Joint Modeling of Survival Time and Longitudinal Data with Subject-specific Changepoints in the Covariates

Abstract

Joint models are frequently used in survival analysis to assess the relationship between time-to-event data and time-dependent covariates, which are measured longitudinally but often with errors. Routinely, a linear mixed-effects model is used to describe the longitudinal data process, while the survival times are assumed to follow the proportional hazards model. However, in some practical situations, individual covariate profiles may contain changepoints. In this article, we assume a two-phase polynomial random effects with subject-specific changepoint model for the longitudinal data process and the proportional hazards model for the survival times. Our main interest is in the estimation of the parameter in the hazards model. We incorporate a smooth transition function into the changepoint model for the longitudinal data and develop the corrected score and conditional score estimators, which do not require any assumption regarding the underlying distribution of the random effects or that of the changepoints. The estimators are shown to be asymptotically equivalent and their finite-sample performance is examined via simulations. The methods are applied to AIDS clinical trial data.

1 Introduction

An ordinary objective in many medical studies is to investigate how survival times, which are usually censored are related to some time-independent or time-varying covariates. When the true values of the time-varying covariates are always observed, the Cox [1] proportional hazards model is a powerful tool which is commonly used to describe the relationship. However, instead of observing the true time-dependent covariate process, some measurements are taken longitudinally. These measured values of the covariate are usually error-prone and their replacements for the true covariate values in the proportional hazards model may cause biased estimation [2].

A popular approach for the analysis of survival time and longitudinal data measured with error is to jointly model the time-to-event data and the covariate process. This makes possible the exploitation of the information contained in both data in dealing with the measurement errors. Joint modeling often assumes a proportional hazards model for the survival times and a linear mixed-effects model for the longitudinal data. The literature is rich in methods developed under this framework. These approaches include parametric methods with an assumption about the distribution function of the random effects and that of the measurement errors [3–5], semiparametric methods such as the conditional score approach [6,7], the corrected score method [8] and the time-varying coefficients proportional hazards model [9].

However, in many studies the trajectories of individual longitudinal data may show some nonlinearity, indicating that a linear random effects model may not be suitable for the covariate process [10]. A motivating example arises from the AIDS clinical trial ACTG-175 data in which the CD4 cells counts tend to increase initially and drop down later for some individuals. Nonlinearity sometimes may be captured by a changepoint model. Models of this type are often of interest in the analysis of data from clinical studies. Kiuchi et al. [11] used a piecewise linear model for the T4 counts to analyze data from homosexual men who were mostly HIV-infected. Their goal was to estimate the distribution of the time of the change that occurs in the T4 counts and the parameters were estimated based on empirical and hierarchical Bayes methods. In a joint modeling with change point framework, Faucett et al. [12] assumed a piecewise linear random effects model for the logCD4 and the proportional hazards model for the time to AIDS for the analysis of data from the AIDS clinical trial ACTG-019 with the purpose of comparing zidovudine (zdv) treatment and placebo. The zdv treatment is believed to cause an initial increase in the CD4 count that tends to later decline. In their model, the random effects are normally distributed and the changepoint follows a lognormal distribution. Multiple-imputation techniques were used to estimate the parameters in the model which was aimed to characterize the association between the CD4 counts and the survival time. In another context, Jacqmin-Gadda et al. [13] jointly modeled cognitive decline and risk of dementia assuming a piecewise polynomial mixed-effects model with random changepoints to describe the cognitive decline and a log-normal model for time-to-dementia. The parameters were estimated using a maximum likelihood-based method. These approaches are very restrictive as they require specification of the underlying distribution functions of the random effects and changepoints.

In this paper, we propose a more flexible joint model of survival time and error-prone longitudinal covariate data, with less restrictive estimation techniques. More explicitly, we assume a subject-specific changepoint model for the longitudinal covariate data and the proportional hazards model is considered for the time-to-event data. The proposed nonlinear random effects model is different and more flexible compared with the aforementioned models. A sharp transition changepoint model is assumed to characterize the longitudinal covariate process. However, asymptotic results are not straightforward for such model as pointed out by Seber and Wild [14]. This has led us to incorporate a smooth transition function into the changepoint model. We concentrate on the estimation of the parameter in the proportional hazards regression model and develop two estimators based on approaches, which leave unspecified the distributions of the random effects and changepoints, namely the conditional score and corrected score methods. These estimators are shown to be asymptotically equivalent.

We describe our model formulation in Section 2 and construct in Section 3 the corrected score and conditional score estimators of the parameter in the model for the survival time data. In Section 4, the performance of the estimators is investigated via simulation studies. The approaches are applied to the ACTG-175 data and the results of the analysis are presented in Section 5. Finally, some concluding remarks are given in Section 6.

2 Model formulation

Suppose there are n subjects, which are followed up over the study period [0, L], where 0 < L < ∞. For each subject i, i = 1, …, n, let $T_i^0$ denote the possibly right-censored survival time and C_i be the censoring time. The observed survival data for subject i are the observed survival time $T_i\equiv min(T_i^0,C_i)$ and the failure time indicator ${\mathrm{\Delta }}_i\equiv I(T_i^0\le C_i)$, where I(.) is an indicator function. Let Z_i(u) denote the true time-dependent covariate process, which is not observable, and Z_0i(u) = {Z_01i(u), …, Z_0K0i(u)}′ be a K₀-dimensional vector of covariates consisting of both time-independent and time-dependent covariates, which can be accurately measured at any time u. Some longitudinal measurements of size m_i are taken on the covariate Z_i at ordered occasions t_ij with t_imi ≤ T_i, j = 1, …, m_i, i = 1, …, n. The univariate assumption for the covariate Z_i(u) is simply for convenience and the general multivariate case can be extended similarly but with more complicated calculations.

2.1 The longitudinal data submodel

In our problem, the true covariate data Z_i(t_ij), i = 1, …, n, j = 1, …, m_i, are subject to measurement error. The longitudinal observations of Z_i are W_ij, j = 1, …, m_i, i = 1, …, n, which are assumed to follow the nonlinear random effects model

$$W_{ij}=\mathbf{D}(t_{ij},{\tau }_i){\mathit{\alpha }}_i+e_{ij},j=1,\dots ,m_i,i=1,\dots ,n,$$ (1)

where D(u, a) ≡ [1, (u − a), …, (u − a)^p, (u − a)sign(u − a), …, (u − a)^psign(u − a)] for any u and a, α_i denotes a (2p+1)-dimensional vector of random coefficients, and τ_i is the subject-specific changepoint occurring in the time trajectory of the covariate Z_i for the ith subject. Here, τ_i and α_i are allowed to be correlated and τ_i should verify t_i2 ≤ τ_i ≤ t_imi−1. The function sign(.) is defined such that sign(u) = −1 for u < 0, sign(0) = 0 and sign(u) = 1 otherwise. Hence, the true process of the possibly mismeasured longitudinal covariate is assumed to be Z_i(u) = D(u, τ_i)α_i, i = 1, …, n. Note that model (1) can be rewritten as

$$W_{ij}=\sum _{k=0}^p{\alpha }_{ij}{(t_{ij}-{\tau }_i)}^k+\sum _{k=1}^p{\alpha }_{ip+k}{(t_{ij}-{\tau }_i)}^k\text{sign}(t_{ij}-{\tau }_i)+e_{ij},j=1,\dots ,m_i,i=1,\dots ,n.$$

This submodel assumes a two-phase polynomial random effects model with a sharp but continuous transition at the changepoint for the process of the longitudinal covariate. It provides a flexible parameterization of the covariate process and the simplest case corresponds to that of p = 1, where the submodel reduces to a two-phase piecewise linear model:

$$Z_i(u)=\{\begin{array}{ll}{\alpha }_{i0}+({\alpha }_{i1}-{\alpha }_{i2})(u-{\tau }_i)\hfill & \text{for}u\le {\tau }_i,\hfill \\ {\alpha }_{i0}+({\alpha }_{i1}+{\alpha }_{i2})(u-{\tau }_i)\hfill & \text{otherwise}.\hfill \end{array}$$

The vector of errors e_i = (e_i1, …, e_imi)′ is assumed to follow a zero-mean multivariate normal distribution with covariance matrix σ²I_mi, where I_mi is the identity matrix of size m_i, and σ² is a nonnegative real number. Let θ_i denote ${({\mathit{\alpha }}_i^{\prime },{\tau }_i)}^{\prime }$ and t_i = (t_i1, …, t_imi)′ be the vector of measurement times for subject i. It is assumed that θ_i and Z_0i are independent, and e_i is independent of ${(T_i,C_i,{\mathrm{\Delta }}_i,m_i,{\mathbf{t}}_i^{\prime },{\mathit{\theta }}_i^{\prime },{\mathbf{Z}}_{0i}^{\prime })}^{\prime }$. Moreover, ${(T_i,C_i,{\mathrm{\Delta }}_i,m_i,{\mathbf{t}}_i^{\prime },{\mathbf{e}}_i^{\prime },{\mathit{\theta }}_i^{\prime },{\mathbf{Z}}_{0i}^{\prime })}^{\prime }$, i = 1, …, n, are independent and identically distributed random variables and the sets of individual observation times possibly differ among subjects. Let μ and ϒ denote the mean and variance covariance matrix of θ₁, respectively. The notations W_i = (W_i1, …, W_imi)′ and η = (σ², μ′, vec(ϒ))′ are used further.

2.2 The submodel for the survival time data

The survival times are linked to the longitudinal covariates via the proportional hazards model. Assuming a non-informative censoring as well as the independence of survival times to measurement times and errors, the hazards function for subject i is defined as follows:

$$\begin{array}{l}{\lambda }_i\{u\mid Z_i,{\mathbf{Z}}_{0i}\}=\underset{du\to 0}{lim}[\text{pr}\{u\le T_i^0<u+du\mid T_i^0\ge u,{\mathit{\alpha }}_i,{\tau }_i,{\mathbf{Z}}_{0i}\}/du]\\ ={\lambda }_0(u)exp\{{\mathit{\beta }}^{\prime }{\mathbf{X}}_i(u)\},\end{array}$$

where ${\mathbf{X}}_i(u)={\{Z_i(u),{\mathbf{Z}}_{0i}^{\prime }(u)\}}^{\prime },\mathit{\beta }={({\beta }_1,{\mathit{\beta }}_2^{\prime })}^{\prime }$ and λ₀(.) is an unspecified baseline hazards function. The main challenge is that Z_i(u) is not observed and has a changepoint. Hence, inference of the parameter estimation will be based on the finite number of longitudinal data W_i, i = 1, …, n.

