Estimation in the single change-point hazard function for interval-censored data with a cure fraction

ABSTRACT

In reliability or survival analysis, the hazard function plays a significant part for it can display the instantaneous failure rate at any time point. In practice, the abrupt change in hazard function at an unknown time point may occur after a maintenance activity or major operation. Under these circumstances, identifying the change point and estimating the size of the change are meaningful. In this paper, we assume that the hazard function is piecewise constant with a single jump at an unknown time. We propose the single change-point model for interval-censored survival data with a cure fraction. Estimation methods for the proposed model are investigated, and large-sample properties of the estimators are established. Simulation studies are carried out to evaluate the performance of the estimating method. The liver cancer data and breast cancer data are analyzed as the applications.

1. Introduction

The change-point problem in distribution arises in quality control problems and has recently received much attention. In survival analysis, it is of great importance to detect a time lag of treatment effect or identify the change point in a hazard function. In medical follow-up studies, after a major operation, e.g. liver transplantation or bone marrow transplantation, the initial risk is usually high and then the risk drops to a lower constant long term risk. In this case, the hazard function with a change point is the commonly employed model:

$$\lambda (t)=\beta +\theta I(t>\tau ),$$ (1)

where β, $\beta +\theta$ and τ are positive constants. In this model, $I(\cdot )$ is the indicator function of an event, β and $\beta +\theta$ are the hazard rates before and after the change point τ, respectively. The jump θ can be either positive or negative, which reflects an increase or decrease in the hazard rate. There exist three main research aspects of this model in the literature. Firstly, fit the model by means of maximum -ikelihood methods (MLE) (see [3,13,15,17]). The second is testing the existence of the change point, considered by [10,16]. The third aspect is obtaining the estimation of the change-point location by the structural properties of (1), explored by [2,6], and a certain function of the estimated cumulative hazard function is needed in their estimation procedures.

For standard survival models, we generally assume that all the patients will die from the event of interest. However, in clinical studies, a substantial proportion of patients who respond favorably to the treatment appear subsequently to be free of any signs or symptoms of the disease and may be considered cured, while the remaining patients may eventually relapse. The standard survival models may be inappropriate for this type of data. To deal with this situation, the cure model was proposed by [1] as follows:

$$S(t)=1-p+p{S}_0(t),$$ (2)

where $S(t)$ is the proportion of people surviving at any given point in time, p is the proportion that is not cured, and $S_0(t)$ represents the survival function of the uncured people. There are several methods proposed to deal with cure models, such as the expectation-maximization algorithm and Markov chain Monte Carlo method [1,19,27]. Survival models with a cure fraction have been extensively studied for decades and many applications have been reported [14], but these models do not consider possible change-point phenomena. In reality, cured patients may well exist in change-point situations. For example, the nonlymphoblastic leukemia data studied in [15] was proved that there exists an abrupt change for the recurrence rate. At the same time, the Kaplan–Meier estimator of the survival function levels off below 1, indicating the presence of cured patients who will never suffer a relapse of leukemia in the data. Inspired by this, we pursue the change-point model (1) with the possible presence of a cure fraction.

Interval censoring is an increasingly common type of censoring and there is voluminous literature on the statistical analysis of interval-censored failure time data (see [9,23,24]). In these studies, we only know that the event occurs within a time interval based on the follow-up visit. In some situations, interval censorship is due to some data-processing programs. The liver cancer data in our real data analysis are from the Surveillance, Epidemiology, and End Results (SEER) cancer incidence public-use database. SEER only publicizes the processed survival month. Thus, we cannot obtain the day-level information, needless to say the exact event time. The survival month (T) after diagnosis is defined by

$$T=\mathrm{floor}\left(\frac{\mathrm{last}\mathrm{contact}\mathrm{date}-\mathrm{diagnosis}\mathrm{date}}{\mathrm{average}\mathrm{days}\mathrm{in}\mathrm{month}}\right),$$

where $\mathrm{floor}(t)=\{n:t\in [n,n+1),n\in N\}$. By the definition of survival month T, we obtain that the exact event time lying on the interval $[T,T+1)$ or $[T,\mathrm{\infty })$ [29]. The breast cancer data in our application come from a retrospective study of 94 patients who received radiation therapy. The clinician, who determined whether or not the retraction had occurred, made a series of observation times to each patient. Hence, the time of retraction is known only to lie between the time of the present and last observation times.

In this article, we focus on modeling the ‘case 2’ interval-censored survival data with the cured patients by model (1). Compared with [28], we deal with different types of data. We replace the Kaplan–Meier method with the ICM method. The NPMLE calculated by ICM can guarantee the consistency which is helpful to get the consistency of the uncured rate's estimation. Additionally, we need some assumptions for interval censoring to obtain asymptotic properties and propose a change-point test procedure.

The rest of the article is organized as follows. In Section 2, we outline the notations and model descriptions of the single change-point hazard model with a cure fraction for interval-censored data. Details of pseudo-maximum-likelihood estimation are presented in Section 3. The procedure of the change-point test is proposed in Section 4. Asymptotic properties are investigated in Section 5. Extensive simulation results are reported in Section 6. In Section 7, we apply the proposed method to the liver cancer data and breast cancer data. Technical proofs are relegated to the supplemental material.

2. Model formulation

Under the cure model (2), let η be the indicator variable with $\eta =1$ if the patient is uncured and 0 if cured, T be the failure time of a patient and $T^{\ast }$ be the failure time of an uncured patient. Define $p=P(\eta =1)$. That is, p is the probability of being uncured. Then the relationship between cumulative distribution functions (c.d.f.) of T and $T^{\ast }$ is

$$\begin{array}{rl}F(t)& =P(T\le t|\eta =1)P(\eta =1)+P(T\le t|\eta =0)P(\eta =0)\\ & =pP(T^{\ast }\le t)=p{F}_0(t),\end{array}$$

where $F(t)$ and $F_0(t)$ are the c.d.f. of T and $T^{\ast }$, respectively. Correspondingly, provided $f(t)$ and $f_0(t)$ are their density functions, the hazard rate function of T is of the form

$$\lambda (t)=\frac{f(t)}{1-F(t)}=\frac{p{f}_0(t)}{1-p{F}_0(t)}.$$ (3)

Note that $\lambda (t)\to 0$ as $t\to \mathrm{\infty }$ for p<1, and hence the hazard rate cannot remain constant with the presence of a cure fraction. A simple example is the exponential lifetime with cured patients, where $T^{\ast }$ is exponentially distributed with a constant hazard rate ψ. Then the hazard rate of T is

$$\lambda (t)=\frac{p\psi \exp (-\psi t)}{1-p+p\exp (-\psi t)},$$

which is no longer constant.

