1.   Inroduction

High dimensionality, heterogeneity, and the existence of outliers make variable selection for censored survival data challenging. There are numerous studies in the literature on variable selection for regression problems with and without censoring. Recently, various regularization methods have been proposed for feature selection in high-dimensional data analysis, which has become increasingly prominent and important across various research fields. These methods include, but are not limited to, the LASSO ^[1], the smoothly clipped absolute deviation (SCAD) ^[2-4], the least angle regression (LARS) algorithm ^[5], the elastic net ^[6-7], the adaptive LASSO ^[8], and the Dantzig selector ^[9]. On the other hand, variable screening methods for high-dimensional survival data are mostly based on the partial-likelihood of the Cox model. For example, Fan et al.^[10] and Sihai-Dave-Zhao and Li^[11] investigated marginal screening based on the Cox proportional hazards model. However, in practice, the true models often remain unknown, and it is unclear whether these methods will perform well under model misspecification. More importantly, these penalized algorithms are effective for mean regressions and parametric models, yet face simultaneous challenges of computational efficiency, statistical accuracy and algorithmic stability when the predictors are ultrahigh dimensional and the sample size is relatively small ^[12].

A computationally simple method for very high-dimensional data that performs well in practice is sure independence screening, as demonstrated in the classical regression context in [13]. In this method, the outcome variable is regressed on each covariate separately. Sure independence screening recruits the features that have the best marginal utility. In the context of least squares regression for a linear model, this corresponds to the largest marginal absolute Pearson correlation between the response and the predictor. Correlation screening is a crude yet effective way to decrease the dimensionality of data. However, as pointed out in [14], the Pearson correlation might not work well for censored survival data because it cannot be reliably estimated, especially when the censoring rate is high. In addition, its performance can be significantly affected by outliers in predictors because correlation is not a robust measure for association. Such outliers pose challenges for theoretical studies of screening methods, most of which require tail probability conditions for the covariates. To address these challenges, ^[14] proposed censored rank independence screening for high-dimensional survival data. However, their method may be adversely affected by the heterogeneity that is often present in high-dimensional data. To this end, ^[15] propose a conditional quantile screening method for high-dimensional survival data with heterogeneity, which enables us to select features that contribute to the conditional quantile of the complete or censored response given the covariates. See [16-22] for further developments. It is worth noting that ^[23] also proposed a quantile adaptive sure independent screening procedure for high-dimensional survival data with heterogeneity. However, compared to ^[23], the computational cost in [15] is significantly lower, as the former involves fitting marginal spline-based quantile regression models, which are quite computationally expensive. In this paper, we propose a censored composite conditional quantile screening (cCCQC-SIS) method for high-dimensional survival data. Our proposed method has several advantages. First, it is robust against the existence of outliers. This robustness is derived from the censored conditional quantile coefficient. Second, it is a non-model-based method, so it works for a wide class of survival models. In particular, the cCCQC makes use of all useful information across quantiles. There have existed several papers which utilize the composite quantile idea in other statistical problems. These methods include, but are not limited to, [24-26]. In this work, we apply the same principle in the context of feature screening for survival data.

The rest of this paper is organized as follows. In Section 2, we introduce the notion of the cCCQC, and the corresponding sure screening property and rank consistency property are rigorously justified. In Section 3, we evaluate the finite sample performance of our proposals through Monte Carlo simulations. The technical details are provided in the Appendix.

2.   Method

2.1   The Censored Composite Conditional Quantile Coefficient

To introduce the notion of the censored composite conditional quantile coefficient (cCCQC), we shall provide a brief discussion on the censored composite conditional quantile coefficient. Let $ Y $ denote the response variable of interest, $ C $ denote the censoring variable, and $ \mathbf{z}=(Z_{1}, Z_{2}, \cdots, Z_{p})^\top $ denote the $ p $-dimensional vector of covariates. Further, define $ X =\min(Y, C) $ and $ \Delta= I(Y<C) $. Here $ I(\cdot) $ denotes the indicator function. The observed data are independent and identically distributed copies of $ \{X, \Delta, (Z_{1}, Z_{2}, \cdots, Z_{p})^\top\} $ and are denoted by $ \{X_i, \Delta_i, (Z_{i1}, Z_{i2}, \cdots, Z_{ip})^\top\}_{i=1}^{n} $. Throughout the paper, we assume that the censoring variable $ C $ is independent of the response $ Y $ and the covariates $ \mathbf{z}. $

