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Semiparametric accelerated failure time model for the analysis of right censored data

Zhezhen Jin

aDepartment of Biostatistics, Columbia University, USA
Correspondence to: 1Department of Biostatistics, Columbia University, 722 W 168th Street, New York, NY 10032, USA. E-mail: zj7@cumc.columbia.edu
Received November 26, 2016; Revised November 27, 2016; Accepted November 27, 2016.
Abstract

The accelerated failure time model or accelerated life model relates the logarithm of the failure time linearly to the covariates. The parameters in the model provides a direct interpretation. In this paper, we review some newly developed practically useful estimation and inference methods for the model in the analysis of right censored data.

Keywords : accelerated failure time model, Buckley-James estimator, censored data, estimating function, Gehan-type estimator, induced smoothing, weighted logrank-type estimator
1. Introduction

The survival data are common in many fields, such as economics, business, industrial engineering and biomedical studies. The nonparametric and semiparametric modeling has been studied extensively because it offers valid estimation and inference with less stringent model assumptions. The Cox proportional hazards model is the most used semiparametric regression models for the analysis of survival data (Cox, 1972) due to the availability of statistical software packages for implementation of the estimation and inference procedures. The accelerated failure time (AFT) models relates the logarithm of the failure time linearly to the covariates. The parameters in the model provides a direct interpretation. However, its estimation and inference procedures are challenging in the presence of censroing. Over the last forty years, there has been considerable research on the AFT models, see Buckely and James (1979), Jin et al. (2003, 2006), Koul et al. (1981), Lai and Ying (1991a, 1991b, 1995), Miller and Halpern (1982), Powell (1984), Prentice (1978), Ritov (1990), Robins and Tsiatis (1992), Tsiatis (1990), Wei et al. (1990), Yang (1997), Ying (1993), Zhou (2005a), Zeng and Lin (2007) and among many others. An earlier review on AFT model can be found in Wei (1992). In this paper, our goal is to review and discuss some newly developed practically useful statistical methods which specifically deal with the semiparametric accelerated failure time model.

2. Semiparametric accelerated failure time model

Let Ti be the failure time for the ith patient, i = 1, …, n. Due to censoring Ci, we observe Yi = min(Ti, Ci) and δi = I{TiCi} which takes value 1 if TiCi and 0 otherwise. The semi-parametric AFT model is of form

$log(Ti)=XiTβ+ɛi,$

where Xi, i = 1, …, n are observed covariates, β is p × 1 unknown parameter vector and εi, i = 1, …, n are independent and unobserved errors (the mean of the εi is not necessarily 0). The semi-parametric AFT model is essentially a regular semiparametric linear regression model and its regression coefficient β is directly quantifies the impact of covariates X on the survival time instead of the more abstract hazard rates. Note that the logarithm in the model may be replaced by any other known strictly increasing functions. However, in the presence of censoring, the intercept parameter is not included in the model specification because the intercept parameter cannot be estimated well as pointed out in Wei (1992).

It is assumed that conditional on the p × 1 covariate vector Xi for the ith subject, the censoring variable Ci and the failure times Ti, i = 1, …, n are independent, and εi, i = 1, …, n are independent and identically distributed random variables whose common distribution function F(·) is completely unspecified. (Cox and Oakes, 1984; Kalbfleisch and Prentice, 2002; Lawless, 2003).

3. Point estimation

Let $ei(β0)=log Yi-XiTβ0=min{log Ti-XiTβ0,log Ci-XiTβ0}=min{ɛi,log Ci-XiTβ0}$, and Ni(β; t) = ΔiI{ei(β) ≤ t}, where I(·) is the indicator function that takes value 1 when the condition is satisfied 0 otherwise.