Now, assume that the hazard function of $T^{\ast }$ is specified as

$${\lambda }_0(t)=\beta +\theta I(t>\tau ).$$

Then, we obtain the corresponding density function and c.d.f. are, respectively,

$$f_0(t)={\lambda }_0(t)e^{-{\int }_0^t{\lambda }_0(u)\mathrm{d}u}=\{\begin{array}{ll}\beta e^{-\beta t},& 0\le t\le \tau ,\\ (\beta +\theta )e^{-\beta t-\theta (t-\tau )},& t>\tau ,\end{array}$$

and

$$F_0(t)=\{\begin{array}{ll}1-e^{-\beta t},& 0\le t\le \tau ,\\ 1-e^{-\beta t-\theta (t-\tau )},& t>\tau .\end{array}$$

By (3), the hazard function of T is

$$\lambda (t)=\{\begin{array}{lll}{\displaystyle {\displaystyle \frac{p\beta e^{-\beta t}}{1-p+p{e}^{-\beta t}}},}& 0\le t\le \tau ,{\displaystyle {\displaystyle \frac{p(\beta +\theta )e^{-\beta t-\theta (t-\tau )}}{1-p+p{e}^{-\beta t-\theta (t-\tau )}}},}& t>\tau .\end{array}$$ (4)

There is a jump at τ of size

$$\frac{p\theta \exp (-\beta \tau )}{1-p+p\exp (-\beta \tau )},$$

which is increasing with respect to p if $\theta >0$ and decreasing for $\theta <0$, and it reaches its maximum (minimum) value θ at p=1 for the case of $\theta >0$ ($\theta <0$). Apparently, there are two different mathematical forms before and after the point τ, and $\lambda (t)\le {\lambda }_0(t)$. As is common in change-point models, we suppose the existence of bounds ${\tau }_1$ and ${\tau }_2$ such that $0<{\tau }_1\le \tau \le {\tau }_2<\mathrm{\infty }$ (see [2,6,16]). In medical research, one often has to deal with interval-censored survival data when patients are assessed only at pre-scheduled visits. If the event has not occurred at one visit but has occurred by the following visit, the time T is known within an interval. Following the usual formulation, let $(T_i,L_i,R_i)$, $1\le i\le n$ be a sample of random variables (r.v.) $(T,L,R)$. We postulate $L_i$, $R_i$ are non-negative, independent with $T_i$, following a joint distribution function H, and $P(R_i-L_i\ge c)=1$ for some positive constant c. Moreover, we assume that H has a density h, satisfying

$$h(l,r)>0,\mathrm{if}0<F_0(l)<F_0(r)<1.$$ (5)

Let ${\mathit{O}}_i=(L_i,R_i,{\delta }_{L,i},{\delta }_{R,i},{\delta }_{I,i})$, $i=1,\dots ,n$ denote the data for subjects, where ${\delta }_{R,i}$, ${\delta }_{L,i}$, and ${\delta }_{I,i}$ are the censoring indicators and satisfy that ${\delta }_{L,i}=1$ when $T_i$ is left censored ($T_i<L_i$), ${\delta }_{R,i}=1$ when $T_i$ is right censored ($T_i\ge R_i$), and ${\delta }_{I,i}=1$ when $T_i$ is interval-censored ($L_i\le T_i<R_i$). Note that ${\delta }_{L,i}+{\delta }_{R,i}+{\delta }_{I,i}=1$ for all i. The observed likelihood function of the parameters β, θ, p and τ under the model (4) for interval-censored data is given by

$$L(\beta ,\theta ,p,\tau |{\mathit{O}}_i,i=1,\dots ,n)=\prod _{i=1}^n\{p{F}_0(L_i){\}}^{{\delta }_{L,i}}\{p{F}_0(R_i)-p{F}_0(L_i){\}}^{{\delta }_{I,i}}\{1-p{F}_0(R_i){\}}^{{\delta }_{R,i}}.$$ (6)

3. Pseudo-Maximum- likelihood estimation

According to the preceding instructions, the log-likelihood function based on the observed data can be written as

$$\begin{array}{rl}{l}_n(\mathit{\varphi })& =\log L(\beta ,\theta ,p,\tau |{\mathit{O}}_i,i=1,\dots ,n)\\ & =\sum _{i=1}^n[{\delta }_{L,i}\{\log (p)+\log F_0(L_i)\}+{\delta }_{I,i}\{\log (p)+\log (F_0(R_i)-F_0(L_i))\}\\ & +{\delta }_{R,i}\log (1-p{F}_0(R_i))]\\ & =\sum _{i=1}^{n}l(\beta ,\theta ,p,\tau |{\mathit{O}}_i),\end{array}$$

where $\mathit{\varphi }=({\mathit{\xi }}^{\prime },\tau {)}^{\prime }$ with $\mathit{\xi }=(\beta ,\theta ,p{)}^{\prime }$ and $l(\beta ,\theta ,p,\tau |{\mathit{O}}_i)$ is the log-likelihood of a single observation ${\mathit{O}}_i$ and given by

$$\begin{array}{rl}l(\beta ,\theta ,p,\tau |{\mathit{O}}_i)& ={\delta }_{L,i}\{\log p+I(L_i\le \tau )\log (1-e^{-\beta L_i})\\ & +I(L_i>\tau )\log (1-e^{-\beta L_i-\theta (L_i-\tau )})\}\\ & +{\delta }_{I,i}\{\log p+I(R_i\le \tau )\log (e^{-\beta L_i}-e^{-\beta R_i})\\ & +I(L_i\le \tau <R_i)\log (e^{-\beta L_i}-e^{-\beta R_i-\theta (R_i-\tau )})\\ & +I(L_i>\tau )\log (e^{-\beta L_i-\theta (L_i-\tau )}-e^{-\beta R_i-\theta (R_i-\tau )})\}\\ & +{\delta }_{R,i}\{I(R_i\le \tau )\log (1-p{e}^{-\beta R_i})\\ & +I(R_i>\tau )\log (1-p(1-e^{-\beta R_i-\theta (R_i-\tau )}))\}.\end{array}$$ (7)

From (7), it is obvious that the function $l(\beta ,\theta ,p,\tau )$ is not continuous at τ. Hence, the sufficient conditions for consistency are not met. The classical maximum likelihood (MLE) is not appropriate to implement. Thus, we resort to the pseudo-likelihood approach, which overcomes this problem.

The pseudo-likelihood approach was proposed by [7] and further studied by others including [11,12]. The key idea is to replace the true (but unknown) ‘nuisance’ parameters p and τ in (7) by their consistent estimators $\hat{p}$ and $\hat{\tau }$, and then treat the log-likelihood function $l_n(\beta ,\theta ,\hat{p},\hat{\tau })$, called the pseudo log-likelihood function, as a usual likelihood function of β and θ to generate the pseudo-MLE $(\hat{\beta },\hat{\theta })$ of $(\beta ,\theta )$.

The consistent estimators of τ and p can be obtained by nonparametric methods as follows. As in [14], let

$$\hat{p}={\hat{F}}_n(Y_{(n)}),$$ (8)

where $Y_{(n)}=max\{R_i;i=1,\dots ,n\}$, and ${\hat{F}}_n(t)$ denotes the nonparametric estimate of the c.d.f. of failure times which is achieved by the Iterative Convex Minorant (ICM) algorithm. The ICM algorithm proposed by [9] is fast in computing the nonparametric maximum-likelihood estimation (NPMLE) of the distribution function for interval-censored data without covariates. We will show that $\hat{p}$ is consistent in Section 5.

To develop the estimator of τ, note that the cumulative hazard function of T can be obtained by

$$\mathrm{\Lambda }(t)={\int }_0^t\lambda (u)\mathrm{d}u=\{\begin{array}{ll}-\log (1-p+p{e}^{-\beta t}),& 0\le t\le \tau ,\\ -\log (1-p+p{e}^{-\beta t-\theta (t-\tau )}),& t>\tau .\end{array}$$

Let

$${\mathrm{\Lambda }}^{\ast }(t)=-\log [\frac{1}{p}\{e^{-\mathrm{\Lambda }(t)}-1+p\}]=\{\begin{array}{ll}\beta t,& 0\le t\le \tau ,\\ \beta t+\theta (t-\tau ),& t>\tau ,\end{array}$$

which is a piecewise linear function of t. Further define

$$X(t)=\{\frac{{\mathrm{\Lambda }}^{\ast }(D)-{\mathrm{\Lambda }}^{\ast }(t)}{D-t}-\frac{{\mathrm{\Lambda }}^{\ast }(t)-{\mathrm{\Lambda }}^{\ast }(0)}{t}\}g\{t(D-t)\},$$ (9)

for 0<t<D, where $D>{\tau }_2$ and $g(x)=x^q$, $0\le q\le 1$. Then we have

$$\begin{array}{rl}X(t)& =\frac{{\mathrm{\Lambda }}^{\ast }(D)g\{t(D-t)\}}{D-t}-\frac{D{\mathrm{\Lambda }}^{\ast }(t)g\{t(D-t)\}}{(D-t)t}\\ & =\theta \frac{D-\tau }{D-t}g\{t(D-t)\}I(t\le \tau )+\theta \frac{\tau }{t}g\{t(D-t)\}I(t>\tau ),\end{array}$$ (10)

which is increasing (decreasing) on $[0,\tau ]$ and decreasing (increasing) on $(\tau ,D]$ for $\theta >0$ $(\theta <0)$. Let $X_n(t)$ be the empirical version of (9) with unknown cumulative hazard function $\mathrm{\Lambda }(t)$ and p replaced by the estimator $\hat{\mathrm{\Lambda }}(t)=-\log \{1-{\hat{F}}_n(t)\}$ and $\hat{p}$ in (8), respectively.