The censored conditional quantile (CCQ) coefficient is given by

where $ \tau\in(0, 1) $, $ F(y)=\mathsf{P}(Y\leq y) $, the weight function

redistributes the masses of censored observations to the right ^{[15, 27]}, $ \widetilde{Z}_{k} $ is i.i.d. copy of $ Z_{k} $ and $ Q_{\tau}(Y) $ is the $ \tau \times 100\% $th quantile of $ Y $.

Motivated by Zou and Yuan^[26], Kong and Xia^[28] and Xu^[29], we here propose the censored composite conditional quantile coefficient (cCCQC), i.e.,

The CCQ in (1) is very useful for handling heterogeneity. However, with a limited sample size, there is variability in the set of selected variables as $ \tau $ changes, even if just slightly. Such variability is clearly undesirable for interpretation. More importantly, some important variables are likely to be missed, simply due to chance, if we perform variable selection at any given $ \tau $. Therefore, one can anticipate that cCCQC is more stable than CCQ, as it takes advantage of all useful information across quantiles to enhance the stability of CCQ.

Let $ \widehat{F}_{n}(y)=1-\widehat{S}_{n}(y), $ where $ \widehat{S}_{n}(y) $ is the Kaplan-Meier estimator of $ Y $ based on $ \{(X_i, \Delta_i)\}_{i=1}^{n} $. The $ \tau $th sample quantile $ \widehat{F}_{n}^{-1}(\tau) $ is an estimator of $ Q_{\tau}(Y) $ when $ Y $ is subject to right censoring. By invoking (2), a natural estimator of cCCQC is given by

where $ \tau_{s}=\frac{s}{n+1}, s=1, \cdots, n $ and $ w_{i\tau_{s}}(\widehat{F}_{n}) $ is denoted in an obvious way. The integral approximation is straightforward by invoking the precursor work of [24, 30]. For the purpose of high-dimensional screening, we focus on rather than the asymptotic properties of $ \mathrm{\widehat{cCCQC}}(X, Z_k) $ but instead the desirable sure screening and rank consistency properties of $ \mathrm{\widehat{cCCQC}}(X, Z_k) $.

Following the work of Kong and Xia^[28], and for the sake of technical convenience, we focus on rather than the case $ (0, 1) $ but instead the following truncated version $ [\delta^{*}, 1-\delta^{*}] $:

and

for some small $ \delta^{*}\in(0, 1). $ This is due to the fact that the uniformity in $ \tau $ of the strong Bahadur-type representation of $ \widehat{F}_{n}^{-1}(\tau) $ cannot be met by all $ \tau\in(0, 1). $ See the proof given in the Appendix for more details. Nevertheless, such truncation need not cause much concern. The reasons are two-fold. On one hand, the integral in (2) is approximated by summing over a sequence of discretized $ \tau $ values. On the other hand, the cCCQC which is derived based on $ (0, 1) $ is expected to closely resemble, if not completely identical to, that based on $ [\delta^{*}, 1-\delta^{*}] $, provided that $ \delta^{*} $ is small enough. In practice, we follow the work by [24, 28] to choose $ \delta^{*}=1/n. $

2.2   A Screening Procedure

In this section we design a sure independence screening procedure based on the cCCQC for high-dimensional survival data. Let $ \mathcal{A} $ denote the index set of the active variables:

With a sample of size $ n $, we aim to select the set of active variables $ \mathbf{z}_{\mathcal{A}} $. The following assumptions are needed.

Assumption 1   The truly important predictors satisfy

Assumption 2   $ F(y) $ is twice differentiable; the density function of $ Y $, $ f(y) $, is uniformly bounded away from zero and infinity, and its derivative $ f'(y) $ is bounded uniformly on $ \left[Q_{\delta^*}(Y)-\varepsilon, Q_{1-\delta^*}(Y)+\varepsilon\right] $ for some $ 0 <\varepsilon<1 $.

