Estimating Regression Parameters in an Extended Proportional Odds Model

Abstract

The proportional odds model may serve as a useful alternative to the Cox proportional hazards model to study association between covariates and their survival functions in medical studies. In this article, we study an extended proportional odds model that incorporates the so-called “external” time-varying covariates. In the extended model, regression parameters have a direct interpretation of comparing survival functions, without specifying the baseline survival odds function. Semiparametric and maximum likelihood estimation procedures are proposed to estimate the extended model. Our methods are demonstrated by Monte-Carlo simulations, and applied to a landmark randomized clinical trial of a short course Nevirapine (NVP) for mother-to-child transmission (MTCT) of human immunodeficiency virus type-1 (HIV-1). Additional application includes analysis of the well-known Veterans Administration (VA) Lung Cancer Trial.

1. INTRODUCTION

Between November, 1997 and January, 2001, a landmark HIV/AIDS randomized prevention trial, namely the HIVNET 012, was conducted to evaluate treatment efficacy and safety of a short course NVP versus a short course zidovudine (AZT) in prevention of MTCT of HIV-1 among infants born to HIV-1 infected pregnant women in Uganda (Jackson et al., 2003). One important secondary objective of this trial is to assess infant survival between treatment arms (The HIVNET/HPTN Group, 2003). In Figure 1, we plot the Kaplan-Meier estimates of infant survival in log-log scale for both of the two treatment arms. It appears that NVP improves infant survival over time.

Figure 1.

Kaplan-Meier estimates of HIVNET 012 infant survival

To estimate a treatment effect, such as comparing infant survival between the NVP and the AZT, the semiparametric Cox proportional hazards model (Cox, 1972) is often used:

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

where Z^* is the treatment indicator of NVP (Z^* = 1) versus AZT (Z^* = 0), λ(·|Z^*) is the hazard function of Z^*, and λ₀(·) is the unspecified baseline hazard function of Z^* = 0. Here, β ∈ ℬ ⊂ R is the regression parameter to measure the treatment effect on hazard functions of infant survival. For the HIVNET 012, the partial-likelihood estimate of β̂ is −0.301 (s.e. = 0.236). It means that NVP would reduce the hazard by 26.0% ((1 − e^−0.301) × 100%), implying a longer infant survival compared with AZT, although such an improvement is not statistically significant (p > 0.20).

A useful alternative to the Cox model is the so-called proportional odds model, which models survival functions directly (Bennett, 1983; Pettitt, 1984):

$$log\frac{S(t\mid Z^{\ast })}{1-S(t\mid Z^{\ast })}=log\frac{S_0(t)}{1-S_0(t)}+\beta Z^{\ast },$$ (2)

where S(·|Z^*) is the survival function of Z^* and S₀(·) is the baseline survival function of Z^* = 0. In this model, the regression parameter β has an interpretation of survival odds ratio. In practice, such an interpretation in survival functions has a direct appeal to measuring disease risks in clinical and epidemiological studies. For example, a generalized Kolmogorov-Smirnov test procedure in Fleming, O’Fallen and O’Brien (1980) was proposed to compare survival functions via Nelson-Aalen estimates (Nelson, 1969). Additional examples of directly modeling survival functions include McCullagh (1980), Jung (1996), and a recent work of Peng and Huang (2007).

In the literature, statistical inferences for the regression parameter of interest, β, in model (2) have been extensively developed. For example, Murphy, Rossini and van der Vaart (1997) studied a maximum likelihood method based on profile likelihood; Yang and Prentice (1999) used estimating equations based on the weighted Nelson-Aalen estimates of baseline survival functions. Other works include Dabrowska and Doksum (1988), Scharfstein et al. (1998) and Zhang and Davidian (2007). Bayesian methods have also been developed (Hanson and Yang, 2007). More notably, some of the works, including Cheng, Wei and Ying (1995), Chen, Jin and Ying (2002), Kosorok, Lee and Fine (2004) and Zeng and Lin (2006, 2007), have studied model (2) under the framework of transformation models. For example, Chen, Jin and Ying (2002) developed simple martingale-based estimating equations to estimate the regression parameter, and Zeng and Lin (2006, 2007) applied a nonparametric maximum likelihood estimation (NPMLE) method to estimate the regression parameter efficiently. More references can also be found therein.

In reality, covariates are not necessarily limited to be time-independent. In the HIVNET 012, for example, follow-up information, such as maternal CD4+ counts and maternal HIV-RNA viral loads (VL) that may affect infant survival through late postnatal breastfeeding, has been collected. Other time-varying covariates, such as treatment-time interaction, can also be of practical interest, since a hypothesis testing on treatment-time interaction may serve the purpose to check the adequacy of model (2). Nevertheless, these time-varying covariates, along with the assigned treatment and other baseline covariates, may all be potentially associated with infant survival. In this paper, we hence aim to extend the proportional odds model (2) to include such time-varying covariates and develop appropriate inference procedures for the regression parameters in analysis of many datasets similar to the HIVNET 012.

We organize the rest of this article as follows. In §2, we propose an extended proportional odds model and examine a few of its properties. In §3, we present two methods, including a simple quasi partial score estimating equation approach, and a maximum likelihood estimation approach, for model inferences. In §4, we present results from our numerical studies, including Monte-Carlo simulations and methods application to the motivating HIVNET 012 Trial and the well-known Veterans Administration (VA) Lung Cancer Trial. More discussion on the extended proportional odds model is in §5. Technical proof is collected in the Appendix.

2. THE EXTENDED PROPORTIONAL ODDS MODEL

The proportional odds model (2) can be considered as a natural extension of the logistic regression model of binary outcomes to time-to-event outcomes. Suppose that T represents a non-negative time-to-event outcome. At a given time t > 0, denote the binary survival indicator of T by D(t) = I(T > t). For Z^*, a usual logistic regression model of D(t) would then assume that

$$\text{logit}\text{pr}\{D(t)=1\mid Z^{\ast }\}=log\frac{\text{pr}\{D(t)=1\mid Z^{\ast }\}}{1-\text{pr}\{D(t)=1\mid Z^{\ast }\}}=\alpha (t)+\beta (t)Z^{\ast },$$ (3)

where α(t) and β(t) are both parameters at t > 0. When β(·) is assumed constant over time, i.e., β(t) ≡ β, then model (3) becomes the proportional odds model (2),

$$\text{logit}S(t\mid Z^{\ast })=\alpha (t)+\beta Z^{\ast },$$ (4)

with α(t) = log[S₀(t)/{1 − S₀(t)}]. When α(·) is unspecified, model (4) is semiparametric.

As discussed earlier, it is of practical interest to include time-varying covariates in the proportional odds model (2), such as maternal CD4+ counts, maternal plasma HIV-RNA VL, or treatment-time interaction in the HIVNET 012 example. Let Z^*(t) be the p–dimensional covariates that may include both baseline and time-varying covariates measured at t. To make Z^*(·)’s association with the survival function S(·) meaningful, we consider the so-called “external” covariates, because pr{T ≥ t |Z^*(t)} ≡ 1 for any “internal” covariates provided that Z^*(·) is not of the baseline level. Formally, the external time-varying covariates shall satisfy that

$$\text{pr}\{{\mathit{Z}}^{\ast }(s)\mid {\mathit{Z}}^{\ast }(t),T\ge t\}=\text{pr}\{{\mathit{Z}}^{\ast }(s)\mid {\mathit{Z}}^{\ast }(t),T=t\},\text{for}\text{any}s\ge t,$$

as in Kalbfleisch and Prentice (2002, pp. 196–198). Intuitively speaking, this condition essentially requires that the occurrence of a failure event at t should not affect the future path of Z^*(·) up to anytime beyond t. This means, in the HIVNET 012 example, whether an infant survives or dies at t shall not affect the mother’s future CD4+ counts or viral loads after t.

Denote Z̃(t) the covariate history of {Z^*(s), 0 ≤ s ≤ t} up to t and Z̃ = Z̃(∞). We consider an extended proportional odds model to include the time-varying covariates Z^*(·), as in

$$\text{logit}S(t\mid \stackrel{\sim }{\mathit{Z}})=-logR(t)+{\mathit{\beta }}^{\text{T}}\mathit{Z}(t),$$ (5)

where $\mathit{Z}(t)={\int }_0^t{\mathit{Z}}^{\ast }(u)d\phi (u,t)$ is a weighted average of Z^*(·) on [0, t], and φ(u; t) is the pre-specified weight function such that ${\int }_0^{t}d\phi (u;t)=1$. Here, R(t) = exp{−α(t)} is the baseline odds function when Z(·) ≡ 0, and the superscript _T defines matrix (vector) transpose.

Apparently in this extended model, with different choices of weight function φ(u; t), Z(t) may cover a wide range of covariate types in practice. Examples include:

1. φ(u; t) = I(u = 0). This means that Z(t) = Z(0) are for time-independent baseline covariates only, and the extended model reduces to model (2);

2. φ(u; t) = I(u = t). This means that Z(t) = Z^*(t) are the value of covariates measured at time t;

3. φ(u; t) = u/t. This means that $\mathit{Z}(t)=1/t{\int }_0^t{\mathit{Z}}^{\ast }(u)du$ are the average value of Z^*(·) on [0, t];

4. Suppose that Z^*(·) are only measured at a finite number of distinct time points, t₍₀₎ < · · · < t_(j−1) < t_(j) < · · · < t_(k). Let $\phi (u;t)={\sum }_{j=1}^{k}I\{t_{(j-1)}\le u=t<t_{(j)}\}$. Then $\mathit{Z}(t)={\int }_0^t{\mathit{Z}}^{\ast }(u)d\phi (u;t)$ are the so-called last-observation-carry-forward covariates.