Then an estimator of τ is given by

$$\hat{\tau }=\{\begin{array}{ll}inf\{t\in [{\tau }_1,{\tau }_2]:X_n(t\pm )=\underset{u\in [{\tau }_1,{\tau }_2]}{sup}{X}_n(u)\},& \mathrm{if}\theta >0,\\ inf\{t\in [{\tau }_1,{\tau }_2]:X_n(t\pm )=\underset{u\in [{\tau }_1,{\tau }_2]}{inf}{X}_n(u)\},& \mathrm{if}\theta <0.\end{array}$$ (11)

In the absence of a cure fraction, Chang et al. [2] showed that $\hat{\tau }$ is a consistent estimator of τ, with $\hat{\tau }-\tau =O_p(n^{-1})$. Using the asymptotic properties of $\hat{\mathrm{\Lambda }}(t)$ and $\hat{p}$, we can also establish the consistency of $\hat{\tau }$.

4. Change-point test

In this section, we propose a test to determine whether there is a change point in the hazard function for the data. We test the null hypothesis ${\mathcal{H}}_0$ : $\theta =0$ or $\tau =\mathrm{\infty }$ which means there is no change point in the survival distribution versus the alternative hypothesis that there is one change point. Following the results of [21,25,26], together with our model (4), we propose two modified test statistics:

$$\begin{array}{rl}{R}_{MF1}& =2\underset{\tau \in \{x_1,\dots ,x_k\}}{sup}\{l_n(\hat{\beta }(\tau ),\hat{\theta }(\tau ),\hat{p},\tau )-l_n({\hat{\beta }}_0,0,\hat{p},\mathrm{\infty })\},\\ R_{MF2}& =\frac{1}{k}\sum _{\tau \in \{x_1,\dots ,x_k\}}\{l_n(\hat{\beta }(\tau ),\hat{\theta }(\tau ),\hat{p},\tau )-l_n({\hat{\beta }}_0,0,\hat{p},\mathrm{\infty })\},\end{array}$$

where $\hat{\beta }(\tau )$ and $\hat{\theta }(\tau )$ are estimators of the parameters corresponding to the model (4) obtained by maximizing the pesudo-likelihood $l_n(\beta ,\theta ,\hat{p},\tau )$, ${\hat{\beta }}_0$ is the MLE when $\theta =0$ or $\tau =\mathrm{\infty }$, $\hat{p}$ is obtained from (8) and $\{x_1,\dots ,x_k\}$ are equally spaced points in the interval $[{\tau }_1,{\tau }_2]$. The asymptotic distributions of $R_{MF1}$ and $R_{MF2}$ under the null hypothesis are complicated. Hence, we apply a resampling procedure to obtain the critical values under ${\mathcal{H}}_0$, taking $R_{MF1}$ as an example:

Step 1. Calculate ${\hat{p}}_0$ by (8), and ${\hat{\beta }}_0$ by maximizing $\log L(\beta ,0,{\hat{p}}_0,\mathrm{\infty }|{\mathit{O}}_i,i=1\dots ,n)$.

Step 2. Obtain a series of observation times $0=s_1<s_2<\cdots <s_m=\mathrm{\infty }$, m<=2n by arranging all the points in the set ${\cup }_{i=1}^n\{L_i,R_i\}$ from small to large, and removing the repeated points.

Step 3. Generate the failure time data $\{{\tilde{T}}_i{\}}_{i=1}^n$ by the model (4) with $p={\hat{p}}_0$, $\beta ={\hat{\beta }}_0$, $\theta =0$ and $\tau =\mathrm{\infty }$. Obtain the interval-censored data $\{{\tilde{L}}_i,{\tilde{R}}_i{\}}_{i=1}^n$ by setting $({\tilde{L}}_i,{\tilde{R}}_i)=(s_j,s_{j+1})$ if $\widetilde{T_i}\in (s_j,s_{j+1}],i=1,\dots ,n,j\in \{1,\dots ,m\}$.

Step 4. Generate a total of B (e.g. B = 500) simulated trials by repeating Step 3. Obtain the likelihood ratio statistics $R_{MF1}^b,b=1,\dots ,B$ for each trial.

Step 5. Reject the null hypothesis if $R_{MF1}$, the likelihood ratio statistic calculated from the original trial, is larger than the 95% percentile of $\{R_{MF1}^b,b=1,\dots ,B\}$.

5. Asymptotic properties

We first introduce the notations. For ease of presentation, we consider the case $\theta >0$ only. Let $\mathit{\varphi }=(\beta ,\theta ,p,\tau {)}^{\prime }\in \mathbf{\Theta }=(0,\mathrm{\infty })\times (0,\mathrm{\infty })\times [{\tau }_1,{\tau }_2]\times (0,1)$, $\mathit{\mu }=(\beta ,\theta {)}^{\prime }$ and $\mathit{\nu }=(\tau ,p{)}^{\prime }$. The true value of $\mathit{\varphi }=({\mathit{\mu }}^{\prime },{\mathit{\nu }}^{\prime }{)}^{\prime }=(\beta ,\theta ,p,\tau {)}^{\prime }$ is denoted by ${\mathit{\varphi }}_0=({\mathit{\mu }}_0^{\prime },{\mathit{\nu }}_0^{\prime }{)}^{\prime }=({\beta }_0,{\theta }_0,p_0,{\tau }_0{)}^{\prime }$. Write $P=P_{{\mathit{\mu }}_0,{\mathit{\nu }}_0}$ for the probability measure of $\mathit{O}=(L,R,{\delta }_L,{\delta }_I,{\delta }_R{)}^{\prime }$ and employ the abbreviation $Pg=\int gdP$ for any measurable function g. Denote $P_n$ as the empirical measure of observations $\{{\mathit{O}}_i,i=1,\dots ,n\}$ and $P_{n}g=\int gd{P}_n=n^{-1}\sum _{i=1}^{n}g({\mathit{O}}_i)$. Define the parameter spaces for $\mathit{\mu }$ and $\mathit{\nu }$ as

$$\begin{array}{rl}{\mathbf{\Theta }}_1& =\{\mathit{\mu }=(\beta ,\theta {)}^{\prime }:\beta \ge A_1,\theta \ge A_2\},\\ {\mathbf{\Theta }}_2& =\{\mathit{\nu }=(\tau ,p{)}^{\prime }:|\tau -{\tau }_0|\le {\eta }_1,|p-p_0|\le {\eta }_2\},\end{array}$$ (12)

