Inference for High-Dimensional Censored Quantile Regression

Abstract

With the availability of high dimensional genetic biomarkers, it is of interest to identify heterogeneous effects of these predictors on patients’ survival, along with proper statistical inference. Censored quantile regression has emerged as a powerful tool for detecting heterogeneous effects of covariates on survival outcomes. To our knowledge, there is little work available to draw inference on the effects of high dimensional predictors for censored quantile regression. This paper proposes a novel procedure to draw inference on all predictors within the framework of global censored quantile regression, which investigates covariate-response associations over an interval of quantile levels, instead of a few discrete values. The proposed estimator combines a sequence of low dimensional model estimates that are based on multi-sample splittings and variable selection. We show that, under some regularity conditions, the estimator is consistent and asymptotically follows a Gaussian process indexed by the quantile level. Simulation studies indicate that our procedure can properly quantify the uncertainty of the estimates in high dimensional settings. We apply our method to analyze the heterogeneous effects of SNPs residing in lung cancer pathways on patients’ survival, using the Boston Lung Cancer Survivor Cohort, a cancer epidemiology study on the molecular mechanism of lung cancer.

1. Introduction

Lung cancer presents much heterogeneity in etiology (McKay et al., 2017; Dong et al., 2012; Huang et al., 2009), and some genetic variants may insert different impacts on different quantile levels of survival time. For example, in the Boston Lung Cancer Survivor Cohort (Christiani, 2017), a cancer epidemiology cohort of over 11,000 lung cancer cases enrolled in the Boston area since 1992, it was found that SNP AX.37793583 (rs115952579), along with age, gender, cancer stage and smoking status, had heterogeneous effects on different quantiles of survival time. A total of 674 patients in the study were genotyped, with the goal of identifying lung cancer survival-predictive SNPs. Target gene approaches, which focus on SNPs residing in cancer-related gene pathways, are appealing for increased statistical power in detecting significant SNPs (Moon et al., 2003; Risch and Plass, 2008; Ho et al., 2019), and the investigators have identified SNPs residing in 14 well-known lung cancer-related genes (Zhu et al., 2017; Korpanty et al., 2014; Yamamoto et al., 2008; Kelley et al., 2001). One goal was to investigate whether and how each SNP might play a different role among the high-risk (i.e. lower quantiles of overall survival) and low-risk (i.e. higher quantiles of overall survival) cancer survivors.

Quantile regression (QR) (Koenker and Bassett Jr, 1978) is a significant extension of classic linear regression. By permitting the effects of active variables to vary across quantile levels, quantile regression can naturally accommodate and examine the heterogeneous impacts of biomarkers on different segments of the response variable’s conditional distribution. As survival data are subject to censoring and may be incomplete, QR methods developed for complete data may be unsuitable. Efforts have been devoted to developing censored quantile regression (CQR) (Powell, 1986; Portnoy, 2003; Peng and Huang, 2008, among others), which has become a useful alternative strategy to traditional survival models, such as the Cox model and accelerated failure time model. QR has also been widely studied to accommodate high dimensional predictors. For example, Wang et al. (2012) dealt with variable selection using non-convex penalization; Zheng et al. (2013) proposed an adaptive penalized quantile regression estimator that can select the true sparse model with high probability; and Fan et al. (2014) studied the penalized quantile regression with a weighted L₁ penalty in an ultra-high dimensional setting. As to high dimensional CQR (HDCQR), He et al. (2013) provided a model-free variable screening procedure for ultra-high dimensional covariates, and Zheng et al. (2018) proposed a penalized HDCQR built upon a stochastic integral based estimating equation. However, most of the existing works in HDCQR were designed to select a subset of predictors and estimate the effects of the selected variables, instead of drawing inference on high dimensional predictors.

Progress in high dimensional inferences has been made for linear and non-linear models (Zhang and Zhang, 2014; Bühlmann et al., 2014; Javanmard and Montanari, 2014; Ning and Liu, 2017; Fei et al., 2019, among others). For example, Meinshausen et al. (2009) proposed to aggregate p-values from multi-sample splittings for high dimensional linear regression. Another line of works referred to as post-selection inference includes Berk et al. (2013), Lee et al. (2016), and Belloni et al. (2019), which recently provided post-selection inference at fixed quantiles for complete data. However, these methods may not handle censored outcomes. For censored median regression, Shows et al. (2010) provided sparse estimation and inference, but it cannot handle high dimensional data.

We propose to draw inference on high dimensional HDCQR based on a splitting and fusing scheme, termed Fused-HDCQR. Utilizing a variable selection procedure for HDCQR such as Zheng et al. (2018), our method operates partial regression followed by smoothing. Specifically, partial regression allows us to estimate the effect of each predictor, regardless whether it is chosen by variable selection or not. The fused estimator aggregates the estimates based on multiple data-splittings and variable selection, with a variance estimator derived by the functional delta method (Efron, 2014; Wager and Athey, 2018). To comprehensively assess the covariate effects on the survival distribution, we adopt a “global” quantile model (Zheng et al., 2015) with the quantile level being over an interval, instead of the local CQR that focuses only on a few pre-specified quantile levels. The global quantile model may indeed address the molecular mechanism of lung cancer, our motivating disease, that hypothesizes that some genetic variants may cause heterogeneous impacts on different but unspecified segments of survival distribution (McKay et al., 2017; Dong et al., 2012; Huang et al., 2009).

Our work presents several advantages. First, compared to high dimensional Cox models (Zhao and Li, 2012; Fang et al., 2017; Kong et al., 2018), the employed HDCQR stems from the accelerated failure time model (Wei, 1992) and offers straightforward interpretations (Hong et al., 2019). Second, utilizing the global conditional quantile regression, it uses various segments of the conditional survival distribution to improve the robustness of variable selection and capture global sparsity. Third, our splitting-and-averaging scheme avoids the technicalities of estimating the precision matrix by inverting the p × p Hessian matrix of the log likelihood, which is a major challenge for debiased-LASSO type methods (Zhang and Zhang, 2014; Van de Geer et al., 2014) and is even more so if we apply debiased-LASSO to the CQR setting. Finally, as opposed to post-selection inferences (Belloni et al., 2019, among others), Fused-HDCQR accounts for variations in model selection and draws inference for all of the predictors.

The rest of the paper is organized as follows. Section 2 introduces the method, and Section 3 details the asymptotic properties. Section 4 derives a non-parametric variance estimator, Section 5 conducts simulation studies, and Section 6 applies the proposed method to analyze the BLCSC data. The technical details, such as proofs and additional lemmas, are relegated to the online Supplementary Material.

2. Model and Method

2.1. High dimensional censored quantile regression

Let T and C denote the survival outcome and censoring time, respectively. We assume that C is independent of T given $\tilde{\mathbf{Z}}$, a (p − 1)-dimensional vector of covariates (p > 1). Let X = min{T, C}, Δ = 1{T ≤ C}, and $\mathbf{Z}={\left(1,{\tilde{\mathbf{Z}}}^{\text{T}}\right)}^{\text{T}}$, where 1{·} is the binary indicator function. The observed data, D⁽ⁿ⁾ = {(X_i,Δ_i,Z_i), i = 1, . . . , n}, are n i.i.d. copies of (X,Δ,Z). With Y = log T, let Q_Y (τ|Z) = inf{t : P(Y ≤ t|Z) ≥ τ} be the τ-th conditional quantile of Y given Z. A global censored quantile regression model stipulates

$$Q_Y(\tau \mid \mathbf{Z})={\mathbf{Z}}^{\text{T}}\mathit{\beta }*(\tau ),\tau \in (0,1),$$ (1)

where β*(τ) is a p-dimensional vector of coefficients at τ. We aim to draw inference on ${\mathit{\beta }}_j^{*}(\tau )$ for each τ ∈ (0, τ_U] and for all j ∈ {1, . . . , p}, where 0 < τ_U < 1 is an upper bound for estimable quantiles subject to identifiability constraint caused by censoring (Peng and Huang, 2008).

Let N(t) = 1{logX ≤ t, Δ = 1}, Λ_T (t|Z) = −log(1 − P(log T ≤ t|Z)), and H(u) = −log(1 − u). Then, M(t) = N(t) − Λ_T (t ∧ logX|Z) is a Martingale process under model (1) (Fleming and Harrington, 2011) and hence E(M(t)|Z) = 0. We use N_i(t) and M_i(t), i = 1, . . . , n, to denote the sample analogs of N(t) and M(t). Let ${\theta }_i(\tau )={\mathbf{Z}}_i^{\text{T}}\mathit{\beta }(\tau )$ and

$${\mathbf{U}}_n(\mathit{\beta },\tau )=n^{-1}\sum _{i=1}^n{\mathbf{Z}}_i\left\{N_i\left({\theta }_i(\tau )\right)-{\int }_0^{\tau }1\left\{\log X_i\ge {\theta }_i(u)\right\}dH(u)\right\}.$$

We denote the expectation of U_n(β, τ) by u(β, τ).