### 3.1. Gehan-type rank estimator

Currently, all available estimation method starts with following estimation equation, based on the Gehan’s statistic:

$UnG(β)=∑i=1n∑j=1nδi (Xi-Xj)I{Yi-Yj-(Xi-Xj)T β≤0}.$

The function $UnG$ is component-wise monotone in β (Fygenson and Ritov, 1994). It was shown that solving Un(β) = 0 or ||Un(β)|| = 0, is equivalent to minimizing Gn(β) (Jin et al., 2003), where

$Gn(β)=∑i=1n∑j=1nδi [Yi-Yj-(Xi-Xj)T β]-,$

where the notation a means the negative part of a (e.g. 3 = 0, (−2) = 2). The minimization of Gn(β) can be done by following linear programming approach: Minimize $∑i=1n∑j=1nΔiuij$ under linear constraints

$Yi-Yj=(Xi-Xj)T β-uij, uij≥0.$

Or alternatively, the minimization of Gn(β) is equivalent to minimization of

$∑i=1n∑j=1nδi|ei(β)-ej(β)|+|M-βT∑k=1n∑l=1nδk(Xl-Xk)|,$

where M is the prespecified extremely large number (Jin et al., 2003). The minimization can be done with many available statistical softwares, such as the function l1fit in R.

Therefore, point estimator of β can be obtained, which is denoted as β̂G. Under regularity conditions, it was shown that β̂G is consistent and $n(β^G-β)~N(0,Γ)$ as n → ∞, where $Γ=AG-1BGAG-1$, the AG and BG are defined in the expression (3.6) and (3.7) with $φ(β,t)=∑j=1nI{ej(β)≥t}$.

### 3.2. Weighted logrank-type rank estimator

The general weighted log-rank estimating function for β takes the form

$Uφ(β)=∑i=1nΔiφ(β;ei(β)) {Xi-∑j=1nXjI{ej(β)≥ei(β)}∑j=1nI{ej(β)≥ei(β)}}$

or

$Uφ(β)=∑i=1n∫-∞∞φ(β;t) {Xi-∑j=1nXjI{ej(β)≥t}∑j=1nI{ej(β)≥t}} dNi(β;t),$

where φ is a possibly data-dependent weight function. When φ = 1, the resulting Uφ(β) corresponds to the log-rank (Mantel, 1966) statistics. When $φ(β,t)=∑j=1nI{ej(β)≥t}$, the resulting Uφ(β) correspond to the $UnG(β)$ in (3.1), which is Gehan (1965) statistics.

Under regularity conditions, the root β̂φ of the estimating function Uφ(β) satisfies that n1/2(β̂φβ) is asymptotically zero-mean normal with covariance matrix $Aφ-1BφAφ-1$, where

$Aφ=limn→∞ n-1∑i=1n∫-∞∞φ(β;t){Xi-∑j=1nXjI{ej(β)≥t}∑j=1nI{ej(β)≥t}}⊗2λ˙(t)λ(t)dNi(β;t),$$Bφ=limn→∞ n-1∑i=1n∫-∞∞φ2(β;t){Xi-∑j=1nXjI{ej(β)≥t}∑j=1nI{ej(β)≥t}}⊗2dNi(β;t),$

A⊗2 = AAT for a matrix A (Ying et al., 1992), λ(·) is the common hazard function of the error terms, and λ̇(t) = (t)/dt (Lai and Ying, 1991b; Tsiatis, 1990; Wei et al., 1990; Wei, 1992; Ying, 1993).

In general, it is difficult to solve the equation Uφ(β) = 0 because Uφ(β) is neither continuous nor componentwise monotone in β. Jin et al. (2003) overcame the difficulty by a class of monotone estimation functions that approximate the general weighted log-rank estimating functions (3.4). The idea is to use the Gehan-type rank estimator β̂G in the Section 3.1 and an iterative algorithm. Specifically, Jin et al. (2003) proposed to use following modified estimating function of (3.4):