Assumption 3   $ G(x)=\mathsf{P}(C\leq x) $ is twice differentiable, the density function of $ C $, $ g(x) $, is uniformly bounded away from zero and infinity, and its derivative $ g'(x) $ is bounded uniformly on $ \left[Q_{\delta^*}(Y)-\varepsilon, Q_{1-\delta^*}(Y)+\varepsilon\right] $ for some $ 0 <\varepsilon<1 $.

Assumption 4   Let $ L $ denote the maximum follow-up variable; then $ \mathsf{P}(L\geq Y)\geq \tau_0>0 $ for some positive constant $ \tau_0 $.

Assumption 1 requires the signals of the important predictors to be strong enough to be detectable by the cCCQC. Similar conditions are widely assumed in the marginal screening literature. See, for example, [14-15]. Assumptions 2–4 are common in the survival analysis literature to ensure that the Kaplan-Meier estimator and its inverse function are well behaved.

If the signal level is not too small, i.e., Assumption 1 is true, we suggest the cCCQC-SIS procedure which retains the predictors indexed by

where $ c $ and $ t $ are specified in Assumption 1.

With the above Assumptions, we can easily establish the desirable sure screening property for the cCCQC-SIS procedure without assuming the marginal distribution functions of either $ \mathbf{z} $ or $ Y $, or both, have exponential tails.

Theorem 1 (Sure Screening Property) Suppose the Assumptions 1–4 hold. Then, we can show that there exists a sufficiently small constant $ s_n $ such that

where $ |\mathcal{A}| $ denotes the cardinality of the index set $ \mathcal{A} $,

One can expect that $ Y $ depends more upon $ \mathbf{z}_{\mathcal{A}} $ than upon $ \mathbf{z}_{\mathcal{A}^{c}} $, though such dependence can be nonlinear. Intuitively speaking, $ \mathrm{cCCQC}_{T}(X, Z_k) $, for $ k\in\mathbf{z}_{\mathcal{A}} $, is greater than $ \mathrm{cCCQC}_{T}(X, Z_k) $, for $ k\in\mathbf{z}_{\mathcal{A}^{c}} $, if we use the $ \mathrm{cCCQC} $ to measure nonlinear dependence. Such an intuition is formulated in the following assumption.

Assumption 5   $ \liminf\limits_{p\rightarrow \infty}\left\{\min\limits_{k\in\mathcal{A}}\mathrm{cCCQC}_{T}(X, Z_k) - \max\limits_{k\in\mathcal{A}^{c}}\mathrm{cCCQC}_{T}(X, Z_k)\right\}\geq d_1, $ where $ d_1 $ is a positive constant.

Assumption 5 imposes an assumption on the gap of signal strength between active and inactive features. With Assumption 5, we can easily establish the ranking consistency property for the cCCQC-SIS procedure.

Theorem 2 (Rank Consistency Property) In addition to the Assumptions 1–5, we further assume that $ p = o\left\{\exp\left( an^{t+\frac{1}{2}}\right) \right\} $ for any fixed $ a>0. $ Then

almost surely.

Theorem 2 ensures that the important predictors will be ranked prior to the unimportant ones with an overwhelming probability, if the signals between the important predictors and the unimportant ones are distinguishable. We shall demonstrate the usefulness of these asymptotic properties in Section 3.

3.   Numerical Studies

In this section, we conduct simulations and a real data illustration to evaluate the empirical performance of the proposed cCCQC-based screening method. Our simulation studies are conducted using Matlab code. We compare our screening procedure (cCCQC-SIS) with the following three competitors: the censored rank independence screening [14; CR-SIS], the censored conditional quantile coefficient based sure independence screening [15; CCQ$ _{\tau} $-SIS] and the sure independent ranking and screening procedure for censored regression [16; cSIRS]. We adopt the following three criteria to compare the performance of different independence screening procedures. These three criteria are generally correlated with each other, so we present the results based on only one or two criteria in some cases to conserve space.