The Martingale property implies u(β*, τ) = 0 with τ ∈ [0, τ_U], entailing the estimating equation with τ ∈ (0, τ_U]:

$$n^{1/2}{\mathbf{U}}_n(\mathit{\beta },\tau )=n^{-1/2}\sum _{i=1}^n{\mathbf{Z}}_i\left\{N_i\left({\theta }_i(\tau )\right)-{\int }_0^{\tau }1\left\{\log X_i\ge {\theta }_i(u)\right\}dH(u)\right\}=0.$$ (2)

The stochastic integral in (2) naturally suggests sequential estimation with respect to τ. We define a grid of quantile values Γ_m = {τ₀, τ₁, . . . , τ_m} to cover the interval [ν, τ_U], where τ₀ = ν and τ_m = τ_U. The assumption on the lower bound ν > 0 is made to circumvent the singularity problem with CQR at τ = 0, as detailed in assumption (A1). In practice, ν is chosen such that only a small proportion of observations are censored below the ν-th quantile.

Then, $\stackrel{\prime }{\mathit{\beta }}\left({\tau }_k\right)$’s, the estimates of β(τ_k)’s, τ_k ∈ Γ_m can be sequentially obtained by solving

$$n^{-1/2}\sum _{i=1}^n{\mathbf{Z}}_i\left(N_i\left({\theta }_i\left({\tau }_k\right)\right)-\sum _{r=0}^{k-1}{\int }_{{\tau }_r}^{{\tau }_{r+1}}1\left\{\log X_i\ge {\stackrel{\prime }{\theta }}_i\left({\tau }_r\right)\right\}dH(u)\right)=0,$$

where ${\stackrel{\prime }{\theta }}_i\left({\tau }_k\right)={\mathbf{Z}}_i^{\text{T}}\stackrel{\prime }{\mathit{\beta }}\left({\tau }_k\right)$. Due to the monotonicity of θ_i(τ) in τ, $\stackrel{\prime }{\mathit{\beta }}\left({\tau }_k\right)$ can be solved efficiently via L₁-minimization. And $\stackrel{\prime }{\mathit{\beta }}\left(\tau \right)$, τ ∈ [ν, τ_U] is defined as a right-continuous piece-wise constant function that only jumps at the grid points. It can be shown that $\stackrel{\prime }{\mathit{\beta }}\left(\tau \right)$ is uniformly consistent and converges weakly to a mean zero Gaussian process for τ ∈ [ν, τ_U] when p = o(n). More importantly, $\stackrel{\prime }{\mathit{\beta }}\left(\tau \right)$ provides a comprehensive understanding of the covariate effects on the conditional survival distribution over the quantile interval [ν, τ_U]. We incorporate this sequential estimating procedure for low dimensional CQR estimation in our proposed method.

In addition, our method requires dimension reduction, which can be accomplished by existing methods, including the screening method proposed by He et al. (2013) and the penalized estimation and selection procedure developed by Zheng et al. (2018). Specifically, Zheng et al. (2018) incorporated an L₁ penalty into the stochastic integral based estimating equation in (2) to obtain an L-HDCQR estimator, which achieves a uniform convergence rate of $\sqrt{q\log (p\vee n)/n}$, and results in “sure screening” variable selection with high probability, where q is defined in condition (A4). Zheng et al. (2018) also proposed an AL-HDCQR estimator by employing the Adaptive Lasso penalties, which attains a uniform convergence rate of $\sqrt{q\log (n)/n}$ and selection consistency.

2.2. Fused-HDCQR estimator

Our proposed Fused-HDCQR procedure consists of multiple data splitting, selecting variables, fitting low dimensional CQRs with partitioned data, applying append-and-estimate to all predictors, and aggregating those estimates.

1. With the full data D⁽ⁿ⁾, determine via cross-validation the tuning parameter(s) λ_n of $\mathcal{S}$, an HDCQR variable selection method.

2. Let B be a large positive number. For each b = 1, . . . ,B,

   1. randomly split the data into equal halves $D_1^b$ and $D_2^b$;
   2. on $D_1^b$, apply the selection procedure $\mathcal{S}$ with λ_n on [ν, τ_U], to select a subset of predictors, denoted by ${\widehat{S}}_{{\lambda }_n}^b$, or ${\widehat{S}}^b$ for short;
   3. on $D_2^b$, for each j = 1, . . . , p, fit the partial CQR using the subset of covariates ${\widehat{S}}_{+j}^b=\left\{j\right\}\cup {\widehat{S}}^b$, and denote the estimator by ${\tilde{\mathit{\beta }}}_{{\widehat{S}}_{+j}^b}(\tau )$, τ ∈ [ν, τ_U]. ${\tilde{\mathit{\beta }}}_{{\widehat{S}}_{+j}^b}(\tau )$ is a right-continuous piecewise-constant function that only jumps at the grid points at τ_k ∈ Γ_m;
   4. denote the entry in ${\tilde{\mathit{\beta }}}_{{\widehat{S}}_{+j}^b}(\tau )$ corresponding to Z_j by ${\tilde{\mathit{\beta }}}_j^b(\tau )={\left({\tilde{\mathit{\beta }}}_{{\widehat{S}}_{+j}^b}(\tau )\right)}_j$.

3. Fusing: the final estimator of ${\beta }_j^{*}(\tau )$, τ ∈ [ν, τ_U], j = 1, . . . , p is

$${\widehat{\beta }}_j(\tau )=\frac{1}{B}\sum _{b=1}^B{\tilde{\beta }}_j^b(\tau ).$$ (3)

Remark 1.

We could select different tuning parameters for $\mathcal{S}$ in each data split, but with much added computation. Our numerical evidence seemed to suggest that a globally chosen λ_n work well.

Remark 2.

Our procedure needs a variable selection procedure to reduce dimension. For example, L-HDCQR selects the subset $\left\{j\in \{2,\cdots ,p\}:{\max }_k\left|{\widehat{\gamma }}_j\left({\tau }_k\right)\right|>a_0,{\tau }_k\in {\mathrm{\Gamma }}_m\right\}$, where ${\widehat{\gamma }}_j\left({\tau }_k\right)$‘s are the L-HDCQR estimates, and a₀ > 0 is a predetermined threshold. We start j with 2 as the intercept term (corresponding to j = 1) is always included in the model. In regards to the choice of variable selection methods, based on our experience, we can adopt the screening method in He et al. (2013) for fast computation, use L-HDCQR for detecting any non-zero effects in the quantile interval [ν, τ_U], and choose AL-HDCQR if we opt to select fewer predictors.

Remark 3.

With the censored outcomes, we have used the deviance residual to define the K-fold cross-validation criterion as in Zheng et al. (2018) and selected λ_n by minimizing it. Specifically, we partition the data to K folds, and let ${\widehat{\mathit{\beta }}}_{\lambda }^{(-k)}(\tau )$ be the penalized estimate of β(τ) using all of the data excluding the k-th fold with a tuning parameter λ and τ ∈ [ν, τ_U], where k = 1, . . . ,K. Under the global CQR model (1), we define the cross-validation error as

$$\text{CV}\mathrm{Error}(\lambda )=\sum _{k=1}^K\sum _{i\in \text{fold}k}{\int }_{\nu }^{{\tau }_U}|D_i[{\widehat{\mathit{\beta }}}_{\lambda }^{(-k)}(\tau )]|d\tau ,$$ (4)

where

$$D_i[\mathit{\beta }(\tau )]=\mathrm{sign}\left\{M_i(\mathit{\beta }(\tau ))\right\}\sqrt{-2M_i(\mathit{\beta }(\tau ))+{\mathrm{\Delta }}_i\log \left\{{\mathrm{\Delta }}_i-M_i(\mathit{\beta }(\tau ))\right\}}$$

with $M_i(\mathit{\beta }(\tau ))=N_i\left({\mathbf{Z}}_i^T\mathit{\beta }(\tau )\right)-{\int }_{\nu }^{\tau }1\left\{\log X_i\ge N_i\left({\mathbf{Z}}_i^T\mathbf{\beta }(u)\right)\right\}dH(u)-\nu$. Here, H(u) = −log(1 − u), N_i(·) is the counting process, and M_i(β(τ)) is the Martingale residual under model (1) (Zheng et al., 2018).

