Additive hazards regression with censoring indicators missing at random

Abstract

In this article, the authors consider a semiparametric additive hazards regression model for right-censored data that allows some censoring indicators to be missing at random. They develop a class of estimating equations and use an inverse probability weighted approach to estimate the regression parameters. Nonparametric smoothing techniques are employed to estimate the probability of non-missingness and the conditional probability of an uncensored observation. The asymptotic properties of the resulting estimators are derived. Simulation studies show that the proposed estimators perform well. They motivate and illustrate their methods with data from a brain cancer clinical trial.

1. INTRODUCTION

In the analysis of failure time data, the cause of failure may be unknown for some subjects for a variety of reasons (e.g., autopsies were not performed or medical records were missing). We motivate and illustrate our methods with data on patients from a brain cancer clinical trial, where we evaluate the effect of two potential explanatory variables on a measure of quality of life. All patients were initially ambulatory, but over time some lost their mobility, some had a progression of their cancer, and some experienced both events. To assess quality of life, we define “survival time” as the time to non-ambulatory progression. Thus, patients who progressed and were no longer ambulatory contributed uncensored times, patients who progressed but were still ambulatory or who had not progressed by the end of the study contributed censored times, and patients who progressed but whose ambulatory status was unknown contributed times with missing censoring indicators. We apply our regression analysis to evaluate the effects of sex and age on the time to non-ambulatory progression.

Specifically, let T be the failure time, let Z be a p×1 vector of covariates, and let C be a censoring time that is assumed to be conditionally independent of T given Z. Data are available on Z and X = T ^C, but the censoring indicator δ = I(T ≤ C) may be missing. If the probability that δ is missing does not depend on either the true value of δ or the values of X and Z, then a missing δ is said to be missing completely at random (MCAR). Alternatively, if the probability that δ is missing depends on the values of X and Z but not on the true value of δ, then a missing δ is said to be missing at random (MAR); see Little & Rubin (1987).

Under the MCAR assumption and in the absence of covariates, Dinse (1982) obtained a nonparametric maximum likelihood estimator (NPMLE) of the survival function using an EM algorithm. Lo (1991) proved that there are infinitely many NPMLEs and some of them may be inconsistent; he consequently constructed a consistent and asymptotically normal estimator. Gijbels, Lin & Ying (1993, 2007) and McKeague & Subramanian (1998) proposed further improvements on these estimators. When covariates are present, Gijbels, Lin & Ying (1993) initiated research on estimation under the Cox model. McKeague & Subramanian (1998) provided an alternative approach to estimation. Subramanian (2000) considered estimation under proportionality of conditional hazards. Zhou & Sun (2003) studied the additive hazards regression model.

Under the MAR assumption, van der Laan & McKeague (1998) first addressed efficient estimation of the survival function and proposed a sieved nonparametric maximum likelihood estimator. Further developments along the lines of efficient estimation can be found in Subramanian (2004, 2006) and Wang & Ng (2008). Goetghebeur & Ryan (1995) and Lu & Tsiatis (2001) analyzed competing risks data with missing cause of failure under proportional hazards regression models. Gao & Tsiatis (2005) considered the linear transformation competing risks model with missing cause of failure. Recently, Lu & Liang (2008) studied competing risks data with missing cause of failure under the semiparametric additive hazards model, and suggested the inverse probability weighted (IPW) and double robust (DR) estimators. To obtain these estimators, however, they imposed parametric models for two components: the probability that the censoring indicator is not missing and the conditional probability of a given failure type.

In this article, we propose estimators for the regression parameters in a semiparametric additive hazards model, where the failure times are subject to right censoring and some censoring indicators are missing at random. We provide simple and fully augmented weighted estimators that incorporate incomplete data nonparametrically. Unlike Lu & Liang (2008), no parametric models are assumed for the missingness probability or the conditional probability of an uncensored observation; instead, we use nonparametric kernel smoothing techniques to estimate these probabilities. The resulting estimators have closed forms and are easy to implement. Under the usual MAR assumption, both the simple and fully augmented weighted estimators are consistent and asymptotically equivalent, i.e., they have the same asymptotic normal distribution. In addition, the asymptotic properties of the estimated baseline cumulative hazard function are also established for the model.

The remainder of the paper is organized as follows. Section 2 presents the simple and fully augmented weighted estimators and their asymptotic properties under the MAR assumption. Section 3 reports simulation results that show the proposed estimators perform well. In Section 4, our methods are applied to analyze the brain cancer data described earlier. Our concluding remarks follow in Section 5 and technical proofs are relegated to the Appendix.

2. ESTIMATION PROCEDURE

Under an additive hazards model, the hazard function for failure time T given covariate Z is assumed to be of the form

$$\lambda (t\mid Z)={\lambda }_0(t)+{\beta }_0^{\prime }Z,$$ (1)

where λ₀(t) is an unspecified baseline hazard function and β₀ is a p-vector of unknown regression parameters. In the case where all data are observed, Lin & Ying (1994) introduced a pseudoscore function for the parameter vector β₀ and showed that the resulting estimator is consistent and asymptotically normal, with an easily estimated covariance matrix.

When censoring indicators are missing for right-censored data, we observe n independent and identically distributed vectors (X_i, ξ_i, ξ_iδ_i, Z_i, R_i) (i = 1, …, n), where ξ_i is an indicator that δ_i is not missing, and R_i is an auxiliary covariate that is not used to model the hazard but may be used to describe the probability that δ_i is missing. The probability that δ_i is missing is characterized by the distribution of ξ_i given δ_i and W_i = (X_i, Z_i, R_i), which is Bernoulli with probability P{ξ_i = 1|δ_i, W_i = w}. Under the MAR assumption (Little & Rubin 1987), we have

$$P\{{\xi }_i=1\mid {\delta }_i,W_i=w\}=P\{{\xi }_i=1\mid W_i=w\}\equiv \rho (w).$$ (2)

Another function of interest is π(w) = P{δ_i = 1|W_i = w, ξ_i = 1}, which is the conditional probability of an uncensored observation, given that δ_i is observed and W_i = w.

A naive method for estimating β₀ is to simply ignore the missing data and to apply the pseudoscore function of Lin & Ying (1994) to the complete data only. Such a procedure (called the complete case estimator) may not only lose efficiency due to discarding incomplete observations, but may also generate biased estimators, even when the censoring indicators are MAR. If either ρ(w) or π(w) is modeled correctly, we can use the approach of Lu & Liang (2008) to obtain the IPW and DR estimators. In many situations, however, knowledge of ρ(w) and π(w) is limited, and thus both models may be misspecified. In this article, no parametric models are assumed for these two probabilities; rather, both are estimated nonparametrically by kernel smoothers. We begin by introducing the simple weighted estimator, which is derived under the MAR assumption.