The minimal model size which is required to include all truly important covariates. We denote this quantity by $ \mathscr{S} $. If an independent screening procedure has the sure screening property, $ \mathscr{S} $ is expected to be close to the number of truly important predictors. We report the minimum, the first quartile, the median, the third quartile and the maximum number of $ \mathscr{S} $ for each independence screening method out of $ 1000 $ replications.

The selection probability that all active predictors are ranked in either the top $ [n/\log n] $ or $ (n-1) $ positions, where $ [a] $ denotes the integer part of $ a $. We denote this quantity by $ {\mathscr{P}}_A $. This measurement counts the proportion that all truly important predictors are selected out of 1000 replications. If an independence screening procedure has the sure screening property, $ {\mathscr{P}}_A $ is expected to be close to 1.

The selection probability that an individual important predictor is ranked in either of the top $ [n/\log n] $ or $ (n-1) $ positions. We denote this quantity by $ {\mathscr{P}}_S $. It can also be used to assess the sure screening property. In addition, it is helpful to understand which predictors are mostly likely missed by a specific independent screening procedure. We expect the value of $ {\mathscr{P}}_S $ to be close to 1 if an independent screening procedure is able to identify each important covariate.

Example 1   Consider the simple linear model:

The high-dimensional covariates $ \mathbf{z}_i = (Z_{i, 1}, Z_{i, 2}, \cdots, Z_{i, p})^\top $ is generated from a multivariate normal population with mean zero and covariance matrix $ \Sigma = \left(0.8^{|k-k'|}\right)_{p\times p} $. The error term $ \varepsilon_i $ is drawn from the standard normal or standard cauchy distribution. We consider a sample size of $ n =100 $ and set the number of covariates to $ p=1000 $. We take the censoring variable $ C $ to be $ \min(\widetilde{C}, L) $, where $ \widetilde{C} $ is generated from $ Un(1, L+2) $ with $ L $ being the study duration variable, which is chosen to yield a censoring rate of about 30%. We consider two quantile levels $ \tau=0.50 $ and $ \tau=0.75, $ respectively.

It can be seen from Tables 1 and 2 that in most scenarios, our proposed method performs the best for example 1, followed by CCQ$ _{0.50} $-SIS, cSRIS, CCQ$ _{0.75} $-SIS and CR-SIS. However, the difference among them is small. This indicates that cCCQC-SIS, cSRIS, CCQ$ _{\tau} $-SIS and CR-SIS are all capable of detecting the linear relationship.

Table  1.  The quantiles of the minimum model size $ \mathscr{S} $ for Examples 1, 2 and 3

Model	Error	Method	min	$ 25\% $	$ 50\% $	$ 75\% $	95%	99%	max
Model (3.1)	Normal	cCCQC-SIS	5	5	5	5	5	5	16
		CR-SIS	5	5	5	5	21	441	838
		CCQ$ _{0.50} $-SIS	5	5	5	5	5	7	28
		CCQ$ _{0.75} $-SIS	5	5	5	5	5	12	52
		cSRIS	5	5	5	5	5	5	19
	Cauchy	cCCQC-SIS	5	5	5	5	6	27	132
		CR-SIS	5	5	7	87	503	942	997
		CCQ$ _{0.50} $-SIS	5	5	5	5	8	33	136
		CCQ$ _{0.75} $-SIS	5	5	5	7	21	182	782
		cSRIS	5	5	5	5	9	40	167
Model (3.2)	Normal	cCCQC-SIS	8	9	20	49	114	437	938
		CR-SIS	9	141	616	871	959	994	1000
		CCQ$ _{0.40} $-SIS	30	114	274	482	712	921	986
		CCQ$ _{0.50} $-SIS	13	72	150	372	636	905	977
		CCQ$ _{0.75} $-SIS	8	9	19	61	213	845	966
		cSRIS	8	11	29	88	305	910	978
	Cauchy	cCCQC-SIS	8	8	22	56	144	581	945
		CR-SIS	24	161	717	926	979	998	1000
		CCQ$ _{0.40} $-SIS	8	184	405	548	772	961	998
		CCQ$ _{0.75} $-SIS	8	8	28	81	281	801	959
		cSRIS	8	12	32	94	326	922	989
Model (3.3)	Normal	cCCQC-SIS	6	6	10	26	60	404	845
		CR-SIS	7	18	166	658	909	991	1000
		CCQ$ _{0.50} $-SIS	6	8	19	61	151	493	861
		CCQ$ _{0.75} $-SIS	6	6	13	60	210	614	969
		cSRIS	6	6	18	44	89	453	890
	Cauchy	cCCQC-SIS	6	6	21	50	133	446	958
		CR-SIS	6	12	101	553	895	992	1000
		CCQ$ _{0.50} $-SIS	6	11	48	105	275	674	987
		CCQ$ _{0.75} $-SIS	6	7	23	106	321	740	956
		cSRIS	6	9	39	90	117	522	980

