Model-free feature screening via distance correlation for ultrahigh dimensional survival data

Abstract

With the explosion of ultrahigh dimensional data in various fields, many sure independent screening methods have been proposed to reduce the dimensionality of data from a large scale to a relatively moderate scale. For censored survival data, the existing screening methods mainly adopt the Kaplan–Meier estimator to handle censoring, which may not perform well for heavy censoring cases. In this article, we propose a novel sure independent screening procedure based on distance correlation after standardizing marginal variables for ultrahigh dimensional survival data. It is a model-free approach and does not involve the Kaplan–Meier estimator, thus its performance is much more robust than the existing methods. Furthermore, our proposed method enjoys other advantages: it avoids the complication to specify an actual model from large number of covariates; it enjoys the sure screening property and the ranking consistency under some mild regularity conditions; it does not require any complicated numerical optimization, so the corresponding calculation is very simple and fast. Extensive numerical studies demonstrate that the proposed method has favorable exhibition over the existing methods. As an illustration, we apply the proposed method to a gene expression data set.

Appendix: Theoretical proofs

Proof of Theorem 1

To prove Theorem 1, we need to prove the uniform consistency of the denominator and the numerator of $\widehat{\omega }_k$ respectively. Because the denominator of $\widehat{\omega }_k$ has a similar form as the numerator, we only deal with its numerator $\widehat{\mathrm{dcov}}^2(Z_k,Y)$ below.

It follows from the definitions of distance covariance and sample distance covariance that

$$\begin{aligned} {\mathrm{dcov}}^2(Z_k,Y) = {S}_{k1}+{S}_{k2}-2{S}_{k3},\quad \widehat{\mathrm{dcov}}^2(Z_k,Y) = \widehat{S}_{k1}+\widehat{S}_{k2}-2\widehat{S}_{k3}, \end{aligned}$$

where $S_{k1} =\mathrm{E}(\Vert Z_k-\widetilde{Z}_k\Vert _1\Vert Y-\widetilde{Y}\Vert _2)$, $S_{k2} =\mathrm{E}(\Vert Z_k-\widetilde{Z}_k\Vert _1)\mathrm{E}(\Vert Y-\widetilde{Y}\Vert _2)$, $S_{k3}=\mathrm{E}\{ \mathrm{E}(\Vert Z_k-\widetilde{Z}_k\Vert _1|Z_k)\mathrm{E}(\Vert Y-\widetilde{Y}\Vert _2|Y)\}$, $\widehat{S}_{k1}=\frac{1}{n^2}\sum \nolimits _{i=1}^n\sum \nolimits _{j=1}^n|Z_{ki}-Z_{kj}|\Vert \widehat{Y}_i-\widehat{Y}_j\Vert _2$, $\widehat{S}_{k2}=\frac{1}{n^2}\sum \nolimits _{i=1}^n\sum \nolimits _{j=1}^n|Z_{ki}-Z_{kj}| \cdot \frac{1}{n^2}\sum \nolimits _{i=1}^n\sum \nolimits _{j=1}^n\Vert \widehat{Y}_i-\widehat{Y}_j\Vert _2$, $\widehat{S}_{k3} =\frac{1}{n^3}\sum \nolimits _{i=1}^n\sum \nolimits _{j=1}^n\sum \nolimits _{l=1}^n|Z_{ki}-Z_{kl}|\Vert \widehat{Y}_j-\widehat{Y}_l\Vert _2$, $(\widetilde{Z}_k,\widetilde{Y})$ is an independent copy of $({Z_k},{Y})$. Similarly, we define

$$\begin{aligned} \widetilde{\mathrm{dcov}}^2(Z_k,Y) = \widetilde{S}_{k1}+\widetilde{S}_{k2}-2\widetilde{S}_{k3}, \end{aligned}$$