Whatever φ(u; ·) is chosen in Z(·), β’s interpretation is straightforward: it is the odds ratio of survival probabilities associated with per unit change in Z(·). When β > 0 (< 0), it means that larger Z(·) are associated with increased (decreased) survival and hence decreased (increased) risk; when β = 0, there is no survival risk associated with Z(·).

Two properties of model (5) are summarized as follows:

Property 1

Under the assumed extended proportional odds model (5),

1. when Z(t) ≡ Z(0) for any t ≥ 0, the family of log-logistic survival distributions = {S(·; θ); S(t; θ) = θ₁t^−θ₂(1 + θ₁t^−θ₂)⁻¹, θ₁, θ₂ > 0} is closed for model (5). That is, if a baseline survival function S₀ ∈ and model (5) holds, then S(·|Z̃) ∈ ;

2. let λ₀(·) be the baseline hazard function of Z (·) ≡ 0 and λ(·|Z̃) be the hazard function of Z̃, then lim_t→0 λ(t|Z̃)/λ₀(t) = exp{β^TZ^*(0)}, and lim_t→∞ λ(t|Z̃)/λ₀(t) = 1, when Z(·) have bounded variations.

Proof of these two properties is relatively straightforward. For Property 1a, it is easy to verify that, if S₀(t) = θ₁t^−θ₂ (1 + θ₁t^−θ₂)⁻¹, then

under model (4). Property 1b follows an application of the usual l’Hôpital’s rule.

These two properties provide some basic characterization of the extended proportional odds model. Property 1a shows that the regression parameter of the proportional odds model (2) would only affect the scale parameter in baseline log-logistic distribution functions, in the same way that the proportional hazards model would do to baseline Weibull distribution functions. Property 1b shows the so-called “converging-hazards” feature in the extended proportional odds model: under the extended model, hazard functions differ by a ratio of β at the initial time zero, and then converge to each other as time increases. This converging-hazards feature has been well recognized (Murphy, Rossini and van der Vaart, 1997). It can be particularly useful if the assumption of constant proportionality seems restrictive for the Cox model, as recommended by Yang and Prentice (1999).

3. INFERENCE PROCEDURES

Consider that study data are collected in a typical setting of censored time-to-event outcomes with time-varying covariates. Suppose that there are n subjects in the dataset, and let T_i and C_i be the failure and censoring times, respectively, for i = 1, 2, …, n. Given Z̃_i, T_i and C_i are assumed to be independent. An observed dataset usually consists of n independent and identically distributed (iid) copies of {(X_i, Δ_i, Z̃_i(X_i)); i = 1, 2, …, n}, where X_i = min(T_i, C_i), Δ_i = I (T_i ≤ C_i), and Z̃_i(X_i) is the covariate history of the ith subject up to X_i. Let the true values of β and R(·) be β₀ and R₀(·), respectively, in the extended model (5). Subject-specific subscripts may be occasionally suppressed for their general use.

3.1 Method of moment-based estimating equations

We first consider a moment-based semiparametric estimation procedure to estimate the regression parameters in the extended model (5). Our approach is to develop a closed-form estimator for the baseline odds function R(·), based on a martingale representation similar to that in Chen and Jewell (2001) and Chen, Jin and Ying (2002) at the true β₀, and then develop a class of log-rank type of estimating equations to estimate β in (5).

Consider N_i(t) = I(X_i ≤ t, Δ_i = 1) and Y_i(t) = I (X_i ≥ t), and denote a filtration by ℱ_t = σ{N_i(u), Y_i(u), Z̃_i; 0 ≤ u ≤ t, i = 1, 2, …, n}. Then we know that

$$E\{{dN}_i(t)\mid {\mathcal{F}}_{t-};{\mathit{\beta }}_0,R_0\}=\frac{Y_i(t)}{B_i(t;{\mathit{\beta }}_0)+R_0(t)}\{{dR}_0(t)-R_0(t)dlog{B}_i(t;{\mathit{\beta }}_0)\},$$ (6)

where B_i(t; β) = exp{β^TZ_i(t)}. Furthermore, let $M_i(t;\mathit{\beta },R)=N_i(t)-{\int }_0^t{Y}_i(u){\{B_i(u;\mathit{\beta })+R(u)\}}^{-1}\{dR(u)-R(u)dlog{B}_i(u;\mathit{\beta })\}$ for i = 1, 2, …, n. Then {M_i(t; β₀, R₀); t ≥ 0} are martingales with respect to the filtration ℱ_t. In addition,

$$\sum _{i=1}^n\{B_i(t;{\mathit{\beta }}_0)+R_0(t)\}{dN}_i(t)-\sum _{i=1}^n{Y}_i(t)\{{dR}_0(t)-R_0(t)dlog{B}_i(t;{\mathit{\beta }}_0)\}=\sum _{i=1}^n\{B_i(t;{\mathit{\beta }}_0)+R_0(t)\}{dM}_i(t;{\mathit{\beta }}_0,R_0).$$ (7)

Since E[{B_i(t; β₀) + R₀(t)}dM_i(t; β₀, R₀)] = 0 by equation (6), we thus consider the following estimating equation of R(t) for any t such that Σ_i Y_i(t) > 0, as if β were known:

$$\sum _{i=1}^n[\{B_i(t;\mathit{\beta })+R(t)\}{dN}_i(t)-Y_i(t)\{dR(t)-R(t)dlog{B}_i(t;\mathit{\beta })\}]=0.$$ (8)

This estimating equation is indeed a first-order stochastic differential equation of R(·), i.e.,

$$dR(t)+R(t)dlog{P}_n(t;\mathit{\beta })={dQ}_n(t;\mathit{\beta }),$$ (9)

where

$$\begin{array}{l}{P}_n(t;\mathit{\beta })=exp\left[-{\int }_0^t\sum _{i=1}^n\{{dN}_i(u)+Y_i(u)dlog{B}_i(u;\mathit{\beta })\}/\sum _{i=1}^n{Y}_i(u)\right],\text{and}\\ Q_n(t;\mathit{\beta })={\int }_0^t\sum _{i=1}^n{B}_i(u;\mathit{\beta }){dN}_i(u)/\sum _{i=1}^n{Y}_i(u).\end{array}$$

Thus, we solve (9) to obtain a closed-form solution for an estimator of R(·),

$${\widehat{R}}_n(t;\mathit{\beta })=P_n{(t;\mathit{\beta })}^{-1}{\int }_0^t{P}_n(u-;\mathit{\beta }){dQ}_n(u;\mathit{\beta }).$$ (10)

Here, P_n(t−) denotes the left-continuous version of P_n(t). Comparing equations (7) and (8), it is true that

$${\widehat{R}}_n(t;{\mathit{\beta }}_0)-R_0(t)=-P_n{(t;{\mathit{\beta }}_0)}^{-1}\sum _{i=1}^n{\int }_0^t\frac{P_n(u-;{\mathit{\beta }}_0)\{B_i(u;{\mathit{\beta }}_0)+R_0(u)\}}{{\sum }_j{Y}_j(u)}{dM}_i(u;{\mathit{\beta }}_0,R_0).$$ (11)

Consider that 0 < τ = inf{t: pr(X > t) = 0} < ∞. We have the following asymptotic properties for this estimator of R(·):

Lemma 2

For a fixed constant t ∈ [0, τ), under the conditions specified in the Appendix, as n → ∞,

1. R̂_n(·; β₀) is consistent on [0, t] almost surely. That is, ||R̂_n (β₀) − R₀|| = sup_u∈[0,t] |R̂_n(u; β₀) − R₀(u)| converges to 0 almost surely.

2. n^1/2{R̂_n(t; β₀) − R₀(t)} is asymptotically mean-zero normal.

Proof of this Lemma follows the standard martingale theory of counting processes. When the covariates are all time-independent, R̂_n(·) in (10) is equivalent to the ones in Yang and Prentice (1999) for estimating the baseline odds function, except that P_n(·) in (10) is a Nelson-Aalen type of estimator, instead of the Kaplan-Meier estimator. Nevertheless, R̂_n(·) in (10) maintains the same meaning of weighted empirical odds functions even in the presence of time-varying covariates.

In order to estimate the regression parameter β in model (5), we consider the following estimating functions:

$$S_n(\mathit{\beta },R_0)=\sum _{i=1}^n{\int }_0^{\tau }{\mathit{Z}}_i(t)[\{B_i(t)+R_0(t;\mathit{\beta })\}{dN}_i(t)-Y_i(t)\{{dR}_0(t)-R_0(t)dlog{B}_i(t;\mathit{\beta })\}].$$

They are unbiased at the true parameter β₀, given the fact of (6). By replacing R₀(·) with its estimator R̂_n(·; β) and some algebra shown in the Appendix, we obtain these estimating equations for β₀:

$$S_n(\mathit{\beta })=\sum _{i=1}^n{\int }_0^{\tau }\{{\mathit{Z}}_i(t)-\overline{\mathit{Z}}(t)\}\left[\left\{B_i(t;\mathit{\beta })+{\widehat{R}}_n(t;\mathit{\beta })\right\}{dN}_i(t)+Y_i(t){\widehat{R}}_n(t;\mathit{\beta })dlog{B}_i(t;\mathit{\beta })\right],$$

where Z̄(t) = Σ_i Y_i(t)Z_i(t)/Σ_i Y_i(t). Apparently, S_n(β) are not necessarily continuous or monotone, particularly when Z(·) are not differentiable. Hence, we consider the parameter estimator β̂_n to satisfy that S_n(β̂_n−)S_n(β̂_n+) ≤ 0, as defined for the estimating equations of rank-type in Tsiatis (1990).