3. Theoretical Studies

3.1. Notation and regularity conditions

For any vector δ ∈ R^p and a subset S ⊂ {1, . . . , p}, denote by S^C the complementary set and define ‖δ‖_r,S = ‖δ_S‖_r, the l_r-norm of the sub-vector δ_S, in which δ_jS = δ_j if j ∈ S and δ_jS = 0 if j ∉ S. We set the following conditions.

(A1) There exists a quantile ν and some constant c > 0 such that

$$n^{-1}\sum _{i=1}^{n}1\left\{\log C_i\le {\mathbf{Z}}_i^{\text{T}}\mathit{\beta }*(\nu )\right\}\left(1-{\mathrm{\Delta }}_i\right)\le c{n}^{-1/2}$$

holds for sufficiently large n.

(A2) (Bounded observations) ‖Z‖_∞ ≤ C₀. Without loss of generality, we assume C₀ = 1. In addition, E|logX| < ∞.

(A3) (Bounded densities) Let F_T (t|Z) = P(log T ≤ t|Z), Λ_T (t|Z) = −log(1 − F_T (t|Z)), F(t|Z) = P(logX ≤ t|Z), and G(t|Z) = P(logX ≤ t,Δ = 1|Z). Also, define f(t|Z) = dF(t|Z)/dt, and g(t|Z) = dG(t|Z)/dt.

• aThere exist constants $\underset{\_}{f}$, $\overline{f}$, $\underset{\_}{g}$ and $\overline{g}$ such that

$$\underset{\_}{f}\le \underset{\mathbf{z},\tau \in \left[\nu ,{\tau }_U\right]}{\inf }f\left({\mathbf{z}}^{\text{T}}\mathit{\beta }*(\tau )\mid \mathbf{z}\right)\le \underset{\mathbf{z},\tau \in \left[\nu ,{\tau }_U\right]}{\sup }f\left({\mathbf{z}}^{\text{T}}\mathit{\beta }*(\tau )\mid \mathbf{z}\right)\le \overline{f},$$

$$\underset{\_}{g}\le \underset{\mathbf{z},\tau \in \left[\nu ,{\tau }_U\right]}{\inf }g\left({\mathbf{z}}^{\text{T}}\mathit{\beta }*(\tau )\mid \mathbf{z}\right)\le \underset{\mathbf{z},\tau \in \left[\nu ,{\tau }_U\right]}{\sup }g\left({\mathbf{z}}^{\text{T}}\mathit{\beta }*(\tau )\mid \mathbf{z}\right)\le \overline{g}.$$

• bThere exist constant κ > 0 and A such that ∀|t| ≤ κ,

$$\underset{\mathbf{z},\tau \in \left[\nu ,{\tau }_U\right]}{\sup }|f\left({\mathbf{z}}^{\text{T}}\mathit{\beta }*(\tau )+t\mid \mathbf{z}\right)-f\left({\mathbf{z}}^{\text{T}}\mathit{\beta }*(\tau )\mid \mathbf{z}\right)|\le A|t|,$$

$$\underset{\mathbf{z},\tau \in \left[\nu ,{\tau }_U\right]}{\sup }|g\left({\mathbf{z}}^{\text{T}}\mathit{\beta }*(\tau )+t\mid \mathbf{z}\right)-g\left({\mathbf{z}}^{\text{T}}\mathit{\beta }*(\tau )\mid \mathbf{z}\right)|\le A|t|.$$

(A4) (Sparsity) Assume log p = o(n^1/2), let

$$S_{\tau }=\left\{j:{\mathit{\beta }}_j^{*}(\tau )\ne 0\right\},S*=\underset{\tau \in \left[\nu ,{\tau }_U\right]}{\cup }{S}_{\tau }=\left\{j:\underset{\tau \in \left[\nu ,{\tau }_U\right]}{\sup }\left|{\mathit{\beta }}_j^{*}(\tau )\right|>0\right\},\text{and}q=\left|S*\right|.$$

Let $\widehat{S}$ be the index set of covariates selected by $\mathcal{S}$ with a tuning parameter λ_n. There exist constants 0 ≤ c₁ < 1/3, c₂, K₁, K₂ > 0 such that $q\le K_1{n}^{c_1}$, $|\widehat{S}|\le K_1{n}^{c_1}$, and

$$\text{P}\left(S*\subseteq \widehat{S}\right)\ge 1-K_2{(p\vee n)}^{-1-c_2}.$$

(A5) Let $\tilde{\mu }\left(\tau \right)=E\left[1\right\{\log X>{\mathbf{Z}}^{\mathbf{T}}\mathit{\beta }*\left(\tau \right)\left\}\right]$. There exists a positive constant L, such that $\left|{\mathit{\beta }}_j^{*}\left({\tau }_1\right)-{\mathit{\beta }}_j^{*}\left({\tau }_2\right)\right|\le L|{\tau }_1-{\tau }_2|$ and $|\tilde{\mu }\left({\tau }_1\right)-\tilde{\mu }\left({\tau }_2\right)|\le L|{\tau }_1-{\tau }_2|$, for all τ₁, τ₂ ∈ (ν, τ_U] and 1 ≤ j ≤ p.

(A6) (Eigenvalues) ${\delta }^{\text{T}}E\left[{\mathbf{Z}}_i{\mathbf{Z}}_i^{\text{T}}\right]\delta /\Vert \delta {\Vert }^2$ is bounded below and above by λ_min and λ_max, respectively, over $\Vert \delta {\Vert }_0\le K_1{n}^{c_1}$, δ ≠ 0, where 0 < λ_min < λ_max. (nonlinear impact)$c_2:={\inf }_{\Vert \delta {\Vert }_0\le K_1{n}^{c_1,\delta \ne \mathbf{0}}}E{\left[{\left({\mathbf{Z}}_i^{\text{T}}\delta \right)}^2\right]}^{3/2}/E\left[{\left|{\mathbf{Z}}_i^{\text{T}}\delta \right|}^3\right]>0$.

(A7) Γ_m is equally gridded with τ_k − τ_k−1 = ϵ_n for τ_k ∈ Γ_m, k = 1, . . . ,m. The grid size satisfies ϵ_n = c₀n⁻¹ for some constant c₀.

Assumption (A1) requires that the number of censored observations below the ν-th quantile does not exceed cn^1/2, which is satisfied if the lower bound of the censoring time C’s support is greater than 0 and seems reasonable in real applications. As recommended in Zheng et al. (2018), ν is chosen such that only a small proportion of the observed survival times below the ν-th quantile are censored. (A2) assumes that the covariates are uniformly bounded. As pointed out by Zheng et al. (2015), the global linear quantile regression model is most meaningful when the covariates are confined to a compact set to avoid crossing of the quantile functions. (A3) ensures the positiveness of f(t|Z) between Z^Tβ*(ν) and Z^Tβ*(τ_U), which is essential for the identifiability of β*(τ) for τ < τ_U. (A4) restricts the order of data dimensions, as well as the sparsity of β*(τ), which is necessary for the convergence of the low dimensional estimator in (2) (Condition C4 in Wang et al. (2012)). (A4) also characterizes the “sure screening” property by S. The asymptotic property does not assess the variability of selection with a finite sample. For high dimensional inference, it is crucial to account for such variability (Fei et al., 2019). Specifically, several variable selection methods for high dimensional CQR satisfy the sure screening property in (A4) with additional mild conditions.

• L-HDCQR: by Corollary 4.1 of Zheng et al. (2018), a Beta-min condition is required in addition to the set of conditions in this paper. Explicitly, there exist constants C₁, C₂ > 0, such that

$$\underset{j\in S*}{\inf }\underset{\tau \in \left[{\tau }_L,{\tau }_U\right]}{\sup }\left|{\mathit{\beta }}_j^{*}(\tau )\right|>C_1\exp \left(C_{2}q{\tau }_U\right)\sqrt{q\log (p\vee n)/n}+L\sqrt{q}{ϵ}_n.$$

• AL-HDCQR: by Corollary 4.2 of Zheng et al. (2018), AL-HDCQR achieves the stronger selection consistency property, which implies the sure screening property.

• Quantile-adaptive Screening: by Theorem 3.3 of He et al. (2013), with a proper threshold value in their technical conditions, the screening procedure achieves the sure screening property.