where $A_1$, $A_2$, ${\eta }_1$, and ${\eta }_2$ are some small positive constants. The first partial derivative of $l(\mathit{\mu },\mathit{\nu }|\mathit{O})$ with respect to $\mathit{\mu }$ is donated by ${\dot{l}}_{\mathit{\mu }}(\mathit{\mu },\mathit{\nu }|\mathit{O})=({\dot{l}}_{\beta }(\mathit{\mu },\mathit{\nu }|\mathit{O}),{\dot{l}}_{\theta }(\mathit{\mu },\mathit{\nu }|\mathit{O}){)}^{\prime }$ where

$$\begin{array}{rl}{\dot{l}}_{\beta }(\mathit{\mu },\mathit{\nu }|\mathit{O})& ={\delta }_L\{I(L\le \tau )\frac{L{e}^{-\beta L}}{1-e^{-\beta L}}+I(L>\tau )\frac{L{e}^{-\beta L-\theta (L-\tau )}}{1-e^{-\beta L-\theta (L-\tau )}}\}\\ & +{\delta }_I\{I(R\le \tau )\frac{R{e}^{-\beta R}-L{e}^{-\beta L}}{e^{-\beta L}-e^{-\beta R}}\\ & +I(L\le \tau <R)\frac{R{e}^{-\beta R-\theta (R-\tau )}-L{e}^{-\beta L}}{e^{-\beta L}-e^{-\beta R-\theta (R-\tau )}}\\ & +I(L>\tau )\frac{R{e}^{-\beta R-\theta (R-\tau )}-L{e}^{-\beta L-\theta (L-\tau )}}{e^{-\beta L-\theta (L-\tau )}-e^{-\beta R-\theta (R-\tau )}}\}\\ & +{\delta }_R\{I(R\le \tau )\frac{pR{e}^{-\beta R}}{1-p{e}^{-\beta R}}\\ & +I(R>\tau )\frac{-pR{e}^{-\beta R-\theta (R-\tau )}}{1-p(1-e^{-\beta R-\theta (R-\tau )})}\},\end{array}$$ (13)

and

$$\begin{array}{rl}{\dot{l}}_{\theta }(\mathit{\mu },\mathit{\nu }|\mathit{O})& ={\delta }_{L}I(L>\tau )\frac{(L-\tau )e^{-\beta L-\theta (L-\tau )}}{1-e^{-\beta L-\theta (L-\tau )}}\\ & +{\delta }_I\{I(L\le \tau <R)\frac{(R-\tau )e^{-\beta R-\theta (R-\tau )}}{e^{-\beta L}-e^{-\beta R-\theta (R-\tau )}}\\ & +I(L>\tau )\frac{(R-\tau )e^{-\beta R-\theta (R-\tau )}-(L-\tau )e^{-\beta L-\theta (L-\tau )}}{e^{-\beta L-\theta (L-\tau )}-e^{-\beta R-\theta (R-\tau )}}\}\\ & +{\delta }_{R}I(R>\tau )\frac{-p(R-\tau )e^{-\beta R-\theta (R-\tau )}}{1-p(1-e^{-\beta R-\theta (R-\tau )})}.\end{array}$$ (14)

Denote ${\dot{l}}_{\mathit{\mu }}(\mathit{\mu },\mathit{\nu }|\mathit{O}){\dot{l}}_{\mathit{\mu }}^{\prime }(\mathit{\mu },\mathit{\nu }|\mathit{O})$ by ${\ddot{l}}_{\mathit{\mu }\mathit{\mu }}(\mathit{\mu },\mathit{\nu }|\mathit{O})$. From (13) and (14), we can easily have that

$$P{\dot{l}}_{\mathit{\mu }}(\mathit{\mu },\mathit{\nu }|\mathit{O})<\mathrm{\infty },P{\ddot{l}}_{\mathit{\mu }\mathit{\mu }}(\mathit{\mu },\mathit{\nu }|\mathit{O})<\mathrm{\infty },$$ (15)

on condition that the censoring distribution has a finite variance. Here, a vector or matrix less than infinity means that its all components are less than infinity. Note that ${\mathit{\mu }}_0$ is the unique point such that $P{\dot{l}}_{\mathit{\mu }}(\mathit{\mu },{\mathit{\nu }}_0|\mathit{O})=0$, then we obtain $\hat{\mathit{\mu }}$ by solving $P_n{\dot{l}}_{\mathit{\mu }}(\mathit{\mu },{\mathit{\nu }}_0|{\mathit{O}}_i,i=1,\dots ,n)=0$.

The main results on asymptotic properties of $\hat{p}$, $\hat{\tau }$, $\hat{\beta }$ and $\hat{\theta }$ are presented in the next five theorems, and their proofs are given in supplemental materia. In order to describe the theorems, we need to define the right extreme ${\tau }_{F_0}$ of $F_0$ by

$${\tau }_{F_0}=sup\{t\ge 0:F_0(t)<1\}.$$

Under assumptions of H, we can obtain that $H(\mathrm{\infty },{\tau }_{F_0})<1$, and hence there exists a constant ${\tau }_H$ satisfying ${\tau }_H=sup\{t>{\tau }_{F_0}:H(\mathrm{\infty },t)<1\}$.

Theorem 1 Consistency of $\hat{p}$ —

Suppose that 0<p<1 and F is continuous at ${\tau }_H$ in case ${\tau }_H<\mathrm{\infty }$. Then $\hat{p}\to p$ in probability as $n\to \mathrm{\infty }$.

Remark 1

Theorem 1 is a modification of [14] for interval-censored data. And because the Kaplan–Meier estimation method is not proper for case 2 interval-censored data, in this paper the nonparametric estimation of the c.d.f. of the failure time is achieved by the ICM algorithm. The strong consistency of ${\hat{F}}_n$ obtained by the ICM algorithm is proved by [9], which is sufficient for the proof of Theorem 1.

Using the consistency of $\hat{p}$ and $\hat{\mathrm{\Lambda }}(t)$, we establish the consistency of $\hat{\tau }$ in the following theorem.

Theorem 2 Consistency of $\hat{\tau }$ —

Assume that F is continuous at ${\tau }_H$ in case ${\tau }_H<\mathrm{\infty }$ and ${\tau }_{F_0}>D$. Then the estimator $\hat{\tau }$ of τ defined in (10) is consistent.

Theorem 3 Consistency of $\hat{\mathit{\mu }}$ —

Suppose that $\hat{p}$ and $\hat{\tau }$ are obtained by (8) and (11), respectively. Then $P_n{\dot{l}}_{\mathit{\mu }}(\hat{\mathit{\mu }},\hat{\mathit{\nu }})=o_{p^{\ast }}(n^{-1/2})$ almost surely, and $\hat{\mathit{\mu }}$ converges in outer probability to ${\mathit{\mu }}_0$.

Remark 2

Note that $o_{p^{\ast }}(1)$ in the following representations indicates convergence to zero in outer probability in case that the term involved is not Borel measurable.

Theorem 4 The rate of convergence of $\hat{\mathit{\mu }}$ —

Under the conditions in Theorem 3, $\sqrt{n}(\hat{\mathit{\mu }}-{\mathit{\mu }}_0)=O_{p^{\ast }}(1)$.