The following theorem summarizes the asymptotic properties of β̂_n that can be used in statistical inferences of β₀:

Theorem 3

Under the conditions specified in the Appendix, there exists a neighborhood containing β₀ as an interior point such that, as n → ∞,

1. β̂_n ∈ is strongly consistent;

2. n^1/2(β̂_n − β₀) converges weakly to a mean-zero normal variate with variance-covariance U⁻¹V (U⁻¹)^T, where

   $$\begin{array}{c}U=\underset{n\to \infty }{lim}{n}^{-1}\sum _{i=1}^n{\int }_0^{\tau }\{{\mathit{Z}}_i(t)-\overline{\mathit{Z}}(t)\}{\left[\left\{\frac{\partial {\widehat{R}}_n(t;{\mathit{\beta }}_0)}{\partial \mathit{\beta }}+e^{{\beta }_0^{\text{T}}{Z}_i(t)}{Z}_i(t)\right\}{dN}_i(t)+Y_i(t)\left\{\frac{\partial {\widehat{R}}_n(t;{\mathit{\beta }}_0)}{\partial \mathit{\beta }}{\mathit{\beta }}_0^{\text{T}}+R_0(t)\right\}{d\mathit{Z}}_i(t)\right]}^{\text{T}},\\ V=\underset{n\to \infty }{lim}{\int }_0^{\tau }\frac{1}{n}\sum _{i=1}^n\frac{Y_i(t){\xi }_i{(t)}^{\otimes 2}\{{dR}_0(t)-R_0(t)dlog{B}_i(t;{\mathit{\beta }}_0)\}}{B_i(t;{\mathit{\beta }}_0)+R_0(t)},\end{array}$$
   and

   $$\begin{array}{l}{\xi }_i(t;\mathit{\beta })=\{{\mathit{Z}}_i(t)-\overline{\mathit{Z}}(t)\}\{B_i(t;\mathit{\beta })+R_0(t)\}\\ -\frac{P_n(t-;\mathit{\beta })\{B_i(t;\mathit{\beta })+R_0(t)\}}{{\sum }_j{Y}_j(t)}\left[\sum _{k=1}^n{\int }_t^{\tau }\frac{\{{\mathit{Z}}_k(u)-\overline{\mathit{Z}}(u)\}{Y}_k(u)}{P_n(u;\mathit{\beta })}dlog\{R_0(u)+B_k(u;\mathit{\beta })\}\right];\end{array}$$
3. U and V can be consistently estimated by

   $${\widehat{U}}_n=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }\{{\mathit{Z}}_i(t)-\overline{\mathit{Z}}(t)\}{\left[\left\{\frac{\partial {\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)}{\partial \mathit{\beta }}+e^{{\widehat{\beta }}_n{Z}_i(t)}{\mathit{Z}}_i(t)\right\}{dN}_i(t)+Y_i(t)\left\{\frac{\partial {\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)}{\partial \mathit{\beta }}{\widehat{\mathit{\beta }}}_n^{\text{T}}+{\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)\right\}{d\mathit{Z}}_i(t)\right]}^{\text{T}},$$
   and

   $${\widehat{V}}_n=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }\frac{Y_i(t){\widehat{\xi }}_i{(t)}^{\otimes 2}\{{d\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)-{\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)dlog{B}_i(t;{\widehat{\mathit{\beta }}}_n)\}}{B_i(t;{\widehat{\mathit{\beta }}}_n)+{\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)},$$

   respectively, where ξ̂_i(t) = ξ_i(t; β̂_n, R̂_n).

Proof of this theorem is outlined in the Appendix. According to this theorem, the proposed moment-based estimating equations yield a usual sandwich-type of variance estimate for β̂_n.

In addition, user-defined weight functions can be included in S_n(·) to obtain a class of weighted estimators of β:

$$\sum _{i=1}^n{\int }_0^{\tau }{W}_n(t)\{{\mathit{Z}}_i(t)-\overline{\mathit{Z}}(t)\}\left[\left\{B_i(t;\beta )+{\widehat{R}}_n(t;\mathit{\beta })\right\}{dN}_i(t)+Y_i(t){\widehat{R}}_n(t;\mathit{\beta })dlog{B}_i(t;\mathit{\beta })\right]=0,$$ (12)

where W_n(·) are ℱ_t–predictable weight functions converging to a deterministic function of w(·). For example, a popular choice of weight functions can be a Prentice-Wilcoxon-type of W (t) = Ŝ_KM(t−), which is a left-continuous version of the Kaplan-Meier estimate of the baseline survival function S₀(·), given its semiparametric optimality when Z (·) is time-independent. Denote the solution to the weight estimating equations by β̂_w,n. Then all the asymptotic properties of β̂_w,n can be derived similarly as in Theorem 3:

Corollary 4

Under the same conditions of Theorem 3, as n → ∞, the following hold for some pre-specified weight functions W_n(·) → w(·) almost surely:

1. β̂_w,n ∈ converges to β₀ almost surely;

2. n^1/2(β̂_w,n − β₀) converges weakly to a mean-zero normal variate with variance-covariance $U_w^{-1}{V}_w{(U_w^{-1})}^{\text{T}}$, where

   $$\begin{array}{c}{U}_w=\underset{n\to \infty }{lim}{n}^{-1}\sum _{i=1}^n{\int }_0^{\tau }{W}_n(t)\{{\mathit{Z}}_i(t)-\overline{\mathit{Z}}(t)\}{\left[\left\{\frac{\partial {\widehat{R}}_n(t;{\mathit{\beta }}_0)}{\partial \mathit{\beta }}+e^{{\beta }_0^{\text{T}}{Z}_i(t)}{Z}_i(t)\right\}{dN}_i(t)+Y_i(t)\left\{\frac{\partial {\widehat{R}}_n(t;{\mathit{\beta }}_0)}{\partial \mathit{\beta }}{\mathit{\beta }}_0^{\text{T}}+R_0(t)\right\}{d\mathit{Z}}_i(t)\right]}^{\text{T}},\\ V_w=\underset{n\to \infty }{lim}{\int }_0^{\tau }\frac{1}{n}\sum _{i=1}^n\frac{Y_i(t){\xi }_{w,i}{(t)}^{\otimes 2}\{{dR}_0(t)-R_0(t)dlog{B}_i(t;{\mathit{\beta }}_0)\}}{B_i(t;{\mathit{\beta }}_0)+R_0(t)},\end{array}$$
   and

   $$\begin{array}{l}{\xi }_{w,i}(t;\mathit{\beta })=W_n(t)\{{\mathit{Z}}_i(t)-\overline{\mathit{Z}}(t)\}\{B_i(t;\mathit{\beta })+R_0(t)\}\\ -\frac{P_n(t-;\mathit{\beta })\{B_i(t;\mathit{\beta })+R_0(t)\}}{{\sum }_j{Y}_j(t)}\left[\sum _{k=1}^n{\int }_t^{\tau }\frac{W_n(u)\{{\mathit{Z}}_k(u)-\overline{\mathit{Z}}(u)\}{Y}_k(u)}{P_n(u;\mathit{\beta })}dlog\{R_0(u)+B_k(u;\mathit{\beta })\}\right].\end{array}$$
3. U_w and V_w can be consistently estimated by

   $${\widehat{U}}_{w,n}=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }{W}_n(t)\{{\mathit{Z}}_i(t)-\overline{\mathit{Z}}(t)\}{\left[\left\{\frac{\partial {\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)}{\partial \mathit{\beta }}+e^{{\widehat{\beta }}_n{Z}_i(t)}{\mathit{Z}}_i(t)\right\}{dN}_i(t)+Y_i(t)\left\{\frac{\partial {\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)}{\partial \mathit{\beta }}{\widehat{\mathit{\beta }}}_n^{\text{T}}+{\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)\right\}{d\mathit{Z}}_i(t)\right]}^{\text{T}},$$
   and

   $${\widehat{V}}_{w,n}=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }\frac{Y_i(t){\widehat{\xi }}_{w,i}{(t)}^{\otimes 2}\{{d\widehat{R}}_n(t;{\widehat{\beta }}_n)-{\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)dlog{B}_i(t;{\widehat{\mathit{\beta }}}_n)\}}{B_i(t;{\widehat{\mathit{\beta }}}_n)+{\widehat{R}}_n(t;{\widehat{\mathit{\beta }}}_n)},$$

   respectively, where ξ̂_w,i(t) = ξ_i(t; β̂_w,n, R̂_n).

Apparently, one use of the weighted estimating equations in (12) is that data analysts may be able to choose optimal weight functions to minimize the sandwich estimate of variance to reach semiparametric efficiency, for example, using a “sample-splitting” technique (Lin and Ying, 1994). In general, moment-based semiparametric estimation approaches do not guarantee that optimal weight functions can be easily obtained, particularly in the presence of time-varying covariates. To see this difficulty, consider the hazard function of a univariate covariate Z̃(·):

$$\lambda (t\mid \stackrel{\sim }{Z})=\frac{{\lambda }_0(t)}{1-S_0(t)+S_0(t)exp\{\beta Z(t)\}}-\frac{\{1-S_0(t)\}\beta Z^{\prime }(t)}{1-S_0(t)+S_0(t)exp\{\beta Z(t)\}}.$$ (13)

Thus an optimal weight function w_opt(·) for the weighted estimating equations (12) would nominally satisfy that:

$$w_{\text{opt}}(t)\propto \underset{\beta \to 0}{lim}\frac{1}{\beta }log\left\{\frac{\lambda (t\mid \stackrel{\sim }{Z})}{{\lambda }_0(t)}\right\}=-S_0(t)-\{1-S_0(t)\}{\lambda }_0(t){\{logZ(t)\}}^{\prime }.$$

(Bickel and Kwon, 2001). When Z(·) is time-independent, it is easy to see that an optimal weight function would be proportional to S₀(·), the baseline survival function. However, when Z (·) is time-varying, it is less straightforward to obtain a simple form of w_opt(·) without estimating λ₀(·), although in practice some kernel estimates of λ₀(·) can be used (Tsiatis, 1990).

