Non-marginal feature screening for varying coefficient competing risks model
Introduction
Lifetime data with competing risks called competing risks data are characterized by many contemporary research problems in biomedical studies, reliability testing, empirical health economics, and so on, in which subjects suffer from multiple mutually exclusive failures. Nevertheless, existing literatures pay little attention to related screening for competing risks data, only a few researches focused on this issue (Li et al., 2018, Chen et al., 2020, Chen et al., 2021).
However, these authors only considered the constant coefficient models, which maybe not applicable in practical situations. To further examine the dynamic impacts of the relevant covariates on survival time under the competing risks model, one can apply the varying coefficient setting to the competing risks model, which can be useful to explore nonlinear interaction effects between the primary covariate and other covariates.
But high-dimensional methods of varying coefficient models in survival analysis have only been sparsely studied, including Song et al., 2014b, Yue and Li, 2017, Yue et al., 2018, Yang et al., 2019, Xia et al., 2019, Liu, 2019, and Qu and Sun (2021), and to our knowledge, no literature on varying coefficient competing risks models. Even the model framework proposed by Liu (2019) covers a huge number of commonly used survival models, it does not include competing risks model. Due to the complexity of this model, extending feature screening to varying coefficient competing risks model presents a tremendous novel challenge.
Motivated by the above facts, we propose and investigate thoroughly a non-marginal screening procedure for varying coefficient proportional subdistribution hazards (PSH) model (Fine and Gray, 1999) with the ultra-high dimensional, which is more flexible and useful for modeling the dynamic changes of regression coefficients. Comparing with the screening procedure based on the marginal partial likelihood of individual feature, the non-marginal screening naturally accounts for the joint effects between features by jointly estimating their model coefficients. Our procedure is constructed with the sparsity-restricted maximum partial likelihood estimate (SMPLE) and implemented by iterative group hard threshold (IGHT) algorithm. Although this work shares the same spirit as Yang et al. (2019) and Qu and Sun (2021), it is not just a simple straightforward application of existing methods. Specifically, we propose the non-marginal screening procedure under a different model from Yang et al. (2019) and Qu and Sun (2021), which has rarely been studied. It is challenging to derive the group partial likelihood and its derivative under the varying coefficient PSH model, while the risk set, time-varying weights, and counting process need to be considered. Moreover, we adopt a different estimation method for varying coefficients from Yang et al. (2019), and compared to most joint screening methods (Li et al., 2018, Yang et al., 2016, Yang et al., 2019), the iterative grouping hard thresholding algorithm we employed is computationally fast because it need not compute the block diagonal matrix, as demonstrated by numerical simulations.
The rest of the article is organized as follows. In Section 2, we briefly review the definition of proportional subdistribution hazards model and apply varying coefficient setting to it. Then, we propose a non-marginal screening procedure and discuss the corresponding theoretical properties. Simulation results and real data analysis are given in Section 3, which demonstrate the effectiveness of the proposed procedure. We briefly summarize the article in Section 4.
Section snippets
Some preliminaries
Fine and Gray (1999) advocated using the proportional subdistribution hazards (PSH) model for describing the cumulative incidence function of the event of interest. To be precise, let and be the failure and censoring times, and let be a bounded time-independent covariate vector. Under the competing risks setting, suppose there are causes such that can be observed. For the usual right-censored data, we observe and the censoring indicator , where is
Simulation
In this section, we assess the performance of the proposed non-marginal feature screening procedure for varying-coefficient proportional subdistribution hazards model, referred to as VCR-IGHT, by Monte Carlo simulation. Since there is no competing method to deal with the competing risks data under the varying coefficient model, we compare the simulation results with CR-SJS (Li et al., 2018) based on constant coefficient PSH model and KNOVAS (Qu and Sun, 2021) based on varying coefficient Cox
Discussion
In this paper, we introduced a non-marginal screening method for varying-coefficient competing risks model based on local-likelihood estimator and IGHT algorithm, namely VCR-IGHT. We systematically studied the ascent property and established sure screening property for VCR-IGHT, and conducted several comprehensive simulations to verify its empirical performance. The numerical comparison indicates that VCR-IGHT has significant capability to screen out the important variables in the constant and
Funding
This work was supported by Fundamental Research Funds for Central Universities of the Central South University (2021zzts048), National Statistical Scientific Research Project of China (2022LZ28), and Changsha Municipal Natural Science Foundation (No. kq2202080).
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