Theorem 5 Asymptotic normality —

Under the conditions in Theorem 3, $\sqrt{n}(\hat{\mathit{\mu }}-{\mathit{\mu }}_0)$ is asymptotically normal with mean 0 and variance $\{P{\ddot{l}}_{\mathit{\mu }\mathit{\mu }}({\mathit{\mu }}_0,{\mathit{\nu }}_0){\}}^{-2}\mathit{V},$ where $\mathit{V}=Var\{{\mathbf{\Lambda }}_1+P_0{\ddot{l}}_{\mathit{\mu }\mathit{\nu }}({\mathit{\mu }}_0,{\mathit{\nu }}_0){\mathbf{\Lambda }}_{\mathbf{2}}\},$ and ${\mathbf{\Lambda }}_1$ and ${\mathbf{\Lambda }}_{\mathbf{2}}$ are random vectors satisfying

$$\sqrt{n}\left[\begin{array}{c}(P_n-P){\dot{l}}_{\mathit{\mu }}({\mathit{\mu }}_0,{\mathit{\nu }}_0)\\ \hat{\mathit{\nu }}-{\mathit{\nu }}_0\end{array}\right]\stackrel{d}{\to }\left[\begin{array}{c}{\mathbf{\Lambda }}_{\mathbf{1}}\\ {\mathbf{\Lambda }}_{\mathbf{2}}\end{array}\right].$$

Remark 3

For the asymptotic variance of $\sqrt{n}(\hat{\mathit{\mu }}-{\mathit{\mu }}_0)$ in Theorem 5, a precise representation of $\mathit{V}$ can be found in Corollary 3.1.4 of [11] for i.i.d. setup. In this case, there exists $\gamma (\mathit{O},\mathit{\mu },\mathit{\nu })\ne 0$ satisfying that $\sqrt{n}P{\ddot{l}}_{\mathit{\mu }\mathit{\mu }}({\mathit{\mu }}_0,{\mathit{\nu }}_0)(\hat{\mathit{\nu }}-{\mathit{\nu }}_0)=\sqrt{n}{P}_n\gamma (\mathit{O},{\mathit{\mu }}_0,{\mathit{\nu }}_0)+o_{p\ast }(1)$, which presents $\mathit{V}=Var\{{\dot{l}}_{\mathit{\mu }}({\mathit{\mu }}_0,{\mathit{\nu }}_0|\mathit{O})\}+Var\{\gamma (\mathit{O},{\mathit{\mu }}_0,{\mathit{\nu }}_0)\}$, where $\gamma (\mathit{O},{\mathit{\mu }}_0,{\mathit{\nu }}_0)$ is defined by (3.1.21) in [11]. Without such a $\gamma (\mathit{O},{\mathit{\mu }}_0,{\mathit{\nu }}_0$), a closed form of $\mathit{V}$ is not available, but we can estimate the variance by the bootstrap method as discussed below.

Since the asymptotic variances of $\hat{\mathit{\mu }}$ and $\hat{\mathit{\nu }}$ are often intractable, we resort to the bootstrap method applied in [18,22]. The algorithm proceeds in three steps.

Step 1. Resample the pairs $(L_1,R_1),\dots ,(L_n,R_n)$ with probability 1/n at each pair $(L_i,R_i)$. Denote the resampled data by $a^{\ast }(1),\dots ,a^{\ast }(B)$, for some positive integer B.

Step 2. For each set of bootstrap data $a^{\ast }(b)$, $b=1,\dots ,B$, evaluate the estimates of interest.

Step 3. Calculate the sample means and standard deviations of estimators.

6. Simulation results

To evaluate our approach, we do a lot of simulations through different settings. Particularly, the studies are performed to explore the influences of the value of q in (9), the jump size, the change-point location, the sample size and the censoring level. We also check the power of the test procedure. In all cases, we compute the estimation of p by (8) and τ by (11), and then achieve the estimation of β and θ.

The study considers the data simulated from the hazard function defined at (3) and the corresponding distribution function is

$$F(t)=\{\begin{array}{ll}1-p+p{e}^{-\beta t},& 0\le t\le \tau ,\\ 1-p+p{e}^{-\beta t-\theta (t-\tau )},& t>\tau ,\end{array}$$

where p=0.8, and $\beta =1$. For checking the influence of the jumping size, we fix $\tau =1$, and set $\theta \in \{0.5,1,1.5\}$. And we set $\tau \in \{0.1,0.5,1,1.5\}$ with θ=1 to see the effect of the change-point location. The change-point search range is set to $(0.05,1.75)$. There are six parameter configurations totally. We also let $q\in \{0.25,0.5,0.75,1\}$ for $g(t)$ in (9) to assess the influence of q. Each $T_i$ is generated by solving $F(t)=u_i$ numerically, where $u_i\sim U(0,1)$. The total number of visit times for each subject is generated according to 1 plus a Poisson random variable having mean parameter σ. The first observation times are the sample from $U(0,b)$ where b is a positive constant. The gap times between adjacent visits are sampled according to an exponential distribution with mean 2. Subsequently, the visit times are given by the cumulative sums of the gap times. The observed interval for the ith subject is then determined to be the two consecutive observation times whose interval contained $T_i$, with the convention that if $T_i$ is less (greater) than the smallest (largest) observation time then the lower (upper) bound of the observed interval is 0 $(\mathrm{\infty })$. Different censoring levels can be obtained by adjusting b and σ. We consider four kinds of left and right censoring levels (CL), (30%, 30%), (20%, 20%), (0, 50%) and (0, 30%). For the purposes of this study, 1000 data sets of the form ${(L_i,R_i)}_{i=1}^n$ are generated for each considered parameter configuration where $n\in \{200,400,800\}$.

Firstly, we investigate the effects of the different q in (9). Since the choice of q can not affect $\hat{p}$, $\hat{\beta }$ and $\hat{\theta }$ directly, the result of $\hat{\tau }$ is the only assessment of q. Table 1 displays the empirical biases (bias) and sample standard deviations (SD) of $\hat{\tau }$ under different model scenarios where $\beta =1$, p=0.8, $\theta \in \{0.5,1,1.5\}$, $\tau \in \{0.1,0.5,1,1.5\}$ and the sample size n=800. The results indicate that the proposed method with any $q\in \{0.25,0.5,0.75,1\}$ performs reasonably well. However, smaller sample standard deviations of $\hat{\tau }$ were obtained when q=1 in all situations. Hence, q=1 is suggested.

Table 1. Results of the estimation with $\beta =1$, p=0.8, $\theta \in \{0.5,1,1.5\}$, $\tau \in \{0.1,0.5,1,1.5\}$ and n=800. CL denotes the average rates of both left and right censoring. Denote ${\tau }_{0.25}$, ${\tau }_{0.5}$, ${\tau }_{0.75}$ and ${\tau }_1$ the pseudo-maximum-likelihood method with $q=0.25,0.5,0.75,1$, respectively.