3 Estimation

Our interest focuses on the characterization of the relationship between the survival time and the longitudinal covariate. Hence, we aim to estimate the parameter β in the survival time submodel. However, the true covariate process is unknown and one may like to naively replace the individual random effects and changepoints with their nonlinear least squares estimators, based on the individual longitudinal data. Note that the regression function f(u, θ_i) ≡ D(u, τ_i)α_i is not differentiable with respect to τ_i. This irregularity constitutes a major drawback for model (1) since desirable properties, such as the consistency and normality of the estimators of the random effects and changepoints, may not hold. In addition, the derivation of the variance covariance matrix of the estimator of θ_i, which is crucial in the estimation of β, may be cumbersome. An alternative would be the use of smoothing techniques. Following Bacon and Watts [15], we replace sign(u) in D(u, τ_i) with a smooth transition function of the form turn(u/γ), where γ is a positive smoothing parameter. To mimic sign(u), the function turn(u/γ) has to satisfy conditions including the followings:

$$\begin{array}{l}{lim}_{u\to \infty }\text{turn}(\mid u\mid /\gamma )=1,{lim}_{u\to \infty }u\text{turn}(u/\gamma )=\mid u\mid ,\\ {lim}_{\gamma \to 0}\text{turn}(u/\gamma )=\text{sign}(u),\text{and}\text{turn}(0)=0.\end{array}$$

The parameter γ controls the sharpness of the transition at the changepoint, the smaller value it takes the steeper the change and the more turn(u/γ) and sign(u) agree at a given time point u. Further insight into smooth transition functions can be found in [15]. We consider “smoothing” the changepoint model (1) with the hyperbolic tangent function, tanh(.), and obtain a more general version of model (1) which is given as follows:

$$W_{ij}=\stackrel{\sim }{f}(t_{ij},{\mathit{\theta }}_i)+e_{ij},j=1,\dots ,m_i,i=1,\dots ,n,$$ (2)

where f̃(u, θ_i) ≡ D̃(u, τ_i)α_i and D̃(u, τ_i) is defined similarly as D(u, τ_i), but with sign(u−τ_i) replaced by tanh{(u−τ_i)/γ_i}. The parameter γ_i denotes the ith subject’s specific smoothing parameter, which may also need to be estimated along with θ_i. In such case, the estimation would require an increase in the minimum number of observations per individual. For simplicity, it is assumed here that γ_i is known. Moreover, since the transition at each changepoint is considered abrupt, we further assume the same and small smoothing parameter γ for all individuals. Based on some simulation results, which did not change considerably for values of γ smaller than 0.01, the value of γ is taken to be 0.01. It is worth mentioning that the regression function f̃(.) has the nice property of being continuously differentiable at each component of θ_i, i = 1, …, n. Next, treating θ_i as a vector of fixed parameters, we proceed as follows for its estimation at any given time u. Let $m_i^u$ denote the number of longitudinal observations for subject i by time u. The estimator of θ_i at time u, such that $m_i^u>2p+2$, is taken to be ${\widehat{\mathit{\theta }}}_i^u$, which is the minimizer of $g_u({\mathit{\theta }}_i)\equiv {\sum }_{j=1}^{m_i^u}{\{W_{ij}-\stackrel{\sim }{f}(t_{ij},{\mathit{\theta }}_i)\}}^2$. Let ${\mathit{\theta }}_i^{\ast }$ denote the true value of θ_i. A first order Taylor series expansion of f̃(u, θ_i) about ${\mathit{\theta }}_i^{\ast }$ yields

$$\stackrel{\sim }{f}(u,{\mathit{\theta }}_i)\approx \stackrel{\sim }{f}(u,{\mathit{\theta }}_i^{\ast })+\mathbf{H}(u,{\mathit{\theta }}_i^{\ast })({\mathit{\theta }}_i-{\mathit{\theta }}_i^{\ast }),$$ (3)

where H(u, θ) ≡ (∂/∂θ)f̃(u, θ). Let ${\mathbf{W}}_i^u={(W_{i1},\dots ,W_{{im}_i^u})}^{\prime },{\mathbf{e}}_i^u={(e_{i1},\dots ,e_{{im}_i^u})}^{\prime },{\stackrel{\sim }{\mathbf{f}}}_i(u,\mathit{\theta })\equiv {[\stackrel{\sim }{f}(t_{i1},\mathit{\theta }),\dots ,\stackrel{\sim }{f}(t_{{im}_i^u},\mathit{\theta })]}^{\prime }$. Also, let ${\stackrel{\sim }{\mathbf{W}}}_i^u={\mathbf{W}}_i^u-{\stackrel{\sim }{\mathbf{f}}}_i(u,{\mathit{\theta }}_i^{\ast })+{\stackrel{\sim }{\mathbf{F}}}_i(u,{\mathit{\theta }}_i^{\ast }){\mathit{\theta }}_i^{\ast }$, where ${\stackrel{\sim }{\mathbf{F}}}_i(u,{\mathit{\theta }}_i^{\ast })$ is the $m_i^u\times (2p+2)$-dimensional matrix whose jth row is $\mathbf{H}(t_{ij},{\mathit{\theta }}_i^{\ast })$. Based on (3), it can be obtained that ${\widehat{\mathit{\theta }}}_i^u$ is approximately ${\{{\stackrel{\sim }{\mathbf{F}}}_i^{\prime }(u,{\mathit{\theta }}_i^{\ast }){\stackrel{\sim }{\mathbf{F}}}_i(u,{\mathit{\theta }}_i^{\ast })\}}^{-1}{\stackrel{\sim }{\mathbf{F}}}_i^{\prime }(u,{\mathit{\theta }}_i^{\ast }){\stackrel{\sim }{\mathbf{W}}}_i^u$, and ${\widehat{Z}}_i(u)\equiv \stackrel{\sim }{f}(u,{\widehat{\mathit{\theta }}}_i^u)$ can be approximated as $\stackrel{\sim }{f}(u,{\mathit{\theta }}_i^{\ast })+\mathbf{H}(u,{\mathit{\theta }}_i^{\ast })({\widehat{\mathit{\theta }}}_i^u-{\mathit{\theta }}_i^{\ast })$. With the assumption of normal error, it is straightforward to see that conditionally on θ_i, ${\widehat{\mathit{\theta }}}_i^u$ is approximately normally distributed with mean θ_i and variance ${\sigma }^2{\{{\stackrel{\sim }{\mathbf{F}}}_i^{\prime }(u,{\mathit{\theta }}_i^{\ast }){\stackrel{\sim }{\mathbf{F}}}_i(u,{\mathit{\theta }}_i^{\ast })\}}^{-1}$. Hence, given θ_i and for known σ², Ẑ_i(u) follows an approximate normal distribution with mean Z_i(u) and variance that can be approximated as σ²Ω_i(u), where ${\mathbf{\Omega }}_i(u)\equiv \mathbf{H}(u,{\widehat{\mathit{\theta }}}_i^u){\{{\stackrel{\sim }{\mathbf{F}}}_i^{\prime }(u,{\widehat{\mathit{\theta }}}_i^u){\stackrel{\sim }{\mathbf{F}}}_i(u,{\widehat{\mathit{\theta }}}_i^u)\}}^{-1}{\mathbf{H}}^{\prime }(u,{\widehat{\mathit{\theta }}}_i^u)$. Let ${\widehat{\mathbf{X}}}_i(u)\equiv {\{{\widehat{Z}}_i(u),{\mathbf{Z}}_{0i}^{\prime }(u)\}}^{\prime },{\mathbf{R}}_i(u)\equiv {[\mathbf{H}(u,{\mathit{\theta }}_i^{\ast }){\{{\stackrel{\sim }{\mathbf{F}}}_i^{\prime }(u,{\mathit{\theta }}_i^{\ast }){\stackrel{\sim }{\mathbf{F}}}_i(u,{\mathit{\theta }}_i^{\ast })\}}^{-1}{\stackrel{\sim }{\mathbf{F}}}_i^{\prime }(u,{\mathit{\theta }}_i^{\ast }){\mathbf{e}}_i^u,{\mathbf{0}}_{K_0}^{\prime }]}^{\prime }$, and Σ_i(u) ≡ diag[Ω_i(u), 0_K0×K0], where 0_K0 and 0_K0×K0 are K₀-dimensional zero vector and K₀ × K₀ zero matrix respectively. Based on X̂_i(u), we can formulate the following additive measurement error model:

$$\{\begin{array}{ccc}{\widehat{\mathbf{X}}}_i(u)& \approx & {\mathbf{X}}_i(u)+{\mathbf{R}}_i(u),\\ \text{E}\{{\mathbf{R}}_i(u)\mid {\mathbf{X}}_i(u)\}& =& {\mathbf{0}}_{K_0+1},\\ var\{{\mathbf{R}}_i(u)\mid {\mathbf{X}}_i(u)\}& =& {\sigma }^2{\mathbf{\sum }}_i(u).\end{array}$$

Let Y_i(u) ≡ I(T_i ≥ u, t_i2p+2 ≤ u) denote the at-risk process and define the counting process increment dN_i(u) ≡ I(Δ_i = 1, u ≤ T_i ≤ u + du, t_i2p+2 ≤ u). For the estimation of β, the “ideal” regression corresponds to the case when the true longitudinal covariate process is known. In this case, the usual Cox partial likelihood score would be defined as

$$U_n^I(\mathit{\beta },\mathbf{X})\equiv \frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{{\mathbf{X}}_i(u)-\frac{S_1(u,\mathit{\beta },\mathbf{X})}{S_0(u,\mathit{\beta },\mathbf{X})}\right\}{dN}_i(u),$$

where X denotes the vector ${({\mathbf{X}}_1^{\prime },\dots ,{\mathbf{X}}_n^{\prime })}^{\prime },S_k(u,\mathit{\beta },\mathbf{X})=n^{-1}{\sum }_{i=1}^n{Y}_i(u){\mathbf{X}}_i^k(u)exp\{{\mathit{\beta }}^{\prime }{\mathbf{X}}_i(u)\}$, k = 0, 1. The “ideal” estimator of β is the solution to $U_n^I(\mathit{\beta },\mathbf{X})=0$. As the true longitudinal profile is unobserved for each subject i, a simple approach, which is referred to here as the “naive” regression, consists of using X̂_i(u) in place of X_i(u) in $U_n^I(\mathit{\beta },\mathbf{X})$ at any time u. The “naive” estimator for β, denoted by β̂_na, solves $U_n^I(\mathit{\beta },\widehat{\mathbf{X}})=0$, where $\widehat{\mathbf{X}}\equiv {({\widehat{\mathbf{X}}}_1^{\prime },\dots ,{\widehat{\mathbf{X}}}_n^{\prime })}^{\prime }$. Note that β̂_na is biased for β [2] and the substitution of X̂ for X may lead to erroneous conclusions. In the following, we develop some functional methods based on the corrected and conditional scores approaches.

3.1 Corrected score method

The principle of the corrected score method consists of removing the bias from a biased estimating function by means of some techniques which involve approximating the expectation of the biased estimating function [8,9,16]. The approach is free from a priori assumptions about the distribution of the random effects or that of the changepoints. Nakamura [16]’s method requires all individuals to have an equal number of longitudinal observations. In [8,9] the number of measurements is allowed to vary across subjects, and the time-dependent covariate process is described by a linear random effects model. Here, a changepoint model is considered and starts from a different biased estimating function for the parameter β. We define a biased estimating function for β as follows:

$$U_n^{bs}(\mathit{\beta })\equiv \frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)-\frac{{\stackrel{\sim }{S}}_1(u,\mathit{\beta },{\sigma }^2)}{{\stackrel{\sim }{S}}_0(u,\mathit{\beta },{\sigma }^2)}\right\}{dN}_i(u),$$

where ${\stackrel{\sim }{S}}_k(u,\mathit{\beta },{\sigma }^2)\equiv n^{-1}{\sum }_{i=1}^n{\stackrel{\sim }{S}}_{ki}(u,\mathit{\beta },{\sigma }^2)$ and ${\stackrel{\sim }{S}}_{ki}(u,\mathit{\beta },{\sigma }^2)\equiv Y_i(u){\widehat{\mathbf{X}}}_i^k(u)exp\{{\mathit{\beta }}^{\prime }{\widehat{\mathbf{X}}}_i(u)-{\sigma }^2{\mathit{\beta }}^{\prime }{\mathbf{\sum }}_i(u)\mathit{\beta }/2\}$, k = 0, 1. Next we try to find a corrected score estimating function from $U_n^{bs}(\mathit{\beta })$. For fixed u, β, and σ², let s_k(u, β), and s̃_k(u, β, σ²) denote E{S_k(u, β, X)}, and E{S̃_k(u, β, σ²)}, respectively, k = 0, 1. With the assumption of independence between the errors and the random effects, it can be shown that s̃₀(u, β, σ²) = s₀(u, β) and s̃₁(u, β, σ²) = s₁(u, β) + σ²s₀(u, β)Σ(u, β), where Σ(u, β) is the limit of $n^{-1}{\sum }_{i=1}^n{\mathbf{\sum }}_i(u)\mathit{\beta }$ as n goes to infinity. Letting ẽ(u, β, σ²) and e(u, β) denote s̃₁(u, β, σ²)/s̃₀(u, β, σ²), and s₁(u, β)/s₀(u, β), respectively, it is clear that ẽ(u, β, σ²) = e(u, β) + σ²Σ(u, β). Based on the functional Taylor series expansion of order one of the function (x, y) ↦ x/y, an approximation to E{S̃₁(u, β, σ²)/S̃₀(u, β, σ²)} yields the following corrected score function:

$$U_n^{cr}(\mathit{\beta },{\sigma }^2)\equiv \frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)+{\sigma }^2{\mathbf{\sum }}_i(u)\mathit{\beta }-\frac{{\stackrel{\sim }{S}}_1(u,\mathit{\beta },{\sigma }^2)}{{\stackrel{\sim }{S}}_0(u,\mathit{\beta },{\sigma }^2)}\right\}{dN}_i(u).$$

For known σ², the corrected score estimator of β, denoted by β̂_cr, solves $U_n^{cr}(\mathit{\beta },{\sigma }^2)=0$. Let β₀ denote the true value of β and define

$${\mathit{\omega }}_i(\mathit{\beta },{\sigma }^2)\equiv {\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)+{\sigma }^2{\mathbf{\sum }}_i(u)\mathit{\beta }-\frac{{\stackrel{\sim }{S}}_1(u,\mathit{\beta },{\sigma }^2)}{{\stackrel{\sim }{S}}_0(u,\mathit{\beta },{\sigma }^2)}\right\}\left\{{dN}_i(u)-\frac{{\stackrel{\sim }{S}}_{0i}(u,\mathit{\beta },{\sigma }^2)d\overline{N}(u)}{{\stackrel{\sim }{S}}_0(u,\mathit{\beta },{\sigma }^2)}\right\},$$

where $d\overline{N}(u)\equiv n^{-1}{\sum }_{i=1}^n{dN}_i(u)$. When approximation (3) is accurate, the asymptotic properties of $\sqrt{n}({\widehat{\mathit{\beta }}}_{cr}-{\mathit{\beta }}_0)$ are given in the following.