$U˜φ (β;β^)=∑i=1n∫-∞∞ψ (β^;t+(β-β^)′Xi)S(0)(β;t) {Xi-∑j=1nXjI{ej(β)≥t}∑j=1nI{ej(β)≥t}} dNi(β;t),$

where ψ(b; x) = φ(b; x)/S(0)(b; x), $S(0)(β;t)=∑i=1nI{ei(β)≥t}$ and β̂ is a intial consistent estimator of β0, say β̂G. It is easy to see that

$U˜φ (β;β^)=n-1∑i=1n∑j=1nψ (β^;ei (β^))Δi (Xi-Xj)I{ei(β)≤ej(β)}.$

The Ũφ(β;β̂) is monotone in each component of β and is the gradient of the following function:

$Lφ (β;β^)=n-1∑i=1n∑j=1nψ (β^;ei (β^))Δi {ei(β)-ej(β)}-.$

Similar to (3.2), the minimisation of Lφ(· ;β̂) can be implemented via linear programming.

The iterative algorithm for estimating β starts from setting: β̂(0) = β̂G, and β̂(k) = arg minβLφ(β; β̂(k−1)) (k ≥ 1). If β̂(k) converges to a limit as the number of iterations k → ∞, then the limit must satisfy Uφ(β) = 0. Therefore, the root β̂φ of original estimating equation Uφ(β) = 0 can be obtained by the iterative algorithm. Under regularity conditions, Jin et al. (2003) showed that each estimator obtained in the k iteration β̂k is a legitimate estimator of β (consistent and asymptotically normal) when the initial estimator is consistent and asymptotically normal, and if the β̂k converges as k → ∞, the limit is indeed the solution of the Uφ(β) = 0. Specifically, Jin et al. (2003) showed that β̂k is asymptotically a weighted average of β̂φ and β̂G,

$β^k={(Aφ+Dφ)-1Dφ}k β^G+[I-{(Aφ+Dφ)-1Dφ}k]β^φ+op (n-12),$

where I is the p × p identity matrix,

$Dφ=limn→∞ n-1∑i=1n∫ψ˙0(t)S(0)(β0;t){Xi-∑j=1nXjI{ej(β)≥ei(β)}∑j=1nI{ej(β)≥ei(β)}}⊗2dNi(β0;t),$

and ψ̇0(t) is the derivative of ψ0(t), the limit of ψ(β0; t) as n → ∞.

### 3.3. Least-squares-type estimator

When there is no censoring, Ti, i = 1, …, n are completely observed, the classical least-squares estimator of β is obtained by minimizing $∑i=1n(log Ti-XiTβ)2$ in terms of β. The minimizer is the solution of the equation

$∑i=1n(Xi-X¯)(log Ti-XiTβ)=0,$

where $X¯=n-1∑i=1nXi$.

In the presence of censoring, we can only observe Yi and Equation (3.13) cannot be used directly. Buckly and James (1979) proposed to replace log Ti by following

$Y^i(β)=δi log Yi+(1-δi) {∫ei(β)∞udF^β(u)1-F^β(ei(β))+XiTβ},$

where β(t) is the Kaplan-Meier estimator of F(t) based on {ei(β), δi} (i = 1, …, n), i.e.,

$F^β(t)=1-∏i:ei(β)

The resulting Buckly-James estimator is the solution to the follwoing equation:

$∑i=1n(Xi-X¯)(Y^i(β)-XiTβ)=0.$

The difficulty to solve the Equation (3.15) lies in that the function $∑i=1n(Xi-X¯)(Y^i(β)-XiTβ)$ is neither continuous nor componentwise monotone in β. To overcome these difficulties, Jin et al. (2006) developed an iterative procedure to obtain a class of consistent and asymptotically normal estimators. Let $Y¯(b)=n-1∑i=1nY^i(b)$, and define

