Journal of Statistical Planning and Inference 142 (2012) 1659–1668
Contents lists available at SciVerse ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
New empirical likelihood inference for linear transformation models Hanfang Yang, Yichuan Zhao n Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, United States
a r t i c l e i n f o
abstract
Article history: Received 30 January 2012 Accepted 7 February 2012 Available online 21 February 2012
The transformation model plays an important role in survival analysis. In this paper, we investigate the linear transformation model based on new empirical likelihood. Motivated by Fine et al. (1998) and Yu et al. (2011), we introduce the truncated survival time t 0 and adjust each term of estimating equations to improve the accuracy of coverage probability. We prove that the log-likelihood ratio has the asymptotic distribution 4w2p þ 1 . The new empirical likelihood method avoids estimating the complicated covariance matrix in contrast to normal approximation method and empirical likelihood method developed by Zhao (2010). Moreover, the proposed method enables us to obtain confidence intervals for the component of regression parameters. In the simulation study, our method demonstrates better performance than the traditional method in the small samples. & 2012 Elsevier B.V. All rights reserved.
Keywords: Transformation model Empirical likelihood U-statistics Martingale Counting process
1. Introduction The well-known proportional hazards model was introduced by Cox (1972). Later, Andersen and Gill (1982) explored the Cox model using martingale theory. The Cox model is the most popular method utilized broadly in survival analysis. An alternative method in survival analysis is the proportional odds model (Pettitt, 1982; Bennett, 1983). The transformation model is a natural generalization of those two models and provides many other potential choices. Cheng et al. (1995) derived a limiting theory of the transformation model using martingale theory. Based on new estimation equations, Chen et al. (2002) develop the inference procedure for the linear transformation model. Let T be the failure time; Z, a corresponding p-dimensional covariate; Sz ðÞ is the survival function of T conditioned on covariate Z. Then, the semiparametric transformation model is (see Cheng et al., 1995) gfSz ðtÞg ¼ hðtÞ þ Z T b,
ð1:1Þ
where hðÞ is a strictly increasing unspecified function and gðÞ is a given decreasing function. An alternative expression of (1.1) is (see Cheng et al., 1995) hðtÞ ¼ Z T b þ e,
ð1:2Þ
where e is a random variable independent of covariate Z with the distribution function FðxÞ ¼ 1g 1 ðxÞ. Fine et al. (1998) considered the truncated t 0 to place a finite limit on survival time and guaranteed the uniform convergence of Gaussian processes on interval ½0,t 0 . Other related studies about the transformation model include Cai et al. (2000, 2005). Recently, Kong et al. (2006) investigated the case-cohort problems using semiparametric linear transformation models. n
Corresponding author. E-mail address:
[email protected] (Y. Zhao).
0378-3758/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2012.02.007
1660
H. Yang, Y. Zhao / Journal of Statistical Planning and Inference 142 (2012) 1659–1668
Owen (1988, 1990) introduced empirical likelihood for mean vector on i.i.d. complete data. Qin and Lawless (1994) proposed empirical likelihood for general estimating equations. DiCiccio et al. (1991) showed that empirical likelihood method is Bartlett-correctable. Owen (2001) made a comprehensive illustration about empirical likelihood and its applications. Recently, empirical likelihood method has been extended to some diverging number of dimensionality, such as Hjort et al. (2009) and Chen et al. (2009). Moreover, Zhao (2010) demonstrated that the empirical likelihood method for transformation models can outperform traditional methods in small samples. However, the methodology of Zhao (2010) sacrificed the tremendous computational resource on estimating the covariance matrix. Motivated by Yu et al. (2011) and Zhao and Yang (2012), we can construct new empirical likelihood for the transformation model which avoids estimating the complicated matrix. The rest of the paper is organized as follows. In Section 2, we develop the new empirical likelihood method for the linear transformation model. Then, we report results of simulation studies in terms of coverage probability in Section 3. A discussion is given in Section 4. The proofs are provided in the Appendix. 2. Inference procedure 2.1. Preliminaries Throughout the paper, we use same notations as Fine et al. (1998). Let T i be the failure time which might not be observed fully. The censoring variables C i with distribution function GðtÞ are independent of failure time T i . Define bivariate vector ðX i , di Þ, i ¼ 1, . . . ,n, where X i ¼ minðT i ,C i Þ and di ¼ IðT i r C i Þ. Let fZ i gni¼ 1 be the corresponding covariate vectors, where Z i 2 Rp . We denote Z ij ¼ Z i Z j , i ¼ 1, . . . ,n, j ¼ 1, . . . ,n. Fine et al. (1998) introduced a known constant t 0 , T where PrfminðT,CÞ 4 t 0 g 40. Denote the h0 and a0 as true values of hðÞ and a ¼ hðt 0 Þ. Define y ¼ ða, b ÞT and true T T y0 ¼ ða0 , b0 Þ and
Zij ðy0 Þ ¼ ZðZ Tij b0 ÞPrðT i ZT j Z t0 9Z i ,Z j Þ,
ð2:1Þ
^ where Z e e Let GðÞ be the Kaplan–Meier estimator of G. Combining the positive weights function wij ðÞ, Fine et al. (1998) proposed the following estimating equation U w ðyÞ: 8 9