Property 1

Under conditions (A1)–(A5) in Appendix A, $\sqrt{n}({\widehat{\mathit{\beta }}}_{cr}-{\mathit{\beta }}_0)$ is asymptotically normally distributed with mean zero and variance, which can be estimated by the robust sandwich estimator {ϑ(β̂_cr)}⁻¹ κ(β̂_cr)[{ϑ(β̂_cr)}⁻¹, where $\vartheta (\mathit{\beta })\equiv n^{-1}{\sum }_{i=1}^n(\partial /\partial \mathit{\beta }){\mathit{\omega }}_i(\mathit{\beta },{\sigma }^2),\kappa (\mathit{\beta })\equiv n^{-1}{\sum }_{i=1}^n{\mathit{\omega }}_i{(\mathit{\beta },{\sigma }^2)}^{\otimes 2}$ and C^⊗2 denotes CC′ for any vector C.

We provide a proof of the asymptotic properties of $\sqrt{n}({\widehat{\mathit{\beta }}}_{cr}-{\mathit{\beta }}_0)$ in Appendix A. For notational convenience, let θ̂_i denote ${\widehat{\mathit{\theta }}}_i^{t_{{im}_i}}$ and F_i be F̃_i(t_imi, θ̂_i), i = 1, …, n. When the estimation of the nuisance parameter, η, is of interest an estimating equation for η based on the method of moments and (3) is given by ${\sum }_{i=1}^{n}I(m_i>2p+2){\mathrm{\Phi }}_i(\mathit{\eta })=\mathbf{0}$, where Φ_i(η) = {φ_i1(η), (φ_i2(η))′, vec(φ_i3(η))}′, φ_i1(η) = {W_i − f̃_i(t_imi, θ̂_i)}′ {W_i − f̃_i(t_imi, θ̂_i)} − σ²(m_i − 2p − 2), φ_i2(η) = θ̂_i − μ and ${\phi }_{i3}(\mathit{\eta })={\widehat{\mathit{\theta }}}_i{\widehat{\mathit{\theta }}}_i^{\prime }-\mathit{\mu }{\mathit{\mu }}^{\prime }-{\sigma }^2{({\mathbf{F}}_i^{\prime }{\mathbf{F}}_i)}^{-1}-\mathrm{\Upsilon }$. Denote by η̂ the estimator of η which solves this estimating equation. Also, let Θ = (β′, η′)′, $\widehat{\mathrm{\Theta }}={({\widehat{\mathit{\beta }}}_{cr}^{\prime },{\widehat{\mathit{\eta }}}^{\prime })}^{\prime }$ and ${\mathrm{\Theta }}_0={({\widehat{\mathit{\beta }}}_0^{\prime },{\widehat{\mathit{\eta }}}_0^{\prime })}^{\prime }$, where η₀ is the true value of the nuisance parameter. The asymptotic variance covariance matrix of $\sqrt{n}(\widehat{\mathrm{\Theta }}-{\mathrm{\Theta }}_0)$ can be estimated by the sandwich estimator {ϑ̃(Θ̂)}⁻¹κ̃(Θ̂) [{ϑ̃(Θ̂)}⁻¹]′, where $\stackrel{\sim }{\vartheta }(\mathrm{\Theta })\equiv n^{-1}{\sum }_{i=1}^n(\partial /\partial \mathrm{\Theta }){\mathrm{\Psi }}_i(\mathrm{\Theta }),\stackrel{\sim }{\kappa }(\mathrm{\Theta })\equiv n^{-1}{\sum }_{i=1}^n{\mathrm{\Psi }}_i{(\mathrm{\Theta })}^{\otimes 2}$ and Ψ_i(Θ) ≡ {ω_i(β, σ²), I(m_i > 2p + 2)Φ_i(η)}′. Note that higher order Taylor series expansion of the function (x, y) ↦ x/y can be considered for the approximation to E{S̃₁(u, β, σ²)/S̃₀(u, β, σ²)}.

3.2 Conditional score estimation

The conditional score method was first introduced by Stefanski and Carroll [17] in generalized linear models with covariate measurement error. Tsiatis and Davidian [7] developed the method further in joint modeling. Their model was later extended by Song et al. [6] to account for multiple longitudinal covariates data measured with error. Note that in their method, the longitudinal covariate process follows a linear model with random effects. The idea behind the approach is to adjust the bias by conditioning on a sufficient statistic of the nuisance parameter. Similar to its corrected score counterpart, the conditional score method does not need any specification of the distribution of the random effects or that of the changepoints. In our case, Q_i(u, β, σ²) = X̂_i(u)+ σ²Σ_i(u)βdN_i(u) is a sufficient statistic for X_i(u) and the dependence of the hazards function for subject i on X_i(u) can be removed by conditioning on Q_i(u, β, σ²). Proceeding as in [7] the conditional hazards function can be shown to be

$$\begin{array}{l}{\lambda }_i\{u\mid {\mathbf{Q}}_i(u,\mathit{\beta },{\sigma }^2)\}=\underset{du\to 0}{lim}[\text{pr}\{{dN}_i(u)=1\mid {\mathbf{Q}}_i(u,\mathit{\beta },{\sigma }^2),Y_i(u),{\mathbf{t}}_i^u\}/du]\\ ={\lambda }_0(u)V_{0i}(u,\mathit{\beta },{\sigma }^2),\end{array}$$

where $V_{ki}(u,\mathit{\beta },{\sigma }^2)\equiv Y_i(u){\mathbf{Q}}_i^k(u,\mathit{\beta },{\sigma }^2)exp\left\{{\mathit{\beta }}^{\prime }{\mathbf{Q}}_i(u,\mathit{\beta },{\sigma }^2)-{\sigma }^2{\mathit{\beta }}^{\prime }{\mathbf{\sum }}_i(u)\mathit{\beta }/2\right\}$ for k = 0, 1, and ${\mathbf{t}}_i^u={(t_{i1},\dots ,t_{{im}_i^u})}^{\prime }$. Moreover, the conditional score estimating function for β is

$$U_n^{cs}(\mathit{\beta },{\sigma }^2)\equiv \frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{{\mathbf{Q}}_i(u,\mathit{\beta },{\sigma }^2)-\frac{V_1(u,\mathit{\beta },{\sigma }^2)}{V_0(u,\mathit{\beta },{\sigma }^2)}\right\}{dN}_i(u),$$

where $V_k(u,\mathit{\beta },{\sigma }^2)\equiv n^{-1}{\sum }_{i=1}^n{V}_{ki}(u,\mathit{\beta },{\sigma }^2)$ for k = 0, 1. For known σ², the conditional score estimator, denoted by β̂_cs, solves $U_n^{cs}(\mathit{\beta },{\sigma }^2)=0$. When (3) holds, an interesting relationship between the conditional and corrected scores estimators is given as follows.

Property 2

Under the regularity conditions (A1)–(A5) in Appendix A, the conditional and corrected scores estimators under the changepoint modeling are asymptotically equivalent.

A sketch of the proof of the asymptotic equivalence between the two estimators can be found in Appendix B.

4 Simulation studies

An extensive simulation study is carried out to investigate the performance of the methods presented in Section 3. A single error-prone time-dependent covariate is considered and its process is assumed to follow the two-phase piecewise linear model Z_i(u) = α_0i+α_1i(u−τ_i)+ α_2i|u − τ_i|. The observations of Z_i are generated according to the model W_ij = Z_i(t_ij) + e_ij, where the error e_ij is simulated from a normal distribution with mean 0 and variance σ² = 0.1, 0.2. The measurement are taken at time t_ij in {0, 4, 8, 10, 16, 24, 32, 40, 48, 56, 64, 72, 80} with at least eight measurements per subject. The available longitudinal observations are obtained prior to the observed event times. The survival times are generated according to the hazards function λ_i(t) = λ₀(t) exp {β₁Z_i(t) + β₂Z_0i}, where Z_0i is a Bernoulli random variable with probability of success p = 0.5; β = (β₁, β₂)′ is (−ln(2), 0)′ or (−ln(3), 0.5)′. The baseline hazard function is taken to be a constant λ₀ = 1. A common censoring time is imposed to all individuals such that the censoring rate is 25%. Two scenarios are considered for the distributions of the random effects and changepoints. In the first situation, α_i are generated from a multivariate normal distributions N((μ₁, μ₂, μ₃)′, G), where (μ₁, μ₂, μ₃) = (2, 0.2, −0.5) and G is a diagonal matrix with $\text{diag}(\mathbf{G})=\{{\sigma }_{11}^2,{\sigma }_{22}^2,{\sigma }_{33}^2\}$, which is (1, 0.01, 0.001). The changepoint, τ_i, is generated from a normal distribution $\text{N}({\mu }_4,{\sigma }_{44}^2)$, where μ₄ = 12, and ${\sigma }_{44}^2=1$. The second scenario deals with the situation where the distributions of α_i and τ_i deviate from normality. α_i is drawn from a homoscedastic normal mixture distribution ρN(μ_m1, G)+(1 − ρ)N(μ_m2, G), where ρ = 0.5, μ_m1 = (2, 0.2, −0.5)′, μ_m2 = (2.5, 0.3, −0.6)′, and G is a diagonal matrix with diag(G) = (1, 0.01, 0.001). The changepoint, τ_i, is simulated from $\rho \text{N}({\mu }_{{\tau }_1},{\sigma }_{\tau }^2)+(1-\rho )\text{N}({\mu }_{{\tau }_1},{\sigma }_{\tau }^2)$, where ρ = 0.5, μ_τ1 = 12, μ_τ2 = 10 and ${\sigma }_{\tau }^2=1$.

For each scenario, a total of 500 Monte Carlo samples are generated with a sample size n = 200 and 600. The parameter β = (β₁, β₂)′ is estimated using the methods described in the previous Section. The main results of the simulations are presented in Tables 1 and 2 in which “Bias” denotes the average of the 500 estimates β̂_k of β_k minus β_k, “ASE” stands for the average of 500 estimates of the standard deviations of the estimate β̂_k, “SDE” means the square root of the sample variance of 500 estimates of β_k, k = 1, 2, and “CP” is the coverage probability for the 95% confidence interval.

Table 1.