				${\tau }_{0.25}$		${\tau }_{0.5}$		${\tau }_{0.75}$		${\tau }_1$
Con.	CL	τ	θ	Bias	SD	Bias	SD	Bias	SD	Bias	SD
1	(30%, 30%)	0.1	1	0.050	0.271	0.048	0.256	0.042	0.241	0.039	0.231
2	(30%, 30%)	0.5	1	0.016	0.387	0.010	0.251	0.002	0.161	0.003	0.111
3	(30%, 30%)	1	1	0.007	0.227	0.001	0.195	0.011	0.181	0.003	0.162
4	(30%, 30%)	1.5	1	0.007	0.172	0.017	0.157	0.019	0.168	0.019	0.168
5	(30%, 30%)	1	0.5	0.008	0.373	0.021	0.342	0.020	0.331	0.006	0.337
6	(30%, 30%)	1	1.5	0.016	0.159	0.006	0.144	0.004	0.123	0.001	0.119
7	(20%, 20%)	0.1	1	0.017	0.157	0.015	0.148	0.007	0.151	0.003	0.151
8	(20%, 20%)	0.5	1	0.005	0,213	0.003	0.118	0.005	0.104	0.006	0.096
9	(20%, 20%)	1	1	0.005	0.175	0.006	0.163	0.005	0.158	0.001	0.141
10	(20%, 20%)	1.5	1	0.005	0.144	0.003	0.145	0.009	0.126	0.006	0.119
11	(20%, 20%)	1	0.5	0.003	0.245	0.001	0.237	0.009	0.203	0.002	0.195
12	(20%, 20%)	1	1.5	0.007	0.130	0.007	0.114	0.002	0.101	0.005	0.097
13	(0, 50%)	0.1	1	0.125	0.403	0.124	0.398	0.121	0.393	0.115	0.296
14	(0, 50%)	0.5	1	0.017	0.228	0.005	0.260	0.018	0.196	0.003	0.211
15	(0, 50%)	1	1	0.002	0.308	0.005	0.288	0.004	0.287	0.001	0.270
16	(0, 50%)	1.5	1	0.004	0.220	0.009	0.218	0.004	0.199	0.016	0.216
17	(0, 50%)	1	0.5	0.007	0.374	0.009	0.378	0.007	0.378	0.003	0.367
18	(0, 50%)	1	1.5	0.001	0.193	0.005	0.189	0.006	0.172	0.002	0.157
19	(0, 30%)	0.1	1	0.090	0.304	0.087	0.299	0.083	0.293	0.072	0.282
20	(0, 30%)	0.5	1	0.001	0.133	0.004	0.098	0.007	0.093	0.003	0.089
21	(0, 30%)	1	1	0.011	0.198	0.001	0.191	0.011	0.168	0.008	0.167
22	(0, 30%)	1.5	1	0.006	0.177	0.003	0.169	0.007	0.169	0.001	0.168
23	(0, 30%)	1	0.5	0.001	0.377	0.001	0.297	0.006	0.292	0.007	0.289
24	(0, 30%)	1	1.5	0.002	0.148	0.001	0.140	0.004	0.138	0.001	0.134

Next, we compare our method with the MLE method suggested by [3,15,25,26]. They obtain the estimators as follows: with a fixed τ, let ${\hat{\mathit{\xi }}}_n(\tau )$ be the value of ξ maximizing $l_n(\mathit{\varphi })$. Then τ is estimated by

$$\hat{\tau }=inf\{\tau \in [{\tau }_1,{\tau }_2]:max(l_n({\hat{\mathit{\xi }}}_n(\tau ),\tau ),l_n({\hat{\mathit{\xi }}}_n(\tau \pm ),\tau \pm )=\underset{\tau \in [{\tau }_1,{\tau }_2]}{sup}{l}_n({\hat{\mathit{\xi }}}_n(\tau ),\tau ))\}.$$

Then the maximum-likelihood estimator of $\mathit{\xi }$ is obtained as ${\hat{\mathit{\xi }}}_n={\hat{\mathit{\xi }}}_n(\hat{\tau })$.

Table 2 displays the empirical biases and sample standard deviations of the estimators considering different configurations of model parameters from the sets $\theta \in (0.5,1,1.5)$, $\tau \in (0.1,0.5,1,1.5)$. We also set different censoring levels and consider samples of size 800. From Table 2, the results indicate that both methods perform reasonably well in estimating the model parameters for the cases investigated except the situation $\tau =0.1$ (see the configuration sequence 1–7–13–19), since the number of the failures smaller than 0.1 is not enough to obtain good estimators. The performances of the two methods are generally comparable showing similar improvement or deterioration behaviors as the system parameters change. More specifically, biases and standard deviations of the estimators are smaller with a larger jump size (see configuration sequences 3–5–6, 9–11–12, 15–17–18 and 21–23–24). By comparing configurations 1 to 6 with 7 to 12 in Table 2, we obtain that the biases and standard deviations of the estimators are smaller with a lower left and right censoring level. More specifically, the performances are better with a smaller right censoring rate referring to configurations 13 to 24. Looking at the configuration sequences 2–3–4, 8–9–10, 14–15–16 and 20–21–22, the change-point location has no great influences on the estimations, if there are enough samples on both sides of the change point. We also obtain that our method provides smaller biases than the MLE method in most cases. Further, when the jump size is small or the location of the change point is close to 0, our method is overwhelmingly better. In Table 3, we take into account the influence of the sample size. As expected, biases and standard deviations decrease as the sample size increases.

Table 2. Results of the estimation with $\beta =1$, p=0.8, $\theta \in \{0.5,1,1.5\}$, $\tau \in \{0.1,0.5,1,1.5\}$ and n=800. PMLE denotes the pseudo-maximum-likelihood method with q=1 and CL denotes the level of right and left censoring.

					PMLE				MLE
Con.	CL	τ	θ		$\hat{p}$	$\hat{\tau }$	$\hat{\beta }$	$\hat{\theta }$	$\hat{p}$	$\hat{\tau }$	$\hat{\beta }$	$\hat{\theta }$
1	(30%, 30%)	0.1	1	Bias	0.005	0.039	0.087	0.089	0.001	0.116	0.087	0.229
				SD	0.022	0.231	0.433	0.451	0.020	0.341	0.496	0.415
2	(30%, 30%)	0.5	1	Bias	0.001	0.003	0.003	0.014	0.003	0.015	0.008	0.046
				SD	0.020	0.111	0.113	0.222	0.019	0.093	0.109	0.229
3	(30%, 30%)	1	1	Bias	0.001	0.003	0.002	0.012	0.001	0.015	0.001	0.040
				SD	0.022	0.111	0.067	0.298	0.021	0.125	0.069	0.320
4	(30%, 30%)	1.5	1	Bias	0.001	0.019	0.001	0.011	0.002	0.074	0.005	0.113
				SD	0.023	0.168	0.062	0.414	0.020	0.284	0.075	0.582
5	(30%, 30%)	1	0.5	Bias	0.002	0.006	0.019	0.038	0.002	0.015	0.014	0.105
				SD	0.024	0.337	0.063	0.237	0.021	0.261	0.084	0.337
6	(30%, 30%)	1	1.5	Bias	0.001	0.001	0.003	0.010	0.001	0.011	0.004	0.040
				SD	0.019	0.119	0.066	0.349	0.019	0.071	0.062	0.411
7	(20%, 20%)	0.1	1	Bias	0.001	0.003	0.100	0.103	0.001	0.037	0.079	0.116
				SD	0.015	0.151	0.379	0.363	0.015	0.185	0.401	0.371
8	(20%, 20%)	0.5	1	Bias	0.001	0.006	0.011	0.024	0.001	0.004	0.015	0.026
				SD	0.015	0.096	0.091	0.142	0.016	0.079	0.092	0.135
9	(20%, 20%)	1	1	Bias	0.001	0.001	0.001	0.007	0.001	0.014	0.002	0.018
				SD	0.015	0.141	0.063	0.212	0.015	0.125	0.066	0.202
10	(20%, 20%)	1.5	1	Bias	0.001	0.006	0.002	0.024	0.001	0.018	0.002	0.027
				SD	0.015	0.119	0.054	0.236	0.015	0.159	0.056	0.263
11	(20%, 20%)	1	0.5	Bias	0.001	0.002	0.002	0.019	0.001	0.023	0.002	0.052
				SD	0.015	0.195	0.061	0.139	0.015	0.210	0.063	0.161
12	(20%, 20%)	1	1.5	Bias	0.001	0.005	0.001	0.017	0.001	0.005	0.001	0.024
				SD	0.015	0.097	0.062	0.247	0.015	0.097	0.062	0.229
13	(0, 50%)	0.1	1	Bias	0.010	0.115	0.002	0.065	0.001	0.130	0.032	0.252
				SD	0.032	0.296	0.413	0.478	0.028	0.410	0.500	0.447
14	(0, 50%)	0.5	1	Bias	0.008	0.003	0.046	0.004	0.001	0.001	0.029	0.135
				SD	0.033	0.211	0.119	0.317	0.029	0.218	0.123	0.351
15	(0, 50%)	1	1	Bias	0.009	0.001	0.022	0.044	0.002	0.002	0.008	0.251
				SD	0.037	0.270	0.075	0.479	0.027	0.245	0.091	0.667
16	(0, 50%)	1.5	1	Bias	0.005	0.016	0.006	0.045	0.006	0.128	0.001	0.319
				SD	0.035	0.216	0.085	0.591	0.032	0.358	0.092	0.868
17	(0, 50%)	1	0.5	Bias	0.002	0.003	0.014	0.108	0.009	0.023	0.003	0.399
				SD	0.038	0.367	0.083	0.438	0.034	0.363	0.107	0.622
18	(0, 50%)	1	1.5	Bias	0.005	0.002	0.006	0.027	0.001	0.020	0.005	0.084
				SD	0.025	0.157	0.077	0.440	0.025	0.134	0.088	0.655
19	(0, 30%)	0.1	1	Bias	0.004	0.072	0.045	0.034	0.001	0.120	0.053	0.073
				SD	0.023	0.282	0.366	0.392	0.018	0.294	0.436	0.400
20	(0, 30%)	0.5	1	Bias	0.001	0.003	0.020	0.028	0.001	0.001	0.023	0.057
				SD	0.019	0.089	0.091	0.199	0.019	0.090	0.101	0.207
21	(0, 30%)	1	1	Bias	0.001	0.008	0.003	0.045	0.001	0.007	0.005	0.082
				SD	0.020	0.167	0.069	0.284	0.019	0.149	0.068	0.276
22	(0, 30%)	1.5	1	Bias	0.001	0.001	0.003	0.018	0.001	0.040	0.008	0.075
				SD	0.020	0.168	0.062	0.349	0.019	0.231	0.069	0.454
23	(0, 30%)	1	0.5	Bias	0.003	0.007	0.012	0.047	0.001	0.008	0.012	0.078
				SD	0.022	0.289	0.062	0.221	0.020	0.265	0.074	0.217
24	(0, 30%)	1	1.5	Bias	0.001	0.001	0.005	0.016	0.001	0.005	0.009	0.088
				SD	0.017	0.134	0.069	0.301	0.017	0.098	0.070	0.416