(A5) characterizes the smoothness of β*(τ). (A6) is analogous to the assumptions on the covariance structure in the high dimensional literature (Zhao and Yu, 2006; Belloni and Chernozhukov, 2011; Fan et al., 2014; Van de Geer et al., 2014). As an extension to Condition C4 in Peng and Huang (2008), it ensures the convergence of low dimensional CQR but with a diverging number of covariates. (A7) details the fineness of Γ_m, which renders an adequate approximation to the stochastic integration in (2).

3.2. Theoretical properties of Fused-HDCQR

We first extend the results in Peng and Huang (2008) from a fixed p to a p-diverges-butless-than-n case. They are novel and critical extensions since we allow the true model size q = |S*| to increase with n, while the selected ${\widehat{S}}^b$’s in the fused procedure vary around S*. Specifically, we assume a subset S ⊂ {1, . . . , p} in Theorems 1 and 2, where $|S|\le K_1{n}^{c_1}$, 0 ≤ c₁ < 1/3 and K₁ > 0. Let ${\stackrel{\prime }{\mathit{\beta }}}_S(\tau )$, τ ∈ [ν, τ_U] be the estimator from Peng and Huang (2008) of fitting the CQR with Z_S over the τ-grid Γ_m.

Theorem 1.

(Consistency with a diverging number of predictors) Under Conditions (A1) – (A7) and given a subset S ⊂ {1, · · · , p} such that S* ⊆ S and $|S|\le K_1{n}^{c_1}$, there exist positive constants ζ₁ and ζ₂ such that

$$\underset{\nu \le \tau \le {\tau }_U}{\sup }\Vert {\stackrel{\prime }{\mathit{\beta }}}_S(\tau )-{\mathit{\beta }}^{*}(\tau )\Vert \le {\zeta }_1\exp ({\zeta }_2)(K_1{n}^{c_1-1}\log n{\text{)}}^{\text{1/2}}$$

with probability at least $1-20c_0^{-2}{K}_1{n}^{c_1-2}$.

Remark 4.

From the proofs of Propositions 1 and 2, it can be seen that ζ₁ and ζ₂ do not depend on the choice of S or n. Thus, ζ₁ and ζ₂ are universal for all possible S satisfying S* ⊆ S and $|S|\le K_1{n}^{c_1}$.

Next, we derive the weak convergence of ${\stackrel{\text{'}}{\mathit{\beta }}}_j$ for any j ∈ S.

Theorem 2.

(Weak convergence with a diverging number of covariates) Suppose Conditions (A1) – (A7) hold. Given a S ⊂ {1, · · · , p} such that S* ⊆ S and $|S|\le K_1{n}^{c_1}$, for any j ∈ S,

$$\sqrt{n}\left({\stackrel{\prime }{\mathit{\beta }}}_j(\tau )-{\mathit{\beta }}_j^{*}(\tau )\right)$$

converges weakly to a mean zero Gaussian process for τ ∈ [ν, τ_U].

In high dimensional settings, the next theorem shows that the fused estimator enjoys desirable theoretical properties.

Theorem 3.

Consider the Fused-HDCQR estimator in (3). Under assumptions (A1) – (A7), for any j ∈ {1, . . . , p},

$$\sqrt{n}\left({\widehat{\beta }}_j(\tau )-{\mathit{\beta }}_j^{*}(\tau )\right)$$

converges weakly to a mean zero Gaussian process for τ ∈ [ν, τ_U].

Our framework enables us to obtain the joint distribution of K-dimensional estimated coefficients, where K is a finite number. Let $\mathcal{K}$ be the collection of the indices of K covariates of interest. We can show that the weak convergence result of ${\stackrel{\prime }{\mathit{\beta }}}_{\mathcal{K}}(\tau )$, a K-dimensional subvector of the oracle estimator, still holds for τ ∈ [ν, τ_U], that is, $\sqrt{n}({\stackrel{\prime }{\mathit{\beta }}}_{\mathcal{K}}(\tau )-{\mathit{\beta }}_{\mathcal{K}}^{*}(\tau ))$, τ ∈ [ν, τ_U] converges to a K-dimensional Gaussian distribution at any τ ∈ [ν, τ_U]. We only need to replace ${\stackrel{\prime }{\mathit{\beta }}}_j(\tau )$ by ${\stackrel{\prime }{\mathit{\beta }}}_{\mathcal{K}}(\tau )$ in the proof of Theorem 2 in the Appendix and slightly modify the arguments accordingly. Consequently, the term I in the proof of Theorem 3 still converges weakly to a mean zero Gaussian distribution, while the norms of items II and III are still o_p(1). Therefore, Theorem 3 still holds for any K-dimensional subvector of ${\widehat{\beta }}_{\mathcal{K}}(\tau )$, i.e., $\sqrt{n}({\widehat{\beta }}_{\mathcal{K}}(\tau )-{\mathit{\beta }}_{\mathcal{K}}^{*}(\tau ))$ converges to a mean zero K-dimensional Gaussian distribution at any τ ∈ [ν, τ_U].

As shown in the proof, the covariance function of ${\widehat{\beta }}_j(\tau )$ depends on the unknown active set S*, the unknown conditional density functions f(t|Z) and g(t|Z), and other unknown quantities. Thus, it is not calculable. The next section proposes an alternative model-free variance estimator based on functional delta method and multi-sample splitting properties (Efron, 2014; Fei et al., 2019).

4. A Variance Estimator via the Functional Delta Method

Let J_bi ∈ {0, 1} be the indicator of whether i^th observation is in the b^th sub-sample $D_2^b$, and $J_{.i}=B^{-1}{\sum }_{b=1}^B{J}_{bi}$. We define the re-sampling covariances between J_bi and ${\tilde{\mathit{\beta }}}_j^b\left({\tau }_k\right)$ at τ_k ∈ Γ_m for each i = 1, . . . , n as

$${\mathbf{s}}_{ij}\left({\tau }_k\right)=\frac{1}{B}\sum _{b=1}^B\left(J_{bi}-J_i\right)\left({\tilde{\mathit{\beta }}}_j^b\left({\tau }_k\right)-{\widehat{\mathit{\beta }}}_j\left({\tau }_k\right)\right);$$

$${\mathbf{S}}_j\left({\tau }_k\right)={\left({\mathbf{s}}_{1j}\left({\tau }_k\right),{\mathbf{s}}_{2j}\left({\tau }_k\right),\dots ,{\mathbf{s}}_{nj}\left({\tau }_k\right)\right)}^{\text{T}}.$$

Let $n_1=\left|D_2^b\right|$. The covariance between ${\widehat{\mathit{\beta }}}_j\left({\tau }_k\right)$ and ${\widehat{\mathit{\beta }}}_j\left({\tau }_{\ell }\right)$ is estimated by

$${\widehat{\mathrm{Cov}}}_j\left({\tau }_k,{\tau }_{\ell }\right)=\frac{n-1}{n}{\left(\frac{n}{n-n_1}\right)}^2\sum _{i=1}^n{\mathbf{s}}_{ij}\left({\tau }_k\right){\mathbf{s}}_{ij}\left({\tau }_{\ell }\right)=\frac{n(n-1)}{{\left(n-n_1\right)}^2}{\mathbf{S}}_j^{\text{T}}\left({\tau }_k\right){\mathbf{S}}_j\left({\tau }_{\ell }\right),$$

where the multiplier n(n − 1)/(n − n₁)² is a finite-sample correction for the sub-sampling (Wager and Athey, 2018). Thus a variance estimator for ${\widehat{\beta }}_j\left({\tau }_k\right)$ is

$${\widehat{V}}_j\left({\tau }_k\right)=\frac{n(n-1)}{{\left(n-n_1\right)}^2}{\mathbf{S}}_j^{\text{T}}\left({\tau }_k\right){\mathbf{S}}_j\left({\tau }_k\right).$$ (5)

It is shown in Wager and Athey (2018) that (5) is consistent, i.e., ${\widehat{V}}_j\left({\tau }_k\right)/\mathrm{Var}\left({\widehat{\beta }}_j\left({\tau }_k\right)\right)\stackrel{p}{\to }1$ as n,B → ∞. Furthermore, for a finite B, we propose a bias corrected version of (5):

$${\widehat{V}}_j^B\left({\tau }_k\right)={\widehat{V}}_j\left({\tau }_k\right)-\frac{n{n}_1}{B\left(n-n_1\right)}\left\{B^{-1}\sum _{b=1}^B{\left({\tilde{\mathit{\beta }}}_j^b\left({\tau }_k\right)-{\widehat{\mathit{\beta }}}_j\left({\tau }_k\right)\right)}^2\right\},{\tau }_k\in {\mathrm{\Gamma }}_m.$$ (6)

The correction term in (6) is a suitable multiplier of the re-sampling variance of ${\tilde{\mathit{\beta }}}_j^b\left({\tau }_k\right)$‘s, which converges to zero as n → ∞ and n₁ = O(n), and the two variance estimators in (5) and (6) are asymptotically equivalent. However, ${\widehat{V}}_j\left({\tau }_k\right)$ in (5) requires B to be of order n^3/2 to reduce the Monte Carlo noise below the sampling noise, while ${\widehat{V}}_j^B\left({\tau }_k\right)$ in (6) only requires B to be of order n to achieve the same (Wager et al., 2014).