Because ρ(W_i) is a function of continuous variables such as X_i, we estimate it with the Nadaraya-Watson estimator based on complete observations. Specifically, let d denote the number of continuous elements of W_i and let K be an rth-order (r > d) kernel function of d variables with finite support that satisfies ∫K(u)du = 1, ∫u^mK(u)du = 0, m = 1, …, r − 1, ∫u^rK(u)du ≠ 0, and ∫K(u)²du < ∞, where u can be a scalar or a vector. If u is a vector, say u = (u₁, …, u_d)′, then u^m denotes ${(u_1^m,\dots ,u_d^m)}^{\prime }$. The motivation for using higher-order kernels is to reduce the order of magnitude of the bias of the curve estimator, leading to a faster rate of convergence of the mean integrated squared error (Wand & Schucany 1990). This type of kernel function may be constructed in various manners. For instance, Wand & Schucany (1990) gave a univariate Gaussian-based kernel of order 2r:

$$K(u_1)=\frac{{(-1)}^r{\phi }^{(2r-1)}(u_1)}{2^{r-1}(r-1)!u_1},$$

where φ^(2r−1)(u₁) is the (2r − 1)-th derivative of the standard normal density function φ(u₁). Hall & Marron (1988) proposed a class of univariate kernels of order r:

$$K(u_1)={\pi }^{-1}{\int }_0^{\infty }cos({tu}_1)exp(-t^r)dt.$$

Some higher-order polynomial kernels can be found in Müller (1984) and Gasser, Müller & Mammitzsch (1985).

Define K_h(·) = K(·/h), where h is a bandwidth sequence, and K(u/h) = K(u₁/h, …, u_d/h) for u = (u₁, …, u_d)′. Write W_i = (W_1i, W_2i), where W_1i and W_2i include all continuous and discrete elements of W_i, respectively. Then the Nadaraya-Watson estimator of ρ(w) is given by

$$\widehat{\rho }(w)=\frac{{\sum }_{i=1}^n{\xi }_i{K}_h(w_1-W_{1i})I(W_{2i}=w_2)}{{\sum }_{i=1}^n{K}_h(w_1-W_{1i})I(W_{2i}=w_2)},$$ (3)

where w = (w₁, w₂). The choice of the kernel function K usually has little effect on the estimator ρ̂(w), and thus the estimator of β₀, but the bandwidth sequence h typically does influence these estimators, both theoretically and practically. We assume that h satisfies nh^2r → 0 and nh^2d → ∞ as n → ∞. If h = O(n^−1/p) for some integer p > 2d, then a reasonable choice for r is the smallest even integer such that r ≥ p − d (Qi, Wang & Prentice 2005). For example, when d = 2, we might choose p = 5 and r = 4. In a similar manner, we can estimate π(w) by

$$\widehat{\pi }(w)=\frac{{\sum }_{i=1}^n{\xi }_i{\delta }_i{K}_h(w_1-W_{1i})I(W_{2i}=w_2)}{{\sum }_{i=1}^n{\xi }_i{K}_h(w_1-W_{1i})I(W_{2i}=w_2)}.$$ (4)

Note that the kernel function K and bandwidth sequence h used in (3) need not be identical to those used in (4), and the bandwidth can be different for each component of W_1i. For example, we can define h = (h₁, …, h_d)′ for different bandwidths, and write K(u/h) = K(u₁/h₁, …, u_d/h_d). Here, we use the same K and h in both for notational convenience.

Let ${\mathrm{\Lambda }}_0(t)={\int }_0^t{\lambda }_0(s)ds$ denote the baseline cumulative hazard function. Using the inverse probability weighted approach, consider the following estimating equations for β₀ and Λ₀:

$$\sum _{i=1}^n{\int }_0^{\tau }\frac{{\xi }_i}{\widehat{\rho }(W_i)}{Z}_i\left[d{N}_i^u(t)-Y_i(t){\beta }^{\prime }{Z}_{i}dt-Y_i(t)d{\mathrm{\Lambda }}_0(t)\right]=0,$$ (5)

$$\sum _{i=1}^n\frac{{\xi }_i}{\widehat{\rho }(W_i)}\left[d{N}_i^u(t)-Y_i(t){\beta }^{\prime }{Z}_{i}dt-Y_i(t)d{\mathrm{\Lambda }}_0(t)\right]=0,$$ (6)

where $N_i^u(t)=I(X_i\le t,{\delta }_i=1)$, Y_i(t) = I(X_i ≥ t), and τ is a prespecified positive constant such that P(X_i ≥ τ) > 0. The resulting simple weighted estimators for β₀ and Λ₀ have the following closed forms:

$$\widehat{\beta }={\left[\sum _{i=1}^n{\int }_0^{\tau }\frac{{\xi }_i}{\widehat{\rho }(W_i)}{Y}_i(t){\{Z_i-\overline{Z}(t)\}}^{\otimes 2}dt\right]}^{-1}\sum _{i=1}^n{\int }_0^{\tau }\frac{{\xi }_i}{\widehat{\rho }(W_i)}\{Z_i-\overline{Z}(t)\}d{N}_i^u(t)$$

and

$${\widehat{\mathrm{\Lambda }}}_0(t)={\int }_0^t\frac{{\sum }_{i=1}^n\widehat{\rho }{(W_i)}^{-1}{\xi }_i[d{N}_i^u(s)-Y_i(s){\widehat{\beta }}^{\prime }{Z}_{i}ds]}{{\sum }_{i=1}^n\widehat{\rho }{(W_i)}^{-1}{\xi }_i{Y}_i(s)},$$

where a^⊗2 = aa′ for any vector a and

$$\overline{Z}(t)=\frac{{\sum }_{i=1}^n\widehat{\rho }{(W_i)}^{-1}{\xi }_i{Y}_i(t)Z_i}{{\sum }_{i=1}^n\widehat{\rho }{(W_i)}^{-1}{\xi }_i{Y}_i(t)}.$$

In practice, we often choose τ to be the largest observation time, say τ = max{X_i}.

Let z̄(t) = E[Y_i(t)Z_i(t)]/E[Y_i(t)]. Define N_i(t) = I(X_i ≤ t) and $B_i={\int }_0^{\tau }\{Z_i-\overline{z}(t)\}d{N}_i(t)$. The asymptotic properties of β̂ are given in the following theorem.

Theorem 1

Under regularity conditions (C1)–(C6), which are stated in the Appendix, β̂ is consistent and n^1/2(β̂−β₀) is asymptotically normal with mean zero and covariance matrix V = A⁻¹ΣA⁻¹ + A⁻¹Σ^*A⁻¹, where

$$\begin{array}{l}\mathrm{\sum }=E\left[{\int }_0^{\tau }{\{Z_i-\overline{z}(t)\}}^{\otimes 2}d{N}_i^u(t)\right],\\ {\mathrm{\sum }}^{\ast }=E\left[\pi (W_i)(1-\pi (W_i))(1-\rho (W_i))\rho {(W_i)}^{-1}{B}_i^{\otimes 2}\right],\end{array}$$

and

$$A=E\left[{\int }_0^{\tau }{Y}_i(t){\{Z_i-\overline{z}(t)\}}^{\otimes 2}dt\right].$$

Note that the first term in V is the asymptotic variance of the Lin & Ying (1994) estimator based only on the complete data (ξ_i ≡ 1) and the second term represents the effect of the missing censoring indicators. If we let ${\widehat{B}}_i={\int }_0^{\tau }\{Z_i-\overline{Z}(t)\}d{N}_i(t)$, then the covariance matrix V can be consistently estimated by V̂ = Â⁻¹(Σ̂ + Σ̂^*)Â⁻¹, where

$$\begin{array}{l}\widehat{\mathrm{\sum }}=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }\frac{{\xi }_i}{\widehat{\rho }(W_i)}{\{Z_i-\overline{Z}(t)\}}^{\otimes 2}d{N}_i^u(t),\\ {\widehat{\mathrm{\sum }}}^{\ast }=n^{-1}\sum _{i=1}^n\widehat{\pi }(W_i)(1-\widehat{\pi }(W_i))(1-\widehat{\rho }(W_i))\widehat{\rho }{(W_i)}^{-1}{\widehat{B}}_i^{\otimes 2},\end{array}$$

and

$$\widehat{A}=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }\frac{{\xi }_i}{\widehat{\rho }(W_i)}{Y}_i(t){\{Z_i-\overline{Z}(t)\}}^{\otimes 2}dt.$$

Define $d(t)={\int }_0^t\overline{z}(s)ds$ and

$$D(t)={\int }_0^t\left\{\frac{E[Y_i(s)Z_i^{\otimes 2}]}{E[Y_i(s)]}-\overline{z}{(s)}^{\otimes 2}\right\}{\beta }_{0}ds.$$

The asymptotic properties of Λ̂₀(t) are given in the next theorem.

