NONPARAMETRIC ESTIMATION AND TESTING FOR PANEL COUNT DATA WITH INFORMATIVE TERMINAL EVENT

Abstract

Informative terminal events often occur in the long term recurrent event follow-up studies. To reflect their effects on recurrent event processes explicitly, we propose a reversed nonparametric mean model for panel count data with a terminal event subject to right censoring. This model enjoys meaningful interpretation for applications and robustness for statistical inference. Treating the distribution of the right-censored terminal event time as a nuisance functional parameter, we develop a two-stage estimation procedure by combining the Kaplan-Meier estimator and nonparametric sieve estimation techniques. The consistency, convergence rate and asymptotic normality of the proposed nonparametric estimator are established. Then we construct a class of new statistics for two-sample test. The asymptotic properties of the new tests are established and evaluated by extensive simulation studies. Panel count data from Chinese Longitudinal Healthy Longevity study are analyzed using the proposed method for illustration.

1. Introduction

In longitudinal follow-up studies, recurrent events are the same type events that individuals may experience repeatedly. However, exact recurrent event times may not be observed in many practical applications. Instead, individuals can be examined at distinct time points, and the number of recurrent events between two observation time points are collected. Such data are socalled panel count data which frequently occur in medical follow-up studies and clinical trials. Many methods have been developed for the analysis of panel count data. For example, Sun and Kalbfeisch (1995), Wellner and Zhang (2000), Hu, Lagakos, and Lockhart (2009), and Lu, Zhang, and Huang (2007) studied nonparametric estimation of mean function of a underlying counting process with panel count data; Thall and Lachin (1988), Sun and Fang (2003), Zhang (2006), Balakrishnan and Zhao (2009), and Zhao and Zhang (2017) developed some nonparametric tests for comparison of mean functions of counting processes based on panel count data. Among others, Cheng and Wei (2000), Sun and Wei (2000), Hu, Sun, and Wei (2003), Sun, Tong, and He (2007), Wellner and Zhang (2007), Lu, Zhang, and Huang (2009), He et al. (2017), and Jiang, Su, and Zhao (2020) studied semiparametric regression models with panel count data, while Zhao et al. (2019) investigated a nonparametric regression model for panel count data analysis.

The recurrent events are often truncated by a terminal event such as death. A commonly used approach is a joint modeling approach through a shared frailty variable to deal with the correlation between the terminal event and recurrent event processes; see for example, Liu, Wolfe, and Huang (2004), Zhao, Zhou, and Sun (2011), Sun et al. (2012), Zhao, Li and Sun (2013a,b), and Zhou et al. (2017). However, these approaches do not exhibit an explicit effect of the terminal event on recurrent events, which is often the study of interest in scientific investigations. For example, Chan et al. (1995) found that the occurrences of AIDS-defending events were accumulated sharply before the death of the HIV-infected individuals; Lunney et al. (2003) claimed that the functional decline changed tremendously in the last year of the HIV-infected individuals. Directly modeling the effect of terminal event on recurrent events is a challenging task. Recently, Chan and Wang (2010) developed a backward model to study the behavior of recurrent events prior to a terminal event. Chan and Wang (2017) considered effects of covariates and extended the backward model to a semiparametric model. Kong et al. (2018) proposed a conditional model for longitudinal data by treating the terminal event time as a covariate. Our goal is to propose a reversed mean model for panel count data with an informative terminal event, and then to develop nonparametric inference procedures for comparing the mean functions of recurrent event processes that are likely truncated by a terminal event.

The main contributions of this work are fourfold. First, we propose a reversed nonparametric mean model for panel count data with a right-censored terminal event, where the nonparametric mean function is increasingly dependent on the terminal event time. Thus, the proposed model provides an intuitive interpretation of the effect of terminal event on recurrent event processes. Second, we develop a two-stage sieve-based nonparametric estimation procedure by treating the distribution function of the terminal event time as a nuisance functional parameter. Third, we establish the asymptotic properties of the proposed estimator by overcoming the challenges from both a nuisance functional parameter and a nonparametric estimator with a convergence rate slower than the standard rate $n^{-1/2}$. Fourth, we develop a class of nonparametric tests for comparison of mean functions of recurrent event processes with panel count data in the presence of informative terminal event.