Since ${\widehat{\mathit{\beta }}}_j(\tau )$ converges weakly to a Gaussian process by Theorem 3, and our variance estimators are consistent on the grid points, we define the asymptotic 100(1 − α)% local confidence intervals for ${\mathit{\beta }}_j^{*}\left({\tau }_k\right)$ at any τ_k ∈ Γ_m as

$$\left({\widehat{\mathit{\beta }}}_j\left({\tau }_k\right)-{\mathrm{\Phi }}^{-1}(1-\alpha /2)\sqrt{{\widehat{V}}_j^B\left({\tau }_k\right)},{\widehat{\mathit{\beta }}}_j\left({\tau }_k\right)+{\mathrm{\Phi }}^{-1}(1-\alpha /2)\sqrt{{\widehat{V}}_j^B\left({\tau }_k\right)}\right),$$

where ${\widehat{V}}_j^B\left({\tau }_k\right)$ is the variance estimator in (6), and Φ is the standard normal cumulative distribution function. The p-value of testing $H_0:{\beta }_j^{*}\left({\tau }_k\right)=0$ for each τ_k ∈ Γ_m is

$$2\times \left\{1-\mathrm{\Phi }\left(\left|{\widehat{\mathit{\beta }}}_j\left({\tau }_k\right)\right|/\sqrt{{\widehat{V}}_j^B\left({\tau }_k\right)}\right)\right\}.$$

5. Simulation Studies

In various settings, we have compared the proposed method, Fused-HDCQR (referred to as “Fused” in the tables and figures hereafter), with some competing methods in quantile regression or high dimensional inference. These methods include Wang et al. (2012) (“W12”) and Fan et al. (2014) (“F14”) for quantile regression; Zheng et al. (2018) (“Z18”) for censored quantile regression; and Meinshausen et al. (2009) (“M09”) for inference with aggregated p-values from multi-sample splittings.

In the simulations and the data analysis, we choose L-HDCQR described in Section 3 as the variable selection tool for Fused-HDCQR. We also explore the feasibility of using other alternatives for variable selection, such as Fan et al. (2009) (“F09”) and M09.

When implementing Fused-HDCQR, we specify the number of splits as B = 300, the quantile interval as [ν, τ_U] = [0.1, 0.8], and the grid length as m = n/log p. The tuning parameter is chosen by minimizing the 5-fold cross-validation error as in (4). We study the following examples with sparse non-zero effects, some of which are heterogeneous.

Example 1.

The event times are generated by

$$\log T_i={\tilde{\mathbf{Z}}}_i^{\text{T}}\mathbf{b}+{\epsilon }_i,i=1,\dots ,n,$$

where the coefficient vector b are sparse with b₂₀ = 0.5, b₄₀ = 1, b₆₀ = 1.5, b_j = 0 for all other j’s, and ε_i ~ N(0, 1). Therefore, the true coefficients are β*(τ) = (Q_ε(τ), b^T)^T for all τ ∈ (0, 1), where Q_ε(τ), τ-th quantile of the distribution of ε, is the intercept. ${\tilde{Z}}_{j,i}$‘s are i.i.d. Unif(−1, 1) for j ∈ {1, . . . , p}. The censoring time is generated independently as log C_i = N(0, 16) + N(−5, 1) + N(8, 0.25), which gives a censoring rate around 25%.

Example 2.

The event times follow

$$\log T_i={\tilde{\mathbf{Z}}}_i^{\text{T}}\mathbf{b}+1.5{\tilde{Z}}_{3,i}{\epsilon }_i,$$ (7)

where b₂₀ = 1, b₄₀ = 1.5, b₆₀ = 2, b_j = 0 for all other j’s, and ε_i ~ N(0, 1). We first generate ${\stackrel{\prime }{\mathbf{Z}}}_i~N_p(0,\mathbf{\Sigma })$ with Σ = (σ_kℓ)_p×p, σ_kℓ = 0.3^|k−ℓ| the AR(1) correlation structure, and then let ${\tilde{\mathbf{Z}}}_i={\stackrel{\prime }{\mathbf{Z}}}_i$, except for the third covariate ${\tilde{Z}}_{3,i}=|{\stackrel{\text{'}}{Z}}_{3,i}|+0.5$. Therefore ${\mathit{\beta }}_1^{*}(\tau )=\text{0}$, ${\mathit{\beta }}_4^{*}(\tau )=1.5Q_{\epsilon }(\tau )$, and ${\mathit{\beta }}_j^{*}(\tau )=b_{j+1}$, for all other j’s. The censoring time is generated independently as log C_i = N(0, 16) + N(−4, 1) + N(8, 0.25), which gives a censoring rate around 23%.

Example 3.

The event times follow

$$\log T_i={\tilde{\mathbf{Z}}}_i^{\text{T}}\mathbf{b}+{\varphi }_1\left({\xi }_i\right){\tilde{Z}}_{1,i}+{\varphi }_{10}\left({\xi }_i\right){\tilde{Z}}_{4,i},$$

where b₂₀ = 1, b₄₀ = 1.5, b₆₀ = 2, b_j = 0 for all other j’s, ξ_i ~ N(0, 1), and ϕ₁, ϕ₁₀ are monotone functions as the dashed lines in Figure 1, both are continuous with zero and non-zero pieces over τ. We first generate ${\stackrel{\prime }{\mathbf{Z}}}_i~N_p(0,\mathbf{\Sigma })$ as in Example 2, and then let ${\tilde{\mathbf{Z}}}_i={\stackrel{\text{'}}{\mathbf{Z}}}_i$, except ${\tilde{Z}}_{1,i}=|{\stackrel{\prime }{Z}}_{1,i}|+0.5$ and ${\tilde{Z}}_{10,i}=|{\stackrel{\prime }{Z}}_{10,i}|+0.5$. Therefore ${\mathit{\beta }}_1^{*}(\tau )=0$, ${\mathit{\beta }}_2^{*}(\tau )={\varphi }_1(\tau )$, ${\mathit{\beta }}_{11}^{*}(\tau )={\varphi }_{10}(\tau )$, and ${\mathit{\beta }}_j^{*}(\tau )=b_{j+1}$, for all other j’s. The censoring time is generated independently as log C_i = N(0, 16) + N(−4, 1) + N(10, 0.25), which gives a censoring rate around 20%.

Figure 1:

Estimated heterogeneous effects and confidence intervals of Fused-HDCQR using Example 3: ${\mathit{\beta }}_2^{*}(\cdot )$(left panel) and ${\mathit{\beta }}_5^{*}(\cdot )$ (right panel). From the top to the bottom are the plots for (n, p) = (300, 1000), (700, 1000) and (700, 2000), respectively.

For each of these examples, we set (n, p)=(300, 1000) and (700, 1000) to study the impacts of the sample size and the number of variables and how the methods fare when p > n. In Example 3, which mimics the real data example in Section 6 most closely, we have also explored (n, p) = (700, 2000), which is roughly equal to the dimension of the real dataset. For every parameter configuration, a total of 100 independent datasets are generated, and we report the averaged results from these replications, unless specified otherwise. The number of 100 is chosen because the penalized methods for high dimensional CQR are in general computationally intensive and take much computing time for one simulated dataset (Table 5).

Table 5:

Comparisons of average computing time (in seconds) when performing Example 1.

	Fused	Z18	W12	F14	M09
(n,p) = (300,1000)	888	853	509	390	170
(n,p) = (700,1000)	3,108	1,812	2,230	1,231	440

Note: see the footnote of Table 2.

We first evaluate the feasibility of using various variable selection tools for our proposed method. Comparisons of true positives and false negatives among F09, M09, and L-HDCQR under Examples 1–3 are reported in Table 1. F09 presents a subpar performance because, by taking intersections of variables selected from different partitions of data, it tends to miss out some true signals and thus have fewer true positives. In contrast, L-HDCQR retains more true positives than both F09 and M09, while having larger false positives. Because our method requires the variable selection step to include the true signals with high probability, even at the cost of some false positives, we have opted to use L-HDCQR as the screening tool for our method.