Theorem 2

Under the assumptions of Theorem 1, Λ̂₀(t) converges in probability to Λ₀(t) uniformly in t ∈ [0, τ], and n^1/2{Λ̂₀(t) − Λ₀(t)g converges weakly on [0, τ] to a zero-mean Gaussian process with covariance function at (t, s) (t ≤ s) equal to

$$\begin{array}{l}\mathrm{\Gamma }(t,s)={\int }_0^t\frac{dE\{N_i^u(u)\}}{{(E\{Y_i(u)\})}^2}+E\left\{\pi (W_i)(1-\pi (W_i))(1-\rho (W_i))\rho {(W_i)}^{-1}{\int }_0^t\frac{d{N}_i(u)}{{(E\{Y_i(u)\})}^2}\right\}-d{(t)}^{\prime }{A}^{-1}\left\{\pi (W_i)(1-\pi (W_i))(1-\rho (W_i))\rho {(W_i)}^{-1}{\int }_0^s\frac{(Z_i-\overline{z}(u))d{N}_i(u)}{E\{Y_i(u)\}}\right\}\\ -d{(s)}^{\prime }{A}^{-1}\left\{\pi (W_i)(1-\pi (W_i))(1-\rho (W_i))\rho {(W_i)}^{-1}{\int }_0^t\frac{(Z_i-\overline{z}(u))d{N}_i(u)}{E\{Y_i(u)\}}\right\}+d{(t)}^{\prime }{A}^{-1}(\mathrm{\sum }+{\mathrm{\sum }}^{\ast })A^{-1}d(s)-d{(t)}^{\prime }{A}^{-1}D(s)-d{(s)}^{\prime }{A}^{-1}D(t).\end{array}$$

The covariance function Γ(t, s) can be consistently estimated by substituting β̂, ρ̂ and π̂ for the unknowns β₀, ρ and π in the appropriate empirical estimators, and by replacing the (unobserved) processes $N_i^u$ with $\widehat{\rho }{(W_i)}^{-1}{\xi }_i{N}_i^u$. For an individual with a given covariate vector z₀, the corresponding estimator of the survival function S(t, z₀) is

$$\widehat{S}(t,z_0)=exp\{-{\widehat{\mathrm{\Lambda }}}_0(t)-{\widehat{\beta }}^{\prime }{z}_{0}t\}.$$

Using the functional delta-method and Theorem 2, we can obtain the asymptotic properties of Ŝ(t, z₀), which can be applied to construct confidence bands for S(t, z₀).

When the missingness probability ρ(w) is known or a parametric model is specified for ρ(w), the simple weighted estimator uses only the complete case data (i.e., only individuals with ξ_i = 1), and the fully augmented weighted estimator (also called the double robust estimator) incorporates contributions from the incomplete observations (i.e., individuals with ξ_i = 0), thus the fully augmented weighted estimator is more efficient than the corresponding simple weighted estimator (Lu & Liang 2008). In addition, the fully augmented weighted estimator has the so-called double-robustness property; that is, the estimator is consistent if one can correctly specify either the missingness probability ρ(w) or the conditional probability of an uncensored observation π(w) (Wang & Chen 2001). However, estimating ρ(w) nonparametrically enables the simple weighted estimator β̂ to follow the same asymptotic distribution as the fully augmented weighted estimator β̂_a (described next). This indicates that β̂ is equivalent asymptotically to β̂_a. These conclusions are consistent with the results of Qi, Wang & Prentice (2005) for proportional hazards regression with missing covariates.

The fully augmented weighted estimators for β₀ and Λ₀ are the solutions to the following estimating equations:

$$\sum _{i=1}^n{\int }_0^{\tau }{Z}_i\left[\frac{{\xi }_i}{\widehat{\rho }(W_i)}d{N}_i^u(t)+\left(1-\frac{{\xi }_i}{\widehat{\rho }(W_i)}\right)\widehat{\pi }(W_i)d{N}_i(t)-Y_i(t){\beta }^{\prime }{Z}_{i}dt-Y_i(t)d{\mathrm{\Lambda }}_0(t)\right]=0,$$ (7)

$$\sum _{i=1}^n\left[\frac{{\xi }_i}{\widehat{\rho }(W_i)}d{N}_i^u(t)+\left(1-\frac{{\xi }_i}{\widehat{\rho }(W_i)}\right)\widehat{\pi }(W_i)d{N}_i(t)-Y_i(t){\beta }^{\prime }{Z}_{i}dt-Y_i(t)d{\mathrm{\Lambda }}_0(t)\right]=0.$$ (8)

The resulting fully augmented weighted estimators for β₀ and Λ₀ have the following closed forms:

$${\widehat{\beta }}_a={\left[\sum _{i=1}^n{\int }_0^{\tau }{Y}_i(t){\{Z_i-{\overline{Z}}^{\ast }(t)\}}^{\otimes 2}dt\right]}^{-1}\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-{\overline{Z}}^{\ast }(t)\}\times \left[\frac{{\xi }_i}{\widehat{\rho }(W_i)}d{N}_i^u(t)+\left(1-\frac{{\xi }_i}{\widehat{\rho }(W_i)}\right)\widehat{\pi }(W_i)d{N}_i(t)\right]$$

and

$${\widehat{\mathrm{\Lambda }}}_a(t)={\int }_0^t\frac{{\sum }_{i=1}^n\left[\widehat{\rho }{(W_i)}^{-1}{\xi }_{i}d{N}_i^u(s)+(1-\widehat{\rho }{(W_i)}^{-1}{\xi }_i)\widehat{\pi }(W_i)d{N}_i(s)-Y_i(s){\widehat{\beta }}_a^{\prime }{Z}_{i}ds\right]}{{\sum }_{i=1}^n{Y}_i(s)},$$

where

$${\overline{Z}}^{\ast }(t)=\frac{{\sum }_{i=1}^n{Z}_i{Y}_i(t)}{{\sum }_{i=1}^n{Y}_i(t)}.$$

Similar to Theorems 1 and 2, the asymptotic properties of β̂_a and Λ̂_a are given in the following theorem.

Theorem 3

Under the assumptions of Theorem 1, we have:

1. β̂_a is consistent and n^1/2(β̂_a − β₀) is asymptotically normal with mean zero and covariance matrix V:

2. Λ̂_a(t) converges in probability to Λ₀(t) uniformly in t ∈ [0, τ], and n^1/2{Λ̂_a(t) − Λ₀(t)} converges weakly on [0, τ] to a zero-mean Gaussian process with covariance function Γ(t, s) at (t, s) (t ≤ s).