where $\widetilde{S}_{k1}$, $\widetilde{S}_{k2}$ and $\widetilde{S}_{k3}$ are obtained by replacing $\widehat{Y}_i$ $(i=1,\ldots ,n)$ in $\widehat{S}_{k1}$, $\widehat{S}_{k2}$ and $\widehat{S}_{k3}$ with ${Y}_i$ $(i=1,\ldots ,n)$ respectively. Specifically, $\widetilde{S}_{k1}=1/{n^2}\sum _{i=1}^n\sum _{j=1}^n|Z_{ki}-Z_{kj}|\Vert Y_i-Y_j\Vert _2$, $\widetilde{S}_{k2}=1/{n^2}\sum _{i=1}^n\sum _{j=1}^n|Z_{ki}-Z_{kj}|\cdot 1/{n^2}\sum _{i=1}^n\sum _{j=1}^n\Vert Y_i-Y_j\Vert _2$, $\widetilde{S}_{k3}=1/{n^3}\sum _{i=1}^n\sum _{j=1}^n\sum _{l=1}^n|Z_{ki}-Z_{kl}|\Vert Y_j-Y_l\Vert _2$. Under the discussion in Li et al. (2012b), for any $\epsilon >0$ and $0<\gamma <1/2-\kappa$, there exist positive constants $C_1$ and $C_2$ such that

$$\begin{aligned}&P\left( |\mathrm{dcov}^2(Z_k,Y)-\widetilde{\mathrm{dcov}}^2(Z_k,Y)|\ge \epsilon \right) \nonumber \\&\quad \le O\left\{ \exp (-C_1\epsilon ^2n^{1-2\gamma })+n\exp (-C_2n^\gamma )\right\} . \end{aligned}$$

(4)

In order to obtain the exponential tail probability bound of $P(|\mathrm{dcov}^2(Z_k,Y)-\widehat{\mathrm{dcov}}^2(Z_k,Y)|\ge \epsilon )$, we only need to compute $P(|\widehat{\mathrm{dcov}}^2(Z_k,Y)-\widetilde{\mathrm{dcov}}^2(Z_k,Y)|\ge \epsilon )$. By the definition of $\widehat{\mathrm{dcov}}^2(Z_k,Y)$ and $\widetilde{\mathrm{dcov}}^2(Z_k,Y)$, we have

$$\begin{aligned} \left| \widehat{\mathrm{dcov}}^2(Z_k,Y)-\widetilde{\mathrm{dcov}}^2(Z_k,Y)\right| \le \left| \widehat{S}_{k1}-\widetilde{S}_{k1}\right| +\left| \widehat{S}_{k2}-\widetilde{S}_{k2}\right| +2\left| \widehat{S}_{k3}-\widetilde{S}_{k3}\right| . \end{aligned}$$

(5)

We first deal with $|\widehat{S}_{k1}-\widetilde{S}_{k1}|$, straightforward calculations entail that

$$\begin{aligned} \left| \widehat{S}_{k1}-\widetilde{S}_{k1}\right|= & {} \frac{1}{n^2}\sum _{i=1}^n\sum _{j=1}^n|Z_{ki}-Z_{kj}| \left( \Vert \widehat{Y}_i-\widehat{Y}_j\Vert _2-\Vert Y_i-Y_j\Vert _2\right) \\\le & {} \frac{1}{n^2}\sum _{i=1}^n\sum _{j=1}^n(|Z_{ki}|+|Z_{kj}|)\Big ( L_{ij}(\widehat{\sigma }_1,\widehat{\sigma }_1;\widehat{\sigma }_2,\widehat{\sigma }_2)-L_{ij}(\sigma _1,\sigma _1;\sigma _2,\sigma _2)\Big )\\\le & {} \frac{1}{n^2}\sum _{i=1}^n\sum _{j=1}^n(|Z_{ki}|+|Z_{kj}|)\Big ( L_{ii}(\widehat{\sigma }_1,\sigma _1;\widehat{\sigma }_2,\sigma _2)+L_{jj}(\widehat{\sigma }_1,\sigma _1;\widehat{\sigma }_2,\sigma _2)\Big ), \end{aligned}$$