Table 3. Results of the estimation with $\beta =1$, p=0.8, $\tau =1$, $\theta \in \{0.5,1.5\}$ and $n\in \{200,400,800\}$. PMLE denotes the pseudo-maximum-likelihood method with q=1, and CL denotes the level of right and left censoring.

					PMLE				MLE
Con.	CL	θ	n		$\hat{p}$	$\hat{\tau }$	$\hat{\beta }$	$\hat{\theta }$	$\hat{p}$	$\hat{\tau }$	$\hat{\beta }$	$\hat{\theta }$
5	(30%, 30%)	0.5	200	Bias	0.001	0.046	0.081	0.127	0.006	0.055	0.035	0.586
				SD	0.035	0.159	0.206	0.378	0.037	0.439	0.172	0.730
			400	Bias	0.004	0.011	0.031	0.105	0.001	0.020	0.026	0.214
				SD	0.028	0.139	0.158	0.293	0.028	0.353	0.112	0.364
			800	Bias	0.002	0.006	0.019	0.038	0.002	0.005	0.014	0.105
				SD	0.024	0.337	0.063	0.237	0.021	0.120	0.084	0.337
6	(30%, 30%)	1.5	200	Bias	0.001	0.003	0.002	0.089	0.002	0.021	0.005	0.497
				SD	0.036	0.215	0.116	0.433	0.037	0.199	0.120	0.775
			400	Bias	0.002	0.005	0.001	0.052	0.001	0.012	0.005	0.276
				SD	0.029	0.167	0.088	0.387	0.027	0.157	0.093	0.593
			800	Bias	0.001	0.001	0.003	0.010	0.001	0.011	0.004	0.040
				SD	0.019	0.119	0.066	0.349	0.019	0.071	0.062	0.411
11	(20%, 20%)	0.5	200	Bias	0.002	0.013	0.012	0.099	0.003	0.071	0.017	0.282
				SD	0.027	0.372	0.095	0.279	0.028	0.443	0.118	0.504
			400	Bias	0.001	0.024	0.003	0.043	0.001	0.069	0.003	0.122
				SD	0.022	0.313	0.078	0.219	0.022	0.348	0.081	0.290
			800	Bias	0.001	0.002	0.002	0.019	0.001	0.023	0.002	0.052
				SD	0.015	0.195	0.061	0.139	0.015	0.210	0.063	0.161
12	(20%, 20%)	1.5	200	Bias	0.001	0.014	0.005	0.035	0.002	0.057	0.014	0.318
				SD	0.027	0.203	0.100	0.413	0.027	0.203	0.107	0.579
			400	Bias	0.001	0.005	0.006	0.020	0.002	0.019	0.007	0.058
				SD	0.022	0.134	0.077	0.340	0.022	0.113	0.082	0.364
			800	Bias	0.001	0.005	0.001	0.017	0.001	0.005	0.001	0.024
				SD	0.015	0.097	0.062	0.247	0.015	0.097	0.062	0.229
17	(0, 50%)	0.5	200	Bias	0.012	0.012	0.060	0.395	0.006	0.005	0.048	1.013
				SD	0.053	0.451	0.167	0.657	0.052	0.471	0.208	1.075
			400	Bias	0.012	0.026	0.045	0.185	0.005	0.016	0.026	0.636
				SD	0.048	0.451	0.105	0.512	0.038	0.416	0.135	0.856
			800	Bias	0.002	0.003	0.014	0.108	0.009	0.023	0.003	0.399
				SD	0.038	0.367	0.083	0.438	0.034	0.363	0.107	0.622
18	(0, 50%)	1.5	200	Bias	0.007	0.009	0.001	0.034	0.001	0.013	0.011	0.598
				SD	0.051	0.297	0.148	0.592	0.045	0.318	0.181	0.935
			400	Bias	0.009	0.002	0.014	0.029	0.002	0.002	0.018	0.386
				SD	0.042	0.270	0.108	0.541	0.036	0.256	0.127	0.837
			800	Bias	0.005	0.002	0.006	0.027	0.001	0.020	0.005	0.084
				SD	0.025	0.157	0.077	0.440	0.025	0.080	0.088	0.655
23	(0, 30%)	0.5	200	Bias	0.004	0.006	0.030	0.232	0.002	0.109	0.027	0.492
				SD	0.038	0.356	0.109	0.434	0.034	0.414	0.137	0.614
			400	Bias	0.003	0.024	0.014	0.145	0.001	0.036	0.018	0.311
				SD	0.027	0.349	0.094	0.368	0.026	0.394	0.121	0.492
			800	Bias	0.003	0.007	0.012	0.047	0.001	0.008	0.012	0.078
				SD	0.022	0.289	0.062	0.221	0.020	0.265	0.074	0.217
24	(0, 30%)	1.5	200	Bias	0.002	0.015	0.004	0.023	0.001	0.019	0.006	0.351
				SD	0.035	0.260	0.128	0.361	0.034	0.267	0.136	0.748
			400	Bias	0.001	0.008	0.002	0.016	0.001	0.013	0.003	0.180
				SD	0.026	0.158	0.095	0.383	0.026	0.164	0.104	0.540
			800	Bias	0.001	0.001	0.005	0.016	0.001	0.005	0.009	0.016
				SD	0.017	0.134	0.069	0.301	0.017	0.098	0.070	0.310

To assess the powers of our proposed change-point test statistics, we explore the percentages of one change point detected in simulated trials. We consider the samples of size 800, $\beta =1$, p=0.8, $\theta \in \{0,0.1,0.2,0.3,0.4,0.5\}$, $\tau \in \{0.1,0.5,1,1.5\}$ and the left and right censoring level is $(20\mathrm{\%},20\mathrm{\%})$. The number of equally spaced points in the interval $[{\tau }_1,{\tau }_2]$ is 500. The results are listed in Table 4. We obtain that the powers of the two statistics are larger with a bigger jump size and smaller when the change-point location is close to 0 or goes to infinity. For each configuration, the power of $R_{MF1}$ is slightly larger than that of $R_{MF2}$. Looking at the configuration $\theta =0$, we also had false positive discovery rates of 5% and 4.8% for $R_{MF1}$ and $R_{MF2}$, respectively, which shows that our methods maintain the overall type I error of $\alpha =0.05$ when the true model has no change point. Overall, the results demonstrate that our methods have a good performance in identifying the true model and estimating the parameters.