For the fully augmented weighted method, the covariance matrix V and covariance function Γ(t, s) can be consistently estimated by substituting β̂_a, ρ̂ and π̂ for β₀, ρ and π in the appropriate empirical estimators, and by replacing the processes $N_i^u$ with $\widehat{\rho }{(W_i)}^{-1}{\xi }_i{N}_i^u+(1-{\xi }_i\widehat{\rho }{(W_i)}^{-1})\widehat{\pi }(W_i)N_i$.

Theorems 1, 2 and 3 show that both the simple and fully augmented weighted estimators have the same asymptotic normal distribution, and the resulting estimators of the baseline cumulative hazard function converge to the same Gaussian process. This means that the simple weighted estimators with nonparametric ρ̂(w) are as efficient as the kernel-assisted fully augmented weighted estimators. One intuitive explanation for this is that the incomplete observations are indirectly incorporated in the simple weighted estimator by using the inverse of ρ̂(w) as a weight.

Note that Λ̂₀(t) and Λ̂_a(t) may not always be monotonic in t, in which case simple modifications such as those discussed in Lin & Ying (1994) can be made to ensure monotonicity while preserving asymptotic properties.

3. SIMULATION STUDIES

We conducted simulation studies to examine and compare the finite-sample performance of the simple and fully augmented weighted estimators proposed in Section 2, and also to compare their performance with that of the full data and complete-case analyses under the MAR model. In these studies, we considered three situations for the covariate Z: (a) Z was assumed to follow a Bernoulli distribution with success probability 0.5; (b) Z was generated from a uniform distribution on (0, 1); (c) Z = (Z₁, Z₂)′, where Z₁ follows a uniform distribution on (0, 1) and Z₂ follows a Bernoulli distribution with success probability 0.5. The underlying additive hazards model for the failure time T was taken to be $\lambda (t\mid Z)=1+{\beta }_0^{\prime }Z$, where β₀ = 0, 0.5 and 1 for the case Z is a scalar, and β₀ = (0, 0)′ and β₀ = (1, −1)′ for the two-dimensional covariate. The censoring time C was generated from a uniform distribution on (0, c), where c was selected to give a censoring rate of either 15% or 55%.

The missingness indicators were generated from the logistic model

$$\rho (W)=\frac{exp({\theta }^{\prime }W)}{1+exp({\theta }^{\prime }W)},$$ (9)

where W = (X, Z), X = T ^ C, and θ was chosen to produce a missingness rate of 50% under each censoring level. When Z was a Bernoulli random variable, there was only one (d = 1) continuous element in W, and we used the univariate Gaussian kernel function K(u) = (2π)^−1/2 exp(−u²/2) and a bandwidth of h = 0.5n^−1/3, with sample size of n = 100. When Z was a uniform random variable or a two-dimensional covariate as in (c), there were two (d = 2) continuous elements in W, and we used the bivariate Gaussian-based kernel function of order 4 (Wand & Schucany 1990)

$$K(u_1,u_2)=\frac{1}{8\pi }(3-u_1^2)(3-u_2^2)exp(-(u_1^2+u_2^2)/2)$$ (10)

and a bandwidth vector of h = (h₁, h₂)′ = (1.5n^−1/5, n^−1/5)′, with sample size of n = 400. We took τ to be the largest observed value of X, so that all data were used in the analysis. All simulation studies were based on 1000 replications for each combination of parameters.

Our simulation results are summarized in Tables 1 and 2. In these tables, Bias is the sample mean of the estimate minus the true value; MSE is the sample mean of the squared differences between the estimate and the true value; and CP is the 95% empirical coverage probability for β₀ based on a normal approximation. Similar summaries for the full-data and complete-case estimators are calculated for comparison.

Table 1.

Simulation results for one covariate with a missingness rate of 50%

	Parameters	β0 = 1			β0 = 0.5			β0 = 0
CR	Estimate	Bias	MSE	CP	Bias	MSE	CP	Bias	MSE	CP
		Z ~ Bernoulli(0.5) with n=100
15%	Full	0.0322	0.1304	0.952	−0.0076	0.0816	0.948	0.0009	0.0506	0.960
	SWE	−0.0205	0.1563	0.939	−0.0033	0.1066	0.948	−0.0168	0.0603	0.943
	FAWE	0.0336	0.1569	0.945	−0.0049	0.0992	0.948	0.0020	0.0607	0.952
	CC	−0.3630	0.3356	0.795	−0.1729	0.1461	0.882	−0.0865	0.0836	0.936
55%	Full	−0.0154	0.2061	0.956	−0.0067	0.1459	0.944	−0.0027	0.0914	0.954
	SWE	0.0030	0.3652	0.948	0.0138	0.2194	0.941	0.0093	0.1631	0.951
	FAWE	0.0042	0.3536	0.950	0.0056	0.2199	0.945	−0.0010	0.1451	0.953
	CC	−0.1918	0.4250	0.913	−0.0788	0.2489	0.923	−0.1122	0.1805	0.932
		Z ~ uniform(0, 1) with n=400
15%	Full	0.0063	0.0810	0.958	0.0164	0.0565	0.961	0.0058	0.0359	0.953
	SWE	0.0133	0.1009	0.946	−0.0094	0.0622	0.939	0.0049	0.0401	0.934
	FAWE	0.0049	0.0957	0.941	0.0187	0.0615	0.956	0.0039	0.0395	0.942
	CC	−0.2304	0.1608	0.866	−0.1510	0.0988	0.918	−0.0962	0.0592	0.912
55%	Full	0.0199	0.1436	0.948	0.0061	0.0845	0.964	0.0052	0.0666	0.947
	SWE	−0.0052	0.2263	0.943	−0.0094	0.1288	0.947	0.0050	0.1130	0.933
	FAWE	0.0195	0.2258	0.950	−0.0034	0.1248	0.952	0.0002	0.1099	0.934
	CC	−0.2371	0.2695	0.910	−0.1839	0.1541	0.910	0.0624	0.1175	0.925

CR stands for censoring rate, Full stands for full data estimator, SWE stands for simple weighted estimator, FAWE stands for fully augmented weighted estimator, and CC stands for complete case estimator.

Table 2.

Simulation results for a two-dimensional covariate vector with a missingness rate of 50%

CR	Estimate	Bias	MSE	CP	Bias	MSE	CP
		β10 = 0			β20 = 0
15%	Full	0.0103	0.0369	0.947	−0.0050	0.0122	0.957
	SWE	−0.0129	0.0526	0.942	0.0092	0.0144	0.949
	FAWE	0.0091	0.0512	0.946	−0.0077	0.0145	0.938
	CC	−0.2222	0.0932	0.805	0.1091	0.0261	0.836
55%	Full	−0.0161	0.0640	0.958	0.0010	0.0212	0.956
	SWE	−0.0352	0.1044	0.949	0.0048	0.0346	0.940
	FAWE	−0.0231	0.1057	0.955	0.0014	0.0322	0.946
	CC	−0.1290	0.1122	0.809	0.1701	0.0605	0.835
		β10 = 1			β20 = −1
15%	Full	−0.0001	0.0537	0.957	−0.0053	0.0221	0.952
	SWE	−0.0100	0.0784	0.957	0.0062	0.0297	0.940
	FAWE	−0.0161	0.0805	0.950	0.0027	0.0302	0.955
	CC	−0.0951	0.0950	0.902	0.0797	0.0460	0.891
55%	Full	0.0113	0.0541	0.949	−0.0078	0.0221	0.953
	SWE	−0.0135	0.0796	0.961	0.0057	0.0309	0.933
	FAWE	0.0086	0.0810	0.948	−0.0014	0.0335	0.955
	CC	−0.0806	0.0930	0.908	0.0714	0.0450	0.887

CR stands for censoring rate, Full stands for full data estimator, SWE stands for simple weighted estimator, FAWE stands for fully augmented weighted estimator, and CC stands for complete case estimator.

