \{\xi h_{2}({{\mathcal {O}}}_{ik};{{\mathcal {O}}}_{jk};{{\mathcal {O}}}_{lk};G,H)\}]$. Jensen’s inequality yields that

$$\begin{aligned} E\left\{ \exp \left( \xi \widetilde{D}_{k2}\right) \right\}= & {} E\left[ \exp \left\{ \xi (n!)^{-1}\Sigma _{n!}D_{2}\left( {{\mathcal {O}}}_{1k}; \cdots ;{{\mathcal {O}}}_{nk};G,H\right) \right\} \right] \nonumber \\\le & {} (n!)^{-1}\Sigma _{n!}E\left[ \exp \left\{ \xi D_{2}\left( {{\mathcal {O}}}_{1k}; \cdots ;{{\mathcal {O}}}_{nk};G,H\right) \right\} \right] \nonumber \\= & {} \psi ^{m}(\xi /m). \end{aligned}$$

As a result,

$$\begin{aligned}&P\left( \widetilde{D}_{k2}-\left\| d_k\right\| _n^2\ge \epsilon \right) \le \exp (-\xi \epsilon )\exp \left( -\xi \left\| d_k\right\| _n^2\right) \psi ^{m}(\xi /m) \\&\quad = \exp (-\xi \epsilon )\left\{ E\left( \exp \left[ m^{-1}\xi \left\{ h_{2} \left( {{\mathcal {O}}}_{ik};{{\mathcal {O}}}_{jk};{{\mathcal {O}}}_{lk}; G,H\right) -\left\| d_k\right\| _n^2\right\} \right] \right) \right\} ^{m}. \end{aligned}$$

Under condition C1, there exists a positive constant $c_{5}$ such that $P(|h_2|<c_{5})=1$. It follows from Lemma 1 in Li et al. (2012) that

$$\begin{aligned} E\left\{ \exp \left[ m^{-1}\xi \left\{ h_{2} \left( {{\mathcal {O}}}_{ik};{{\mathcal {O}}}_{jk}; {{\mathcal {O}}}_{lk};G,H\right) -\left\| d_k\right\| _n^2\right\} \right] \right\} \le \exp \left\{ c_{5}^2 \xi ^2/(2m^2)\right\} , \end{aligned}$$

which immediately entails that

$$\begin{aligned} P\left( \widetilde{D}_{k2}-\left\| d_k\right\| _n^2\ge \epsilon \right) \le \exp \left( -\frac{\epsilon ^2 m}{2c_{5}^2}\right) , \end{aligned}$$

by choosing $\xi =\epsilon m/c_{5}^2$. It further follows from the symmetry of the U-statistic that

$$\begin{aligned} P\left( \left| \widetilde{D}_{k2}-\left\| d_k\right\| _n^2\right| \ge \epsilon \right) \le 2\exp \left( -\frac{\epsilon ^2 m}{2c_{5}^2}\right) . \end{aligned}$$

Using the similar argument, we also have

$$\begin{aligned} P\left( \left| \widetilde{D}_{k1}-E(\widetilde{D}_{k1})\right| \ge \epsilon \right) \le 2\exp \left( -\frac{\epsilon ^2 m^{*}}{2c_{6}^2}\right) , \end{aligned}$$

where $c_{6}$ is a positive constant such that $P(|h_1|< c_{6})=1$ and $m^{*}=[n/2]$. Obviously, under condition C1, there exist constants $c_{7}$ and $c_{8}$ such that $0\le \Vert d_k\Vert _n^2=E(\widetilde{D}_{k2})\le E|\widetilde{D}_{k2}|\le c_{7}$ and $0\le E(\widetilde{D}_{k1})\le E|\widetilde{D}_{k1}|\le c_{8}$ for any $1\le k \le p_n$. Taking $\epsilon =\upsilon n^{-\alpha }$ and n large enough such that $(3n-2)n^{-2}E(\widetilde{D}_{k2})<\upsilon n^{-\alpha }$ and $(n-1)n^{-2}E(\widetilde{D}_{k1})<\upsilon n^{-\alpha }$, we have

