Communicaiions in Statistics—Simulation and Computation®, 34: 135-143, 2005 Copyright © Taylor & Francis, Inc. ISSN: 0361-0918 print/1532-4141 online DOI: 10.1081/SAC-200047114
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Survival Analysis
Semiparametric Analysis for Additive Risk Model via Empirical Likelihood YICHUAN ZHAO AND YU-SHENG HSU Department of Mathematics and Statistics, Georgia State University, Atlanta, Georgia, USA Cox's proportional hazards model has been the most popular model for the regression analysis of censored survival data. However, the additive risk model provides a useful alternative. In this article, we apply the empirical likelihood ratio method to the additive risk model with right censoring and derive its limiting distribution. Based on the result, we construct a confidence region for the regression parameter. Useful extensions are discussed. Simulation results are presented to compare the proposed method with that method based on normal approximation. Keywords Counting process; Martingale; Normal likelihood; Regression; Right censoring.
approximation;
Partial
Mathematics Subject Classification Primary 62J99; Secondary 62P10.
1. Introduction In medical studies, we are interested in a regression model for the survival rate which incorporates information from covariates. Although the Cox model has been considered as a major tool for regression analysis of survival data, the proportional hazards assumption may not be appropriate for some data analyses. The additive risk model (Cox and Oakes, 1984) provides a useful alternative to Cox's (1972) proportional hazards model for studying the association between failure time and risk factors through hazard functions. This valuable model for the analysis of survival data seems simple and easy to interpret for medical researchers. Additive risk models in various forms have been successfully utilized by many authors, e.g., Aalen (1980), Buckley (1984), Aalen (1989), Huffer and McKeague (1991), McKeague and Sasieni (1994) among others. Lin and Ying (1994) proposed a simple semiparametric estimating function for regression coefficient. The approach of Lin and Ying (1994) has been applied to successfully predict the survival probability of patients. For instance. Song et al. (1997) derived simultaneous Received September 3, 2003; Accepted October 27, 2004 Address correspondence to Yichuan Zhao, Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303, USA; E-mail:
[email protected] 135
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confidence bands for survival functions under the additive risk model; Shen and Cheng (1999) obtained confidence band for cumulative incidence curves under the additive risk model. Recently, Lin et al. (1998) applied the additive risk model to current status data. Chen et al. (2002) considered more general additive hazards model with latent treatment effectiveness lag time in comparative randomized clinical trials. The empirical likelihood method is a powerful nonparametric method. It holds some unique features, such as range respecting, transformation-preserving, asymmetric confidence interval, and Bartlett correctability. The use of empirical likelihood (EL) in survival analysis traces back to Thomas and Grunkemeier (1975) who derived pointwise confidence intervals for survival function with right-censored data (see also Li, 1995 and Murphy, 1995). For right-censoring data, this approach has been used in the construction of simultaneous confidence band under a variety of settings (see Hollander et al., 1997; Einmahl and McKeague, 1999; Li and Van Keilegom, 2002; McKeague and Zhao, 2002; among others). Owen (1988, 1990) introduced EL confidence regions for the mean of a random vector based on i.i.d. complete data. Since then, the EL has been widely applied to statistical models to do inferences for the parameter of interest. Owen (1991) and Chen (1993, 1994) derived EL inference procedures for linear model. Wang and Rao (2001, 2002) extended it to linear models for missing data. Kolaczyk (1994) and Chen and Cui (2003) considered EL for generalized linear models based on constraints derived from the score function of the quasi-likelihood. The partial linear regression model was investigated by Wang and Jing (1999) and Shi and Lau (2000). Linear regression with right censoring was studied by Qin and Jing (2001) and Li and Wang (2003). The Cox proportional hazard model under right censoring was studied by Qin and Jing (2001). Qin and Jing (2001) and Wang and Li (2002) developed EL methods for regression coefficients in the partial linear regression model with right-censored data. However, to the best of our knowledge, the inference for the parameter under semiparametric additive risk model has not been developed yet via EL. Based on the idea of EL (cf. Owen, 1988, 1990), estimating equation concerning with the regression parameter is essential. In the present article, we focus on the semiparametric additive risk model under right censoring, make full use of the estimating function of Lin and Ying (1994), and find one tractable likelihoodratio based confidence region for the unknown regression parameter. The proposed confidence region and main asymptotic result are presented in Sec. 2. Extensions are discussed in Sec. 3. In Sec. 4, we conduct simulation to compare the EL method with normal approximation based method. Proof is given in the Appendix. 2. 2.1.
Main Results Preliminaries
Let X{t\Z) denote the hazard function for the life time T under covariate Z{t). The semiparametric additive risk model of Lin and Ying (1994) has the following form (2.1)
Semiparametric Additive Risk Model
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where Z(t) is a p-vector of possibly time-varying covariates, ^Q is a p-vector of unknown regression parameters, and Ao(O is an unspecified baseline hazard function. Let C denote the censoring time. Assume T and C are conditionally independent given covariate Z(t). Suppose that data consist of n iid samples of (X,, (5,, Z,), where X,. = min(r;, C,), ^, = /(r,. < C,). Let A?,(0 = ^,/(X,. < f)(i = 1 , . . . , n) be a counting process for the /th subject, which indicates that the failure time of the ith subject is observed up to time t. Let Yi(t) = /(X, > t) denote the predictable indicator process indicating whether or not the ith subject is at risk just before time t. Let i satisfy , > T) > 0. Lin and Ying (1994) proposed the following estimating function
ioiP, t) - Yi{t)P'Zi{t)dt],
(2.2)
1=1
where
Ao(i?, t ) = f —^ (2.2) is equivalent to
i=l
"
where
The regression coefficients are estimated by solving the equation uQ) — 0. The resulting estimator ^ is of the form
where a®^ = aa'. Under model (2.1), the counting process A',(f) can be uniquely decomposed so that for every i and t, NM = MM + f Y,{s)l{s\Z)ds, where M,(f) is a local square integrable martingale.
(2.4)
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Zhao and Hsu It follows that from (2.3) and (2.4), (2.5) (=1
which is a martingale. The random vector n^l^{]i - p^) converges weakly to a normal distribution with niean zero and the covariance matrix which can be consistently estimated by A~^BA~^, where
i=\
and
Then an asymptotic 100(1 - a)% confidence region for p based on the above normal approximation is given by
where xl{^) is the upper a-quantile of the chi-squared distribution with degrees of freedom p. 2.2. EL Confidence Region It is clear that EU{PQ) = 0 since Uiji^) is a martingale from the estimating Eq. (2.5). Thus, we can apply standard EL method (see Owen, 1988, 1990) to construct confidence region for regression parameter fi. For 1 < / < n, we define
where a,(/) = «-' E , Z f i ^ ( O with a®" = 1, a®' = a. Therefore, an EL at the true value ^Q is given by L()?o) - sup \\\pr.
ZPi = 1' Pi > 0, ipiW^, = o).
Let p = (pi,..., p J be a probability vector, i.e., Yl"=\ Pi = 1 and p, > 0 for 1 < i < n. Note that n"=i Pi attains its maximum at p, = I/n. Thus EL ratio at the PQ is defined by
o) = sup I n "A : E/'/ = 1' A > 0. EAM'., = 0j. I =l
1=1
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By using Lagrange multipliers, we know that R(lio) is maximized when p.= }-{l+X'Wj-\ n
i=\,...,n,
where A = (A,,..., 2.^)' satisfies the equation 1 "
W