Tables 1 and 2 show that the complete case estimator is highly biased in all situations, with coverage probabilities that are too small, whereas the simple and fully augmented weighted estimators are nearly unbiased, with very reasonable coverage probabilities. Furthermore, the simple and fully augmented weighted estimators have similar MSE values, which are only slightly larger than those of the full data estimator and are often much smaller than those of the complete case estimator. These results suggest that our proposed estimators are more efficient than the complete case estimator and are adequate for practical use. We also simulated data under different parameter configurations and obtained similar results.

We compared the proposed methods and the parametric approach of Lu & Liang (2008) under MAR and MCAR assumptions. Data were simulated under correctly and incorrectly specified parametric models, using the same setup as in Table 1 with a censoring rate of 55% and a missingness rate of 50%, where Z follows a Bernoulli distribution with a sample size of n = 200 and β₀ = 0 and 1. The results are presented in Table 3. In Table 3, LIPW1 and LIPW2 stand for the inverse probability weighted (IPW) estimators of Lu & Liang (2008) when using the logistic model and the constant model for ρ(w), respectively; LDR1 and LDR2 stand for the double robust (DR) estimators of Lu & Liang (2008) when using the logistic model and the constant model for ρ(w), respectively. In all cases, we used a constant model for π(w), which is misspecified.

Table 3.

Comparison of the proposed method with the parametric approach of Lu and Liang (2008) under MAR and MCAR for a missingness rate of 50%

	Parameters	β0 = 0			β0 = 1
	Estimates	Bias	MSE	CP	Bias	MSE	CP
MAR	Full	0.0040	0.0448	0.948	0.0063	0.1071	0.950
	SWE	0.0018	0.0669	0.955	0.0099	0.1670	0.945
	FAWE	0.0051	0.0677	0.953	0.0011	0.1669	0.947
	LIPW1	0.0077	0.0909	0.945	0.0013	0.2192	0.950
	LDR1	0.0031	0.0675	0.950	0.0006	0.1683	0.947
	LIPW2	0.6229	0.4311	0.139	1.1296	1.3814	0.087
	LDR2	0.2121	0.0843	0.813	0.5999	0.4515	0.471
MCAR	Full	0.0060	0.0447	0.961	0.0108	0.1079	0.958
	SWE	0.0089	0.0694	0.950	0.0080	0.1679	0.954
	FAWE	0.0104	0.0698	0.946	0.0083	0.1661	0.950
	LIPW2	0.0004	0.0906	0.944	0.0298	0.2099	0.942
	LDR2	0.0089	0.0686	0.946	0.0209	0.1686	0.954
	CC	0.0041	0.0932	0.955	0.0041	0.2191	0.955

CR stands for censoring rate, Full stands for full data estimator, CC stands for complete case estimator, SWE stands for simple weighted estimator, FAWE stands for fully augmented weighted estimator, LIPW1 and LIPW2 stand for the inverse probability weighted estimators of Lu and Liang (2008) when using the logistic model and the constant model for ρ(w), respectively, LDR1 and LDR2 stand for the double robust estimators of Lu and Liang (2008) when using the logistic model and the constant model for ρ(w), respectively, and in all cases a constant model is used for π(w).

It can be seen from Table 3 that the proposed methods are essentially unbiased in all the settings, and the parametric approach of Lu & Liang (2008) is also unbiased when the parametric model for ρ(w) is correctly specified. Furthermore, the proposed estimators are as efficient as the DR estimator of Lu & Liang (2008), and are more efficient than the IPW estimator of Lu & Liang (2008). When both ρ(w) and π(w) are misspecified, however, both the IPW and DR estimators of Lu & Liang (2008) are biased under MAR. The key advantage of our method is that it provides reasonable estimation without making parametric modeling assumptions about ρ(w) and π(w). Rather than assuming parametric models, our approach uses nonparametric smoothing techniques to estimate these probabilities. In addition, the proposed estimators are more efficient than the complete case estimator under MCAR. So, if MCAR is true, our proposed approach still works well and does not lose efficiency.

We also conducted simulation studies to examine the performance of the proposed methods when MNAR (missing not at random) is true. In the study, the setup was the same as in Table 1, where Z follows a uniform distribution on (0, 1) with β₀ = 0 and n = 400, except that the censoring rate was set to be 20%, and the missingness probability was given by

$$\rho (W,\delta )=\frac{exp({\theta }_1^{\prime }W+{\theta }_2\delta )}{1+exp({\theta }_1^{\prime }W+{\theta }_2\delta )},$$

where θ₁ and θ₂ were chosen to produce a missingness rate of either 20% or 50%. The results are summarized in Table 4. It can be seen from Table 4 that the proposed estimation procedures perform well when the missingness rate is low (say, 20%), but when the missingness rate is high (say, 50%), the proposed estimators are a little biased. However, the biases are relatively small compared to those of the complete case estimator.

Table 4.

Simulation results for the proposed method under MNAR

MR	Estimate	Full	SWE	FAWE	CC
20%	Bias	0.0023	0.0103	0.0184	−0.0119
	MSE	0.0385	0.0416	0.0424	0.0430
	CP	0.936	0.935	0.938	0.938
50%	Bias	−0.0012	0.0167	0.0562	−0.0891
	MSE	0.0383	0.0434	0.0491	0.0716
	CP	0.938	0.936	0.930	0.926

MR stands for missingness rate, Full stands for full data estimator, SWE stands for simple weighted estimator, FAWE stands for fully augmented weighted estimator, and CC stands for complete case estimator.

4. EXAMPLE: ANALYSIS OF BRAIN CANCER DATA

We applied our methods to the brain cancer data mentioned earlier. We analyzed the data on all 387 patients who entered the clinical trial with a form of brain cancer known as glioblastoma. Dinse (1982) used a subset of these data to illustrate his nonparametric maximum likelihood analysis, which did not account for covariates. All patients were ambulatory when they entered the trial, but over time some lost their mobility, some had a progression of their cancer, and some experienced both events. As a measure of quality of life, we defined “survival time” as the time to non-ambulatory progression, and we evaluated the effects of sex and age on this event time.

Of the 387 patients, 86 progressed and were non-ambulatory, 24 progressed but were still ambulatory, 220 did not progress by the end of the study, and 57 progressed but had an unknown ambulatory status. Thus, our analysis treated these outcomes as 86 uncensored times, 244 censored times, and 57 times with a missing censoring indicator. There were 144 women and 243 men, ranging in age from 14 to 74 years, and the length of time on study (or until progression) varied from 2 to 1088 days.