Simulation results for normally distributed random effects and change points; Ideal, “ideal” regression; Naive, “naive” regression; CDS, conditional score approach; CRS, corrected score method.

n	β			σ2 = 0.10				σ2 = 0.20
				Ideal	Naive	CDS	CRS	Ideal	Naive	CDS	CRS
200	(−ln(2), 0)	β1	Bias	−0.0054	0.0114	−0.0058	−0.0072	−0.0054	0.0269	−0.0065	−0.0092
			ASE	0.0507	0.0497	0.0521	0.0527	0.0507	0.0487	0.0543	0.0557
			SDE	0.0524	0.0517	0.0550	0.0554	0.0524	0.0513	0.0580	0.0589
			CP	0.9460	0.9460	0.9460	0.9460	0.9460	0.8920	0.9460	0.9420
		β2	Bias	−0.0052	−0.0036	−0.0035	−0.0035	−0.0052	−0.0029	−0.0028	−0.0028
			ASE	0.1665	0.1664	0.1674	0.1677	0.1665	0.1664	0.1677	0.1682
			SDE	0.1697	0.1693	0.1728	0.1731	0.1697	0.1696	0.1768	0.1774
			CP	0.9480	0.9560	0.9600	0.9600	0.9480	0.9520	0.9480	0.9480
	(−ln(3), 0.5)	β1	Bias	−0.0076	0.0441	−0.0100	−0.0178	−0.0076	0.0890	−0.0126	−0.0298
			ASE	0.0745	0.0709	0.0887	0.0936	0.0745	0.0679	0.0969	0.1100
			SDE	0.0759	0.0743	0.0858	0.0883	0.0759	0.0729	0.0965	0.1034
			CP	0.9500	0.8600	0.9660	0.9700	0.9500	0.6980	0.9640	0.9720
		β2	Bias	−0.0038	−0.0215	−0.0003	0.0030	−0.0038	−0.0379	0.0019	0.0092
			ASE	0.1711	0.1707	0.1791	0.1812	0.1711	0.1703	0.1809	0.1862
			SDE	0.1739	0.1748	0.1845	0.1860	0.1739	0.1764	0.1961	0.1999
			CP	0.9420	0.9380	0.9480	0.9500	0.9420	0.9380	0.9440	0.9480
600	(−ln(2), 0)	β1	Bias 1	−0.002	0.0142	−0.0029	−0.0033	−0.0021	0.0298	−0.0033	−0.0042
			ASE	0.0287	0.0281	0.0296	0.0298	0.0287	0.0276	0.0309	0.0314
			SDE	0.0280	0.0280	0.0297	0.0298	0.0280	0.0277	0.0313	0.0314
			CP	0.9620	0.9140	0.9560	0.9580	0.9620	0.7820	0.9520	0.9560
		β2	Bias	0.0003	0.0009	0.0010	0.0010	0.0003	0.0010	0.0012	0.0012
			ASE	0.0949	0.0949	0.0951	0.0951	0.0949	0.0949	0.0952	0.0953
			SDE	0.0971	0.0993	0.1012	0.1012	0.0971	0.1006	0.1045	0.1046
			CP	0.9520	0.9500	0.9460	0.9460	0.9520	0.9480	0.9380	0.9380
	(−ln(3), 0.5)	β1	Bias	−0.0042	0.0481	−0.0062	−0.0090	−0.0042	0.0940	−0.0075	−0.0134
			ASE	0.0419	0.0398	0.0513	0.0531	0.0419	0.0381	0.0566	0.0614
			SDE	0.0415	0.0408	0.0472	0.0476	0.0415	0.0397	0.0526	0.0538
			CP	0.9540	0.7560	0.9660	0.9680	0.9540	0.3240	0.9620	0.9740
		β2	Bias	0.0027	−0.0171	0.0044	0.0056	0.0027	−0.0347	0.0053	0.0078
			ASE	0.0969	0.0967	0.1010	0.1017	0.0969	0.0965	0.1018	0.1033
			SDE	0.0972	0.1016	0.1067	0.1069	0.0972	0.1042	0.1149	0.1157
			CP	0.9600	0.9360	0.9400	0.9400	0.9600	0.9220	0.9300	0.9320

Table 2.

Simulation results when the random effects and changepoints follow bimodal distributions; Ideal, “ideal” regression; Naive, “naive” regression; CDS, conditional score approach; CRS, corrected score method.

n	β			σ2 = 0.10				σ2 = 0.20
				Ideal	Naive	CDS	CRS	Ideal	Naive	CDS	CRS
200	(−ln(2), 0)	β1	Bias	−0.0044	0.0110	−0.0062	−0.0075	−0.0044	0.0261	−0.0073	−0.0100
			ASE	0.0507	0.0497	0.0524	0.0530	0.0507	0.0488	0.0546	0.0559
			SDE	0.0508	0.0502	0.0533	0.0537	0.0508	0.0493	0.0557	0.0565
			CP	0.9540	0.9280	0.9420	0.9420	0.9540	0.8940	0.9480	0.9440
		β2	Bias	0.0164	0.0144	0.0147	0.0147	0.0164	0.0136	0.0140	0.0141
			ASE	0.1664	0.1663	0.1671	0.1674	0.1664	0.1663	0.1674	0.1680
			SDE	0.1651	0.1658	0.1689	0.1692	0.1651	0.1665	0.1729	0.1735
			CP	0.9460	0.9440	0.9420	0.9420	0.9460	0.9480	0.9360	0.9360
	(−ln(3), 0.5)	β1	Bias	−0.0076	0.0418	−0.0125	−0.0204	−0.0076	0.0865	−0.0159	−0.0333
			ASE	0.0745	0.0711	0.0901	0.0950	0.0745	0.0681	0.0983	0.1115
			SDE	0.0746	0.0726	0.0834	0.0859	0.0746	0.0702	0.0923	0.0987
			CP	0.9620	0.8880	0.9640	0.9680	0.9620	0.7120	0.9600	0.9720
		β2	Bias	0.0203	−0.0024	0.0194	0.0228	0.0203	−0.0209	0.0198	0.0272
			ASE	0.1712	0.1708	0.1793	0.1815	0.1712	0.1704	0.1812	0.1866
			SDE	0.1726	0.1736	0.1824	0.1838	0.1726	0.1753	0.1935	0.1972
			CP	0.9420	0.9480	0.9500	0.9540	0.9420	0.9480	0.9400	0.9520
600	(−ln(2), 0)	β1	Bias	−0.0029	0.0135	−0.0036	−0.0040	−0.0029	0.0291	−0.0040	−0.0049
			ASE	0.0287	0.0281	0.0298	0.0300	0.0287	0.0275	0.0311	0.0316
			SDE	0.0281	0.0280	0.0298	0.0299	0.0281	0.0277	0.0313	0.0315
			CP	0.9600	0.9300	0.9460	0.9500	0.9600	0.7920	0.9520	0.9500
		β2	Bias	0.0019	0.0019	0.0019	0.0019	0.0019	0.0019	0.0019	0.0020
			ASE	0.0949	0.0949	0.0950	0.0951	0.0949	0.0949	0.0951	0.0952
			SDE	0.0950	0.0963	0.0980	0.0981	0.0950	0.0971	0.1007	0.1008
			CP	0.9520	0.9520	0.9460	0.9460	0.9520	0.9420	0.9360	0.9360
	(−ln(3), 0.5)	β1	Bias	−0.0052	0.0469	−0.0075	−0.0103	−0.0052	0.0928	−0.0090	−0.0150
			ASE	0.0418	0.0398	0.0516	0.0535	0.0418	0.0380	0.0569	0.0617
			SDE	0.0401	0.0400	0.0464	0.0469	0.0401	0.0390	0.0522	0.0535
			CP	0.9660	0.7340	0.9700	0.9740	0.9660	0.3080	0.9720	0.9740
		β2	Bias	0.0022	−0.0183	0.0031	0.0042	0.0022	−0.0362	0.0036	0.0061
			ASE	0.0969	0.0967	0.1010	0.1016	0.0969	0.0965	0.1017	0.1032
			SDE	0.0971	0.0999	0.1048	0.1051	0.0971	0.1016	0.1120	0.1127
			CP	0.9540	0.9460	0.9540	0.9540	0.9540	0.9260	0.9260	0.9340

Table 1 shows the results of the simulation from scenario 1. It can be observed from this table that for β₁ the “ideal” estimator, which is not applicable in real problems, outperforms all the other estimators with regard to the bias. Its coverage probability is also close to the nominal 95% level. The bias correction methods hold a clear advantage over the “naive” estimator in terms of bias and coverage probability. They display smaller biases and larger coverage probabilities, close to the nominal level. However, the standard deviations of the “naive” estimator are smaller than the corrected and conditional scores counterparts. Hence, the bias reduction by both functional methods leads to a variance inflation. Also, the standard deviations and biases of both corrected score and conditional score estimators increase with |β₁| and σ². The same phenomenon was reported in different settings by [8], Song and Wang [9] and Kong and Gu [18]. In addition, it can be noted that the conditional score estimator shows smaller bias and standard deviation compared with the corrected score estimator when n = 200. Therefore, the conditional score approach has better performance than the corrected score method under small sample size situations. But, the advantage diminishes when n = 600 as the two estimators are asymptotically equivalent. It can also be seen that the biases and standard deviations of all the estimators, except those for the “naive” estimator, are generally decreasing functions of the sample size, n.

The results of the simulation from the second scenario are presented in Table 2. They are similar to those reported in Table 1, suggesting that the performance of the conditional score and corrected score approaches are neither related to the distribution function of the changepoints nor to that of the random effects. Furthermore, it can be observed from Tables 1 and 2 that the estimation of β₂ is also affected by large σ² and |β₂| although, the effect is less pronounced than that on β₁. In addition to estimating the parameter of interest, we have also estimated the nuisance parameter $\mathit{\eta }={({\sigma }^2,{\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4,{\sigma }_{11}^2,{\sigma }_{22}^2,{\sigma }_{33}^2,{\sigma }_{44}^2)}^{\prime }$ under the first scenario. The results of the estimation are given in Table 3, where it can be noted that the measurement error clearly affects the estimation of the nuisance parameter. The standard deviations of the estimates of the components of η are increasing functions of the variance of the measurement error and decreasing functions of the sample size, n. Additionally, the effect of the smoothing parameter on the results from the estimation of the parameter of interest has been investigated via simulations, under the same settings as those of the first scenario with γ = 0.5, 0.1, 0.01 and 0.001. The results of these simulations are presented in Table 4 and can be seen to be almost the same across the two γ values, 0.01 and 0.001.

Table 3.

Simulation results regarding the estimation of the nuisance parameter under the first scenario.

n	Parameter	σ2 = 0.10				σ2 = 0.20
		Bias	ASE	SDE	CP	Bias	ASE	SDE	CP
200	σ2	−0.0002	0.0049	0.0047	0.9600	−0.0010	0.0099	0.0094	0.9580
	μ1	−0.0045	0.0725	0.0708	0.9580	−0.0089	0.0746	0.0734	0.9600
	μ2	−0.0004	0.0072	0.0070	0.9560	0.0000	0.0074	0.0071	0.9560
	μ3	−0.0004	0.0027	0.0028	0.9340	−0.0008	0.0032	0.0032	0.9280
	μ4	0.0050	0.0779	0.0803	0.9280	0.0079	0.0856	0.0877	0.9260
	${\sigma }_{11}^2$	−0.0066	0.1044	0.1077	0.9260	−0.0055	0.1107	0.1134	0.9280
	${\sigma }_{22}^2$	−0.0002	0.0010	0.0010	0.9220	−0.0002	0.0011	0.0011	0.9220
	${\sigma }_{33}^2$	0.0000	0.0002	0.0002	0.9340	0.0000	0.0002	0.0002	0.9260
	${\sigma }_{44}^2$	0.0093	0.1235	0.1309	0.9340	0.0509	0.1572	0.1719	0.9320
600	σ2	−0.0001	0.0029	0.0029	0.9620	−0.0007	0.0057	0.0058	0.9440
	μ1	−0.0052	0.0421	0.0401	0.9700	−0.0098	0.0433	0.0415	0.9660
	μ2	−0.0011	0.0041	0.0042	0.9380	−0.0007	0.0043	0.0043	0.9340
	μ3	−0.0002	0.0016	0.0016	0.9480	−0.0006	0.0018	0.0018	0.9400
	μ4	0.0077	0.0450	0.0428	0.9620	0.0108	0.0494	0.0474	0.9440
	${\sigma }_{11}^2$	0.0018	0.0615	0.0620	0.9500	0.0032	0.0652	0.0660	0.9440
	${\sigma }_{22}^2$	−0.0002	0.0006	0.0006	0.9240	−0.0002	0.0006	0.0006	0.9340
	${\sigma }_{33}^2$	0.0000	0.0001	0.0001	0.9600	0.0000	0.0001	0.0001	0.9640
	${\sigma }_{44}^2$	0.0052	0.0719	0.0691	0.9520	0.0428	0.0916	0.0890	0.9520

Table 4.