$U(β,b)=∑i=1n(Xi-X¯)(Y^i(b)-XiTβ),$

or equivalently

$U(β,b)=∑i=1n(Xi-X¯){Y^i(b)-Y¯(b)-(Xi-X¯)T β}.$

Set U(β, b) = 0, which yields

$β=Q(b)={∑i=1n(Xi-X¯)⊗2}-1 [∑i=1n(Xi-X¯){Y^i(b)-Y¯(b)}].$

It leads to an iterative algorithm

$β^(m)=Q(β^(m-1)), m≥1.$

With the consistent initial estimator β̂(0) = β̂G, the Gehan-type estimator in the Section 3.1, Jin et al. (2006) showed that β̂m is consistent and asymptotically normal for every m ≥ 1, and β̂(m) is asymptotically a weighted average of the Buckley-James estimator β̂BJ and β̂G,

$β^m=(I-D-1A)m β^G+[I-(I-D-1A)m]β^BJ+op (n-12),$

where $D=limn→∞(1/n)∑i=1n(Xi-X¯)⊗2$ and A is the slope matrix of the Buckley-James estimating function.

### 3.4. Kernel-smoothed profile likelihood estimator

Zeng and Lin (2007) proposed a kernel-smoothed profile likelihood estimator by an approximate nonparametric maximum likelihood method. Based on the semiparametric AFT model (2.1), the observed data (Yi, δi, Xi), i = 1, …, n yields the following log-likelihood function of β and hazard function λ(·) of the error ε:

$loglik=1n∑i=1n[-δiXiTβ+δiλ(ei(β))-Λ (ei(β))],$

Zeng and Lin (2007) showed that the direct maximization of the log-likelihood does not yield a solution. It also does not lead to the maximization of a profile likelihood obtained by replacing the λ(ei(β)) by an estimator $δi/∑j=1nI{ej(β)≥ei(β)}$ nor the maximization of a sieve profile likelihood obtained by replacing the λ(·) by a piecewise constant approximation. On the other hand, an approximation of

$dP (δ=1,e(β)≤t)/dtP (e(β)≥t)$

by

$1t(nan)-1∑i=1nδiK (ei(β)-log tan)∫log(t)∞(nan)-1∑i=1nδiK (ei(β)-san) ds,$

where K(·) is a kernel function (such as the standard normal density function) and an is a bandwidth, yields a kernel-smoothed approximation of the likelihood of β,

$lns(β)=-1n∑i=1nδiXiTβ-1n∑i=1nδiei(β)+1n∑i=1nδi log {1nan∑j=1nδjK (ej(β)-ei(β)an)}-1n∑i=1nδi log {1n∑j=1n∫-∞ej(β)-ei(β)anK(s)ds}.$

The kernel-smoothed profile likelihood estimator β̂pl is obtained by maximizing the $lns(β)$ and its variance-covariance estimator is obtained by the use of the information matrix of the $lns(β)$.

Given β̂pl, the λ(·) and Λ(·) can be estimated by

$λ^n(t)=(nant)-1∑i=1nδiK (ei (β^pl)-log tan)(n)-1∑i=1n∫-∞ei (β^pl)-log t)anK(u)du,$$Λ^n(t)=∫-∞log t(nan)-1∑i=1nδiK (ei(β^pl)-log tan)(n)-1∑i=1n∫-∞ei (β^pl)-log tanK(u)duds.$

Under regularity conditions, Zeng and Lin (2007) showed that the estimator β̂pl achieves semiparametric efficiency. The challenge for the practical use is the choice of the Kernel function, bandwidth, and the stability of the $lns(β)$. Currently, there is no statistical software that implements the procedure.

4. Variance-covariance estimation

Except the kernel-smoothed profile likelihood estimator, the variance-covariance estimation of the Gehan-type rank estimator, weighted logrank-type estimator, and least-squares-type estimator is difficult because the corresponding variance-covariance matrices involve nonparametric estimation of the underlying probability density function. Here two computationally feasible approaches will be presented: resampling method and induced smoothing method.