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Table  2.  The empirical probabilities $ {\mathscr{P}}_S $ and $ {\mathscr{P}}_A $ for Example 1

			$ {\mathscr{P}}_S $					$ {\mathscr{P}}_A $
Model Size	Error	Method	$ X_1 $	$ X_2 $	$ X_3 $	$ X_4 $	$ X_5 $
$ (n-1) $	Normal	cCCQC-SIS	1.00	1.00	1.00	1.00	1.00	1.00
		CR-SIS	1.00	1.00	0.98	0.95	0.92	0.88
		CCQ$ _{0.50} $-SIS	1.00	1.00	1.00	1.00	1.00	1.00
		CCQ$ _{0.75} $-SIS	1.00	1.00	1.00	1.00	1.00	1.00
		cSRIS	1.00	1.00	1.00	1.00	1.00	1.00
	Cauchy	cCCQC-SIS	1.00	1.00	1.00	1.00	0.96	0.96
		CR-SIS	1.00	0.95	0.90	0.86	0.80	0.71
		CCQ$ _{0.50} $-SIS	1.00	1.00	1.00	0.97	0.94	0.89
		CCQ$ _{0.75} $-SIS	1.00	1.00	0.98	0.93	0.86	0.78
		cSRIS	1.00	1.00	1.00	0.97	0.93	0.91

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Example 2   Consider the following linear model with heterogeneity:

The covariates and the error term are simulated as in model (7). The $ L $ is also chosen to yield a censoring rate of about 30%. However, to accommodate the heterogeneity, we consider two quantile levels $ \tau=0.40 $ and $ \tau=0.75, $ respectively.

It can be clearly seen from Tables 1 and 3 that the proposed cCCQC-SIS performs the best for Example 2. In particular, Table 3 indicates that our proposal can detect the heteroscedasitic errors with an over-whelming probability. As expected, cCCQC-SIS is more stable than CCQ$ _{\tau} $-SIS in that the former takes advantage of all useful information across quantiles. Also, the CR-SIS and cSRIS have unsatisfactory performance in this example due to the heterogeneity.

Table  3.  The empirical probabilities $ {\mathscr{P}}_S $ and $ {\mathscr{P}}_A $ for Example 2

			$ {\mathscr{P}}_S $								$ {\mathscr{P}}_A $
Model Size	Error	Method	$ X_1 $	$ X_2 $	$ X_3 $	$ X_4 $	$ X_5 $	$ X_6 $	$ X_7 $	$ X_8 $
$ (n-1) $	Normal	cCCQC-SIS	1.00	1.00	1.00	1.00	1.00	0.98	0.91	0.80	0.74
		CR-SIS	0.69	0.76	0.73	0.58	0.38	0.19	0.14	0.11	0.04
		CCQ$ _{0.40} $-SIS	1.00	1.00	0.99	0.95	0.64	0.31	0.24	0.25	0.05
		CCQ$ _{0.50} $-SIS	1.00	1.00	0.98	0.97	0.73	0.49	0.46	0.38	0.16
		CCQ$ _{0.75} $-SIS	0.98	1.00	1.00	0.99	0.96	0.90	0.80	0.68	0.63
		cSRIS	1.00	1.00	1.00	1.00	0.97	0.95	0.88	0.75	0.64
	Cauchy	cCCQC-SIS	0.98	0.98	0.98	0.96	0.95	0.95	0.94	0.88	0.70
		CR-SIS	0.52	0.55	0.48	0.35	0.27	0.18	0.09	0.08	0.03
		CCQ$ _{0.40} $-SIS	0.98	0.98	0.95	0.80	0.43	0.27	0.28	0.32	0.03
		CCQ$ _{0.50} $-SIS	0.94	0.95	0.92	0.86	0.47	0.35	0.32	0.36	0.09
		CCQ$ _{0.75} $-SIS	0.89	0.90	0.91	0.90	0.87	0.89	0.74	0.63	0.53
		cSRIS	0.97	0.96	0.97	0.94	0.94	0.92	0.87	0.72	0.57