Simulation results for various values of γ under the settings of the first scenario when the random effects and changepoints are normally distributed; β = (−log(3), 0.5)′; σ² = 0.20; n = 200; Ideal, “ideal” regression; Naive, “naive” regression; CDS, conditional score approach; CRS, corrected score method.

		γ = 0.5				γ = 0.10
		Bias	ASE	SDE	CP	Bias	ASE	SDE	CP
Ideal	β1	−0.0076	0.0745	0.0759	0.9500	−0.0076	0.0745	0.0759	0.9500
	β2	−0.0038	0.1711	0.1739	0.9420	−0.0038	0.1711	0.1739	0.9420
Naive	β1	0.0893	0.0678	0.0729	0.6980	0.0891	0.0679	0.0729	0.7020
	β2	−0.0380	0.1703	0.1763	0.9380	−0.0379	0.1703	0.1764	0.9380
CDS	β1	−0.0121	0.0968	0.0964	0.9620	−0.0126	0.0969	0.0965	0.9620
	β2	0.0017	0.1809	0.1959	0.9480	0.0018	0.1809	0.1961	0.9440
CRS	β1	−0.0291	0.1098	0.1032	0.9740	−0.0297	0.1099	0.1034	0.9720
	β2	0.0090	0.1861	0.1997	0.9480	0.0092	0.1862	0.1999	0.9480
		Nuisance parameter
	σ2	−0.0009	0.0099	0.0094	0.9580	−0.0010	0.0099	0.0094	0.9560
	μ1	−0.0254	0.0747	0.0735	0.9480	−0.0090	0.0747	0.0734	0.9600
	μ2	−0.0016	0.0073	0.0071	0.9520	0.0000	0.0074	0.0071	0.9560
	μ3	0.0012	0.0032	0.0032	0.9260	−0.0008	0.0032	0.0032	0.9280
	μ4	0.0316	0.0841	0.0869	0.9200	0.0079	0.0856	0.0877	0.9260
	${\sigma }_{11}^2$	−0.0034	0.1109	0.1137	0.9300	−0.0054	0.1107	0.1134	0.9280
	${\sigma }_{22}^2$	−0.0002	0.0010	0.0010	0.9320	−0.0002	0.0011	0.0011	0.9260
	${\sigma }_{33}^2$	0.0000	0.0002	0.0002	0.9300	0.0000	0.0002	0.0002	0.9240
	${\sigma }_{44}^2$	0.0261	0.1553	0.1707	0.9260	0.0511	0.1571	0.1719	0.9300
		γ = 0.01				γ = 0.001
		Bias	ASE	SDE	CP	Bias	ASE	SDE	CP
Ideal	β1	−0.0076	0.0745	0.0759	0.9500	−0.0076	0.0745	0.0759	0.9500
	β2	−0.0038	0.1711	0.1739	0.9420	−0.0038	0.1711	0.1739	0.9420
Naive	β1	0.0891	0.0679	0.0729	0.7020	0.0890	0.0679	0.0729	0.6980
	β2	−0.0380	0.1703	0.1763	0.9380	−0.0379	0.1703	0.1764	0.9380
CDS	β1	−0.0124	0.0969	0.0965	0.9640	−0.0126	0.0969	0.0965	0.9640
	β2	0.0017	0.1809	0.1959	0.9440	0.0019	0.1809	0.1962	0.9440
CRS	β1	−0.0294	0.1098	0.1034	0.9720	−0.0298	0.1100	0.1034	0.9720
	β2	0.0090	0.1861	0.1997	0.9480	0.0092	0.1862	0.1999	0.9480
		Nuisance parameter
	σ2	−0.0012	0.0099	0.0094	0.9560	−0.0010	0.0099	0.0094	0.9580
	μ1	−0.0114	0.0747	0.0734	0.9580	−0.0089	0.0746	0.0734	0.9600
	μ2	−0.0002	0.0074	0.0071	0.9560	0.0000	0.0074	0.0071	0.9560
	μ3	−0.0005	0.0032	0.0032	0.9360	−0.0008	0.0032	0.0032	0.9280
	μt	0.0104	0.0854	0.0878	0.9260	0.0079	0.0856	0.0878	0.9260
	${\sigma }_{11}^2$	−0.0048	0.1108	0.1134	0.9300	−0.0055	0.1107	0.1134	0.9260
	${\sigma }_{22}^2$	−0.0002	0.0011	0.0011	0.9280	−0.0002	0.0011	0.0011	0.9220
	${\sigma }_{33}^2$	0.0000	0.0002	0.0002	0.9280	0.0000	0.0002	0.0002	0.9260
	${\sigma }_{44}^2$	0.0533	0.1577	0.1720	0.9380	0.0510	0.1572	0.1720	0.9320

5 ACTG 175 data analysis

5.1 Data

The AIDS Clinical Trial Group Study 175 (ACTG-175) is a double-blind randomized trial comparing the efficacy of four different antiretroviral treatments: zidovudine (zdv) monotherapy, the combination of didanosine with zidovudine (zdv/ddl), zidovudine plus zalcitabine (zdv/ddc) and didanosine (ddl) alone. The study involved 2467 HIV-infected patients who were enrolled in the clinical trial units between December 1991 and October 1992 and followed longitudinally for almost three years (until November 1994). For each patient, some baseline characteristics (e.g. gender, race, age) were recorded along with the censoring time or the time to the event of interest, which is death or diagnosis with AIDS disease. The time-dependent covariate was the CD4 cell count, which was measured intermittently and approximately at 0, 8, 20, 32, …, 168 weeks after randomization. The number of longitudinal observations at each individual level ranges from 1 to 19 with 13 as median. Note that the data on the CD4 count were collected for a secondary objective which was the investigation of a possible link between this covariate and the time to AIDS disease development or death. More details on the clinical trial can be found in [19]. The time is rescaled with origin at the time of randomization because the subjects entered the study at different times. In the following analysis, we include individuals with more than 5 longitudinal observations, resulting in a subset of n = 2027 subjects and a total of N_T = 24333 CD4 count measurements. Individual observations obtained after diagnosis of AIDS disease were not included. In the considered subset, 255 individuals were diagnosed with AIDS or died in the study period. The censoring was due to administrative censoring in most cases.

5.2 Analysis models

The primary interest in the ACTG-175 study was to compare the four treatments, and the zidovudine momotherapy was found to be the least efficient treatment. Here, we focus on the secondary objective, which is the characterization of the relationship between the CD4 count covariate and the time to AIDS disease progression or death. In the analysis, we use the logarithmic transformation to stabilize the variance and normalize the within-subject measurement error for the CD4 counts.

Figure 1 displays the log₁₀-transformed CD4 count trajectories for ten subjects from the analyzed data set. It can be observed that the log₁₀CD4 initially increases and later drops down for some individuals, showing evidence of changepoints for those individuals. We then consider using the following changepoint model for individual log₁₀ CD4 profile:

$${log}_{10}\text{CD}{4}_{ij}={\alpha }_{i1}+{\alpha }_{i2}(t_{ij}-{\tau }_i)+{\alpha }_{i3}\mid t_{ij}-{\tau }_i\mid +e_{ij},$$ (4)

where e_ij, the measurement error at time t_ij, is assumed to follow a zero-mean normal distribution with variance σ², i = 1, …, n, j = 1, …, m_i. Note that measurements on CD4 counts are usually error-prone and subject to individual biological variation. Moreover, observe that model (4) is a two-phase piecewise linear model with an abrupt transition at the subject-specific changepoint. For subject i, the slopes before and after the changepoint, τ_i, are α_i2 − α_i3 and α_i2 + α_i3, respectively, i = 1, …, n. In the analysis, model (4) is approximated by the hyperbolic tangent model, which consists of replacing |t_ij − τ_i| in (4) with (t_ij − τ_i) tanh{(t_ij − τ_i)/γ} similarly as in (2). The value of the smoothing parameter is chosen as γ = 0.01 and the choice of an appropriate value of γ is discussed later. The simple linear and quadratic models are routinely used to model the log-transformed CD4 count in many applications. We compare the two-phase piecewise linear model with these models based on the Akaike information criterion (AIC), which is defined as N ln(RSS/N)+2(q+1), and the root mean square error (RMSE), which is $\sqrt{\mathit{RSS}/(N-q)}$. Here RSS is short for the sum of squared residuals of the model, q denotes the number of parameters and N is the number of observations in the model. The AIC for the two-phase piecewise linear model is found to be −106706.3, while those for the simple linear and quadratic models are −99863.1 and −104073.5, respectively. Moreover, the two-phase piecewise linear model has the smallest RMSE (0.09796) compared with the simple linear and quadratic models, whose RSME’s are 0.11913 and 0.10596, respectively. Clearly, both criteria favor the two-phase piecewise linear model, which we adopt afterward to describe the process of log₁₀ CD4. Figure 2 depicts the fitted profiles using the two-phase piecewise linear model, the simple linear and quadratic models, along with the observed log₁₀ CD4 profiles of the ten individuals in Figure 1. It is seen that the simple linear or quadratic model does not fit as well as the changepoint model among several individuals.

Figure 1.

log₁₀ CD4 profiles of ten individuals from the analyzed data set.

Figure 2.

Observed log₁₀ CD4 for the ten individuals in Figure 1 with the fitted profiles using the two-phase piecewise linear, simple linear and quadratic models.

We characterize the relationship between the time to AIDS disease development or death and the covariates through the proportional hazard function

$${\lambda }_i(t)={\lambda }_0(t)exp\{{\beta }_1{Z}_{1i}(t)+{\beta }_2{Z}_{2i}+{\mathit{\beta }}_3^{\prime }{\mathbf{Z}}_{3i}+{\mathit{\beta }}_4^{\prime }{\mathbf{Z}}_{4i}\},$$

where Z₁(t) represents the process of log₁₀CD4, Z₂ is the gender variable, taking 1 for female and 0 for male, Z₃ = (Z₃₁, Z₃₂, Z₃₃)′ and Z₄ = (Z₄₁, Z₄₂, Z₄₃)′ are vectors of dummy variables for treatment (zdv, zdv/ddl, zdv/ddc, ddl) and race (non-hispanic white, non-hispanic black, hispanic, other), respectively. The reference category for treatment is zdv and that for race is non-hispanic white. The parameter $\mathit{\beta }={({\beta }_1,{\beta }_2,{\mathit{\beta }}_3^{\prime },{\mathit{\beta }}_4^{\prime })}^{\prime }$ is estimated using the “naive” regression, the corrected and conditional scores methods. Furthermore, the p-values for the significance of the parameters are based on Wald statistics.

5.3 Results

The main results of the analysis are presented Table 5. It appears that the estimates of β₁ by all methods are significant (p-values are smaller than 0.001) at the significance level of 5%, indicating an association between log₁₀CD4 and the time to AIDS disease progression or death. The negative sign of the estimates of β₁ implies that the hazards function is a decreasing function of the CD4 counts. The “naive” regression leads to a smaller absolute value of the estimates of β₁ as well as smaller standard errors compared with the corrected or conditional score counterparts. For β₂ and the components of β₃, none of the estimates is significant (p-values are greater than 0.4), suggesting that gender and treatment are not associated with the time to AIDS progression or death when log₁₀CD4 and race variables are included in the proportional hazards model. Regarding β₄, the “naive” estimation of β₄₁ yields a non-significant estimate (p-value is 0.0741), while the results ( p-values are smaller than 0.01) of the conditional and corrected scores estimations signal an association between race and time to AIDS or death. Non-hispanic black patients have higher hazards in comparison with non-hispanic white patients according to both bias-correction estimations. Meanwhile, there is no evidence of significant difference in the hazards between patients in the non-hispanic white, hispanic and other groups.