### 4.1. Resampling method

The resampling method is similar to the resampling scheme similar to those in Rao and Zhao (1992), Parzen et al. (1994) and Jin et al. (2001). The basic idea is to perturb the objective functions that are minimized or estimating equations that are solved. Let Zi (i = 1, …, n) be n independent positive random variables satisfying E(Zi) = Var(Zi) = 1 and be independent of the observed data (i, δi, Xi) (i = 1, …, n). The Zi can be generated by a number of known distributions, such as the standard exponential distribution or Poisson distribution Poisson(1).

4.1.1. Variance-covariance estimation for Gehan-type estimator

Jin et al. (2003) perturbed the objective function (3.2) and used following function

$Gn*(β)=n-1∑i=1n∑j=1nδi{ei(β)-ej(β)}- Zi.$

The minimisation of $Gn*$ is also a linear programming problem and can be implemented with many existing softwares. Let $β^G*$ be the minimiser of $Gn*$, it can be shown that $β^G*$ is the root of the estimating function

$UG*(β)=∑i=1n∑j=1nδi(Xi-Xj)I{ei(β)-ej(β)≤0}Zi.$

Jin et al. (2003) showed that the asymptotic distribution of n1/2(β̂Gβ0) can be approximated by the conditional distribution of $n1/2(β^G*-β^G)$ given the data (i, δi, Xi) (i = 1, …, n). Consequently, the distribution of β̂G, can be approximated by a large number of realizations of $β^G*$ by repeatedly generating the random sample (Z1, …, Zn) with the observed data (i, δi, Xi) (i = 1, …, n). The empirical variance-covariance matrix of $β^G*$ provides an estimate of the variance-covariance matrix of β̂G. The confidence intervals and hypothesis testing for individual β can be carried out based on the empirical distribution of $β^G*$ or by the Wald method.

The developed methods are implemented in the Splus/R package lss and the design and application features of lss are provided in Huang and Jin (2007).

4.1.2. Variance-covariance estimation for weighted logrank-type rank estimator

Due to the similarity between the objective function (3.10) for general weighted logrank-type rank estimator and the objective function (3.2) for Gehan-type estimator, the variance-covariance matrix of the β̂(k) can be obtained by a similar resampling approach. Specifically, we can construct the perturbed objective function

$Lφ*(β;b):=n-1∑i=1n∑j=1nψ(b;ei(b))δi{ei(β)-ej(β)}- Zi,$

and define $β^0*=β^G*$ and $β^k*=arg minβ Lφ*(β;β^k-1*) (k≥1)$. Jin et al. (2003) showed that the asymptotic distribution of n1/2(β̂kβ) can be approximted by the conditional distribution of $n1/2(β^k*-β^k)$ given the data (i, δi, Xi) (i = 1, …, n). Inference on β can then be carried out on the basis of the empirical distribution of $β^k*$.

4.1.3. Variance-covariance estimation for least-squares-type estimator

Jin et al. (2006) showed that the covariance matrix of β̂m can be approximated by a resampling procedure. A theoretical derivation showed that both the Kaplan-Meier estimator used for the Ŷi(b) and the estimating function (3.17) are required to be perturbed by the same Zis, (i = 1, …, n). Specifically,

$F^b*(t)=1-∏i:ei(b)

and

$Y^i*(b)=δiY˜i+(1-δi) {∫ei(b)∞udF^b*(u)1-F^b*(ei(b))+XiTb},$$Q*(b)={∑i=1nZi(Xi-X¯)⊗2}-1 [∑i=1nZi (Xi-X¯){Y^i*(b)-Y¯*(b)}].$

Then, Q*(b) leads to an iterative process $β^m*=Q*(β^m-1*)$, m ≥ 1. By setting the initial value $β^0*=β^G*$, the iteration procedure $β^k*=L*(β^k-1*)$ yields a $β^k*(1≤k≤m)$. Jin et al. (2006) showed that the asymptotic distribution of n1/2(β̂kβ) can be approximted by the conditional distribution of $n1/2(β^k*-β^k)$ given the data (i, δi, Xi) (i = 1, …, n). Therefore, by generating random samples of (Z1, …, Zn) repeatedly N times, we can obtain N realizations of $β^m*$, denoted by $β^m,j* (j=1,…,N)$. For each m ≥ 1, the covariance matrix of β̂m can be estimated by

$s2=1N-1∑j=1N(β^m,j*-β¯m*)(β^m,j*-β¯m*)T,$

where $β¯m*=(1/N)∑j=1Nβ^m,j*$.