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Example 3   Consider the following nonlinear model including a three-way interaction term:

We keep the rest of the set-up the same as in model (7). From Tables 1 and 4, it is evident that the proposed cCCQC-SIS performs best for Example 3 in comparison with the existing counterparts. However, the differences among them are substantial. This indicates that compared to the existing choices, cCCQC-SIS has an excellent capability of identifying the interactions.

Table  4.  The empirical probabilities $ {\mathscr{P}}_S $ and $ {\mathscr{P}}_A $ for Example 2 with $ p=4000 $, cauchy error, model size $ n-1 $ and the active covariates spread out

	$ {\mathscr{P}}_S $								$ {\mathscr{P}}_A $
Method	$ X_{2001} $	$ X_{2002} $	$ X_{2003} $	$ X_{2004} $	$ X_{2005} $	$ X_{2006} $	$ X_{2007} $	$ X_{2008} $
cCCQC-SIS	0.95	0.97	0.96	0.95	0.91	0.92	0.90	0.89	0.76
CR-SIS	0.50	0.52	0.49	0.37	0.23	0.22	0.14	0.13	0.06
CCQ$ _{0.40} $-SIS	0.96	0.95	0.96	0.86	0.45	0.30	0.29	0.34	0.11
CCQ$ _{0.50} $-SIS	0.95	0.95	0.94	0.89	0.46	0.37	0.36	0.40	0.18
CCQ$ _{0.75} $-SIS	0.91	0.92	0.92	0.93	0.88	0.90	0.78	0.66	0.59
cSRIS	0.96	0.94	0.96	0.95	0.90	0.91	0.89	0.75	0.65

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Example 4   Upon the suggestion of a reviewer, we reconsider the model (9) using $ \log(Y_i) $ instead of $ Y_i $. A small $ \mathscr{S} $ tends to be associated with high proportions for $ {\mathscr{P}}_A $ and $ {\mathscr{P}}_S $. So we present the results based on the criterion $ \mathscr{S} $ in this example to conserve space. From the simulation results summarized in Tables 4–6, we can draw similar conclusions to Example 1.

Table  5.  The empirical probabilities $ {\mathscr{P}}_S $ and $ {\mathscr{P}}_A $ for Example 2 with the heavy censoring case, normal error and model size $ n-1 $

	$ {\mathscr{P}}_S $								$ {\mathscr{P}}_A $
Method	$ X_1 $	$ X_2 $	$ X_3 $	$ X_4 $	$ X_5 $	$ X_6 $	$ X_7 $	$ X_8 $
cCCQC-SIS	0.89	0.90	0.86	0.91	0.94	0.86	0.84	0.77	0.62
CR-SIS	0.54	0.60	0.61	0.49	0.30	0.13	0.09	0.08	0.01
CCQ$ _{0.40} $-SIS	0.88	0.87	0.85	0.90	0.52	0.20	0.15	0.13	0.02
CCQ$ _{0.50} $-SIS	0.90	0.92	0.86	0.90	0.64	0.40	0.37	0.28	0.10
CCQ$ _{0.75} $-SIS	0.87	0.89	0.85	0.89	0.92	0.82	0.63	0.56	0.41
cSRIS	0.81	0.85	0.83	0.88	0.90	0.81	0.76	0.63	0.49

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Table  6.  The empirical probabilities $ {\mathscr{P}}_S $ and $ {\mathscr{P}}_A $ for Example 3