Table 5.

Results of the analysis of the ACTG-175 data; Naive, “naive” regression; CDS, conditional score method; CRS, corrected score method; Est, estimate; SE, standard error.

β	Naive			CDS			CRS
	Est	SE	p-value	Est	SE	p-value	Est	SE	p-value
β1	−0.9500	0.0412	< 0.0001	−1.3559	0.1354	< 0.0001	−1.4410	0.1723	< 0.0001
β2	−0.1038	0.2003	0.6041	−0.1383	0.2202	0.5301	−0.1495	0.2101	0.4766
β31	−0.1314	0.1828	0.4722	−0.0947	0.2262	0.6755	−0.0823	0.1904	0.6655
β32	0.0198	0.1794	0.9122	0.0884	0.2121	0.6767	0.1128	0.1987	0.5704
β33	−0.1406	0.1742	0.4194	0.0386	0.2480	0.8764	0.0523	0.1955	0.7891
β41	0.3855	0.2159	0.0741	0.5583	0.2324	0.0163	0.5454	0.2313	0.0184
β42	0.2126	0.2902	0.4636	0.2961	0.3022	0.3271	0.2744	0.3040	0.3668
β43	−1.4172	1.0267	0.1675	−0.8480	1.0367	0.4134	−0.8362	1.0286	0.4162

η	Nuisance parameter
	Est	SE	p-value
σ2	0.0096	0.0004	< 0.0001
μ1	2.4886	0.0063	0.0002
μ2	−0.0017	0.0005	< 0.0001
μ3	−0.0029	0.0004	< 0.0001
μ4	62.1790	0.7461	< 0.0001
${\sigma }_{11}^2$	0.0759	0.0068	< 0.0001
σ12	0.0011	0.0001	< 0.0001
σ13	−0.0003	0.0001	0.0553
σ14	−1.2269	0.2093	< 0.0001
${\sigma }_{22}^2$	0.0004	0.0003	0.1926
σ23	−0.0003	0.0003	0.3087
σ24	−0.0905	0.0257	0.0004
${\sigma }_{33}^2$	0.0004	0.0003	0.2018
σ34	0.0350	0.0256	0.1711
${\sigma }_{44}^2$	1127.0142	27.1854	< 0.0001

Note: $\mathit{\beta }={({\beta }_1,{\beta }_2,{\mathit{\beta }}_3^{\prime },{\mathit{\beta }}_4^{\prime })}^{\prime }$ represents the vector of the parameters of interest which characterizes the relationship between the time to AIDS or death and the covariates; β₁, β₂, β₃ and β₄ are the coefficients of log₁₀ CD4 count, gender and the vectors of dummy variables for treatment and race, respectively; η is the nuisance parameter, whose components are the variance of the error, the mean of the random effects and the non-redundant elements of the covariance matrix of the random effects.

Table 5 also shows the results of the estimation of the nuisance parameter, denoted by η = (σ², μ′, vech(ϒ))′, where μ = (μ₁, μ₂, μ₃, μ₄)′ and ϒ are the mean and variance covariance matrix of the vector θ_i = (α_i1, α_i2, α_i3, τ_i)′, respectively. Here, vech(ϒ) denotes ${({\sigma }_{11}^2,{\sigma }_{12},{\sigma }_{13},{\sigma }_{14},{\sigma }_{22}^2,{\sigma }_{23},{\sigma }_{24},{\sigma }_{33}^2,{\sigma }_{34},{\sigma }_{44}^2)}^{\prime }$. The results of the estimation show a significant (p-value is smaller than 0.0001) estimate of σ², pointing out evidence that measurement error is present in the log₁₀CD4. Moreover, the estimates of all the components of μ are significant (p-value is smaller than 0.0001), indicating a change in the mean slope of log₁₀CD4 over time. The estimate of the mean slope in the first phase (μ₂ − μ₃) is positive (0.0012), suggesting a somewhat initial rise in the mean log₁₀CD4 probably due to the treatments. Meanwhile, the mean slope in the second phase (μ₂ + μ₃) has a negative estimate (−0.0046), which suggests a decline in the mean log₁₀CD4 following the initial increase. The estimated mean changepoint is 62.179 weeks. Moreover, the estimate of σ₄₄, which is the standard deviation of the changepoints, is significant (p-value is smaller than 0.0001) and large (33.571 weeks), indicating a large amount of heterogeneity in the individual changepoints. This is understandable since observation times vary widely across individuals.

5.4 Choice of the smoothing parameter

To estimate the parameter of interest in the analysis, it is necessary to specify the value of the smoothing parameter γ, which governs the sharpness of the hyperbolic tangent model. This model becomes sharper as γ approaches zero [14,15]. Hence, it is advisable to choose a small value of γ in order for the hyperbolic tangent model to be almost identical to the original model (4), which is an abrupt changepoint model. To select γ, an approach is to consider a decreasing sequence (γ_k)_k≥0 of positive real numbers, such that γ_k converges to zero as k goes to infinity. Then, for a given k, one has to carry out the estimation of the parameter of interest, β, using γ_k as the value of γ. Letting β̂_k denote the estimate of β using γ_k, a plausible value of γ is chosen as γ_k^* such that k^* = min{k: ||β̂_k+1 − β̂_k|| < ε, k ≥ 0}, where ||C|| = max{|C₁|, …, |C_q|} for any vector C = (C₁, …, C_q)′, and ε is a small tolerance. In the analysis, we set ε at 10⁻² and consider γ_k = γ₀a^−k for any k, where γ₀ = 1 and a = 10. The value of k^* is found to be 2 and therefore, γ is chosen as 0.01. We note that the results do not vary considerably for values of γ smaller than 0.01. In Table 6, we report the results of the analysis using several values of the smoothing parameter. It can be observed from this table that the discrepancy between the results for γ = 0.01 and γ = 0.001 is small.

Table 6.

Results of the ACTG-175 data analysis for various values of the smoothing parameter; Naive, “naive” regression; CDS, conditional score method; CRS, corrected score method; Param, parameter; Est, estimate; SE, standard error.

Param	γ = 1		γ = 0.1		γ = 0.01		γ = 0.001
	Est	SE	Est	SE	Est	SE	Est	SE
	Naive
β1	−0.9913	0.0437	−0.9508	0.0412	−0.9500	0.0412	−0.9500	0.0412
β2	−0.1016	0.2002	−0.1038	0.2003	−0.1038	0.2003	−0.1038	0.2003
β31	−0.1308	0.1825	−0.1313	0.1827	−0.1314	0.1828	−0.1314	0.1828
β32	0.0092	0.1795	0.0199	0.1794	0.0198	0.1794	0.0198	0.1794
β33	−0.1237	0.1742	−0.1410	0.1742	−0.1406	0.1742	−0.1406	0.1742
β41	0.3958	0.2159	0.3855	0.2159	0.3855	0.2159	0.3855	0.2159
β42	0.2197	0.2903	0.2128	0.2902	0.2126	0.2902	0.2126	0.2902
β43	−1.0850	1.0213	−1.3893	1.0260	−1.4172	1.0267	−1.4171	1.0267
	CDS
β1	−1.5121	0.1662	−1.3524	0.1372	−1.3559	0.1354	−1.3574	0.1386
β2	−0.1554	0.2218	−0.1377	0.2205	−0.1383	0.2202	−0.1395	0.2202
β31	−0.1174	0.2495	−0.0937	0.2265	−0.0947	0.2262	−0.0970	0.2270
β32	0.0831	0.2446	0.0891	0.2122	0.0884	0.2121	0.0859	0.2128
β33	0.0163	0.2750	0.0395	0.2483	0.0386	0.2480	0.0362	0.2490
β41	0.6353	0.2369	0.5566	0.2325	0.5583	0.2324	0.5594	0.2326
β42	0.3305	0.3159	0.2953	0.3020	0.2961	0.3022	0.2955	0.3023
β43	−0.7134	1.0363	−0.8707	1.0352	−0.8480	1.0367	−0.8475	1.0371
	CRS
β1	−1.5992	0.1959	−1.4378	0.1732	−1.4410	0.1723	−1.4431	0.1713
β2	−0.1690	0.2141	−0.1489	0.2101	−0.1495	0.2101	−0.1511	0.2101
β31	−0.1122	0.2084	−0.0812	0.1903	−0.0823	0.1904	−0.0854	0.1899
β32	0.1034	0.2379	0.1133	0.1983	0.1128	0.1987	0.1096	0.1977
β33	0.0239	0.2306	0.0533	0.1951	0.0523	0.1955	0.0491	0.1950
β41	0.6323	0.2399	0.5436	0.2313	0.5454	0.2313	0.5470	0.2316
β42	0.3112	0.3238	0.2737	0.3038	0.2744	0.3040	0.2735	0.3043
β43	−0.6885	1.0351	−0.8588	1.0279	−0.8362	1.0286	−0.8353	1.0286
	Nuisance parameter
μ1	2.4887	0.0063	2.4886	0.0063	2.4886	0.0063	2.4886	0.0063
μ2	−0.0017	0.0004	−0.0017	0.0004	−0.0017	0.0005	−0.0017	0.0005
μ3	−0.0029	0.0004	−0.0029	0.0004	−0.0029	0.0004	−0.0029	0.0004
μ4	62.1867	0.7458	62.1779	0.7462	62.1790	0.7462	62.1789	0.7461
σ2	0.0096	0.0004	0.0096	0.0004	0.0096	0.0004	0.0096	0.0004
${\sigma }_{11}^2$	0.0756	0.0068	0.0759	0.0068	0.0759	0.0068	0.0758	0.0068
σ12	0.0011	0.0001	0.0011	0.0001	0.0011	0.0001	0.0011	0.0001
σ13	−0.0003	0.0001	−0.0003	0.0001	−0.0003	0.0001	−0.0003	0.0001
σ14	−1.2277	0.2092	−1.2278	0.2093	−1.2269	0.2093	−1.2270	0.2093
${\sigma }_{22}^2$	0.0004	0.0003	0.0004	0.0003	0.0004	0.0003	0.0004	0.0003
σ23	−0.0003	0.0003	−0.0003	0.0003	−0.0003	0.0003	−0.0003	0.0003
σ24	−0.0891	0.0260	−0.0905	0.0255	−0.0905	0.0256	−0.0904	0.0257
${\sigma }_{33}^2$	0.0004	0.0003	0.0004	0.0003	0.0004	0.0003	0.0004	0.0003
σ34	0.0343	0.0259	0.0349	0.0254	0.0350	0.0256	0.0349	0.0256
${\sigma }_{44}^2$	1124.4563	27.1443	1127.3108	27.1906	1127.0142	27.1845	1126.8695	27.1854

6 Conclusion

The paper presents a new joint model of survival time and longitudinal covariate data. A subject-specific changepoint model is proposed for the longitudinal covariate process and the time-to-event data are modeled through the Cox proportional hazards model. Under the changepoint model framework, two estimators based on the corrected score and the conditional score methods are developed for the regression parameter in the proportional hazards model. These estimators perform well and are shown to be asymptotically equivalent. However, the conditional score estimator has a better finite sample size performance.

Spline is commonly used to model nonlinear longitudinal profile [20–23], and it has been used in joint modeling of survival times and longitudinal data [24]. Our approach for longitudinal measurements is different from nonparametric smoothing spline, which generally assumes smoothness and requires a large sample size for longitudinal measurements. On the other hand, our approach may be somewhat similar to regression spline. Nevertheless, the change point model that we propose in the paper does not require continuous derivatives, which are generally assumed in say cubic splines. Hence, our change point model is less restrictive in terms of dealing with abrupt changes.