The developed methods are implemented in the Splus/R package lss and the design and application features of lss are provided in Huang and Jin (2007).

### 4.2. Induced smoothing method

Brown and Wang (2005) proposed a general variance estimation procedure based on an induced smoothing for non-smooth estimating functions. The approach is computationally efficient and easy to implement when the smoothing in terms of integration with respect to a Gaussian distribution has an explicit form. For the Gehan-type rank estimator, the induced smoothing in Brown and Wang (2005) yields an explicit form and its varaince-covariance matrix can be easily estimated as shown in Brown and Wang (2007). However, the induced smoothing in Brown and Wang (2005) for the general logrank-weighted estimator and least-squares-type estimator does not yield an explicit form and cannot be used directly. To overcome the difficulty, Jin et al. (2015) developed a general Monte carlo approach for variance estimation for estimators based on induced smoothing which can be applied to both the general logrank-weighted estimator and least-squares-type estimator.

4.2.1. Induced smoothing for Gehan-type rank estimator

Brown and Wang (2007) and Johnson and Strawderman (2009) applied the induced smoothing method in Brown and Wang (2005) to the Gehan-type rank estimator β̂G. Because $n(β^G-β)~N(0,AG-1BGAG-1)$ as n → ∞, Brown and Wang (2007) set $β=β^G+n-1/2[AG-1BGAG-1]1/2W$, where W ~ N(0, I), and introduced following induced smoothing estimating function

$U˜nG(V;β)=EWUnG (β+n-12V12W),$

where EW denotes the expectation with respect to W. Brown and Wang (2007) showed that the expression (4.8) has an explicit closed form, specifically,

$U˜nG(V;β)=∑j=1nδj∑i=1n(Xi-Xj)Φ(n(ei(β)-ej(β))(Xi-Xj)T V(Xi-Xj)),$

where Φ(·) is the cumulative distribution function of N(0, 1). The resulting induced smoothed estimating function $U˜nG(V;β)$ is continuous and differentiable function of β and $U˜nG(V;β)=0$ can be solved iteratively with the initial values of β̂G and V0 = I. Therefore, they suggested to obtain β̂ and by jointly solving $U˜nG(V;β)=0$ and the following equation

$V=A˜-1(V;β^)B^GA˜-1(V;β^),$

where $A˜(V;β)=∂U˜nG(V;β)/∂β$ and G is a consistent estimator of BG. Brown and Wang (2007) showed that the resulting parameter estimates and their covariance matrix usually converge in a few steps.

4.2.2. Induced smoothing for weighted logrank-type rank estimator

For the weighted logrank-type rank estimator β̂φ, n1/2(β̂φβ) is asymptotically zero-mean normal with covariance matrix $Aφ-1BφAφ-1$. Ying et al. (1992) showed that

$B^φ=1n∑i=1n{φi (β^)}2δi (∑j=1n1{ej (β^)≥ei (β^)}Xj⊗2∑j=1n1{ej (β^)≥ei (β^)}-[∑j=1n1{ej (β^)≥ei (β^)}Xj∑j=1n1{ej (β^)≥ei (β^)}]⊗2).$

The challenging part is to estimate Aφ. To estimate Aφ, we can introduce following induced smoothing estimating function similar to (4.8),

$U˜φ(V;β)=EWUφ (β+n-12V12W),$

where EW denotes the expectation with respect to W. However, the expression (4.11) is too complicated to be written out in simple analytic forms. Consequently, the estimation of Ãφ(V; β) = ∂Ũφ(V; β)/∂β is difficult to obtain. Using Stein’s identity, Jin et al. (2015) showed that

$A˜φ(V;β)=n-12 [EWUφ (β+n-12V12W) WT] V-12.$

Based on the result (4.12), Jin et al. (2015) showed a general Monte carlo approach that can be used to estimate Ãφ(V; β) as following:

• Step 1: Calculate a consistent estimator φ of the matrix Bφ.