Let X be the observed time (in days), measured from the beginning of the trial, and let δ indicate whether the patient had progressed and was non-ambulatory. We defined Z₁ to be a binary indicator of the patient’s sex, which was 1 for men and 0 for women, and Z₂ to be the age at trial entry (in years), which was treated as a continuous covariate. Since W = (X, Z₁, Z₂) contains two continuous elements, we used the bivariate Gaussian-based kernel function of order 4 for K, as defined in (10), with a bandwidth vector of h = (h₁, h₂)′ = (34, 10)′. We used τ = 1088, which was the largest observed value of X.

The analysis of the brain cancer data is summarized in Table 5, which gives the results for our simple weighted estimator (SWE) and our fully augmented weighted estimator (FAWE). For comparison, Table 5 also gives the results of the complete case (CC) analysis. None of the three methods suggested that men and women had different hazard rates for non-ambulatory progression. On the other hand, our two estimators showed that age is important (p = 0.037 for SWE and p = 0.011 for FAWE), but the CC analysis did not (p = 0.367). Specifically, the hazard rate for non-ambulatory progression increased as patients grew older, which is consistent with worsening quality of life. The age coefficients were of similar magnitude for all three methods, but the standard error was much larger for the CC analysis than for our SWE and FAWE analyses. Thus, as a result of excluding data, the complete case analysis missed the age effect on non-ambulatory progression that our approaches appropriately identified.

Table 5.

Estimates of regression coefficients for sex and age (in years), along with estimated standard errors and significance levels, from the analysis of time (in days) to non-ambulatory progression for patients in the brain cancer clinical trial.

	CC		SWE		FAWE
	sex	age	sex	age	sex	age
β̂ × 104	1.570	0.081	1.589	0.077	1.708	0.087
SE(β̂) × 104	2.445	0.090	2.306	0.037	2.241	0.034
P-Value	0.521	0.367	0.491	0.037	0.446	0.011

Note: The sample size is n = 387; the bandwidth vector is h = (h₁, h₂)′ = (34, 10)′ CC denotes the complete case estimator; SWE denotes the simple weighted estimator; FAWE denotes the fully augmented weighted estimator.

5. CONCLUDING REMARKS

Model (1) has the limitation that the linear predictor ${\beta }_0^{\prime }Z$ needs to be constrained to ensure non-negativity for the right side of (1). One may avoid this constraint by using a nonnegative link function, such as ${\lambda }_0(t)+exp({\beta }_0^{\prime }Z)$. The ideas presented in this paper can be applied to any regression function $g({\beta }_0^{\prime }Z)$, where g(·) is a known link function. In addition, Our approach can be extended to incorporate missing covariates (Qi, Wang & Prentice 2005) in the situation where both the failure indicators and the covariates are partially observed.

Nonparametric kernel estimation can be done for a small number of continuous covariates, but for categorical covariates, it would usually require stratified kernel estimation within each strata defined by the categorical covariates. In practice, when there are too many categories, it may be desirable to specify a more flexible model for the missingness probability, such as a partially linear additive model, and then use local kernel regression to estimate the missingness probability. Here we focus on a kernel estimation approach for ρ(w) and π(w). Of course, other smoothing techniques such as the local polynomial method (Fan & Gijbels 1996) may be used and require the same assumptions. Furthermore, n^1/2-rate asymptotic normality of the proposed estimators indicates that an appropriate choice for the bandwidth sequence h depends only on the second order terms of the mean square error of the estimators, and thus bandwidth selection may not be critical for estimating β₀ and Λ₀.

Since the estimating functions in (5) to (8) were obtained in a somewhat ad hoc fashion, it might be worthwhile investigating possible improvements that could result from other approaches, such as the one suggested by McKeague & Sasieni (1994) or perhaps a nonparametric maximum likelihood approach. Alternatively, estimation procedures based on the general Aalen additive model (Aalen 1980) or the linear transformation model (Gao & Tsiatis 2005) with missing censoring information might also be worthy of investigation.

Another limitation of the approach given here is that the covariates Z are time-invariant. In some applications, we might want to incorporate time-dependent covariates. Thus, a more general approach might extend model (1) to a time-varying version:

$$\lambda (t\mid Z(t))={\lambda }_0(t)+{\beta }_0{(t)}^{\prime }Z(t),$$

where β₀(t) is an unknown p-vector of time-varying regression coefficients and Z(t) is a vector of covariates that may depend on time. However, the proposed estimation procedure cannot be extended in a straightforward manner to deal with time-dependent covariates because of the curse of dimensionality created by Z(t) and a need for alternative smoothing techniques for estimating β₀(t). In addition, when the dimension of Z(t) is high, the probabilities ρ(w) and π(w) can be modeled parametrically (Lu & Liang 2008). As a different approach, perhaps dimension-reduction techniques could be extended in conjunction with a partially linear model (Liang, Härdle & Carroll 1999) for ρ(w) and π(w).

APPENDIX

We will use the same notation defined in the previous sections and assume that the following regularity conditions hold:

• (C1) Λ₀(τ) < ∞, Pr(X ≥ τ) > 0, Z is bounded, and ${inf}_{0\le t\le \tau }[{\lambda }_0(t)+{\beta }_0^{\prime }Z]>0$ a.e.

• (C2) The probability (density) f(w) of W_i is bounded away from 0, and has r continuous and bounded partial derivatives with respect to the continuous components of W_i a.e.

• (C3) The missingness probability ρ(w) is bounded away from 0, and has r continuous and bounded partial derivatives with respect to the continuous components of W_i a.e.

• (C4) The conditional probability π(w) has r continuous and bounded partial derivatives with respect to the continuous components of W_i a.e.

• (C5) $A=E[{\int }_0^{\tau }{Y}_i(t){\{Z_i-\overline{z}(t)\}}^{\otimes 2}dt]$ is nonsingular.

• (C6) nh^2r → 0 and nh^2d → ∞, as n → ∞.

We give the proof of Theorem 3 and outline the proof of Theorem 1; Theorem 2 can be proven in the same manner. For notational convenience, we assume that all components of W_i are continuous in the following proof.

Proof of Theorem 3(i)

Substituting Λ̂_a into equation (7), we find that β̂_a is the solution to U (β) = 0, where

$$U(\beta )=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-{\overline{Z}}^{\ast }(t)\}\left[\frac{{\xi }_i}{\widehat{\rho }(W_i)}d{N}_i^u(t)+\left(1-\frac{{\xi }_i}{\widehat{\rho }(W_i)}\right)\widehat{\pi }(W_i)d{N}_i(t)-Y_i(t){\beta }^{\prime }{Z}_{i}dt\right].$$

Let $M_i(t)=N_i^u(t)-{\int }_0^t{Y}_i(s)\{{\lambda }_0(s)+{\beta }_0^{\prime }{Z}_i\}ds$. Then we can write

$$U({\beta }_0)=U_1({\beta }_0)+U_2({\beta }_0)+U_3({\beta }_0),$$ (A.1)

where

$$\begin{array}{l}{U}_1({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-\overline{Z}(t)\}d{M}_i(t),\\ U_2({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-\overline{Z}(t)\}\left(\frac{{\xi }_i}{\widehat{\rho }(W_i)}-1\right)d{N}_i^u(t),\end{array}$$

and

$$U_3({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-\overline{Z}(t)\}\left(1-\frac{{\xi }_i}{\widehat{\rho }(W_i)}\right)\widehat{\pi }(W_i)d{N}_i(t).$$

Note that U₁(β₀) is a martingale integral. Thus, it follows that

$$U_1({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-\overline{z}(t)\}d{M}_i(t)+o_p(n^{1/2}).$$ (A.2)