The proposed estimators are not sensitive to the choice of the transition function for the two-phase piecewise linear model as long as the function satisfies the conditions given in Section 3. However, their performance is affected by large variance of the measurement error, or by large |β₁|. Nonetheless, these estimators are more reliable than the “naive” estimator in any situation. An important feature of the proposed methods is their ability to account for the nonlinearity in the longitudinal covariate trajectories without any assumption regarding the distribution function of the random effects or that of the change points.

There may be applications where the proportional hazards model does not hold, and under this situation, a change point in the survival function may be considered. We are currently investigating the piecewise proportional hazards model for time-to-event data. Another interesting situation to explore is the consideration of a more than two-phase change point model, which may be relevant for the longitudinal covariates in some applications. This will obviously give rise to more computational complexity.

Appendix A: Proof of the asymptotic properties of the corrected score estimator

Regularity conditions

• (A1) Pr {Y_i(L) = 1} > 0 for any t ∈[0, L], i = 1, …, n.

• (A2) Z_i(u) is bounded almost surely.

• (A3) m_i and t_ij are random variables i = 1, …, n; j = 1, …, m_i.

• (A4) Σ_i(u) is bounded and continuously differentiable function of u, i = 1, …, n.

• (A5) $\mathbf{\Gamma }(\mathit{\beta })\equiv {\int }_0^L[s_2(u,\mathit{\beta })/s_0(u,\mathit{\beta })-{\{s_1(u,\mathit{\beta })/s_0(u,\mathit{\beta })\}}^{\otimes 2}]s_0(u,\mathit{\beta }){\lambda }_0(u)du$ is positive definite, where s₂(u, β) is the limit of $n^{-1}{\sum }_{i=1}^n{Y}_i(u)X_i{(u)}^{\otimes 2}exp\{{\mathit{\beta }}^{\prime }{X}_i(u)\}$ as n goes to infinity.

Consistency

For simplicity we assume that σ² is known. Note that the corrected score estimating function can be expressed as follows:

$$\begin{array}{l}{U}_n^{cr}(\mathit{\beta },{\sigma }^2)=\frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)+{\sigma }^2{\mathbf{\sum }}_i(u)\mathit{\beta }-\stackrel{\sim }{E}(u,\mathit{\beta },{\sigma }^2)\right\}{dM}_i(u)+\frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)+{\sigma }^2{\mathbf{\sum }}_i(u)\mathit{\beta }-\stackrel{\sim }{E}(u,\mathit{\beta },{\sigma }^2)\right\}{Y}_i(u)exp\{{\mathit{\beta }}_0^{\prime }{\mathbf{X}}_i(u)\}{\lambda }_0(u)du\\ =A_2+A_3,\end{array}$$

where ${dM}_i(u)={dN}_i(u)-Y_i(u)exp\{{\mathit{\beta }}_0^{\prime }{\mathbf{X}}_i(u)\}{\lambda }_0(u)du$ is a local-square martingale, and Ẽ(u, β, σ²) denotes S̃₁(u, β, σ²)/S̃₀(u, β, σ²). Define the process $\widehat{\mathrm{\sum }}(u,\mathit{\beta })\equiv n^{-1}{\sum }_{i=1}^n{\mathbf{\sum }}_i(u)\mathit{\beta }$. Under assumptions (A1)–(A4), and by Anderson and Gill [25], S̃₀(u, β, σ²), S̃₁(u, β, σ²), S₀(u, β, X), S₁(u, β, X) and Σ̂(u, β) converge in probability to s̃₀(u, β, σ²), s̃₁(u, β, σ²), s₀(u, β), s₁(u, β) and Σ(u, β), respectively, and uniformly in u and β. It can be noted that s̃₀(u, β, σ²) = s₀(u, β) and s̃₁(u, β, σ²) = s₁(u, β) + σ²Σ(u, β)s₀(u, β). Therefore, it is not difficult to see that Ẽ (u, β, σ²) has limit e(u, β) + σ²Σ(u, β), where e(u, β) = s₁(u, β)/s₀(u, β).

$$\begin{array}{l}{A}_2=\frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)+{\sigma }^2{\mathbf{\sum }}_i(u)\mathit{\beta }-\stackrel{\sim }{E}(u,\mathit{\beta },{\sigma }^2)\right\}{dM}_i(u)\\ =\frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)-e(u,\mathit{\beta })\right\}{dM}_i(u)\end{array}$$ (5)

$$+\frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{e(u,\mathit{\beta })+{\sigma }^2\widehat{\mathrm{\sum }}(u,\mathit{\beta })-\stackrel{\sim }{E}(u,\mathit{\beta },{\sigma }^2)\right\}{dM}_i(u).$$ (6)

Note that (5) converges to zero in probability since it is a local-square integrable martingale with variance that converges to zero. Furthermore, using similar arguments as in [8], it can be shown that sup_u,β ||e(u, β) + σ²Σ̂ (u, β) − Ẽ (u, β, σ²)|| = o_p(1). As a result, (6) is o_p(1) and so is A₂.

$$\begin{array}{l}{A}_3=\frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)+{\sigma }^2{\mathbf{\sum }}_i(u)\mathit{\beta }-\stackrel{\sim }{E}(u,\mathit{\beta },{\sigma }^2)\right\}{Y}_i(u)exp\{{\mathit{\beta }}_0^{\prime }{\mathbf{X}}_i(u)\}{\lambda }_0(u)du\\ =\frac{1}{n}\sum _{i=1}^n{\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)-e(u,\mathit{\beta })\right\}{Y}_i(u)exp\{{\mathit{\beta }}_0^{\prime }{\mathbf{X}}_i(u)\}{\lambda }_0(u)du+o_p(1)\\ ={\int }_0^L\{S_1(u,{\mathit{\beta }}_0,\mathbf{X})-e(u,\mathit{\beta })S_0(u,{\mathit{\beta }}_0,\mathbf{X})\}{\lambda }_0(u)du+A_4+o_p(1),\end{array}$$

where $A_4=n^{-1}{\sum }_{i=1}^n{\int }_0^L{R}_i(u)Y_i(u)exp\{{\mathit{\beta }}_0^{\prime }{\mathbf{X}}_i(u)\}{\lambda }_0(u)du$, which is a sum of independent and identically distributed random variables with mean zero. Hence, A₄ converges in probability to zero and A₃ = U (β) + o_p(1), where $U(\mathit{\beta })={\int }_0^L\{s_1(u,{\mathit{\beta }}_0)-e(u,\mathit{\beta })s_0(u,{\mathit{\beta }}_0)\}{\lambda }_0(u)du$. As a result, $U_n^{cr}(\mathit{\beta },{\sigma }^2)$ converges in probability to U(β) uniformly in β. Hence, U(β̂_cr) converges to 0 in probability. Since U(.) is concave and β₀ is the unique solution to U (β) = 0, the consistency of β̂_cr follows readily from the inverse function theorem [26]. In the following, we prove that $\sqrt{n}({\widehat{\mathit{\beta }}}_{cr}-{\mathit{\beta }}_0)$ asymptotically follows a normal distribution.

Asymptotic distribution

By the mean value theorem, $\sqrt{n}({\widehat{\mathit{\beta }}}_{cr}-{\mathit{\beta }}_0)={\{{\widehat{\mathbf{\Gamma }}}_n^{cr}({\mathit{\beta }}^{\ast },{\sigma }^2)\}}^{-1}\sqrt{n}{U}_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)$, where ${\widehat{\mathbf{\Gamma }}}_n^{cr}(\mathit{\beta },{\sigma }^2)=-\partial U_n^{cr}({\mathit{\beta }}^{\ast },{\sigma }^2)/\partial \mathit{\beta }$ and β^* is in the line segment joining β̂_cr and β₀. With similar arguments as in the proof of the convergence in probability of $U_n^{cr}(\mathit{\beta },{\sigma }^2)$ to U(β) uniformly in β, we can also show that ${\widehat{\mathbf{\Gamma }}}_n^{cr}(\mathit{\beta },{\sigma }^2)$ converges in probability to

$$\mathbf{\Gamma }(\mathit{\beta })\equiv {\int }_0^L[s_2(u,\mathit{\beta })/s_0(u,\mathit{\beta })-{\{s_1(u,\mathit{\beta })/s_0(u,\mathit{\beta })\}}^{\otimes 2}]s_0(u,\mathit{\beta }){\lambda }_0(u)du,$$

uniformly in β, under conditions (A1)–(A5). With the consistency of β̂_cr and the continuity of Γ(.) at β₀, the convergence in probability of ${\widehat{\mathbf{\Gamma }}}_n^{cr}({\mathit{\beta }}_{\ast },{\sigma }^2)$ to Γ(β₀) is straightforward. In the following, we show that $\sqrt{n}{U}_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)$ is asymptotically normally distributed. We first express $U_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)$ as a mapping of empirical processes.

$$\begin{array}{l}{U}_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)={\int }_0^L\frac{1}{n}\sum _{i=1}^n{\mathbf{K}}_i(u,{\mathit{\beta }}_0,{\sigma }^2){dN}_i(u)-\frac{n^{-1}{\sum }_{i=1}^n{\stackrel{\sim }{S}}_{1i}(u,{\mathit{\beta }}_0,{\sigma }^2)}{n^{-1}{\sum }_{i=1}^n{\stackrel{\sim }{S}}_{0i}(u,{\mathit{\beta }}_0,{\sigma }^2)}\frac{1}{n}\sum _{i=1}^n{dN}_i(u)\\ ={\int }_0^{L}d\widehat{E}\{\mathbf{K}(u,{\mathit{\beta }}_0,{\sigma }^2)N(u)\}-\widehat{E}\left\{\frac{d\mathbf{K}(u,{\mathit{\beta }}_0,{\sigma }^2)}{du}N(u)\right\}du-{\int }_0^L\frac{\widehat{E}\left\{{\stackrel{\sim }{S}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}}{\widehat{E}\left\{{\stackrel{\sim }{S}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}}d\stackrel{\sim }{E}\{N(u)\},\end{array}$$

where K_i(u, β₀, σ²) ≡ X̂_i(u) + σ² Σ_i(u)β₀ and $\widehat{E}(c)=n^{-1}{\sum }_{i=1}^n{c}_i$ for all c. It appears that $U_n^{cr}(\mathit{\beta },{\sigma }^2)$ is a function of five empirical processes and for any (F, G, P₀, P₁, N) in D⁵[0, L] the function is defined as follows:

$$\mathit{\phi }(F,G,P_0,P_1,N)={\int }_0^{L}dF(u)-G(u)du-\frac{P_1(u)}{P_0(u)}dN(u).$$

With conditions (A1)–(A4), the limit of each process exists and by the extended strong law of large number stated in [20], it follows that each process converges almost surely to its limit uniformly in u ∈ [0, L]. Now, let E {K(u, β₀, σ²)N(u)}, $E\left\{\frac{d\mathbf{K}(u,{\mathit{\beta }}_0,{\sigma }^2)}{du}N(u)\right\}$, E{S̃₁(u, β₀, σ²)}, E{S̃₀(u, β₀, σ²)}, E {N(u)} denote the limit of Ê{K(u, β₀, σ²)N(u)}, $\widehat{E}\left\{\frac{d\mathbf{K}(u,{\mathit{\beta }}_0,{\sigma }^2)}{du}N(u)\right\}$, Ê{S̃₁(u, β₀, σ²)}, Ê {S̃₀(u, β₀, σ²)}, Ê {N(u)} respectively. Furthermore, for each u, let F_n(u), G_n(u), P_n0(u), P_n1(u), N_n(u) be defined as follows:

$$\begin{array}{l}{F}_n(u)=\widehat{E}\{\mathbf{K}(u,{\mathit{\beta }}_0,{\sigma }^2)N(u)\}-E\{\mathbf{K}(u,{\mathit{\beta }}_0,{\sigma }^2)N(u)\},\\ G_n(u)=\widehat{E}\left\{\frac{d\mathbf{K}(u,{\mathit{\beta }}_0,{\sigma }^2)}{du}N(u)\right\}-E\left\{\frac{d\mathbf{K}(u,{\mathit{\beta }}_0,{\sigma }^2)}{du}N(u)\right\},\\ P_{0n}(u)=\widehat{E}\left\{{\stackrel{\sim }{S}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}-E\left\{{\stackrel{\sim }{S}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)\right\},\\ P_{1n}(u)=\widehat{E}\left\{{\stackrel{\sim }{S}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}-E\left\{{\stackrel{\sim }{S}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)\right\},\\ N_n(u)=\widehat{E}\{N(u)\}-E\{N(u)\}.\end{array}$$