3.2 Maximum likelihood method

Nevertheless, we can apply a nonparametric maximum likelihood estimation (NPMLE) procedure to estimate the extended model, following the approach recently advocated by Zeng and Lin (2006, 2007). Denote Λ₀(t) the cumulative hazard function of λ₀(t), i.e., ${\mathrm{\Lambda }}_0(t)={\int }_0^t{\lambda }_0(u)du$. Then under the extended model, it is straightforward to write out a likelihood function for β and Λ₀, which is proportional to

$$\mathcal{L}=\prod _{i=1}^n\prod _{t\le \tau }{\left\{Y_i(t){\mathrm{\Lambda }}^{\prime }(t\mid {\stackrel{\sim }{\mathit{Z}}}_i)\right\}}^{{dN}_i(t)}exp\left\{-{\int }_0^{\tau }{Y}_i(u)d\mathrm{\Lambda }(u\mid {\stackrel{\sim }{\mathit{Z}}}_i)\right\},$$ (14)

where

$${\mathrm{\Lambda }}^{\prime }(t\mid {\stackrel{\sim }{\mathit{Z}}}_i)={\{-logS(t\mid {\stackrel{\sim }{\mathit{Z}}}_i)\}}_t^{\prime }=\frac{{\mathrm{\Lambda }}_0^{\prime }(t)-[1-exp\{-{\mathrm{\Lambda }}_0(t)\}]{\mathit{\beta }}^{\text{T}}{d\mathit{Z}}_i(t)}{1-exp\{-{\mathrm{\Lambda }}_0(t)\}+exp\{-{\mathrm{\Lambda }}_0(t)+{\mathit{\beta }}^{\text{T}}{\mathit{Z}}_i(t)\}}.$$

Apparently, the maximum of ℒ does not exist if Λ₀ are restricted to be absolutely continuous. Instead, we allow Λ₀ to be discrete and replace dΛ₀(t) by the jump size of Λ₀ at t, Λ₀{t}. Then calculation of the NPMLE of β and Λ₀ is equivalent to maximizing ℒ with respect to β and Λ₀(·) being the step functions with jumps at observed event times X_i, i = 1, 2, …, n. Denote the NPMLE of β and Λ₀(·) by β̌_n and Λ̌_0,n(·), respectively. Then the NPMLEs of β̌_n and Λ̌_0,n(·) have the following asymptotic properties:

Theorem 5

Under the conditions specified in the Appendix,

1. |β̌_n − β₀| + sup_t∈[0,τ] |Λ̌_0,n(t) − Λ₀(t)| converges to 0 almost surely;

2. n^1/2(β̌_n −β₀, Λ̌_0,n − Λ₀) converges weakly to a zero-mean Gaussian process in ℛ^p × (ℋ), where ℋ = {h(u) of bounded variation on [0, τ]: ||h(u)|| ≤ 1}. And the limiting covariance matrix of n^1/2(β̌_n − β₀) reaches the semiparametric efficiency bound.

The NPMLE for the extended proportional odds model is a special example under the general NPMLE theory for semiparametric regression models in Zeng and Lin (2007). So it is straightforward to follow the general theory to establish the desired asymptotic properties in Theorem 5, which is omitted here due to space limit. Nevertheless, details of these proofs can be found in Appendix B of Zeng and Lin (2007).

As shown in Theorems 3 and 5, both the moment-based estimation procedure and the NPMLE procedure provide asymptotically tractable estimators for the regression parameters. In particular, when the extended model is correctly specified, the NPMLE procedure yields asymptotically semiparametric efficient estimators, which may have a great advantage in theory and in practical inferences. Nevertheless, the moment-based procedure would yield a simpler closed-form estimator of the baseline odds function and generally robust sandwich estimates of variance. In fact, through appropriate weight functions, moment-based estimators may gain substantial efficiency.

4. NUMERICAL STUDIES

4.1 Computing Algorithms

For either estimation procedure proposed earlier, computing can be challenging. For the moment-based estimating equations, they are not necessarily always smooth or monotone, standard numerical approaches, such as Newton-Raphson, do not work well, particularly when sample size is small. For lower dimensions, a direct grid search or bisection methods can be used to estimate the regression parameters. For higher dimensions, random search technique such as the simulated annealing has been suggested more effective (Lin and Ying, 1995). In our calculation, we use a recursive bisection method proposed by Huang (2002) to search for roots. That is, having solved for the first k components of β, (β₁, β₂, …, β_k), we will use a one-dimensional bisection algorithm to solve for β_k+1. Details on the recursive bisection algorithm is implemented can be found in Chen and Jewell (2001).

For the NPMLE procedure, computing is more demanding because maximization would occur at all of the observed event times. In our calculation, we use an optimization algorithm called fmi-nunc in MATLAB for unconstrained nonlinear optimization, suggested by Zeng and Lin (2007). For this optimization algorithm, we also provide Hessian matrix so that the algorithm converges faster and more reliably, although the Hessian matrix is not required.

In both calculations, we use initial values from MLE of parametric extended proportional odds models, with baseline survival functions chosen from various parametric families, such as exponential and log-logistic. With these initial values, we find that calculation of moment-based estimates is quite reliable and much faster than the NPMLE’s, similar to the experience in Yang and Prentice (1999), which also reported that moment-based estimation showed great reduction in computing time relative to the NPMLE.

4.2 Simulation Studies

Simulation studies are conducted to assess the validity of our proposed inference procedures. In these simulation studies, we choose our simulation parameters similar to those in Zucker & Yang (2006). In each simulated dataset, n observations are generated. For each observation, we generate a time-independent covariate Z₁ that follows a continuous uniform distribution Unif[0, 4]. For the purpose of demonstration, time-varying covariates are calculated by the interaction between Z₁ and time t: Z₂(t) = Z₁ · t. For the proportional odds model that would generate failure times, we consider a baseline odds function F (t |Z = 0)/{1 − F (t |Z = 0)} = 0.01 · t. Censoring times are generated such that the censoring proportion of each simulated data is around 30%. We present our simulation results in Tables 1 and 2. Each cell in the tables is based on 10,000 simulated datasets. Sample size n is selected to be 100, 300 or 500, representing relatively small, medium or large sample sizes, respectively.

Table 1.

Summary of simulation studies with time-independent covariates only

Sample Size	β̂
n	β0	Method	Bias	SSE	MSE	CP
100	0	ME-0	0.0006	0.294	0.299	0.951
100	0	ME-W	0.0007	0.246	0.248	0.947
100	0	NPMLE	−0.0005	0.245	0.255	0.955
100	1	ME-0	0.0003	0.308	0.296	0.953
100	1	ME-W	−0.0003	0.255	0.247	0.954
100	1	NPMLE	0.0009	0.253	0.244	0.952
300	0	ME-0	0.0004	0.132	0.139	0.957
300	0	ME-W	−0.0001	0.110	0.116	0.949
300	0	NPMLE	0.0008	0.109	0.114	0.951
300	1	ME-0	0.0009	0.180	0.176	0.949
300	1	ME-W	0.0002	0.151	0.149	0.950
300	1	NPMLE	−0.0004	0.149	0.148	0.951
500	0	ME-0	−0.0002	0.107	0.108	0.947
500	0	ME-W	0.0001	0.090	0.090	0.951
500	0	NPMLE	0.0006	0.090	0.092	0.948
500	1	ME-0	0.0005	0.128	0.125	0.951
500	1	ME-W	−0.0003	0.108	0.103	0.950
500	1	NPMLE	0.0002	0.107	0.104	0.952

SSE, sample standard error of estimates; ME-0, unweighted moment-based estimates; MEW, weighted moment-based estimates; NPMLE, nonparametric maximium likelihood estimates; MSE, mean of estimated standard errors; CP, 95% nominal coverage probabilities.

Table 2.