Table 1:

Summary of variable selection results based on the simulated datasets.

				TP			FP
	(n,p)	CR	q	L-HDCQR	M09	F09	L-HDCQR	M09	F09
Example 1	(300,1000)	0.25	3	2.67	2.12	1.64	7.95	0.00	0.19
	(700,1000)	0.25	3	2.98	2.78	2.27	13.08	0.01	0.34
Example 2	(300,1000)	0.22	4	3.60	3.58	2.22	12.45	0.00	0.22
	(700,1000)	0.23	4	3.99	3.99	3.54	11.29	0.00	0.64
Example 3	(300,1000)	0.20	5	3.82	3.63	1.91	10.00	0.00	0.17
	(700,1000)	0.20	5	4.81	4.77	4.35	11.73	0.01	0.54
	(700,2000)	0.19	5	4.78	4.76	4.17	16.34	0.00	0.47

Note: CR, average censoring rate; q = |S*|; TP, average true positives; FP, average false positives; M09, Meinshausen et al. (2009); F09, Fan et al. (2009); L-HDCQR, Zheng et al. (2018).

We next compare the performance of Fused-HDCQR with other high dimensional quantile regression methods at τ = .25, .5, .75 under Example 1. As a benchmark for comparisons, we also compute the oracle estimates based on the true model (with S* known). As W12, F14, and Z18 provide coefficient estimates without standard errors (SEs), only the estimation biases are reported for them, while the average SEs, empirical standard deviations (SDs) and coverage probabilities of the confidence intervals are reported for our method. Table 2 shows that Fused-HDCQR presents the smallest biases, which are comparable to those of the oracle estimates. In contrast, Z18 has smaller biases when the sample size is large, and larger biases otherwise, while W12 and F14 incur substantial biases since they are not designed for censored data. Moreover, the SEs based on Fused-HDCQR agree with the empirical SDs of the estimates. The consistent estimates of coefficients and SEs obtained by Fused-HDCQR lead to proper coverage probabilities around the 0.95 nominal level. In addition, the coverage probabilities improve as n increases.

Table 2:

Results of Example 1 based on the simulated datasets.

	Bias					EmpSD	SE	Cov	Power
	Oracle	Fused	Z18	F14	W12	Fused			Fused	M09
n = 300, p = 1000
	0.02	0.02	−0.38	−0.50	−0.48	0.14	0.13	0.93	0.97	0.06
β21 = 0.5	0.02	0.01	−0.24	−0.49	−0.48	0.12	0.13	0.95	0.98	0.04
	0.01	0.01	−0.13	−0.50	−0.48	0.12	0.13	0.96	1.00	0.02
	−0.01	−0.01	−0.02	−0.91	−0.33	0.14	0.13	0.92	1.00	0.99
β41 = 1	−0.00	−0.00	−0.03	−0.68	−0.32	0.14	0.12	0.92	1.00	0.98
	0.02	0.01	−0.01	−0.70	−0.30	0.17	0.14	0.93	1.00	0.94
	−0.00	0.01	0.00	−0.92	−0.24	0.12	0.13	0.92	1.00	1.00
β61 = 1.5	0.00	0.01	0.01	−0.64	−0.25	0.11	0.13	0.97	1.00	1.00
	0.02	0.01	0.02	−0.70	−0.25	0.13	0.14	0.95	1.00	1.00
n = 700, p = 1000
	−0.02	−0.01	−0.01	−0.47	−0.23	0.09	0.08	0.92	1.00	0.56
β21 = 0.5	−0.01	−0.01	−0.01	−0.39	−0.22	0.08	0.08	0.89	1.00	0.65
	−0.01	−0.01	−0.01	−0.40	−0.23	0.10	0.09	0.89	1.00	0.44
	0.00	0.00	0.04	−0.53	−0.17	0.09	0.08	0.91	1.00	1.00
β41 = 1	−0.00	0.00	0.03	−0.49	−0.19	0.09	0.08	0.90	1.00	1.00
	−0.01	−0.01	0.01	−0.53	−0.18	0.08	0.10	0.87	1.00	1.00
	0.01	0.01	0.06	−0.54	−0.21	0.10	0.09	0.93	1.00	1.00
β61 = 1.5	0.01	0.01	0.03	−0.62	−0.21	0.08	0.08	0.93	1.00	1.00
	−0.00	0.00	0.03	−0.71	−0.21	0.07	0.09	0.94	1.00	1.00

Note: Each β has three rows corresponding to τ = .25,.5,.75 from the top to bottom; EmpSD, empirical standard deviation; SE, average standard error; Cov, coverage probability; Oracle, Oracle estimator; Z18, Zheng et al. (2018).F14, Fan et al. (2014); W12, Wang et al. (2012); M09, Meinshausen et al. (2009).

Table 2 also concerns the power for detection of signals. Since W12, F14, and Z18 cannot draw inference and, in general, there is lack of literature that deals with inference for HDCQR, we compare our method with the aggregated p-value approach (M09) in the quantile setting, though M09 originated from linear regression. The results indicate that Fused-HDCQR outperforms M09, and presents adequate testing power when the effect size is moderate or large.

Table 3 summarizes the results from Example 2 with the heterogeneous effect β₄ varying with τ. We compare the estimation accuracy between Fused-HDCQR and Z18, as well as the statistical power between Fused-HDCQR and M09. Again, Fused-HDCQR presents smaller biases than Z18 and a higher power than M09. To assess whether the tuning parameters selected as in Remark 3 help the variable selection method (L-HDCQR) used by Fused-HDCQR satisfy assumption (A4) in Section 3, we report the selection frequency of each signal variable in Table 3 (and also in Table 4), and observe that the selection frequency increases as the sample size increases, hinting that assumption (A4) may be satisfied with these selected tuning parameters.

Table 3:

Results of Example 2 based on the simulated datasets.

	Bias			EmpSD	SE	Cov	Freq	Power
	Oracle	Fused	Z18	Fused				Fused	M09
n = 300, p = 1000
	0.01	0.13	0.29	0.32	0.31	0.88		0.82	0.16
β4 = 1.5Qε(τ )	−0.05	−0.07	0.06	0.33	0.29	0.90	0.73	0.11	0.00
	0.01	−0.14	−0.05	0.31	0.34	0.82		0.62	0.10
	−0.01	−0.01	−0.01	0.14	0.13	0.90		1.00	0.88
β21 = 1	−0.03	−0.01	−0.05	0.12	0.12	0.91	0.69	1.00	0.92
	−0.01	−0.00	−0.02	0.14	0.13	0.92		1.00	0.84
	0.01	0.01	0.03	0.13	0.13	0.90		1.00	1.00
β41 = 1.5	−0.01	0.01	0.03	0.12	0.13	0.93	0.99	1.00	1.00
	−0.00	0.02	−0.02	0.13	0.14	0.93		1.00	1.00
	−0.03	−0.03	0.04	0.13	0.13	0.91		1.00	1.00
β61 = 2	−0.03	−0.02	0.03	0.11	0.13	0.92	1.00	1.00	1.00
	−0.01	−0.01	−0.00	0.12	0.15	0.95		1.00	1.00
n = 700, p = 1000
	0.03	0.08	0.19	0.19	0.21	0.92		0.99	0.61
β4 = 1.5Qε(τ )	0.02	0.03	0.14	0.18	0.19	0.89	0.89	0.11	0.00
	0.04	−0.03	−0.01	0.21	0.23	0.92		0.97	0.56
	0.01	0.01	0.05	0.09	0.08	0.94		1.00	1.00
β21 = 1	0.01	0.01	0.01	0.08	0.08	0.87	0.99	1.00	1.00
	0.01	0.01	0.05	0.10	0.09	0.89		1.00	1.00
	−0.01	0.00	0.08	0.08	0.08	0.94		1.00	1.00
β41 = 1.5	−0.00	0.00	0.05	0.09	0.08	0.92	1.00	1.00	1.00
	0.00	0.01	0.04	0.09	0.09	0.95		1.00	1.00
	−0.01	−0.01	0.10	0.08	0.09	0.93		1.00	1.00
β61 = 2	−0.01	−0.01	0.06	0.08	0.09	0.91	1.00	1.00	1.00
	−0.00	−0.00	0.07	0.09	0.10	0.90		1.00	1.00

Note: See the footnote of Table 2; Freq, average selection frequency in B splits.

Table 4:

Results of Example 3 based on the simulated datasets.