where

$$\begin{aligned} L_{ij}(\widehat{\sigma }_1,\sigma _1;\widehat{\sigma }_2,\sigma _2) = \sqrt{\left( \frac{X_i}{\widehat{\sigma }_1}-\frac{X_j}{\sigma _1}\right) ^2+ \left( \frac{\varDelta _i}{\widehat{\sigma }_2}-\frac{\varDelta _j}{\sigma _2}\right) ^2}, \end{aligned}$$

and the last inequality obtained by using the triangle inequality. As we all know, for any events $H_1$ and $H_2$, it holds

$$\begin{aligned} P(H_1)=P(H_1|H_2)P(H_2)+P(H_1|H_2^\mathrm{c})P(H_2^\mathrm{c})\le P(H_1|H_2)+P(H_2^\mathrm{c}). \end{aligned}$$

(6)

Define events $A_i=\{|Z_{ki}|< n^{\gamma }\}$ $(i=1,\ldots ,n)$, by (6), condition C1, Bonferroni’s inequality and Markov’s inequality, we obtain

$$\begin{aligned}&P\left( |\widehat{S}_{k1}-\widetilde{S}_{k1}|\ge \epsilon \right) \nonumber \\&\quad \le P\Big (\frac{1}{n^2}\sum _{i=1}^n\sum _{j=1}^n(|Z_{ki}|+|Z_{kj}|) \big (L_{ii}(\widehat{\sigma }_1,\sigma _1;\widehat{\sigma }_2,\sigma _2)+L_{jj}(\widehat{\sigma }_1,\sigma _1;\widehat{\sigma }_2,\sigma _2)\big )\nonumber \\&\quad \ge \epsilon \left| \bigcap _{i=1}^nA_i\right. \Big )+P\Big (\Big (\bigcap _{i=1}^nA_i\Big )^\mathrm{c}\Big ) \nonumber \\&\quad \le P\left( 4n^{\gamma }\cdot \frac{1}{n}\sum _{i=1}^n L_{ii}(\widehat{\sigma }_1,\sigma _1;\widehat{\sigma }_2,\sigma _2) \ge \epsilon \right) +C_3n\exp (-n^{\gamma })\nonumber \\&\quad \le P\left( \frac{1}{n}\sum _{i=1}^n\left( \left| \frac{X_i}{\widehat{\sigma }_1}-\frac{X_i}{{\sigma }_1}\right| +\left| \frac{\varDelta _i}{\widehat{\sigma }_2}-\frac{\varDelta _i}{{\sigma }_2}\right| \right) \ge \frac{\epsilon }{4n^{\gamma }}\right) +C_3n\exp (-n^{\gamma }), \end{aligned}$$

(7)

where $C_3$ is a positive constant. Straightforward calculations entail that

$$\begin{aligned}&P\left( \frac{1}{n}\sum _{i=1}^n\left( \left| \frac{X_i}{\widehat{\sigma }_1}-\frac{X_i}{{\sigma }_1}\right| +\left| \frac{\varDelta _i}{\widehat{\sigma }_2}-\frac{\varDelta _i}{{\sigma }_2}\right| \right) \ge \frac{\epsilon }{4n^{\gamma }}\right) \nonumber \\&\quad \le P\left( \frac{1}{n}\sum _{i=1}^n\left| \frac{X_i}{\widehat{\sigma }_1}-\frac{X_i}{{\sigma }_1}\right| \ge \frac{\epsilon }{8n^{\gamma }}\right) + P\left( \frac{1}{n}\sum _{i=1}^n\left| \frac{\varDelta _i}{\widehat{\sigma }_2}-\frac{\varDelta _i}{{\sigma }_2}\right| \ge \frac{\epsilon }{8n^{\gamma }}\right) \nonumber \\&\quad \le P\left( \left| \frac{1}{\widehat{\sigma }_1}-\frac{1}{{\sigma }_1}\right| \cdot \frac{1}{n}\sum _{i=1}^nX_i \ge \frac{\epsilon }{8n^{\gamma }}\right) + P\left( \left| \frac{1}{\widehat{\sigma }_2}-\frac{1}{{\sigma }_2}\right| \cdot \frac{1}{n}\sum _{i=1}^n\varDelta _i \ge \frac{\epsilon }{8n^{\gamma }}\right) \nonumber \\&\quad \equiv I_1+I_2. \end{aligned}$$