Summary of simulation studies with time-varying covariates

Sample Size	β̂ (Time-invariant)						γ̂ (Time-variant)
n	β0	Method	Bias	SSE	MSE	CP	γ0	Method	Bias	SSE	MSE	CP
100	0	ME-0	−0.0002	0.175	0.172	0.942	0	ME-0	0.0004	0.173	0.179	0.966
100	0	ME-W	0.0008	0.170	0.169	0.957	0	ME-W	0.0002	0.159	0.167	0.960
100	0	NPMLE	0.0006	0.168	0.163	0.954	0	NPMLE	0.0003	0.158	0.164	0.948
100	1	ME-0	−0.0009	0.548	0.518	0.959	0	ME-0	0.0008	0.270	0.266	0.951
100	1	ME-W	−0.0007	0.529	0.497	0.949	0	ME-W	0.0008	0.233	0.251	0.957
100	1	NPMLE	0.0001	0.509	0.487	0.943	0	NPMLE	−0.0010	0.240	0.246	0.947
100	0	ME-0	−0.0008	0.403	0.395	0.941	1	ME-0	0.0002	0.346	0.345	0.954
100	0	ME-W	0.0007	0.388	0.386	0.936	1	ME-W	−0.0006	0.320	0.320	0.947
100	0	NPMLE	0.0006	0.383	0.379	0.955	1	NPMLE	0.0005	0.315	0.315	0.929
100	1	ME-0	0.0009	0.609	0.596	0.940	1	ME-0	−0.0008	0.355	0.357	0.945
100	1	ME-W	0.0004	0.578	0.576	0.947	1	ME-W	0.0009	0.331	0.331	0.955
100	1	NPMLE	0.0005	0.583	0.562	0.948	1	NPMLE	0.0005	0.326	0.320	0.949
300	0	ME-0	0.0001	0.020	0.018	0.955	0	ME-0	0.0004	0.050	0.058	0.957
300	0	ME-W	0.0004	0.019	0.017	0.957	0	ME-W	0.0003	0.047	0.055	0.936
300	0	NPMLE	0.0003	0.019	0.017	0.951	0	NPMLE	0.0002	0.045	0.053	0.939
300	1	ME-0	−0.0005	0.428	0.429	0.951	0	ME-0	0.0010	0.243	0.250	0.945
300	1	ME-W	0.0007	0.420	0.422	0.956	0	ME-W	0.0006	0.225	0.236	0.956
300	1	NPMLE	0.0002	0.412	0.473	0.946	0	NPMLE	0.0001	0.220	0.227	0.950
300	0	ME-0	0.0005	0.303	0.308	0.952	1	ME-0	−0.0001	0.162	0.169	0.958
300	0	ME-W	0.0001	0.291	0.303	0.939	1	ME-W	0.0004	0.152	0.158	0.923
300	0	NPMLE	0.0004	0.291	0.297	0.948	1	NPMLE	0.0008	0.147	0.153	0.956
300	1	ME-0	0.0001	0.351	0.356	0.953	1	ME-0	−0.0000	0.228	0.228	0.940
300	1	ME-W	0.0009	0.345	0.347	0.945	1	ME-W	0.0007	0.212	0.213	0.957
300	1	NPMLE	−0.0008	0.332	0.337	0.952	1	NPMLE	−0.0003	0.208	0.208	0.937
500	0	ME-0	0.0000	0.089	0.091	0.945	0	ME-0	0.0001	0.041	0.037	0.950
500	0	ME-W	−0.0003	0.085	0.090	0.961	0	ME-W	0.0009	0.038	0.034	0.946
500	0	NPMLE	0.0006	0.084	0.087	0.954	0	NPMLE	0.0008	0.037	0.034	0.969
500	1	ME-0	0.0002	0.352	0.360	0.950	0	ME-0	0.0006	0.221	0.223	0.955
500	1	ME-W	0.0003	0.348	0.353	0.942	0	ME-W	0.0002	0.210	0.209	0.936
500	1	NPMLE	−0.0007	0.337	0.341	0.951	0	NPMLE	0.0003	0.200	0.202	0.948
500	0	ME-0	0.0006	0.251	0.257	0.949	1	ME-0	−0.0005	0.253	0.252	0.948
500	0	ME-W	0.0003	0.242	0.247	0.948	1	ME-W	0.0006	0.236	0.234	0.950
500	0	NPMLE	0.0005	0.241	0.246	0.956	1	NPMLE	0.0007	0.232	0.227	0.962
500	1	ME-0	0.0000	0.302	0.300	0.951	1	ME-0	−0.0002	0.187	0.183	0.946
500	1	ME-W	0.0008	0.292	0.291	0.947	1	ME-W	0.0003	0.177	0.170	0.949
500	1	NPMLE	−0.0001	0.289	0.283	0.943	1	NPMLE	0.0004	0.168	0.166	0.952

SSE, sample standard error of estimates; ME-0, unweighted moment-based estimates; MEW, weighted moment-based estimates; NPMLE, nonparametric maximium likelihood estimates; MSE, mean of estimated standard errors; CP, 95% nominal coverage probabilities.

In Table 1, we present simulation results from fitting the extended proportional odds model with Z₁ only, i.e., logit S(t | Z₁) = −log R(t) + βZ₁. This is indeed a proportional odds model with time-independent covariates. Three estimation procedures are used to evaluate their performance in estimating the regression parameter for Z₁:

1. ME-0: solving unweighted moment-based estimating equations S_n(β) = 0;

2. ME-W: solving weighted moment-based estimating equations S_n,w (β) = 0, where the weight function is chosen to be the Kaplan-Meier estimate of the baseline survival function;

3. NPMLE: maximizing nonparametric likelihood function ℒ.

As shown in the table, all these three estimation procedures for this model with time-independent covariates yield virtually unbiased estimates, and reasonable coverage probabilities. Among them, NPMLE estimates apparently outperform the unweighted moment-based estimation approach uniformly. However, when the weight function is incorporated in the moment-based estimating equations, efficiency of the moment-based estimates is greatly improved, almost as good as their NPMLE counterparts.

In Table 2, we present simulation results from the extended proportional odds model that include both Z₁ and Z₂(·) as covariates, i.e., logit S(t | Z) = −log R(t) + βZ₁ + γZ₂(t). As shown in Table 2, both estimates for the time-independent and the time-varying covariates are also virtually unbiased for all of the three estimating procedures, regardless of different sample sizes and true parameter values. However, when sample size is relatively small, the discrepancy between empirical and large-sample approximation variance estimates tends to be big. For relatively large sample sizes, our large-sample approximation appears to behave well: variance estimates are consistent between the empirical and their large-sample approximation ones, and coverage probabilities are around their nominal value of 95%. Again, NPMLE estimates outperform unweighted moment-based estimation procedure by a noticeable margin in their variances. Such an advantage is less noticeable when weighted moment-based estimation procedure is used, even though the chosen weight function is not necessarily the one to reach the semiparametric efficiency, as shown in our earlier calculation for the extended model with time-dependent covariates.

4.3 Data Analyses

In this section, we demonstrate our proposed methods by analyzing the motivating example of the HIVNET 012 trial, and the well-known VA Lung Cancer Trial (Prentice, 1973).

Example 1

In the HIVNET 012 Trial, a total of 626 HIV-1 infected pregnant women in Uganda were recruited and randomized to receive either NVP or AZT at more than 36 weeks of gestation. Complete medical histories and physical examination of all participants were collected before their entry to the study, on enrollment, at delivery, at discharge from hospital and at 7 days and 6 weeks after delivery. Among those 619 women analyzed in the primary analysis (Jackson et al., 2003), 308 women were randomized to receive AZT, and 311 were randomized to receive NVP. Their median ages were 25 and 24 (p > 0.1000); their CD4+ counts measured at the baseline were 426 and 459 (p > 0.2500); and their HIV-1 RNA viral copies were 27,800 and 25,247 (p > 0.7000), all respectively. The total follow-up time is 18 months. At the end of the trial, the primary efficacy analysis showed that the HIV-1 transmission risks in the AZT and the NVP groups were 10.4% and 8.2% (p > 0.3000) at birth, 21.3% and 11.9% (p < 0.0300) by age 6–8 weeks, and 25.1% and 13.1% (p < 0.0001) by age 14–16 weeks, which supported that NVP significantly lowered the vertical transmission risk in less-developed countries.

According to the HIVNET 012 study design, the NVP was administered only once in order to maintain certain plasma drug concentration for up to 7 days. Preliminary studies showed that its potent antiviral effect usually persisted for 1 to 2 weeks, followed by a rapid development of detectable viral resistance to the NVP by 6 to 8 weeks after ingestion (Eshleman et al., 2001). Its long-term effect to suppress the maternal viral load could be diminished, which might lead to a decreasing effect on the ultimate infant survival, as time progresses. Therefore, the proportional odds model may serve as a reasonable tool given its feature of modeling converging hazard functions.

In our analysis, we first fit the Cox model with these covariates: the treatment indicator of NVP versus AZT, the treatment-time interaction, and maternal CD4+ counts and viral loads (VL). Here, maternal CD4+ counts and VLs are grouped into three categories: low maternal risk category (CD4+> 350 and VL≤ 50, 000), intermediate maternal risk category (CD4+≤ 350 and VL≤ 50, 000, or CD4+> 350 and VL> 50, 000), and high maternal risk category (CD4+≤ 350 and VL> 50, 000). An estimate of the regression parameter for treatment-time interaction is −0.003 (s.e. = 0.0003) with a 95% CI (−0.004, −0.003), which is statistically significantly different from no interaction (p < 0.001). As a result, the NVP administration is not considered to have a constant effect on the hazard functions of infant survival. The comparison of their respective survival functions would then depend on both the regression coefficients and the baseline hazard function, which shall lead to a complicated interpretation of the NVP’s effect in terms of the infants’ survival functions. Nevertheless, the negative interaction means that NVP and time are potentially antagonistic in hazard functions. This may mean that the NVP’s effect on the infants’ hazard functions may be reduced during the observation period.

We also fit an extended proportional odds model (2) with the same set of covariates. The estimates of regression parameters are tabulated in Table 3. As shown in Table 3, the interaction between time and the administration of NVP to HIV-infected mothers during labor and delivery is not statistically significant (p = 0.08). The main effect of the NVP administration is nevertheless statistically significant. That is, under the assumed proportional odds model, the NVP administration would improve the odds of infants’ overall survival by e^0.265 − 1 = 30.3% (p = 0.01) uniformly over time. For the maternal CD4+ counts and VLs, when compared with the low maternal risk category, the intermediate and high maternal risk categories appear to be associated with 1 − e^−0.026 = 2.6% and 1 − e^−0.052 = 5.1% of loss of infant survival, respectively. Additional NPMLE of fitting the same model yields a similar result: the administration of NVP would significantly improve the odds of infant survival by e^0.276 − 1 = 31.7%. This significant improvement of infant survival due to NVP has a profound impact: its broader administration shall improve the overall human life expectancy, especially in resource-limited settings.

Table 3.

Summary of fitting proportional odds models for HIVNET012 data

Parm	Est.	s.d.	CI
NVP	0.265	0.121	(0.027, 0.502)
NVP×Time	−8.52×10−5	6.20×10−5	(−2.10×10−4, 4.00×10−5)
CD4+ and VL versus Low risk
Intermediate risk	−0.026	0.018	(−0.061, 0.009)
High risk	−0.052	0.033	(−0.116, 0.012)

NVP, indicator of Nevirapine (versus AZT); Est., parameter estimate; s.d., standard error; CI, 95% confidence interval; Low risk, CD4+>350, VL≤50,000; Intermediate risk, CD4+>350, VL>50,000 or CD4+≤350, VL≤50,000; High risk, CD4+≤350, VL>50,000.