$$\begin{aligned}&P\left( \left| \left\| \widetilde{d_k}\right\| _n^2-\left\| d_k\right\| _n^2\right| \ge 4 \upsilon n^{-\alpha }\right) \nonumber \\&\quad =P\left\{ \left| \frac{(n-1)(n-2)}{n^2}\left( \widetilde{D}_{k2} -\left\| d_k\right\| _n^2\right) -\frac{3n-2}{n^2}E(\widetilde{D}_{k2}) \right. \right. \nonumber \\&\qquad \left. \left. +\frac{n-1}{n^2}\left\{ \widetilde{D}_{k1}-E\left( \widetilde{D}_{k1}\right) \right\} +\frac{n-1}{n^2}E(\widetilde{D}_{k1})\right| \ge 4 \upsilon n^{-\alpha }\right\} \nonumber \\&\quad \le P\left( \left| \widetilde{D}_{k1}-E(\widetilde{D}_{k1})\right| \ge \upsilon n^{1-\alpha }\right) + P\left( \left| \widetilde{D}_{k2}-\left\| d_k\right\| _n^2\right| \ge \upsilon n^{-\alpha }\right) \nonumber \\&\quad \le 2\exp \left( -\frac{\upsilon ^2 n^{2-2\alpha } m^{*}}{2c_{6}^2}\right) + 2\exp \left( -\frac{\upsilon ^2 n^{-2\alpha }m}{2c_{5}^2}\right) \nonumber \\&\quad \le 2\exp \left( -c_{9}\upsilon ^2 n^{3-2\alpha }\right) + 2\exp \left( -c_{10}\upsilon ^2 n^{1-2\alpha }\right) , \end{aligned}$$

(A.7)

by noting that $m^*\ge m\ge n/4$, where $c_{9}=1/(8c_{6}^2)$ and $c_{10}=1/(8c_{5}^2)$. It follows from (A.6) and (A.7) that

$$\begin{aligned}&P\left( \left| \left\| \widehat{d_k}\right\| _n^2-\Vert d_k\Vert _n^2\right| \ge 8 \upsilon n^{-\alpha }\right) \nonumber \\&\quad \le P\left( \left| \left\| \widehat{d_k}\right\| _n^2-\left\| \widetilde{d_k}\right\| _n^2\right| \ge 4 \upsilon n^{-\alpha }\right) + P\left( \left| \big \Vert \widetilde{d_k}\big \Vert _n^2-\big \Vert d_k\big \Vert _n^2\right| \ge 4 \upsilon n^{-\alpha }\right) \nonumber \\&\quad \le 5\exp \left( -2c_3^{-2}\upsilon ^{2}n^{3-2\alpha } +\mu _1 c_3^{-1}\upsilon n^{3/2-\alpha }\right) \nonumber \\&\qquad + 5\exp \left( -2c_{4}^{-2}\upsilon ^{2}n^{1-2\alpha } +\mu _2 c_{4}^{-1}\upsilon n^{1/2-\alpha }\right) \nonumber \\&\qquad +2\exp \left( -c_{9}\upsilon ^2 n^{3-2\alpha }\right) + 2\exp \left( -c_{10}\upsilon ^2 n^{1-2\alpha }\right) \nonumber \\&\quad \le O\left\{ \exp \left( -\eta n^{1-2\alpha }\right) \right\} , \end{aligned}$$

(A.8)

where $\eta =\min \{2c_{4}^{-2}\upsilon ^{2},c_{10}\upsilon ^2\}$. Immediately, we have

$$\begin{aligned} P\left( \max _{1\le k \le p_n}\left| \left\| \widehat{d_k}\right\| _n^2-\left\| d_k\right\| _n^2\right| \ge 8 \upsilon n^{-\alpha }\right) \le O\left\{ p_n\exp \left( -\eta n^{1-2\alpha }\right) \right\} , \end{aligned}$$

(A.9)

which proves the first part of Theorem 1 by taking $c=8\upsilon$.

If $\mathcal {A}\nsubseteq \widehat{\mathcal {A}}$, then there must exist some $k\in \mathcal {A}$ such that $\Vert \widehat{d_k}\Vert _n^2< cn^{-\alpha }$. It follows from condition C2 that $|\Vert \widehat{d_k}\Vert _n^2-\Vert d_k\Vert _n^2|>cn^{-\alpha }$ for some $k\in \mathcal {A}$, which implies that $\{\mathcal {A}\nsubseteq \widehat{\mathcal {A}}\}\subseteq \{|\Vert \widehat{d_k}\Vert _n^2-\Vert d_k\Vert _n^2|>cn^{-\alpha }$ for some $k\;\in \mathcal {A}\}$. As a result, $\{\max _{k\in \mathcal {A}}|\Vert \widehat{d_k}\Vert _n^2-\Vert d_k\Vert _n^2|\le cn^{-\alpha }\}\subseteq \{\mathcal {A}\subseteq \widehat{\mathcal {A}}\}$. Using (A.8), we have