Under conditions (A1)–(A4), the multivariate central limit theorem leads to the convergence in probability of $\sqrt{n}(F_n,G_n,P_{n0},P_{n1},N_n)$ to a zero-mean Gaussian process. Moreover, under the bounded variability condition the limiting distribution of $\sqrt{n}(F_n,G_n,P_{n0},P_{n1},N_n)$ is tight and therefore $\sqrt{n}(F_n,G_n,P_{n0},P_{n1},N_n)$ converges weakly to a zero-mean Gaussian process with a continuous path. Also, the mapping ϕ is Hadamard differentiable and

$$d{\mathit{\phi }}_{(F,G,P_0,P_1,N)}(\delta ,\gamma ,{\pi }_0,{\pi }_1,\mathit{\eta })={\int }_0^{L}d\delta (u)-\gamma (u)-\frac{P_1(u)}{P_0(u)}d\mathit{\eta }(u)-\left\{\frac{{\pi }_1(u)}{P_0(u)}-\frac{{\pi }_0(u)P_1(u)}{P_0{(u)}^2}\right\}dN(u).$$

Then, a functional Taylor series expansion of $\sqrt{n}{U}_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)$ is the following:

$$\begin{array}{l}\sqrt{n}{U}_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)=\sqrt{n}\{U_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)-U({\mathit{\beta }}_0)\}\\ ={\int }_0^{L}d\widehat{E}\{\mathbf{K}(u)N(u)\}-\widehat{E}\left\{\frac{d\mathbf{K}(u)}{du}N(u)\right\}du-\frac{E\left\{{\stackrel{\sim }{S}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}}{E\left\{{\stackrel{\sim }{S}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}}d\widehat{E}\{N(u)\}-\left[\frac{\widehat{E}\left\{{\stackrel{\sim }{S}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}}{E\left\{{\stackrel{\sim }{S}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}}-\frac{\widehat{E}\left\{{\stackrel{\sim }{S}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}E\left\{{\stackrel{\sim }{S}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}}{E{\left\{{\stackrel{\sim }{S}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)\right\}}^2}\right]dE\{N(u)\}+o_p(1)\\ =\frac{1}{\sqrt{n}}\sum _{i=1}^n{\int }_0^{L}d\{{\mathbf{K}}_i(u)N_i(u)\}-\frac{d{\mathbf{K}}_i(u)}{du}{N}_i(u)du-\frac{{\stackrel{\sim }{s}}_1(u,{\mathit{\beta }}_0)}{{\stackrel{\sim }{s}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)}{dN}_i(u)-\left[\frac{{\stackrel{\sim }{S}}_{1i}(u,{\mathit{\beta }}_0,{\sigma }^2)}{{\stackrel{\sim }{s}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)}-\frac{{\stackrel{\sim }{S}}_{0i}(u,{\mathit{\beta }}_0,{\sigma }^2){\stackrel{\sim }{s}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)}{{\stackrel{\sim }{s}}_0{(u,{\mathit{\beta }}_0,{\sigma }^2)}^2}\right]dE\{N(u)\}+o_p(1),\end{array}$$

where K(u) and K_i(u) denote K(u, β₀, σ²) and K_i(u, β₀, σ²), respectively, for notational convenience. After rearranging the above expression we obtain the following:

$$\begin{array}{l}\sqrt{n}{U}_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)=\frac{1}{\sqrt{n}}\sum _{i=1}^n{\int }_0^L\left\{{\mathbf{K}}_i(u,{\mathit{\beta }}_0,{\sigma }^2)-\frac{{\stackrel{\sim }{s}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)}{{\stackrel{\sim }{s}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)}\right\}{dN}_i(u)-\left[\frac{{\stackrel{\sim }{S}}_{1i}(u,{\mathit{\beta }}_0,{\sigma }^2)}{{\stackrel{\sim }{s}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)}-\frac{{\stackrel{\sim }{S}}_{0i}(u,{\mathit{\beta }}_0,{\sigma }^2){\stackrel{\sim }{s}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)}{{\stackrel{\sim }{s}}_0{(u,{\mathit{\beta }}_0,{\sigma }^2)}^2}\right]dE\{N(u)\}+o_p(1)\\ =\frac{1}{\sqrt{n}}\sum _{i=1}^n{\mathbf{\Psi }}_i({\mathit{\beta }}_0,{\sigma }^2)+o_p(1),\end{array}$$

where

$${\mathbf{\Psi }}_i(\mathit{\beta },{\sigma }^2)\equiv {\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)+{\sigma }^2{\mathbf{\sum }}_i(u)\mathit{\beta }-\frac{{\stackrel{\sim }{s}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)}{{\stackrel{\sim }{s}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)}\right\}\left\{{dN}_i(u)-{\stackrel{\sim }{S}}_{0i}(u,\mathit{\beta },{\sigma }^2)\frac{dE\{N(u)\}}{{\stackrel{\sim }{s}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)}\right\}.$$

Hence, $\sqrt{n}{U}_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)$ is a normalized sum of independent and identically distributed random variables, and it follows from the central limit theorem that it is asymptotically normally distributed. We then conclude that $\sqrt{n}({\widehat{\mathit{\beta }}}_{cr}-{\mathit{\beta }}_0)$ has an asymptotic normal distribution with mean zero and variance {Γ(β₀)}⁻¹ var {Ψ₁(β₀, σ²)} {Γ(β₀)}⁻¹]′. Note that var {Ψ₁(β₀, σ ²)} can be consistently estimated by $n^{-1}{\sum }_{i=1}^n{\widehat{\mathrm{\Psi }}}_i{({\mathit{\beta }}_0,{\sigma }^2)}^{\otimes 2}$, where

$${\widehat{\mathrm{\Psi }}}_i(\mathit{\beta },{\sigma }^2)\equiv {\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)+{\sigma }^2{\mathbf{\sum }}_i(u)\mathit{\beta }-\frac{{\stackrel{\sim }{S}}_1(u,\mathit{\beta },{\sigma }^2)}{{\stackrel{\sim }{S}}_0(u,\mathit{\beta },{\sigma }^2)}\right\}\left\{{dN}_i(u)-\frac{{\stackrel{\sim }{S}}_{0i}(u,\mathit{\beta },{\sigma }^2)n^{-1}{\sum }_{i=1}^n{dN}_i(u)}{{\stackrel{\sim }{S}}_0(u,\mathit{\beta },{\sigma }^2)}\right\}.$$

Appendix B: Proof of the asymptotic equivalence between the corrected score and conditional score estimators

In the following, we assume for simplicity that σ² is known and use similar arguments as in [27]. Note that

$$\sqrt{n}({\widehat{\mathit{\beta }}}_{cs}-{\widehat{\mathit{\beta }}}_{cr})=-\sqrt{n}\left[{\{{\widehat{\mathbf{\Gamma }}}_n^{cs}({\mathit{\beta }}_0,{\sigma }^2)\}}^{-1}{U}_n^{cs}({\mathit{\beta }}_0,{\sigma }^2)-{\{{\widehat{\mathbf{\Gamma }}}_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)\}}^{-1}{U}_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)\right]+o_P(1),$$

where ${\widehat{\mathbf{\Gamma }}}_n^{cs}({\mathit{\beta }}_0,{\sigma }^2)\equiv (\partial /\partial \mathit{\beta })U_n^{cs}({\mathit{\beta }}_0,{\sigma }^2)$, and ${\widehat{\mathbf{\Gamma }}}_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)\equiv (\partial /\partial \mathit{\beta })U_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)$. Using similar arguments as in the proof that $U_n^{cr}(\mathit{\beta },{\sigma }^2)$ converges in probability to U(β), and uniformly in β, it can be shown that, under conditions (A1)–(A5), both ${\widehat{\mathbf{\Gamma }}}_n^{cr}(\mathit{\beta },{\sigma }^2)$ and ${\widehat{\mathbf{\Gamma }}}_n^{cs}(\mathit{\beta },{\sigma }^2)$ converge in probability to Γ(β) uniformly in β. Therefore,

$$\sqrt{n}({\widehat{\mathit{\beta }}}_{cs}-{\widehat{\mathit{\beta }}}_{cr})=-\sqrt{n}{\{\mathbf{\Gamma }({\mathit{\beta }}_0)\}}^{-1}[U_n^{cs}({\mathit{\beta }}_0,{\sigma }^2)-U_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)]+o_P(1).$$

Moreover, the argument of the functional delta method can be applied once again to show that $\sqrt{n}{U}_n^{cs}({\mathit{\beta }}_0,{\sigma }^2)=1/\sqrt{n}{\sum }_{i=1}^n{\stackrel{\sim }{\mathbf{\Psi }}}_i(\mathit{\beta },{\sigma }^2)+o_p(1)$, where Ψ̃_i(β₀, σ²) is given by

$${\int }_0^L\left\{{\mathbf{Q}}_i(u,{\mathit{\beta }}_0,{\sigma }^2)-\frac{v_1(u,{\mathit{\beta }}_0,{\sigma }^2)}{v_0(u,{\mathit{\beta }}_0,{\sigma }^2)}\right\}\left\{{dN}_i(u)-V_{0i}(u,{\mathit{\beta }}_0,{\sigma }^2)\frac{dE\{N(u)\}}{v_0(u,{\mathit{\beta }}_0,{\sigma }^2)}\right\},$$

dN(u) is the limit of $n^{-1}{\sum }_{i=1}^n{dN}_i(u)$, and v_k(u, β₀, σ²) = E{V_k(u, β₀, σ²)}. Using the Lebesgue measure, it can be obtained that V_ki(u, β₀, σ²) = S̃_ki(u, β₀, σ²), i = 1, …, n, k = 0, 1, almost surely, for fixed u. It follows that v_k(u, β₀, σ²) = s̃_k(u, β₀, σ²) almost surely, for fixed u. Furthermore, we can derive $\sqrt{n}\{U_n^{cs}({\mathit{\beta }}_0,{\sigma }^2)-U_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)\}=1/\sqrt{n}{\sum }_{i=1}^n\mathrm{\Delta }{U}_i+o_p(1)$, where

$$\mathrm{\Delta }{U}_i={\int }_0^L\left\{{\widehat{\mathbf{X}}}_i(u)+{\sigma }^2{\mathbf{\sum }}_i(u){\mathit{\beta }}_0-\frac{{\stackrel{\sim }{s}}_1(u,{\mathit{\beta }}_0,{\sigma }^2)}{{\stackrel{\sim }{s}}_0(u,{\mathit{\beta }}_0,{\sigma }^2)}\right\}\left[\frac{{\stackrel{\sim }{S}}_{0i}(u,{\mathit{\beta }}_0,{\sigma }^2)-V_{0i}(u,{\mathit{\beta }}_0,{\sigma }^2)}{{\stackrel{\sim }{s}}_0(u,\mathit{\beta },{\sigma }^2)}dE\{N(u)\}\right].$$

It is straightforward to obtain that $\sqrt{n}\{U_n^{cs}({\mathit{\beta }}_0,{\sigma }^2)-U_n^{cr}({\mathit{\beta }}_0,{\sigma }^2)\}=o_p(1)$, and as a result, $\sqrt{n}({\widehat{\mathit{\beta }}}_{cs}-{\widehat{\mathit{\beta }}}_{cr})=o_p(1)$. The asymptotic equivalence between the two functional estimators is then proven.