			$ {\mathscr{P}}_S $						$ {\mathscr{P}}_A $
Model Size	Error	Method	$ X_1 $	$ X_2 $	$ X_3 $	$ X_4 $	$ X_5 $	$ X_6 $
$ [n/\log n] $	Normal	cCCQC-SIS	0.59	0.83	0.84	0.87	0.92	0.88	0.42
		CR-SIS	0.26	0.43	0.46	0.44	0.23	0.18	0.06
		CCQ$ _{0.50} $-SIS	0.49	0.70	0.68	0.68	0.82	0.68	0.26
		CCQ$ _{0.75} $-SIS	0.43	0.60	0.69	0.70	0.87	0.70	0.34
		cSRIS	0.44	0.57	0.63	0.69	0.89	0.68	0.30
	Cauchy	cCCQC-SIS	0.44	0.75	0.76	0.70	0.82	0.77	0.31
		CR-SIS	0.37	0.45	0.41	0.42	0.29	0.15	0.11
		CCQ$ _{0.50} $-SIS	0.40	0.60	0.56	0.52	0.65	0.45	0.13
		CCQ$ _{0.75} $-SIS	0.39	0.53	0.55	0.60	0.78	0.65	0.23
		cSRIS	0.36	0.52	0.54	0.57	0.73	0.60	0.19
$ (n-1) $	Normal	cCCQC-SIS	0.87	0.96	0.93	0.96	1.00	0.98	0.82
		CR-SIS	0.42	0.62	0.63	0.62	0.47	0.38	0.18
		CCQ$ _{0.50} $-SIS	0.78	0.90	0.89	0.91	0.97	0.92	0.60
		CCQ$ _{0.75} $-SIS	0.65	0.82	0.86	0.89	0.99	0.96	0.56
		cSRIS	0.66	0.80	0.83	0.85	0.98	0.94	0.52
	Cauchy	cCCQC-SIS	0.76	0.93	0.89	0.95	0.96	0.94	0.69
		CR-SIS	0.47	0.60	0.60	0.55	0.41	0.29	0.24
		CCQ$ _{0.50} $-SIS	0.60	0.80	0.81	0.83	0.92	0.87	0.48
		CCQ$ _{0.75} $-SIS	0.57	0.76	0.79	0.85	0.94	0.90	0.47
		cSRIS	0.62	0.83	0.82	0.84	0.90	0.81	0.48

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Table  7.  The quantiles of the minimum model size $ \mathscr{S} $ for Example 4

Error	Method	min	$ 25% $	$ 50% $	$ 75% $	95%	99%	max
Normal	cCCQC-SIS	6	6	6	12	27	116	622
	CR-SIS	6	8	55	129	420	604	1000
	CCQ$ _{0.50} $-SIS	6	6	10	35	86	216	813
	CCQ$ _{0.75} $-SIS	6	6	7	21	71	189	916
	cSRIS	6	6	11	39	95	289	848
Cauchy	cCCQC-SIS	6	6	7	17	38	224	889
	CR-SIS	6	10	65	198	476	752	1000
	CCQ$ _{0.50} $-SIS	6	8	20	46	98	314	933
	CCQ$ _{0.75} $-SIS	6	8	15	40	85	233	908
	cSRIS	6	8	24	56	102	335	950

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Example 5   As an illustration, we apply the proposed screening method to the analysis of microarray diffuse large-B-cell lymphoma (DLBCL) data of ^[31]. The DLBCL is one of the most common types of lymphoma in adults of United States. However, the survival rate after the standard chemotherapy is only about 35 to 40%. Thus it is of interest in studying how the survival rate depends on an individual's gene information The outcome in the study was the survival variable of $ n $ = 240 DLBCL patients after chemotherapy. Measurements of $ p $ = 7399 genes obtained from cDNA microarrays for each individual patient were the predictors. Given such a large number of predictors and small sample size, feature screening seems a necessary initial step as a prelude to any other sophisticated statistical modeling that does not cope well with such high dimensionality.