• Step 2: Choose V0 = I and a large number m.

• Step 3: For the kth step (k ≥ 1), generate zj, j = 1, …, m from multivariate normal distribution N(0, I). Estimate Ãφ(V; β) by $Ak=A(Vk-1)=n121m∑j=1mU (β^+n-12Vk-1zj)zjTVk-1-12$

• Step 4: Calculate $Vk=Ak-1D^Ak-1$.

• Step 5: Repeat Step 3 and Step 4 for next k until Vk converges.

The covariance matrix of $n(β^-β)$ will be estimated using the Vk at the convergence in the above iterative algorithm.

In Jin et al. (2015), it is also shown that the Step 3 can be replaced by an approach based on Gaussian quadrature weights.

4.2.3. Induced smoothing for least-squares-type estimator

For the least-squares-type estimator β̂m, induced smoothing estimator can developed similar to the weighted logrank-type rank estimator in the Section 4.2.2. Here we present the results for the Buckley-James estimator β̂BJ of the estimating equation (3.15). Ying et al. (1992) showed that n1/2(β̂BJβ) is asymptotically zero-mean normal with covariance matrix $ABJ-1BBJABJ-1$. Ying et al. (1992) also showed that

$B^BJ=1n∫(∑i=1n(Xi-X¯)⊗2 1{ei (β^)≥t}-[∑i=1n(Xi-X¯)1{ei (β^)≥t}]⊗2∑i=1n1{ei (β^)≥t})×(∫(1-F^β^(s))1{s≥t}ds)2(1-F^β^(t))3dF^β^(t).$

Again, the challenging part is to estimate ABJ. To estimate ABJ, we can introduce following induced smoothing estimating function similar to (4.8) and (4.11),

$U˜BJ(V;β)=EWUBJ (β+n-12V12W),$

where EW denotes the expectation with respect to W. The expression (4.14) again does not have a simple analytic form. Consequently, the estimation of Ãφ(V; β) = ∂Ũφ(V; β)/∂β can be done a Monte Carlo method similar to the one used for the expression (4.12), i.e.,

$A˜BJ(V;β)=n12 [EWUBJ (β+n-12V12W) WT] V-12,$

see Jin et al. (2015). The remaining steps are straightforward as in the Section 4.2.2.

5. Other developments and remarks

In this paper, we have reviewed recently developed methods for point and variance estimation for the AFT model in the analysis of right censored data. The development is based on the assumption that errors are independent and identically distributed. The resampling approach offers valid inference but numerically intensive for large datasets, on the other hand, the induced smoothing approach is attractive due to its computational efficiency.

The paper of Jin and Ying (2004) studied asymptotic theory of rank estimation for AFT model under fixed censorship. Zhou (1992) and Jin (2007) studied M-estimation for the AFT models.

For the hypothesis testing framework, empirical likelihood approach has also been developed, see Zhou (2005a, 2005b), Zhou and Li (2008). For the heteroscedastic errors, Stute (1993, 1996) studied convergence properties of weighted estimators and Zhou et al. (2012) developed an empirical likelihood approach under a hypothesis testing framework.

For time-dependent covariates, there has been theoretical development, see Robins and Tsiatis (1992) and Lin and Ying (1995). The profile nonparametric likelihood approach of Zeng and Lin (2007) is applicable theoretically, but there has been no numerical investigation.

There are still many research problems, such as model checking, variable selection and efficient and reliable software development.

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