	Bias			EmpSD	SE	Cov	Freq	Power
	Oracle	Fused	Z18	Fused				Fused	M09
n = 300, p = 1000
	−0.05	0.06	0.59	0.34	0.36	0.94		0.06	0.00
β2 = φ1(τ )	0.11	0.37	1.01	0.52	0.51	0.89	0.71	0.20	0.00
	0.04	−0.20	−0.05	0.80	0.72	0.89		0.87	0.06
	0.08	0.14	0.27	0.65	0.50	0.90		0.77	0.36
β11 = φ10(τ )	0.10	−0.20	−0.36	0.62	0.51	0.91	0.67	0.19	0.00
	0.16	0.06	−0.03	0.56	0.52	0.90		0.10	0.00
β21 = 1.5	0.03	0.03	0.04	0.25	0.23	0.95	0.65	1.00	0.77
β41 = 2	0.00	−0.00	0.02	0.23	0.25	0.93	0.93	1.00	0.99
β61 = 2.5	0.09	0.07	0.19	0.21	0.26	0.94	0.99	1.00	1.00
n = 700, p = 1000
	0.02	0.04	0.27	0.21	0.23	0.94		0.06	0.00
β2 = φ1(τ )	0.17	0.30	0.79	0.37	0.40	0.88	0.96	0.27	0.01
	0.15	0.08	0.35	0.51	0.51	0.90		1.00	0.77
	0.07	0.09	0.18	0.33	0.33	0.91		0.99	0.92
β11 = φ10(τ )	−0.01	−0.19	−0.23	0.35	0.34	0.85	0.92	0.21	0.00
	−0.00	−0.04	−0.08	0.37	0.31	0.94		0.06	0.00
β21 = 1.5	−0.00	0.00	0.04	0.16	0.17	0.97	0.98	1.00	1.00
β41 = 2	−0.03	−0.02	−0.01	0.15	0.18	0.95	1.00	1.00	1.00
β61 = 2.5	0.00	0.00	0.07	0.18	0.18	0.94	1.00	1.00	1.00
n = 700, p = 2000
	0.05	0.11	0.13	0.32	0.32	0.93		0.07	0.00
β2 = φ1(τ )	0.09	0.34	0.87	0.46	0.44	0.91	0.93	0.09	0.02
	0.25	0.36	1.77	0.53	0.46	0.87		0.74	0.58
	0.13	0.25	0.73	0.45	0.35	0.84		1.00	0.83
β11 = φ10(τ )	0.09	−0.02	0.56	0.41	0.36	0.89	0.90	0.76	0.01
	−0.04	−0.30	−0.13	0.36	0.34	0.85		0.15	0.00
β21 = 1.5	0.01	0.01	0.03	0.18	0.21	0.98	0.98	1.00	1.00
β41 = 2	0.01	0.03	−0.07	0.22	0.20	0.91	0.99	1.00	0.98
β61 = 2.5	−0.02	−0.01	−0.05	0.25	0.20	0.94	1.00	1.00	0.98

Note: See the footnote of Tables 2 and 3; For β₂ and β₁₁, the numbers are shown at τ = .25,.5,.75 from the top to the bottom and, for the other β’s, at τ = 0.5.

Table 4 summarizes the results based on Example 3. For the two heterogeneous effects β₂ and β₁₁ that vary with τ, their estimation biases of Fused-HDCQR become smaller and the estimated SEs are closer to the empirical ones as n increases. Figure 1 shows that the Fused-HDCQR estimates agree with the oracle estimates and the truth, except at the change points, and have narrower confidence intervals with a larger n.

Finally, we compare the computation intensity among Z18, M09, W12, F14, and Fused-HDCQR under Example 1 and report in Table 5 the average computing time per dataset. Our method is the most computationally intensive, because it involves multiple data-splittings and draws inferences on all of the p coefficients. However, by utilizing parallel computing, we have managed to reduce the computational time to the same order of Z18, W12, and F14 that are based on penalized regression.

6. Application to the Boston Lung Cancer Survivor Study (BLCSC)

Detection of molecular profiles related to cancer patients’ survival can aid personalized treatment, leading to prolonged survival and improved quality of life. In a subset of BLCSC samples, 674 lung cancer patients were measured with survival times, along with 40, 000 SNPs and clinical indicators, such as lung cancer subtypes (adenocarcinoma, squamous cell carcinoma, or others), cancer stages (1–4), age, gender, education level ( ≤ high school or > high school) and smoking status (active or non-active smokers); see Table 6 for patients’ characteristics. The censoring rate was 23% and a total of 518 deaths were observed during the followup period, with the observed followup time varying from 13 to 8, 584 days.

Table 6:

Patients’ characteristics in the BLCSC samples.

(n = 674)
		Mean (SD)
Age		60 (10.8)
		Count (%)
Female		259 (38)
Education level	≤ High school	264 (39)
	> High school	410 (61)
Smoking	Non-active	418 (62)
	Active	256 (38)
Cancer type	Adenocarcinoma	283 (42)
	Squamous cell	110 (16)
	Other	281 (42)
Cancer stage	1	283 (42)
	2	110 (16)
	3	256 (38)
	4	25 (4)

We could have included all 40,000 SNPs in our analysis. However, for more statistical power, we opt for the targeted gene approach by focusing on 2,002 SNPs residing in 14 genes identified to be cancer related, namely, ALK, BRAF, BRCA1, EGFR, ERBB2, ERCC1, KRAS, MET, PIK3CA, RET, ROS1, RRM1, TP53, and TYMS (Brose et al., 2002; Toyooka et al., 2003; Paez et al., 2004; Soda et al., 2007). Pinpointing the effects of individual loci within the targeted genes is helpful for understanding disease mechanisms (Evans et al., 2011; D’Antonio et al., 2019) and designing gene therapies (Pâques and Duchateau, 2007; Hanawa et al., 2004). We also adjust for patients’ clinical and environmental characteristics listed in Table 6, which gives a total of p = 2, 011 predictors.

We apply Fused-HDCQR to compute the coefficient estimates (3) and variance estimates (6). We set the quantile interval to be [0.2, 0.7], which is wide enough to cover high risk and low risk groups and, in the meantime, ensures the quantile parameters be estimable in the presence of censoring (Zheng et al., 2015). We choose the lower bound τ₀ = ν = 0.1 to circumvent the singularity problem with CQR at τ = 0, because few (< 2%) observations are censored below the ν-th quantile. With ϵ_n = 01, we form the τ-grid Γ_m of length m = 61. We set B = 750 as the number of re-samples, which is sufficiently large based on our numerical experience. To determine the tuning parameter λ_n in L-HDCQR for selection, we use 5-fold cross-validation as specified in Remark 3.

For ease of presentation, we summarize the results evaluated at 6 quantile levels, τ = 0.2, 0.3, . . . , 0.7, instead of the whole grid Γ_m. To highlight the findings of the high risk group, we rank all SNPs based on their p-values at τ = 0.2. After Bonferroni correction for multiple testing, there are 83 significant SNPs with the overall type I error of α = 0.05. Our method estimates the coefficients and the p-values for all predictors, and we only present the results for the patient characteristics, the top 10 significant SNPs, and the 3 least significant SNPs in Figure 2 and Table 7. The estimated coefficient of active smoking drops from −0.42 (p = 0.0011) to −0.53 (p = 0.0005) as τ changes from 0.2 to 0.5, and then increases to −0.31 (p = 0.038) as τ changes to 0.7, suggesting that active smoking might be more harmful to the high or median risk groups than the low risk group of patients. The most significant SNP at τ = 0.2 is AX.37793583 T, which remains significant throughout τ = 0.2 to τ = 0.7. However, its estimated coefficient decreases from 2.75 (τ = 0.2) to 1.39 (τ = 0.7), indicating its heterogeneous impacts on survival, i.e. stronger protective effect at lower quantiles and vice versa.

Figure 2:

Estimated quantile-specific coefficients of the predictors in Table 7.

Table 7:

Analysis of the BLCSC data with Fused-HDCQR. The SNPs are sorted by their p-values at τ = 0:2, corresponding to the high risk groups. Results for the top 10 and the bottom 3 are presented.