Define ${\mathrm{\Phi }}_n(t)=n^{-1}{\sum }_{i=1}^n(\widehat{\rho }{(W_i)}^{-1}{\xi }_i-1)N_i^u(t)$, and write Φ_n(t) = Φ_n1(t) + Φ_n2(t) + Φ_n3(t), where

$$\begin{array}{l}{\mathrm{\Phi }}_{n1}(t)=n^{-1}\sum _{i=1}^n\left({\xi }_i-\rho (W_i)\right)\frac{N_i^u(t)}{\rho (W_i)},\\ {\mathrm{\Phi }}_{n2}(t)=n^{-1}\sum _{i=1}^n\left(\rho (W_i)-\widehat{\rho }(W_i)\right)\frac{N_i^u(t)}{\rho (W_i)},\end{array}$$

and

$${\mathrm{\Phi }}_{n3}(t)=n^{-1}\sum _{i=1}^n\left({\xi }_i-\widehat{\rho }(W_i)\right)\left(\rho (W_i)-\widehat{\rho }(W_i)\right)\frac{N_i^u(t)}{\widehat{\rho }(W_i)\rho (W_i)}.$$

By the uniform strong law of large numbers (Pollard, 1990), sup_0≤t≤τ |Φ_n1(t)| = o(1) almost surely. It can be checked that

$${\mathrm{\Phi }}_{n2}(t)=n^{-2}\sum _{i=1}^n\sum _{j=1}^n\frac{(\rho (W_i)-{\xi }_j)K_h(W_i-W_j)N_i^u(t)}{\rho (W_i)h^d\widehat{f}(W_i)},$$

where f̂(w) = (nh^d)⁻¹K_h(w − W_i), which is a kernel density estimate of f(w). By a Taylor expansion of 1/f̂(W_i) about f (W_i), Φ_n2(t) can be written as Φ_n21(t) − Φ_n22(t) + o_p(1), where

$${\mathrm{\Phi }}_{n21}(t)=n^{-2}\sum _{i=1}^n\sum _{j=1}^n\frac{(\rho (W_i)-{\xi }_j)K_h(W_i-W_j)N_i^u(t)}{\rho (W_i)h^{d}f(W_i)},$$

and

$${\mathrm{\Phi }}_{n22}(t)=n^{-2}\sum _{i=1}^n\sum _{j=1}^n\frac{(\rho (W_i)-{\xi }_j)K_h(W_i-W_j)N_i^u(t)(\widehat{f}(W_i)-f(W_i))}{\rho (W_i)h^{d}f{(W_i)}^2}.$$

A straightforward calculation yields that E{Φ_n21(t)} = O(h^r) → 0, and Var{Φ_n21(t)} = O(h^2r + (nh^2d)⁻¹) → 0, which imply Φ_n21(t) = o_p(1). Similarly, Φ_n22(t) = o_p(1), and thus it follows that Φ_n2(t) = o_p(1). Likewise, Φ_n3(t) = o_p(1). Therefore, we have Φ_n(t) = o_p(1). Note that Φ_n(t) is monotone and bounded in t. Consequently, we obtain

$$\underset{0\le t\le \tau }{sup}\mid {\mathrm{\Phi }}_n(t)\mid =o_p(1).$$ (A.3)

The functional central limit theorem (Pollard 1990) implies that

$$\underset{0\le t\le \tau }{sup}\mid {\overline{Z}}^{\ast }(t)-\overline{z}(t)\mid =O_p(n^{-1/2}).$$ (A.4)

Using (A.3) and (A.4), we have

$${\int }_0^{\tau }\{{\overline{Z}}^{\ast }(t)-\overline{z}(t)\}d{\mathrm{\Phi }}_n(t)=o_p(n^{-1/2}).$$

Hence,

$$\begin{array}{l}{U}_2({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-\overline{z}(t)\}\left(\frac{{\xi }_i}{\widehat{\rho }(W_i)}-1\right)d{N}_i^u(t)-n{\int }_0^{\tau }\{{\overline{Z}}^{\ast }(t)-\overline{z}(t)\}d{\mathrm{\Phi }}_n(t)\\ =\sum _{i=1}^n\left(\frac{{\xi }_i}{\widehat{\rho }(W_i)}-1\right){\delta }_i{B}_i+o_p(n^{1/2}).\end{array}$$ (A.5)

In a similar manner, we obtain

$$U_3({\beta }_0)=\sum _{i=1}^n\left(1-\frac{{\xi }_i}{\widehat{\rho }(W_i)}\right)\widehat{\pi }(W_i)B_i+o_p(n^{1/2}).$$ (A.6)

Thus, it follows from (A.1), (A.2), (A.5) and (A.6) that

$$\begin{array}{l}U({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-\overline{z}(t)\}d{M}_i(t)+\sum _{i=1}^n\left(\frac{{\xi }_i}{\widehat{\rho }(W_i)}-1\right)({\delta }_i-\widehat{\pi }(W_i))B_i+o_p(n^{1/2})\\ =\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-\overline{z}(t)\}d{M}_i(t)+\sum _{i=1}^n\left(\frac{{\xi }_i}{\rho (W_i)}-1\right)({\delta }_i-\pi (W_i))B_i+R_{n1}+R_{n2}+o_p(n^{1/2}),\end{array}$$

where

$$R_{n1}=\sum _{i=1}^n\left(1-\frac{{\xi }_i}{\rho (W_i)}\right)(\widehat{\pi }(W_i)-\pi (W_i))B_i,$$

and

$$R_{n2}=\sum _{i=1}^n\left(\widehat{\rho }(W_i)-\rho (W_i)\right)(\widehat{\pi }(W_i)-{\delta }_i)\frac{{\xi }_i{B}_i}{\widehat{\rho }(W_i)\rho (W_i)}.$$

Let m(w) = ρ(w)f(w) and $\widehat{m}(w)={({nh}^d)}^{-1}{\sum }_{i=1}^n{\xi }_i{K}_h(w-W_i)$. Then by the Taylor expansion of 1/m̂W_i) at m(W_i), we can write

$$\sum _{i=1}^n(\widehat{\pi }(W_i)-\pi (W_i))B_i=R_{n11}+R_{n12}+o_p(n^{1/2}),$$

where

$$R_{n11}=n^{-1}\sum _{i=1}^n\sum _{j=1}^n\frac{{\xi }_j({\delta }_j-\pi (W_i))K_h(W_i-W_j)B_i}{h^{d}m(W_i)},$$

and

$$R_{n12}=-n^{-1}\sum _{i=1}^n\sum _{j=1}^n\frac{{\xi }_j({\delta }_j-\pi (W_i))K_h(W_i-W_j)(\widehat{m}(W_i)-m(W_i))B_i}{h^{d}m{(W_i)}^2}.$$

Define

$$R_{n11}^{\ast }=n^{-1/2}{R}_{n11}-n^{-1/2}\sum _{i=1}^n\frac{{\xi }_i({\delta }_i-\pi (W_i))B_i}{\rho (W_i)}.$$

Some straightforward calculation gives $E\{R_{n11}^{\ast }\}=O(n^{1/2}{h}^r)\to 0$, and $\mathit{Var}\{R_{n11}^{\ast }\}=O({nh}^{2r}+{({nh}^2)}^{-1})\to 0$, which imply that

$$R_{n11}=\sum _{i=1}^n\frac{{\xi }_i({\delta }_i-{\pi }_i(W_i))B_i}{\rho (W_i)}+o_p(n^{1/2}).$$