Table  8.  The p-values of the log-rank test for Example 5 with several combinations of $ (n_1, n_2) $

$ (n_1, n_2) $	cCCQC-SIS	CR-SIS	CCQ$ _{0.50} $-SIS	CCQ$ _{0.75} $-SIS	cSRIS
$ (120, 120) $	0.001	0.034	0.120	0.060	0.021
$ (180, 60) $	0.003	0.019	0.082	0.113	0.009
$ (80, 160) $	0.004	0.134	0.107	0.085	0.115

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In this data set, all gene expression levels are standardized to have mean zero and standard deviation one during the exploratory data analysis. We split these data set into a training set with $ n_1 $ subjects and a test setwith $ n_2 $ subjects. Here $ n_1+n_2=240. $ We first apply the screening procedures to the training data set, and retain $ [n_1/\log n_1] $ covariates during this screening stage. Considering that some truly unimportant covariates are also retained in the screening stage, we next perform the lasso penalization to further remove those irrelevant covariates. We then build an un-penalized Cox proportional hazards model using the selected genes. We next apply the log-rank test to compare the prediction power of different screening methods. Table 6 describes the Kaplan–Meier estimate of survival curves for the two risk groups of patients in the testing data with the log-rank test yielding different p-values. These results indicate our good prediction of the fitted model.

Appdendix A.   Proof of Theorem 1

The following Lemma paves the road for proving Theorem 1. Lemma 3 is the modified version of Lemma S1 of [15]. Hence, the details are omitted here and a detailed technical report is available from the author.

Lemma 3   Let $ \mathcal{F} $ be a class of distribution functions whose support is the same as that of $ F $, and let $ \mathcal{Y} $ be the support of $ Y $. For any $ \varepsilon>0 $, let

where $ Y^* $ follows the distribution $ F^* $. Then

where the constant $ c_{01} $ is independent of $ \tau $.

Proof of Theorem 1   Define

Simple calculations yield

where

and

with $ h_1 $ and $ h_2 $ being the kernels of the U-statistics. Likewise, we have

where $ \widehat{R}_{k1} $ is obtained by replacing $ F $ and $ Q_{\tau}(Y) $ in $ \widetilde{R}_{k1} $ with $ \widehat{F}_{n} $ and $ \widehat{F}_{n}^{-1}(\tau) $, respectively, and similarly for $ \widehat{R}_{k2}. $

Due to the fact $ I(\cdot) $ is uniformly bounded, simple calculations yield

and

Under Assumptions (A.2), (A.3) and (A.5), we have $ \left| \left| \widehat{F}_{n}-F\right| \right| _{\infty}=O\left( n^{-1/2}\left( \log(n)\right) ^{1/2}\right) $ and $ \left| \left| \widehat{F}_{n}^{-1}-Q_{\tau}(Y)\right| \right| _{\infty}=O(n^{-1/2}\left( \log(n))^{1/2}\right) $ almost surely via invoking Lemma 8.4 in [23].

Employing arguments similar to those for dealing with (S11)–(S13) in [15] and combining Lemma 3, we have that there exists a positive constant $ c_1, c_2 $, $ c_3 $, $ c_4 $ and $ c_5 $ such that

Using the result above, we get

On the other hand, invoking the proof of Theorem 2 in [32], we can directly use the theory of U-statistics to establish asymptotic property of $ \mathrm{\widehat{cCCQC}}_{T}(Y, X_k) $. Our following arguments are exactly parallel to those used in the proof of Theorem 2 of ^[32] with a slight modification. Hence, the details are omitted here and a detailed technical report is available from the author. In other words, it is easy to show that there exists a sufficiently small constant $ s_n\in(0, 2n^{\frac{1}{2}-t}) $ such that

Combining (A.6) and (A.7) leads to the desired result.

Appdendix B.   Proof of Theorem 2

Denote $ w_k=\mathrm{cCCQC}(X, Z_k) $ and $ \widehat{w}_k=\mathrm{\widehat{cCCQC}}(X, Z_k). $ The proof of Theorem 2 follows the proofs of Theorem 2.2 in ^[33].

Noting that $ p=o\left( \exp(an^{t+\frac{1}{2}})\right) $, then we have that for some sufficiently large $ N $,

Hence, using Borel-Contelli Lemma leads to the desired result.

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