	Estimator	SE	p-value	Estimator	SE	p-value	Estimator	SE	p-value
τ		0.2			0.3			0.4
Int	6.90	0.25	1.4E–165	7.48	0.28	4.3E–157	7.94	0.24	3.2E–241
Adeno	0.20	0.16	2.1E–01	0.14	0.18	4.5E–01	0.02	0.13	8.7E–01
Squamous	−0.16	0.16	3.0E–01	−0.20	0.16	2.1E–01	−0.34	0.13	1.0E–02
Stage2	−0.82	0.24	6.3E–04	−0.99	0.25	6.0E–05	−0.98	0.24	3.2E–05
Stage3	−0.97	0.17	1.6E–08	−1.04	0.20	2.0E–07	−1.13	0.14	2.0E–15
Stage4	−1.54	0.17	3.0E–20	−1.77	0.20	1.7E–19	−1.86	0.14	2.2E–42
Age	−0.01	0.01	1.5E–02	−0.01	0.01	3.0E–02	−0.02	0.01	1.0E–02
Edu	0.08	0.14	6.0E–01	0.06	0.15	6.9E–01	0.07	0.13	5.8E–01
Female	−0.30	0.13	2.2E–02	−0.35	0.14	1.0E–02	−0.37	0.12	1.6E–03
Smoke	−0.42	0.13	1.1E–03	−0.48	0.14	5.0E–04	−0.52	0.11	3.4E–06
AX.37793583 T	2.75	0.22	3.0E–36	2.61	0.20	4.6E–39	2.39	0.20	3.7E–33
AX.83104700 A	2.32	0.20	4.0E–31	1.91	0.19	6.3E–24	1.54	0.19	1.5E–15
AX.15207405 G	2.03	0.20	1.0E–24	1.59	0.22	9.8E–13	1.17	0.21	3.7E–08
AX.16619495 T	1.79	0.20	3.3E–19	1.36	0.20	1.3E–11	0.97	0.20	1.2E–06
AX.13920550 G	1.93	0.23	2.5E–17	1.41	0.28	5.3E–07	0.87	0.27	1.6E–03
AX.83444620 C	1.39	0.17	7.4E–17	1.05	0.19	6.6E–08	0.71	0.21	8.8E–04
AX.82902859 T	1.58	0.20	8.7E–16	1.19	0.18	2.0E–11	0.90	0.12	3.4E–14
AX.40182999 A	1.50	0.21	9.6E–13	1.01	0.25	3.9E–05	0.64	0.14	6.5E–06
AX.82976133 A	2.32	0.33	3.8E–12	2.02	0.35	6.7E–09	1.58	0.35	6.1E–06
AX.82900605 G	2.21	0.35	1.6E–10	1.91	0.29	9.1E–11	1.54	0.33	2.9E–06
...
AX.41828883 G	1.4E–03	0.34	1.00	−3.2E–02	0.42	0.94	−5.7E–02	0.54	0.92
AX.11293250 T	−3.6E–04	0.14	1.00	6.2E–02	0.15	0.67	5.0E–02	0.12	0.68
AX.37863475 C	−3.1E–04	0.26	1.00	−1.1E–01	0.25	0.68	−1.8E–01	0.24	0.46
Int	8.30	0.27	4.8E-214	8.55	0.30	4.9E-180	8.69	0.35	2.8E-132
Adeno	−0.09	0.15	5.3E–01	−0.09	0.13	4.8E–01	−0.09	0.13	5.1E–01
Squamous	−0.50	0.15	1.0E–03	−0.60	0.16	2.1E–04	−0.50	0.19	7.1E–03
Stage2	−0.88	0.25	5.0E–04	−0.73	0.24	2.1E–03	−0.57	0.19	2.8E–03
Stage3	−1.08	0.17	1.7E–10	−0.91	0.15	6.4E–10	−0.68	0.16	2.0E–05
Stage4	−1.91	0.15	7.0E–38	−1.93	0.14	1.7E–44	−1.69	0.16	2.1E–27
Age	−0.02	0.00	3.3E–05	−0.02	0.01	1.3E–03	−0.02	0.01	1.9E–03
Edu	0.15	0.14	2.7E–01	0.16	0.13	2.2E–01	0.11	0.13	4.0E–01
Female	−0.44	0.11	6.4E–05	−0.47	0.12	1.6E–04	−0.38	0.15	1.3E–02
Smoke	−0.53	0.15	4.9E–04	−0.36	0.16	2.4E–02	−0.31	0.15	3.8E–02
AX.37793583 T	2.16	0.20	4.1E–28	1.84	0.28	2.8E–11	1.39	0.25	4.2E–08
AX.83104700 A	1.15	0.27	1.6E–05	0.58	0.27	3.5E–02	0.13	0.25	6.0E–01
AX.15207405 G	0.75	0.25	2.3E–03	0.34	0.37	3.5E–01	−0.05	0.48	9.2E–01
AX.16619495 T	0.66	0.22	3.1E–03	0.44	0.31	1.5E–01	0.18	0.35	6.1E–01
AX.13920550 G	0.54	0.27	4.3E–02	0.26	0.60	6.7E–01	0.11	0.60	8.6E–01
AX.83444620 C	0.55	0.23	2.0E–02	0.29	0.22	1.8E–01	0.01	0.18	9.7E–01
AX.82902859 T	0.73	0.13	4.2E–08	0.51	0.32	1.1E–01	0.22	0.46	6.3E–01
AX.40182999 A	0.41	0.18	2.6E–02	0.22	0.27	4.1E–01	−0.01	0.30	9.6E–01
AX.82976133 A	1.17	0.42	5.4E–03	0.61	0.52	2.4E–01	0.24	0.46	6.0E–01
AX.82900605 G	1.22	0.35	4.5E–04	0.86	0.34	1.1E–02	0.50	0.31	1.0E–01
...
AX.41828883 G	0.26	0.60	0.66	0.32	0.52	0.54	0.12	0.68	0.86
AX.11293250 T	−0.00	0.12	1.00	−0.09	0.12	0.44	−0.09	0.15	0.56
AX.37863475 C	−0.24	0.20	0.23	−0.37	0.17	0.03	−0.57	0.32	0.08

The effects of some SNPs are nearly zero for higher quantiles. For example, the estimated coefficient of AX.15207405 G decreases from 2.03 (τ = 0.2; p = 10⁻²⁴) to −0.05 (τ = 0.7; p = 0.92), with the estimated standard error increasing from 0.20 to 0.48. Similarly, the estimated coefficient of AX.40182999 A decreases from 1.5 (τ = 0.2; p = 9.6×10⁻¹³) to −0.01 (τ = 0.7; p = 0.96). The results again hint at heterogeneous SNP effects in various risk groups, which cannot be detected using traditional Cox models.

Finally, our results shed light on the roles of SNPs in the high risk group (i.e. lower quantiles). Specifically, we map the 83 SNPs with significant effects at the 0.2-th quantile by Fused-HDCQR to the corresponding genes and rank the genes by the number of significant SNPs (over total number of SNPs for each gene in the parenthesis), which are TP53 (14/321), RRM1 (14/174), ERCC1 (10/167), BRCA1 (10/114), ALK (8/163), ROS1 (5/294), EGFR (5/261), ERBB2 (4/167), and 6 other genes with numbers of significant SNPs less than 4. While these genes were reported to be associated with lung cancer (Toyooka et al., 2003; Takeuchi et al., 2012; Rosell et al., 2011; Lord et al., 2002; Zheng et al., 2007; Sasaki et al., 2006; Brose et al., 2002), our analysis provides more detailed information as to which SNPs and locations of the genes are jointly associated with the lung cancer survival, as well as the estimated effects and uncertainties. Analysis of heterogeneous SNP effects has been gaining increasing research attention in lung cancer research (McKay et al., 2017; Dong et al., 2012; Huang et al., 2009), and beyond it (Garcia-Closas et al., 2008; Cheng et al., 2010; Gulati et al., 2014).

7. Conclusions

Our proposed procedure involves repeated estimates from low dimensional CQRs, which are computationally straightforward and can be efficiently implemented with parallel computing. We require the variable selection to possess a sure screening property as in condition (A4). This seems to be supported by our simulations, which find our procedure works well when the variable selection method can select a superset of the true model with high probability. Our condition is much weaker than a stringent condition of selection consistency as specified in Fei et al. (2019).

In regards to the selection of B, we recommend B to be in the same order of the sample size n. Smaller B might not affect coefficient estimation much, but it would yield biased standard errors for inference. In addition, we opt to define Γ_m by setting the grid as n/log p equally spaced points between τ₀ and τ_U. This may cover the quantile interval well, with reasonable computation efficiency.

There are open questions left to be addressed. First, substantial work is needed when predictors are highly correlated as the performance of our method, like the other competing methods, deteriorates when correlations among predictors become stronger. Second, it is of interest to investigate an alternative method when the sparsity condition fails. For example, it is challenging to find an effective strategy to draw inference when a non-negligible portion of predictors have small but non-zero effects. We will pursue them elsewhere.