Example 2

The well-known VA Lung Cancer Trial dataset in Prentice (1973) has been analyzed extensively in the statistical literature. The originally published dataset contains only two baseline covariates, performance score Z_PS on a scale of 0 to 100, and tumor type Z_TU indicating large, adeno, small and squamous types, and the censored time-to-event outcome of lung cancer survival. Here, Z_TU are 3-dimensional dummy vectors with the large type as the baseline category. We apply our proposed extended model to this dataset to allow comparison between our proposed method and the existing ones in the literature. Moreover, we demonstrate that the extended model can also serve as a goodness-of-fit tool for the usual proportional odds model.

Specifically, we analyze a subgroup of 97 patients without prior therapy to study the association between the lung cancer survival and Z_PS and Z_TU. In Bennett (1983), logit {S(·)} are plotted by Z_PS ≤ 50 and Z_PS > 50, which leads to a visual justification of assuming constant effect of Z_PS in a parametric proportional odds model

$$\text{logit}S(t\mid Z_{\text{PS}})=-logR(t)+{\beta }_1{Z}_{\text{PS}}+{\mathit{\gamma }}^{\text{T}}{\mathit{Z}}_{\text{TU}},$$

where R(·) is of log-logistic distributions. Most subsequent works, including Pettitt (1984), Cheng, Wei and Ying (1995) and Yang and Prentice (1998), all similarly assume that the performance score would have constant effect in their proportional odds models, despite the fact that the cutoff of 50 is chosen for convenience.

In our extended model, we instead use Z_PS of its original scale. Results from a series of model-fitting by the extended proportional odds model are tabulated in Table 4. When Z_PS is the only covariate in the extended model, we obtain a significant parameter estimate, as shown by Model I of Table 4. However, when we include an additional time-varying interaction between Z_PS and t in Model II, the parameter estimate of this interaction is then 0.016 (s.e. = 0.003, p < 0.001), which is highly significant. This means that the difference in log survival odds is not necessarily constant throughout time for Z_PS, if Z_TU are not included. When Z_TU are the only covariates in the extended model, small type, compared with large type, appears significantly associated with the lung cancer survival in Model III, while all of the tumor-time interactions are not significant in Model IV. This may indicate that it is reasonable to assume a constant effect for Z_TU.

Table 4.

Summary of fitting proportional odds models for Veteran Administration data

Parm	Model I			Model II			Model III
	Est.	s.d.	CI	Est.	s.d.	CI	Est.	s.d.	CI
PS	−0.134	0.014	(−0.161, −0.107)	−0.089	0.019	(−0.126, −0.052)	-	-	-
PS×Time	-	-	-	0.016	0.003	(0.010, 0.022)	-	-	-
Tumor type versus Large
AD	-	-	-	-	-	-	1.292	0.367	(0.573, 2.011)
AD×Time	-	-	-	-	-	-	-	-	-
SM	-	-	-	-	-	-	1.972	0.486	(1.019, 2.925)
SM×Time	-	-	-	-	-	-	-	-	-
SQ	-	-	-	-	-	-	−0.105	0.376	(−0.842, 0.632)
SQ×Time	-	-	-	-	-	-	-	-	-
	Model IV			Model V			Model VI
Parm	Est.	s.d.	CI	Est.	s.d.	CI	Est.	s.d.	CI
PS	-	-	-	−0.059	0.012	−0.083–0.035	−0.049	0.021	−0.090–0.008
PS×Time	-	-	-	-	-	-	−0.008	0.005	−0.018 0.002
Tumor type versus Large
AD	0.958	0.520	(−0.061, 1.977)	1.521	0.492	(0.557, 2.485)	1.222	0.625	(−0.003, 2.447)
AD×Time	−0.228	0.127	Z(−0.477, 0.021)	-	-	-	−0.169	0.225	(−0.610, 0.272)
SM	1.210	0.671	(−0.105, 2.525)	1.451	0.614	(0.249, 2.655)	1.141	0.781	(−0.390, 2.672)
SM×Time	−0.054	0.232	(−0.509, 0.401)	-	-	-	−0.011	0.223	(−0.449, 0.425)
SQ	0.072	0.591	(−1.086, 1.230)	−0.009	0.563	(−1.112, 1.094)	0.021	0.603	(−1.161, 1.203)
SQ×Time	0.082	0.092	(−0.098, 0.262)	-	-	-	0.071	0.104	(−0.133, 0.275)

PS, performance score; AD, adeno; SM, small; SQ, squamous; Est., parameter estimate; s.d., standard error; CI, 95% confidence interval; -, corresponding covariates excluded in the model.

Nevertheless, when no interaction is included in the proposed model, all of the estimates we obtain are similar to the ones in the literature, as shown by Model V in Table 4. This validates our proposed inference procedures that are able to replicate what have been in the literature in absence of time-varying covariates. Moreover, when the time-varying interactions of ${(Z_{\text{PS}},Z_{\text{TU}}^{\text{T}})}^{\text{T}}$ and t are included in the proposed extended model, we find that only the interaction of Z_PS and t is marginally significant at α-level 10%. This may suggest it is reasonable to assume constant covariate effect in the proportional odds model.

As a summary, even when all the covariates are only collected at baseline, the extended proportional odds model can serve as a useful tool to assess covariate-time interaction, and leads to an approach of goodness-of-fit assessment for the usual proportional odds model.

5. DISCUSSION

In this article, we study an extended proportional odds model in presence of time-varying covariates. This extended model directly models survival functions, with an appealing interpretation of regression parameters in absolute risk. In the literature, inclusion of time-varying covariates in the usual proportional odds model has been discussed. For example, a specific version of our proposed model (5) was discussed in Yang and Prentice (1999). Although the self-consistency integral equations proposed in Yang and Prentice (1999) work with the usual proportional odds model elegantly, they are yet to be developed for time-varying covariates. Compared with the integral equations, the ordinary differential equations used in this article yield a simple closed-form estimator for the baseline odds function, which eventually makes the estimation of regression parameters more straightforward.

Nevertheless, the popular Cox proportional hazards model can also incorporate time-varying covariates. For example, the Cox proportional hazards model would usually assume that

$$\lambda \{t\mid {\mathit{Z}}^{\ast }(t)\}={\lambda }_0(t)exp\{{\mathit{\beta }}^{\text{T}}{\mathit{Z}}^{\ast }(t)\},$$ (15)

as in Kalbfleisch and Prentice (2002, pp. 96–98). It is then apparent that the association of Z^*(·) in survival functions is not always straightforward in model (15), because it implies that

$$\mathrm{\Lambda }\{t\mid \mathit{Z}(t)\}={\mathrm{\Lambda }}_0(t){\int }_0^{t}exp\{{\mathit{\beta }}^{\text{T}}\mathit{Z}(u)\}\psi (u;t)du,$$

where $\mathrm{\Lambda }(t)={\int }_0^t\lambda (u)du$, and ψ(u; t) = λ₀(u)/Λ₀(t) such that ${\int }_0^t\psi (u;t)du=1$. Mathematically, unless Z(·) are constant almost everywhere on [0, t], the regression parameter β does not yield a direct comparison between survival functions. This has been the same complication in studying association of survival functions for other hazard-based regression models, for example, the joint hazards model of Chen and Jewell (2000), in presence of time-varying covariates.

Without time-varying covariates, both the proportional odds model and the Cox proportional hazards model are special examples of the transformation model (Cheng, Wei and Ying, 1995). Recent effort has been focused on developing transformation models with time-varying covariates (Kosorok, Lee and Fine, 2004; Zeng and Lin, 2007). For example, in the latest Zeng and Lin (2007), an extended transformation model with time-varying covariates assumes that

$$\mathrm{\Lambda }(t\mid \stackrel{\sim }{\mathit{Z}})=G\left[{\mathrm{\Lambda }}_0(t){\int }_0^{t}exp\{{\mathit{\beta }}^{\text{T}}\mathit{Z}(u)\}\psi (u;t)du\right],$$ (16)

where G(·) is a continuously differentiable and strictly increasing transformation function. Similar to the Cox proportional hazards model, however, this model does not provide direct comparison of survival functions either, unless Z(·) are constant almost everywhere.

One direct extension of the transformation model to include time-varying covariates, as suggested by a reviewer, is to consider the following model,

$$S(t\mid \stackrel{\sim }{\mathit{Z}})=exp(-G[{\mathrm{\Lambda }}_0(t)exp\{{\mathit{\beta }}^{\text{T}}\mathit{Z}(t)\}]),$$ (17)

where Λ₀(·) is an unspecified monotonically increasing function with Λ₀(0) = 0, and G(·) is a specified monotonically increasing function with G(0) = 1. As shown in this model, the effect of time-varying covariates are also directly related the survival, which includes the extended proportional odds model and the extended Cox model. This in fact appealing in offering a formal way to justify between these two models. Both martingale based estimating equations and self-consistent integral equations can be used in estimating regression parameters, however, there may not be closed-form solutions for the baseline function. This extension shall be explored in future.

In this article, we mainly focus on estimating the regression parameters in the extended model, given that the major interest of our motivating examples in randomized clinical trials is to compare treatment effect. One particular feature of the extended model, however, is that it may be used to estimate subject-specific survival probabilities given a covariate profile. To serve this purpose, it is also critical to develop more asymptotic properties for estimating the baseline odds function. Conceptually, this shall be straightforward, because of the establishment of consistency and asymptotic normality of the regression parameters, as shown in Cheng, Wei and Ying (1997). Extensive theory calculation and reliable computing algorithms for the asymptotic variance-covariance of regression parameter estimators and baseline odds function estimators are yet to be developed in both moment-based estimating equations and the NPMLE approaches in our future research.