(8)

We first consider $I_1$, defining events $B=\{|\widehat{\sigma }_1-{\sigma }_1|\le \sigma _1/2\}$ and $C=\{|n^{-1}\sum _{i=1}^nX_i-{\mu }_1|\le \mu _1/2\}$, there exist some positive constants $C_4$ and $C_5$ such that

$$\begin{aligned} I_1= & {} P\left( \left| \frac{1}{\widehat{\sigma }_1}-\frac{1}{{\sigma }_1}\right| \cdot \frac{1}{n}\sum _{i=1}^nX_i \ge \frac{\epsilon }{8n^{\gamma }}\right) \nonumber \\\le & {} P\left( \left| \frac{1}{\widehat{\sigma }_1}-\frac{1}{{\sigma }_1}\right| \cdot \frac{1}{n}\sum _{i=1}^nX_i \ge \frac{\epsilon }{8n^{\gamma }}|B,C\right) +P((BC)^{c}) \nonumber \\\le & {} P\left( |\widehat{\sigma }_1^2-{\sigma }_1^2|\ge C_4n^{-\gamma }\epsilon \right) +P(B^{c})+P(C^{c}) \nonumber \\\equiv & {} I_3+P(B^{c})+P(C^{c}) \end{aligned}$$

(9)

for positive constant $C_4$. Since

$$\begin{aligned} |\widehat{\sigma }_1^2-{\sigma }_1^2| =\left| \frac{1}{n}\sum _{i=1}^n(X_i-\widehat{\mu }_1)^2-{\sigma }_1^2\right| \le \left| \frac{1}{n}\sum _{i=1}^n(X_i-{\mu }_1)^2-{\sigma }_1^2\right| + \left| {\widehat{\mu }}_1-{\mu }_1 \right| ^2, \end{aligned}$$

$I_3$ can be deduced as

$$\begin{aligned} I_3\le & {} P\left( \left| \frac{1}{n}\sum _{i=1}^n(X_i-{\mu }_1)^2-{\sigma }_1^2\right| \ge \frac{C_4n^{-\gamma }\epsilon }{2}\right) +P\left( \left| {\widehat{\mu }}_1-{\mu }_1 \right| ^2 \ge \frac{C_4n^{-\gamma }\epsilon }{2}\right) \nonumber \\\le & {} 2\exp (-C_5\epsilon ^2n^{1-2\gamma }) + 2\exp (-C_6\epsilon n^{1-\gamma }) \le 2\exp (-C_7\epsilon ^2n^{1-2\gamma }), \end{aligned}$$

(10)

where $C_5$, $C_6$ and $C_7$ are some positive constants. Furthermore, there exists positive constant $C_8$ such that

$$\begin{aligned} P(B^{c})= & {} P\left( |\widehat{\sigma }_1-{\sigma }_1|\ge \sigma /2\right) \le P\left( |\widehat{\sigma }_1^2-{\sigma }_1^2|\ge \sigma ^2/2\right) \le 2\exp (-C_8n),\nonumber \\ P(C^{c})= & {} P(|n^{-1}\sum _{i=1}^nX_i-{\mu }_1|\ge \mu _1/2)\le 2\exp (-C_8n). \end{aligned}$$

(11)

