The goal of this paper is to investigate the
The goal of this melatonin receptor agonist paper is to investigate the penalized empirical likelihood method for the Cox model, as a competitor to the penalized partial likelihood. Since empirical likelihood (EL) was proposed by Owen, 1988, Owen, 2001, empirical likelihood methods have been investigated well from many statistical perspectives. With regards to the model selection, Variyath et al. (2010) investigated the properties of the information criteria with the empirical likelihood. Tang and Leng (2010) proposed penalized empirical likelihood methods for the mean vector estimation and linear regression with the divergent number of dimensions, in conjunction with appropriate penalty functions. Leng and Tang (2012) extended their work for more general estimation equations with growing dimensionality. Other work about the impact of growing dimensionality on the inference of empirical likelihood can be found in Hjort et al. (2009), Chen and Van Keilegom (2009), and Chen et al. (2009). For right-censored data, Zhao and Huang (2007) proposed the bias-corrected empirical likelihood for the accelerated failure time model based on estimated influence functions. Motivated by Zhao and Huang (2007), Wu et al. (2015) developed the penalized empirical likelihood method for the censored accelerated failure time model. For the Cox’s proportional hazards model, Hou et al. (2014) investigated the performance of the penalized empirical likelihood estimator via the bridge penalty. Their empirical likelihood was derived with known baseline hazard function using the method in Qin and Jing (2001), which makes their approach not suitable or applicable in practice. In this paper, we aim to develop the penalized empirical likelihood (PEL) method for sparse Cox model from the following two vantage points: (1) a bias-corrected empirical likelihood, proposed in Sun et al. (2009), Zhao and Jinnah (2012) and Zheng and Yu (2013) with a plug-in cumulative baseline hazard function, is evaluated in order to facilitate the application of the method to real data, and; (2) more general requirements regarding the penalty function to maintain the sparsity and asymptotic normality of the coefficient estimators are explored, so that penalty functions such as LASSO and SCAD can be applied with no any concerns.
In the rest of intron paper, we study the property of the bias-corrected empirical likelihood in conjunction with appropriate penalty functions for parameter estimation and model selection. In Section 2.1, we define a penalized empirical likelihood after providing some preliminary results about the Cox model. We investigate the asymptotic properties of the penalized empirical likelihood in Section 2.2, and discuss the optimization and tuning procedure in Section 2.3. Results from extensive simulation studies are presented in Section 3 to evaluate the asymptotic properties of the existing methods, the bias-corrected empirical likelihood and penalized empirical likelihood. We illustrate the proposed method by re-analyzing a well-known primary biliary cirrhosis (PBC) data set in Section 3.2, followed by the discussions in Section 4. The proofs of theorems are given in the Appendix.
Methodology We start this section by restating some preliminary results about the Cox model and introducing notations that will be used throughout the paper. Let , , and be the survival time, the censoring time and the associated vector of covariates, respectively. Assume that given , and are independent. Let be the observed time and be the censoring indicator, where is the indicator function. The proportional hazards model proposed by Cox (1972) is given as with the unspecified baseline hazard function and the parameter vector of interest . Define and , as the counting and at-risk process, respectively. The parameter vector is commonly estimated by maximizing the log partial likelihood For the simplicity, we assume no ties in the observed event times. When ties are present, the log partial likelihood at (2.2) can be readily rewritten; see more details in Breslow (1974). For brevity, will be used in the rest of this paper to stand for the time-varying covariate .