$$\begin{aligned} P\big (\mathcal {A}\subseteq \widehat{\mathcal {A}}\big )\ge & {} P\Big (\max _{k\in \mathcal {A}}\big |\big \Vert \widehat{d_k}\big \Vert _n^2-\big \Vert d_k\big \Vert _n^2\big |\le cn^{-\alpha }\Big )\nonumber \\\ge & {} 1-O\big \{a_n\exp \big (-\eta n^{1-2\alpha }\big )\big \}, \end{aligned}$$

where $a_n=|\mathcal {A}|$. Thus, the proof of Theorem 1 is completed. $\square$

Proof of Theorem 2

Under assumption (i), we rewrite

$$\begin{aligned} d_k(t,z)= & {} E\left[ \left\{ \frac{\Delta I(X>t)}{G(X)}-P(T>t)\right\} I(Z_k< z)\right] \nonumber \\= & {} E\left\{ E\left[ \left. \left\{ \frac{\Delta I(X>t)}{G(X)}-P(T>t)\right\} I(Z_k< z)\right| \mathbf{Z}\right] \right\} \nonumber \\= & {} E\left\{ I(Z_k< z)E\left[ \left. \left\{ \frac{\Delta I(X>t)}{G(X)}-P(T>t)\right\} \right| \mathbf{Z}\right] \right\} \nonumber \\= & {} E\left\{ I(Z_k< z)\left[ E\left\{ \left. \frac{\Delta I(X>t)}{G(X)}\right| \mathbf{Z}_{\mathcal {A}} \right\} -P(T>t)\right] \right\} . \end{aligned}$$

If $k\notin \mathcal {A}$, then assumption (ii) implies that

$$\begin{aligned} d_k(t,z)=E\left\{ I(Z_k< z)\right\} E\left[ E\left\{ \left. \frac{\Delta I(X>t)}{G(X)}\right| \mathbf{Z}_{\mathcal {A}} \right\} -P(T>t)\right] =0, \end{aligned}$$

for any t and z. As a result, $\Vert d_k\Vert _n^2=E\{d_k(X,Z_k)^2\}=0$. It follows from condition C2 that $\max _{k\notin \mathcal {A}}\Vert d_k\Vert _n^2 < \min _{k\in \mathcal {A}}\Vert d_k\Vert _n^2$. On the other hand, $\Vert d_k\Vert _n^2=0$ directly implies that $k\notin \mathcal {A}$ under condition C2. Thus, the first part of Theorem 2 is proved.

Under condition C2 and assumptions (i) and (ii), coupled with (A.9), we have

$$\begin{aligned}&P\left( \min _{k\in \mathcal {A}} \left\| \widehat{d_k}\right\| _n^2 \le \max _{k \notin \mathcal {A}} \left\| \widehat{d_k}\right\| _n^2\right) \nonumber \\&\quad =P\left( \max _{k \notin \mathcal {A}} \left\| \widehat{d_k}\right\| _n^2 -\max _{k \notin \mathcal {A}} \Vert {d_k}\Vert _n^2- \min _{k\in \mathcal {A}} \left\| \widehat{d_k}\right\| _n^2+\min _{k\in \mathcal {A}} \left\| d_k\right\| _n^2\ge \min _{k\in \mathcal {A}}\Vert d_k\Vert _n^2\right) \nonumber \\&\quad \le P\left( \max _{k \notin \mathcal {A}}\left| \left\| \widehat{d_k}\right\| _n^2-\left\| {d_k}\right\| _n^2\right| \ge cn^{-\alpha }\right) + P\left( \max _{k \in \mathcal {A}}\left| \left\| \widehat{d_k}\right\| _n^2-\left\| {d_k}\right\| _n^2\right| \ge cn^{-\alpha }\right) \nonumber \\&\quad \le 2P\left( \max _{1\le k \le p_n}\left| \left\| \widehat{d_k}\right\| _n^2-\left\| d_k\right\| _n^2\right| \ge cn^{-\alpha }\right) \nonumber \\&\quad \le O\left\{ p_n\exp \left( -\eta n^{1-2\alpha }\right) \right\} , \end{aligned}$$

which completes the proof of Theorem 2. $\square$