Combining (9), (10) and (11), we obtain

$$\begin{aligned} I_1\le & {} 2\exp (-C_{7}\epsilon ^2n^{1-2\gamma }) +2\exp (-C_8n) +2\exp (-C_8n)\nonumber \\= & {} O \left( \exp (-C_{9}\epsilon ^2n^{1-2\gamma })\right) , \end{aligned}$$

(12)

for positive constant $C_{9}$. Using the similar arguments, there exists positive constant $C_{10}$ such that

$$\begin{aligned} I_2= & {} O\left( \exp (-C_{10}\epsilon ^2n^{1-2\gamma })\right) . \end{aligned}$$

(13)

Following (7), (8), (12) and (13), we have

$$\begin{aligned} P\left( |\widehat{S}_{k1}-\widetilde{S}_{k1}|\ge \epsilon \right)\le & {} O\left( \exp (-C_{9}\epsilon ^2n^{1-2\gamma })\right) \\&\quad +O\left( \exp (-C_{10}\epsilon ^2n^{1-2\gamma })\right) +C_3n\exp (-n^{\gamma })\\= & {} O\left( \exp (-C_{11}\epsilon ^2n^{1-2\gamma }) + n\exp (-n^{\gamma }) \right) . \end{aligned}$$

Using the analogous arguments, along with some tedious calculation, we can obtain similar conclusion of $P\left( |\widehat{S}_{k2}-\widetilde{S}_{k2}|\ge \epsilon \right)$ and $P\left( |\widehat{S}_{k3}-\widetilde{S}_{k3}|\ge \epsilon \right)$. Hence, we conclude that

$$\begin{aligned}&P\left( \left| \widehat{\mathrm{dcov}}^2(Z_k,Y)-\widetilde{\mathrm{dcov}}^2(Z_k,Y)\right| \ge \epsilon \right) \\&\quad \le P\left( |\widehat{S}_{k1}-\widetilde{S}_{k1}|\ge \epsilon /4\right) + P\left( |\widehat{S}_{k2}-\widetilde{S}_{k2}|\ge \epsilon /4\right) + P\left( |\widehat{S}_{k3}-\widetilde{S}_{k3}|\ge \epsilon /4\right) \\&\quad =O\left( \exp (-C_{12}\epsilon ^2n^{1-2\gamma })+n\exp (-n^{\gamma })\right) \end{aligned}$$

for positive constant $C_{12}$. Let $\epsilon =cn^{-\kappa }$, where $\kappa$ satisfies $0<\kappa +\gamma <1/2$. Immediately, there exist some positive constants $c_1$ and $c_2$ such that

$$\begin{aligned}&P\left( \left| {\mathrm{dcov}}^2(Z_k,Y)-\widehat{\mathrm{dcov}}^2(Z_k,Y)\right| \ge cn^{-\kappa }\right) \nonumber \\&\quad \le P\left( \left| {\mathrm{dcov}}^2(Z_k,Y)-\widetilde{\mathrm{dcov}}^2(Z_k,Y)\right| \ge \frac{c}{2} n^{-\kappa }\right) +\, P\left( \left| {\widetilde{\mathrm{dcov}}}^2(Z_k,Y)-\widehat{\mathrm{dcov}}^2(Z_k,Y)\right| \ge \frac{c}{2} n^{-\kappa }\right) \nonumber \\&\quad \le O\left( \exp \left\{ -C_1n^{1-2(\kappa +\gamma )}\right\} +n\exp (-C_2n^\gamma )\right) +\,O\left( \exp \left\{ -C_{12}n^{1-2(\kappa +\gamma )}\right\} +n\exp (-n^{\gamma })\right) \nonumber \\&\quad =O\left( \exp \left\{ -c_1n^{1-2(\kappa +\gamma )}\right\} +n\exp (-c_2n^{\gamma })\right) . \end{aligned}$$

(14)

The convergence rate of the numerator of $\widehat{\omega }_k$ is now achieved. Following similar arguments, we can obtain the same convergence rate of the denominator. Then we have

$$\begin{aligned}&P\left( \max _{1\le k \le p}|\widehat{\omega }_k-{\omega }_k|\ge cn^{-\kappa }\right) \nonumber \\&\quad \le p\max _{1\le k \le p}P\left( |\widehat{\omega }_k-{\omega }_k|\ge cn^{-\kappa }\right) \nonumber \\&\quad = O\left( p\left[ \exp \left\{ -c_1n^{1-2(\kappa +\gamma )}\right\} +n\exp (-c_2n^{\gamma })\right] \right) . \end{aligned}$$

(15)

The first part of Theorem 1 is proved.