Table 4. Size and powers with one or no change point.

		θ
τ		0	0.1	0.2	0.3	0.4	0.5
0.1	$R_{MF1}$	5.0%	12.8%	20.6%	36.4%	48.6%	60.4%
	$R_{MF2}$	4.8%	11.0%	20.0%	30.8%	40.2%	48.0%
0.5	$R_{MF1}$	5.0%	22.6%	64.0%	88.4%	98.2%	100%
	$R_{MF2}$	4.8%	26.0%	54.4%	76.0%	92.2%	100%
1	$R_{MF1}$	5.0%	22.0%	56.0%	84.4%	96.6%	100%
	$R_{MF2}$	4.8%	24.4%	50.2%	84.2%	94.8%	100%
1.5	$R_{MF1}$	5.0%	16.4%	32.4%	64.4%	84.8%	90.4%
	$R_{MF2}$	4.8%	16.8%	32.0%	56.6%	82.6%	86.0%

7. Real data analysis

We are interested in finding whether there exists a change point in the hazard function, whether the cure model is proper and estimating the location of the change point and all other model parameters.

7.1. The liver cancer data

The proposed method is applied to the 2008–2013 Iowa liver cancer data set from the Surveillance, Epidemiology, and End Results cancer incidence public-use database (SEER). Observations with missing values for any of these variables are excluded from this study. People who were lost to follow-up after diagnosis or died immediately after diagnosis are also excluded. The final analytic data set includes 546 patients where a total of $19.6\mathrm{\%}$ of the observations are subjected to right censoring and the others belong to interval censoring. As discussed in Section 1, we calculate the survival month T based on the difference in the last contact date and the diagnosis date, then rounded down. We treat it as interval-censored data and apply the change-point model to analyze this data.

Table 5 records the 5-year relative survival probability for liver cancer, and we can see that with the development of the medical treatment, the improvement of 5-year relative survival probability is obvious. In the live cancer data, there are 107 patients still alive at the last visit time. And the 95% confidential interval of $\hat{p}$ is $(0.926,0.971)$. Hence, it is appropriate to admit the presence of long-term survivors. And our suggested model (4) is proper for the analysis of the data. Based on the change-point model (4), the MLE of the parameters are calculated to be $(\hat{p},\hat{\tau },\hat{\beta },\hat{\theta })=(0.954,55,0.0065,0.0256)$, which implies that the hazard function has a change point around 55 days with a jump of 0.0256.

Table 5. Liver cancer 5-year relative survival probability.

Year	1975	1980	1985	1990	1995	2000	2005	2010
5-year relative survival probability	3.0%	3.2%	7.0%	5.3%	5.7%	11.7%	16.8%	16.8%

Figure 1 shows two modified cumulative hazard functions. The broken line is generated by the estimation of the model (4). The irregular line is the result of NPMLE. In the figure, the increasing ratio of the irregular line changes around 50–60 which is the rough range of the change point. And the dot of the broken line is 55 which is the change point obtained by the model (4). Table 6 records the results of the estimations of each parameter. Since the asymptotic variance of $\hat{\mathit{\mu }}$ is often intractable, we resort to the bootstrap method, which is proposed by [4] to produce the standard deviation (SD) based on 1000 repetitions. The results in Table 6 imply the value of p at around 0.95, indicating a proportion of $5\mathrm{\%}$ for the long-term survivors. The change point τ is estimated as 55 months, with an estimated jump of about 0.0256 for the hazard rate. We fitted the data using the survival cure model (4). And the test procedure in Section 4 was applied to determine whether there is an abrupt change for the recurrence rate of the live cancer. We obtain that $R_{MF1}=265.502$ and $R_{MF2}=135.979$. Under the null hypothesis, the 95% quantiles of $R_{MF1}$ and $R_{MF2}$ are 254.478 and 70.339, respectively. Hence, there is overwhelming evidence to reject the null hypothesis ${\mathcal{H}}_0$ and conclude that a change point does exist for the data.

Figure 1.

Modified cumulative hazard function for liver cancer data where the broken line is the result of the estimation of change point model and the irregular line is the result of NPMLE.

Table 6. Estimates for liver cancer data.

$\hat{p}$	$\mathrm{SD}$	$\hat{\tau }$	$\mathrm{SD}$	$\hat{\beta }$	$\mathrm{SD}$	$\hat{\theta }$	$\mathrm{SD}$
0.954	0.0130	55.00	0.274	0.0065	0.000637	0.0256	0.00179

7.2. The breast cancer data

The breast cancer data set is described and given in [5]. This data set considers the information of time to cosmetic deterioration of the breast for women with Stage 1 breast cancer who have undergone a radiotherapy. The data come from a retrospective study of 94 patients who received radiation. Each woman made a series of visits to a clinician who determined whether or not the retraction had occurred. If the retraction had occurred, the time was known only to lie between the time of the present and last visits. Finally, a total of 40.6% of the observations were subjected to right censoring and 5% belongs to left censoring.

By the bootstrap procedure described in Section 5 and (8), the 95% confidential interval of $\hat{p}$ is $(0.866,0.901)$. Hence the cure model (4) is suitable to analyze the data. And our estimates for $(\tau ,\beta ,\theta ,p)$, together with their sample standard deviations, are shown in Table 7. We apply the test procedures in Section 4 to determine whether there is a change point. This shows the deviances $R_{MF1}=10.157$ and $R_{MF2}=5.878$. The 95% quantiles of $R_{MF1}$ and $R_{MF2}$ under the null hypothesis are 6.804 and 3.987 respectively. Hence, there is a strong evidence to reject the null hypothesis ${\mathcal{H}}_0$ and conclude that a change point does exist for the data.

Table 7. Estimates for breast cancer data.

$\hat{p}$	$\mathrm{SD}$	$\hat{\tau }$	$\mathrm{SD}$	$\hat{\beta }$	$\mathrm{SD}$	$\hat{\theta }$	$\mathrm{SD}$
0.882	0.0170	16.00	0.074	0.017	0.004	0.028	0.010

8. Discussion

In this paper, we develop the pseudo-maximum-likelihood method to handle the single change-point hazard model for interval-censored data with a cure fraction. To guarantee the consistency of the uncured rate estimate, we have applied the ICM method to calculate the NPMLE. Compared with [3], our approach possesses a smaller bias of the change size estimate. The simulation and two real data examples illustrate that the proposed method can effectively deal with interval-censored data with a cure fraction.

In our future work, the multiple change-point hazard model will be the key research object. There is some research work on this model, such as [8,20]; however, they did not consider interval-censored data.

Funding Statement

The research work of Wang is supported by the National Natural Sciences Foundation of China grant 11471065. The research work of Song is supported by the National Natural Sciences Foundation of China grant 11371077 and 61175041.

Disclosure statement

No potential conflict of interest was reported by the authors.