APPENDIX: ASYMPTOTIC PROPERTIES

For ease of presentation, in this Appendix we outline a proof of the stated asymptotic properties in Theorems 3, assuming that Z(·) and regression parameter β are scalar. Following the proof, it is straightforward to establish asymptotic properties in higher dimensions.

Regularity Conditions

We first assume some general regularity conditions for the asymptotic results summarized in Theorem 3, Corollary 4 and Theorem 5. Additional assumptions, if needed for a specific result, are stated in the proof.

• C1β₀ ∈ ℬ ⊂ ℛ lies in the interior of a compact set .
• C2
  The baseline density function f₀ of T and its derivative $f_0^{\prime }$ are bounded, satisfying that

  $${\int }_0^{\infty }{\left\{\frac{f_0^{\prime }(t)}{f_0(t)}\right\}}^2{f}_0(t)dt<\infty ,\text{and}{\int }_0^{\infty }{t}^{\epsilon }{f}_0(t)dt<\infty \text{for}\text{some}\epsilon >0.$$
• C2′Baseline cumulative hazard function Λ₀(·) > 0 is strictly increasing and continuously differentiable.
• C3The density function g of C are uniformly bounded, i.e., sup_t g(t) < ∞.
• C4τ is finite such that pr(T > τ ) > 0 and pr(C = τ ) > 0.
• C5There exists κ₁ > 0 and κ₂ > 0 such that sup_{|s−t|≤n^−κ2} n⁻¹ Σ_i |Z_i(s)−Z_i(t)| = O(n^{−1/2−κ₁}). And, for any positive sequence d_n → 0, there exists κ₃ > 0 such that ${sup}_{\mid s-t\mid \le d_n}{n}^{-1}{\sum }_i\mid Z_i(s)-Z_i(t)\mid =o\{max(d_n^{{\kappa }_3},n^{-{\kappa }_3})\}$.

These conditions are mainly used to facilitate our proofs, which are not necessarily the weakest. Nevertheless, most of these conditions have been commonly used in the literature. Specifically, C1 and C2′ are used in Zeng and Lin (2006) to justify the asymptotics for NPMLE, and C2-C4 are used in Chen, Jin and Ying (2001). C5 is the smoothness condition for time-dependent covariates, which was used in Lin and Ying (1995). C5 is satisfied when Z_i(·) are smooth in the sense that they have uniformly bounded derivatives or when Z_i(·) are step functions with random between-jump times having uniformly bounded densities.

Proof of Theorem 3

To show the asymptotic results in this theorem, we first establish a martingale representation of S_n(β₀).

Consider that

$$\begin{array}{l}{S}_n({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }{Z}_i(t)\left[\left\{B_i(t)+{\widehat{R}}_n(t;{\beta }_0)\right\}{dN}_i(t)-Y_i(t)\left\{{d\widehat{R}}_n(t;{\beta }_0)-{\widehat{R}}_n(t;{\beta }_0)dlog{B}_i(t)\right\}\right]\\ =\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left[\left\{B_i(t)+{\widehat{R}}_n(t;{\beta }_0)\right\}{dN}_i(t)+Y_i(t){\widehat{R}}_n(t;{\beta }_0)dlog{B}_i(t)\right],\end{array}$$

where Z̄(t) =Σ_i Y_i(t)Z_i(t)/Σ_i Y_i(t) as defined in §3. Replace N_i(t) by

$$N_i(t)=M_i(t)+{\int }_0^t{Y}_i(u){\{B_i(u;{\beta }_0)+R_0(u)\}}^{-1}\{{dR}_0(u)-R_0(u)dlog{B}_i(u;{\beta }_0)\},$$

and then we have

$$\begin{array}{l}{S}_n({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}(\left\{B_i(t;{\beta }_0)+{\widehat{R}}_n(t;{\beta }_0)\right\}\\ \times \left[M_i(t)+\frac{Y_i(t)\{{dR}_0(t)-R_0(t)dlog{B}_i(t;{\beta }_0)\}}{B_i(t;{\beta }_0)+R_0(t)}\right]+Y_i(t){\widehat{R}}_n(t;{\beta }_0)dlog{B}_i(t;{\beta }_0))\\ =\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left\{B_i(t;{\beta }_0)+{\widehat{R}}_n(t;{\beta }_0)\right\}{dM}_i(t)\\ +\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left\{B_i(t;{\beta }_0)+{\widehat{R}}_n(t;{\beta }_0)\right\}\left[\frac{Y_i(t)\{{dR}_0(t)-R_0(t)dlog{B}_i(t;{\beta }_0)\}}{B_i(t;{\beta }_0)+R_0(t)}\right]\\ +\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}{Y}_i(t){\widehat{R}}_n(t;{\beta }_0)dlog{B}_i(t;{\beta }_0)\\ =\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left\{B_i(t;{\beta }_0)+{\widehat{R}}_n(t;{\beta }_0)\right\}{dM}_i(t)\\ +\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\frac{{\widehat{R}}_n(t;{\beta }_0)-R_0(t)}{B_i(t;{\beta }_0)+R_0(t)}{Y}_i(t){dR}_0(t)\\ +\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left\{{\widehat{R}}_n(t;{\beta }_0)-R_0(t)\right\}\frac{Y_i(t)B_i(t;{\beta }_0)dlog{B}_i(t;{\beta }_0)}{B_i(t;{\beta }_0)+R_0(t)}.\end{array}$$

Further replace R̂_n(t; β₀) − R₀(t) by its martingale representation in (11), and we have

$$\begin{array}{l}{S}_n({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left\{B_i(t;{\beta }_0)+{\widehat{R}}_n(t;{\beta }_0)\right\}{dM}_i(t)\\ +\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\frac{{\widehat{R}}_n(t;{\beta }_0)-R_0(t)}{B_i(t;{\beta }_0)+R_0(t)}{Y}_i(t)d\{R_0(t)+B_i(t;{\beta }_0)\}\\ =\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left\{B_i(t;{\beta }_0)+{\widehat{R}}_n(t;{\beta }_0)\right\}{dM}_i(t)\\ +\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left[-{\int }_0^t\frac{P_n(u-;{\beta }_0){\sum }_k\{B_k(u;{\beta }_0)+R_0(u)\}}{P_n(t;{\beta }_0){\sum }_j{Y}_j(u)}{dM}_k(u;{\beta }_0,R_0)\right]\\ \times Y_i(t)dlog\{R_0(t)+B_i(t;{\beta }_0)\}.\end{array}$$

Therefore, as a result of integration by parts, we have

$$\begin{array}{l}{S}_n({\beta }_0)=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left\{B_i(t;{\beta }_0)+{\widehat{R}}_n(t;{\beta }_0)\right\}{dM}_i(t),\\ -\sum _{i=1}^n{\int }_0^{\tau }\frac{P_n(t-)\{B_i(t;{\beta }_0)+R_0(t)\}}{{\sum }_j{Y}_j(t)}\left[\sum _{k=1}^n{\int }_t^{\tau }\frac{\{Z_k(u)-\overline{Z}(u)\}{Y}_k(u)}{P_n(u)}dlog\{R_0(u)+B_k(u)\}\right]{dM}_i(t)\\ \bumpeq \sum _{i=1}^n{\int }_0^{\tau }{\xi }_i(t;{\beta }_0){dM}_i(t).\end{array}$$

where

$${\xi }_i(t;\beta )=\{Z_i(t)-\overline{Z}(t)\}\{B_i(t;\beta )+R_0(t)\}-\frac{P_n(t-;\beta )\{B_i(t;\beta )+R_0(t)\}}{{\sum }_j{Y}_j(t)}\left[\sum _{k=1}^n{\int }_t^{\tau }\frac{\{Z_k(u)-\overline{Z}(u)\}{Y}_k(u)}{P_n(u;\beta )}dlog\{R_0(u)+B_k(u;\beta )\}\right],$$

as defined in Theorem 3. By this martingale representation of S_n(β₀), it is apparent that n⁻¹S_n(β₀) →_P 0 by the weak law of large numbers (WLLN).

Let μ(t) = lim_n Σ_i Y_i(t)Z_i(t)/Σ_i Y_i(t) = ℰ {Y (t)Z (t)}/ℰ Y (t) and

$$\begin{array}{l}R(t;\beta )=\underset{n}{lim}{\widehat{R}}_n(t;\beta )=\frac{{lim}_n{\int }_0^t{P}_n(t;\beta ){dQ}_n(t;\beta )}{{lim}_n{P}_n(t;\beta )}\\ =\frac{{\int }_0^{t}exp\left[-{\int }_0^u\mathcal{E}\{dN(s)+\beta Y(s)dZ(s)\}]/\mathcal{E}Y(s)\right]\mathcal{E}[exp\{\beta Z(u)\}dN(u)]/\mathcal{E}Y(u)}{exp\left[-{\int }_0^t\mathcal{E}\{dN(u)+\beta Y(u)dZ(u)\}]/\mathcal{E}Y(u)\right]}.\end{array}$$

It is apparent R₀(t) = R(t; β₀) under the assumed model (5). Furthermore, let

$$H_1(t)={\int }_0^t\frac{\mathcal{E}\{Y(u)dZ(u)\}}{\mathcal{E}Y(u)}\text{and}{dH}_2(t;{\beta }_0)=\frac{H_1(t)\mathcal{E}\{e^{{\beta }_{0}Z(t)}dN(t)\}}{\mathcal{E}Y(t)}-\frac{\mathcal{E}{e}^{\beta Z(t)}Z(t)dN(t)}{\mathcal{E}Y(t)}.$$