We now consider the second part of Theorem 1. If $\mathcal {A}\not \subseteq \widehat{\mathcal {A}}$, then there must exist some $k\in \mathcal {A}$ such that $\widehat{\omega }_k< cn^{-\kappa }$. It follows from condition C2 that $|\widehat{\omega }_k-{\omega }_k|>cn^{-\kappa }$ for some $k\in \mathcal {A}$, which implies that $\{\mathcal {A}\not \subseteq \widehat{\mathcal {A}}\}\subseteq \{|\widehat{\omega }_k-{\omega }_k|>cn^{-\kappa }$ for some $k\;\in \mathcal {A}\}$. As a result, $\{\max _{k\in \mathcal {A}}|\widehat{\omega }_k-{\omega }_k|\le cn^{-\kappa }\}\subseteq \{\mathcal {A}\subseteq \widehat{\mathcal {A}}\}$. Using (15), we have

$$\begin{aligned}&P\left( \mathcal {A}\subseteq \widehat{\mathcal {A}}\right) \ge P\left( \max _{k\in \mathcal {A}}\big |\widehat{\omega }_k-{\omega }_k\big | \le cn^{-\kappa }\right) \\&\quad \ge 1-O\left( a\left[ \exp \left\{ -c_1n^{1-2(\kappa +\gamma )}\right\} +n\exp (-c_2n^{\gamma })\right] \right) , \end{aligned}$$

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

Proof of Theorem 2

Under condition C2, noting that $\min _{k\in \mathcal {A}}\omega _k \ge 2cn^{-\kappa }$ and combining it with (15) and the assumption ${\omega }_k=0$ ($k \notin \mathcal {A}$), we have

$$\begin{aligned}&P\left( \min _{k\in \mathcal {A}} \widehat{\omega }_k \le \max _{k \notin \mathcal {A}} \widehat{\omega }_k\right) \\&\quad =P\left( \max _{k \notin \mathcal {A}} \widehat{\omega }_k-\max _{k \notin \mathcal {A}} {\omega }_k- \min _{k\in \mathcal {A}}\widehat{\omega }_k+\min _{k\in \mathcal {A}}{\omega }_k\ge \min _{k\in \mathcal {A}}{\omega }_k\right) \nonumber \\&\quad \le P\left( \max _{k \notin \mathcal {A}}\left| \widehat{\omega }_k-{\omega }_k\right| \ge cn^{-\kappa }\right) + P\left( \max _{k \in \mathcal {A}}\left| \widehat{\omega }_k-{\omega }_k\right| \ge cn^{-\kappa }\right) \nonumber \\&\quad \le 2P\left( \max _{1\le k \le p}\left| \widehat{\omega }_k-{\omega }_k\right| \ge cn^{-\kappa }\right) \nonumber \\&\quad = O\left( p\left[ \exp \left\{ -c_1n^{1-2(\kappa +\gamma )}\right\} +n\exp (-c_2n^{\gamma })\right] \right) , \end{aligned}$$

so

$$\begin{aligned} P\left( \max _{k \notin \mathcal {A}} \widehat{\omega }_k <\min _{k\in \mathcal {A}} \widehat{\omega }_k\right) \ge 1-O\left( p\left[ \exp \left\{ -c_1n^{1-2(\kappa +\gamma )}\right\} +n\exp (-c_2n^\gamma )\right] \right) , \end{aligned}$$

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