Similarly, we have R_n12 = o_p(n^1/2), and thus

$$\sum _{i=1}^n(\widehat{\pi }(W_i)-\pi (W_i))B_i=\sum _{i=1}^n\frac{{\xi }_i({\delta }_i-{\pi }_i(W_i))B_i}{\rho (W_i)}+o_p(n^{1/2}).$$ (A.7)

In a similar manner, we obtain

$$\sum _{i=1}^n\frac{{\xi }_i}{\rho (W_i)}\left(\widehat{\pi }(W_i)-\pi (W_i)\right)B_i=\sum _{i=1}^n\frac{{\xi }_i({\delta }_i-{\pi }_i(W_i))B_i}{\rho (W_i)}+o_p(n^{1/2}).$$ (A.8)

It follows from (A.7) and (A.8) that R_n1 = o_p(n^1/2). Likewise, R_n2 = o_p(n^1/2). Thus,

$$U({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-\overline{z}(t)\}d{M}_i(t)+\sum _{i=1}^n\left(\frac{{\xi }_i}{\rho (W_i)}-1\right)({\delta }_i-\pi (W_i))B_i+o_p(n^{1/2}).$$ (A.9)

The law of large numbers and the multivariate central limit theorem show that n⁻¹U(β₀) → 0 in probability and n^−1/2U(β₀) converges in distribution to a normal random variable with mean zero and variance matrix Σ + Σ^*. Note that

$${\widehat{\beta }}_a-{\beta }_0=-\frac{\partial U(\beta )}{\partial \beta }U({\beta }_0),$$

and

$$-n^{-1}\frac{\partial U(\beta )}{\partial \beta }=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }{Y}_i(t){\{Z_i-{\overline{Z}}^{\ast }(t)\}}^{\otimes 2}dt\to A$$

almost surely by the uniform strong law of large numbers (Pollard 1990). Then it follows from (A.9) that β̂_a is consistent and n^1/2(β̂_a − β₀) is asymptotically normal with mean zero and covariance matrix V = A⁻¹(Σ + Σ^*)A⁻¹.

Proof of Theorem 3(ii)

First write

$${\widehat{\mathrm{\Lambda }}}_a(t)-{\mathrm{\Lambda }}_0(t)={\int }_0^t\frac{{\sum }_{i=1}^{n}d{M}_i(s)}{{\sum }_{i=1}^n{Y}_i(s)}-{({\widehat{\beta }}_a-{\beta }_0)}^{\prime }{\int }_0^t{\overline{Z}}^{\ast }(s)ds+\sum _{i=1}^n\left(\frac{{\xi }_i}{\widehat{\rho }(W_i)}-1\right)({\delta }_i-\widehat{\pi }(W_i)){\int }_0^t\frac{d{N}_i(s)}{{\sum }_{i=1}^n{Y}_i(s)}.$$

Note that

$$\underset{0\le t\le \tau }{sup}\left|n^{-1}\sum _{i=1}^n{Y}_i(t)-E[Y_1(t)]\right|=O_p(n^{-1/2}).$$

Following similar arguments as in the proof of (i), we obtain

$${\widehat{\mathrm{\Lambda }}}_a(t)-{\mathrm{\Lambda }}_0(t)=n^{-1}{\int }_0^t\frac{{\sum }_{i=1}^{n}d{M}_i(s)}{E\{Y_i(s)\}}-d{(t)}^{\prime }({\widehat{\beta }}_a-{\beta }_0)+n^{-1}\sum _{i=1}^n\left(\frac{{\xi }_i}{\rho (W_i)}-1\right)({\delta }_i-\pi (W_i)){\int }_0^t\frac{d{N}_i(s)}{E\{Y_i(s)\}}+o_p(n^{-1/2})$$ (A.10)

uniformly on [0, τ]. In view of the consistency of β̂_a, it follows from the uniform strong law of large numbers and the multivariate central limit theorem that sup_0≤t≤τ |Λ̂_a(t) − Λ₀(t)| → 0 in probability, and n^1/2{Λ̂_a(t) − Λ₀(t)} converges in finite dimensional distributions to a zero-mean Gaussian process. The first term on the right-hand side of (A.10) is tight as it is a martingale integral. The second term is tight because n^1/2(β̂_a − β₀) converges in distribution and d(t) is a deterministic function. Note that for each i, (ξ_iρ(W_i)⁻¹ − 1) (δ_i − π(W_i)) ${\int }_0^{t}d{N}_i(s)/E\{Y_i(s)\}$ can be written as sums of monotone processes. Then the tightness of the third term follows from Example 2.11.16 of van der Vaart & Wellner (1996). Thus, n^1/2{Λ̂_a(t)− Λ₀(t)} is tight and converges weakly to a zero-mean Gaussian process with covariance function Γ(s, t) at (s, t).

Outlined proof of Theorem 1

Note that β̂ is the solution to U^*(β) = 0, where

$$U^{\ast }(\beta )=\sum _{i=1}^n{\int }_0^{\tau }\frac{{\xi }_i}{\widehat{\rho }(W_i)}\left\{Z_i-\overline{Z}(t)\right\}\left[d{N}_i^u(t)-Y_i(t){\beta }^{\prime }{Z}_{i}dt\right].$$

Then it can be checked that

$$U^{\ast }({\beta }_0)=U_1^{\ast }({\beta }_0)+U_2^{\ast }({\beta }_0),$$ (A.11)

where

$$U_1^{\ast }({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\frac{{\xi }_i}{\rho (W_i)}\left\{Z_i-\overline{Z}(t)\right\}d{M}_i(t),$$

and

$$U_2^{\ast }({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }{\xi }_i\left(\frac{1}{\widehat{\rho }(W_i)}-\frac{1}{\rho (W_i)}\right)\left\{Z_i-\overline{Z}(t)\right\}d{M}_i(t).$$

Similarly to (A.2), we get

$$U_1^{\ast }({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\frac{{\xi }_i}{\rho (W_i)}\left\{Z_i-\overline{z}(t)\right\}d{M}_i(t)+o_p(n^{1/2}).$$ (A.12)

From an argument similar to that in the proof of (A.7), we have

$$U_2^{\ast }({\beta }_0)=-\sum _{i=1}^n{\int }_0^{\tau }\frac{{\xi }_i-\rho (W_i)}{\rho (W_i)}\left\{Z_i-\overline{z}(t)\right\}d{M}_i(t)+\sum _{i=1}^n\left(\frac{{\xi }_i}{\rho (W_i)}-1\right)({\delta }_i-\pi (W_i))B_i+o_p(n^{1/2}).$$ (A.13)

It follows from (A.11)–(A.13) that

$$U^{\ast }({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i-\overline{z}(t)\}d{M}_i(t)+\sum _{i=1}^n\left(\frac{{\xi }_i}{\rho (W_i)}-1\right)({\delta }_i-\pi (W_i))B_i+o_p(n^{1/2}),$$

which implies that n^−1/2U^*(β₀) converges in distribution to a normal random variable with mean zero and variance matrix Σ + Σ^*. Then it follows from the Taylor expansion of U^*(β̂) that n^1/2(β̂ − β₀) is asymptotically normal with mean zero and covariance matrix V = A⁻¹(Σ + Σ^*)A⁻¹.