Then

$$\begin{array}{l}{R}_{\beta }^{\prime }(t;\beta )=\frac{d}{d\beta }\frac{{\int }_0^{t}exp\left[-{\int }_0^u\mathcal{E}\{dN(s)+\beta Y(s)dZ(s)\}]/\mathcal{E}Y(s)\right]\mathcal{E}[exp\{\beta Z(u)\}dN(u)]/\mathcal{E}Y(u)}{exp\left[-{\int }_0^t\mathcal{E}\{dN(u)+\beta Y(u)dZ(u)\}]/\mathcal{E}Y(u)\right]}\\ ={\left\{\frac{1}{exp\left[-{\int }_0^t\mathcal{E}\{dN(u)+\beta Y(u)dZ(u)\}]/\mathcal{E}Y(u)\right]}\right\}}^2\\ \times ({\int }_0^{t}exp\left[-{\int }_0^u\mathcal{E}\{dN(s)+\beta Y(s)dZ(s)\}]/\mathcal{E}Y(s)\right]\left[-{\int }_0^u\mathcal{E}\{Y(s)dZ(s)\}]/\mathcal{E}Y(s)\right]\\ \times \mathcal{E}[exp\{\beta Z(u)\}dN(u)]/\mathcal{E}Y(u)·exp\left[-{\int }_0^t\mathcal{E}\{dN(u)+\beta Y(u)dZ(u)\}]/\mathcal{E}Y(u)\right]\\ +{\int }_0^{t}exp\left[-{\int }_0^u\mathcal{E}\{dN(s)+\beta Y(s)dZ(s)\}]/\mathcal{E}Y(s)\right]\mathcal{E}[exp\{\beta Z(u)\}Z(u)dN(u)]/\mathcal{E}Y(u)\\ \times exp\left[-{\int }_0^t\mathcal{E}\{dN(u)+\beta Y(u)dZ(u)\}]/\mathcal{E}Y(u)\right]\\ -{\int }_0^{t}exp\left[-{\int }_0^u\mathcal{E}\{dN(s)+\beta Y(s)dZ(s)\}]/\mathcal{E}Y(s)\right]\mathcal{E}[exp\{\beta Z(u)\}dN(u)]/\mathcal{E}Y(u)\\ exp\left[-{\int }_0^t\mathcal{E}\{Y(u)dZ(u)\}]/\mathcal{E}Y(u)\right]).\end{array}$$

As a result,

$$R_{\beta }^{\prime }(t;{\beta }_0)=-\frac{1}{S_0(t)}\left[{\int }_0^t{S}_0(u){dH}_2(u)+R_0(u)exp\{-H_1(t)\}\right].$$

We assume an additional regularity condition on $R_{\beta }^{\prime }(t;{\beta }_0)$ (C6):

$$\mid \mathcal{E}\{e^{\beta Z(t)}Z(t)\}+R_{\beta }^{\prime }(t;{\beta }_0)d\mathcal{E}N(t)+{\{\beta R(t;\beta )\}}_{\beta ={\beta }_0}^{\prime }\mathcal{E}Y(t)dZ(t)\mid >0.$$

This condition can is generally satisfactory, in particular when Z(·) is time-independent.

Now consider an arbitrary β ≠ β₀. Denote

$$\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\mu (t)\}[\{B_i(t;\beta )-B_i(t;{\beta }_0)+R(t;\beta )-R_0(t)\}{dN}_i(t)+Y_i(t)\{\beta R(t;\beta )-{\beta }_0{R}_0(t)\}{dZ}_i(t)]$$

by s_n(β). Apparently, s_n(β₀) = 0. Assume that there exists ε > 0 such that (C7):

$$\text{pr}\{\mid Z_i(t)-\mu (t)\mid >\epsilon ,i=1,2,\dots ,\dots ,n\}>0,$$

which essentially means covariate processes cannot be identical for all the study subjects. Then under conditions (C6) and (C7), we know that $\mid {lim}_n{n}^{-1}{s}_n^{\prime }({\beta }_0)\mid >0$. Without loss of generality, we simply assume that $s_n^{\prime }({\beta }_0)>0$. Therefore, there must exist a neighborhood of β₀ and an N₀ > 0, U(β₀) say, such that n > N₀, s_n(β) is monotonically increasing with a unique zero-crossing at β₀ in U (β₀).

We know that

$$\begin{array}{l}{n}^{-1}{S}_n(\beta )-n^{-1}{S}_n({\beta }_0)\\ =n^{-1}\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}[\{B_i(t;\beta )-B_i(t;{\beta }_0)+{\widehat{R}}_n(t;\beta )-{\widehat{R}}_n(t;{\beta }_0)\}{dN}_i(t)\\ +Y_i(t)\{{\widehat{R}}_n(t;\beta )dlog{B}_i(t;\beta )-{\widehat{R}}_n(t;{\beta }_0)dlog{B}_i(t;{\beta }_0)\}]\bumpeq n^{-1}{s}_n(\beta ),\end{array}$$

by the definition of s_n(β). Hence, n⁻¹||S_n(β) − s_n(β)|| → 0. Moreover, since

$$S_n^{\prime }(\beta )=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left[\left\{B_i^{\prime }(t;\beta )+{\widehat{R}}_n(t;\beta )\right\}{dN}_i(t)+Y_i(t){\{\beta {\widehat{R}}_n(t;\beta )\}}_{\beta }^{\prime }{dZ}_i(t)\right],$$

and

$$S_n^{\prime }(\beta )=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\mu (t)\}\left[\{e^{\beta Z_i(t)}{Z}_i(t)+R_{\beta }^{\prime }(t;\beta )\}{dN}_i(t)+Y_i(t){\{\beta R(t;\beta )\}}_{\beta }^{\prime }{dZ}_i(t)\right],$$

with an application of the Glivenko-Cantelli Lemma, we know that $n^{-1}\left|\right|S_n^{\prime }(\beta )-s_n^{\prime }(\beta )\left|\right|{\to }_{P}0$ uniformly in β ∈ U(β₀) under condition (C5). Therefore, for any ε > 0, there exists sufficiently large n and ν > 0, such that S_n(β₀ − ε) · S_n(β₀ + ε) < 0, and pr{β̂_n ∈ (β₀ − ε, β₀ + ε)} > 1 − ν, which then leads to the consistency of β̂_n.

Moreover, by the martingale central limit theorem (MCLT), n^−1/2S₀(β₀) is asymptotically mean-zero normal with variance-covariance matrix

$$\begin{array}{l}V=\underset{n\to \infty }{lim}{V}_n=\underset{n\to \infty }{lim}<n^{-1/2}{S}_n({\beta }_0),n^{-1/2}{S}_n({\beta }_0)>(\tau )=\underset{n\to \infty }{lim}{\int }_0^{\tau }\frac{1}{n}\sum _{i=1}^n{\xi }_i{(u)}^{\otimes 2}{Y}_i(u){d\mathrm{\Lambda }}_i(u)\\ =\underset{n\to \infty }{lim}{\int }_0^{\tau }\frac{1}{n}\sum _{i=1}^n\frac{Y_i(u){\xi }_i{(u)}^{\otimes 2}[\{{dR}_0(u)-R_0(u)dlog{B}_i(u;{\beta }_0)\}]}{B_i(u;{\beta }_0)+R_0(u)},\end{array}$$

as in Yang & Prentice (1999).

For the asymptotic normality of β̂_n, we consider that

$$\begin{array}{l}\frac{\partial S_n({\beta }_0)}{\partial \beta }=\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}[\left\{\frac{\partial B_i(t;{\beta }_0)}{\partial \beta }+\frac{\partial {\widehat{R}}_n(t;{\beta }_0)}{\partial \beta }\right\}{dN}_i(t)\\ {+Y_i(t)\frac{\partial {\widehat{R}}_n(t;{\beta }_0)}{\partial \beta }dlog{B}_i(t;\beta )+Y_i(t){\widehat{R}}_n(t;{\beta }_0)d\frac{\partial log{B}_i(t;{\beta }_0)}{\partial \beta }]}^{\text{T}}\\ =\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}[\left\{e^{{\beta }_0{Z}_i(t)}{Z}_i(t)+\frac{\partial {\widehat{R}}_n(t;{\beta }_0)}{\partial \beta }\right\}{dN}_i(t)\\ {+Y_i(t)\frac{\partial {\widehat{R}}_n(t;{\beta }_0)}{\partial \beta }dlog{B}_i(t;\beta )+Y_i(t){\widehat{R}}_n(t;{\beta }_0){dZ}_i(t)]}^{\text{T}}.\end{array}$$

Under the assumed regularity conditions, it is true that n⁻¹∂S_n(β₀)/∂β converges to

$$U=\underset{n\to \infty }{lim}{n}^{-1}\sum _{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\overline{Z}(t)\}\left[\left\{\frac{\partial {\widehat{R}}_n(t;{\beta }_0)}{\partial \beta }+e^{{\beta }_0{Z}_i(t)}{Z}_i(t)\right\}{dN}_i(t)+Y_i(t)\left\{\frac{\partial {\widehat{R}}_n(t;{\beta }_0)}{\partial \beta }{\beta }_0+R_0(t)\right\}{dZ}_i(t)\right].$$

Via Taylor expansion, we obtain that n^1/2(β̂_n − β₀) ≏ {n⁻¹∂S_n(β₀)/∂β}⁻¹ · n^−1/2S_n(β₀). Therefore, an application of Delta-method coupled with the consistency of β̂_n leads to the conclusions that

$$n^{1/2}({\widehat{\beta }}_n-{\beta }_0)\stackrel{\mathcal{D}}{\to }\mathcal{N}\{0,U^{-1}V{(U^{-1})}^{\text{T}}\},$$

as stated in the theorem.

In addition, since Corollary 4 establishes same asymptotic properties for the weighted estimators, proof of Corollary 4 follow essentially same steps, except that additional weight functions would be included in all calculations. To save space, we here only present the major steps to prove Theorem 3.