The remainder of this paper is organized as follows. In Section 2, we present a reversed mean model anchoring at a terminal event and propose a two-stage nonparametric sieve-based estimation procedure. In Section 3, we establish the asymptotic properties of the proposed estimator. In Section 4, we propose a class of new test statistics for two-sample comparison and establish their asymptotic normality. In Section 5, we conduct simulation studies to demonstrate the finite-sample performance of the proposed methods. In Section 6, we apply the proposed methods to panel count data from Chinese Longitudinal Healthy Longevity study. Some concluding remarks are made in Section 7. All technical proofs are provided in the supplementary materials.

2. Model Setting and Estimation Procedure

Suppose that a counting process $\left\{N\right(t):0\le t\le \tau \}$ denotes the number of recurrent events occurring up to time $t$, where $\tau$ is a fixed time point. Let $\underset{\_}{T}=\left(T_1,T_2,\cdots ,T_K\right)$ be the observation times of $N\left(t\right)$, where $K$ represents the total number of observation times. Then the observed counting process is $\underset{\_}{N}=\left(N_1,N_2,\cdots ,N_K\right)=\left(N\left(T_1\right),N\left(T_2\right),\cdots ,N\left(T_K\right)\right)$. Let $U$ and $C$ be the terminal event time and the censoring time, respectively. The time for the last observation is denoted by $Y=U\wedge C$ and whether the time is the terminal even time is denoted by $\Delta =1_{\{U\le C\}}$. The observed data for subject $i$ consists of $X_i=\left(Y_i,{\Delta }_i,K_i,\underset{\_}{T_i},\underset{\_}{N_i}\right),i=1,\cdots ,n$, where $\underset{\_}{T_i}=\left(T_{i1},T_{i2},\cdots ,T_{i{K}_i}\right)$ and $\underset{\_}{N_i}=\left(N\left(T_{i1}\right),N\left(T_{i2}\right),\cdots ,N\left(T_{i{K}_i}\right)\right)$. Let $𝒳=\left\{X_i,i=1,\dots ,n\right\}$ denote the sample of such panel count data from $n$ subjects.

To investigate the effect of terminal event on the recurrent event process, motivated by Chan and Wang (2010, 2017) and Kong et al. (2018), we consider a counting process $\tilde{N}(t;U)$ denoting the event counts from time $t$ to the terminal event $U$, and propose a reversed nonparametric mean model anchoring at the terminal event:

$$E\left(\tilde{N}\left(t;U\right)∣U=u\right)=\Lambda \left(u-t\right),$$ (2.1)

where $\Lambda (\cdot )$ is an unknown nondecreasing function with $\Lambda \left(0\right)=0$ to ensure the identifiability of this model. In this paper, we propose to model the recurrent events anchoring at the terminal event time through (2.1). The advantage of this approach is to straightforwardly describe the behavior of the stochastic process from the terminal event while all the event counts can still be used for learning the model of the entire process. This model implies that $E\left(\tilde{N}\left(t_1;U\right)\mid U=u\right)-E\left(\tilde{N}\left(t_2;U\right)\mid U=u\right)=\Lambda \left(u-t_1\right)-\Lambda \left(u-t_2\right)$, where $0\le t_1\le t_2\le u$. Noting that $N\left(t_2\right)-N\left(t_1\right)=\tilde{N}\left(t_1;U\right)-\tilde{N}\left(t_2;U\right)$ and $N\left(0\right)=0$, we obtain $E\left(N\right(t)\mid U=u)=\Lambda \left(u\right)-\Lambda (u-t)$.

Let $F\left(u\right)$ denote the underlying distribution function of $U$. To make a valid inference for our proposed model, we consider the following data scenarios: (i) $U$ and $C$ are independent; (ii) The censoring event time $C$ is noninformative to $\Lambda$. (iii) Given $(Y,\Delta )$, the distribution of $(K,\underset{\_}{T})$ is non-informative to $\Lambda$. Define $△N_j=N\left(T_j\right)-N\left(T_{j-1}\right)$ and $△{\Lambda }_j\left(u\right)=\Lambda \left(u-T_{j-1}\right)-\Lambda \left(u-T_j\right)$ for $j=1,\cdots ,K$ with $T_0=0$. A straightforward calculation yields that

$$\begin{gathered} E\left[\sum _{j=1}^K{\left\{△N_j-\Delta {\Lambda }_j\left(U\right)\right\}}^2\mid Y,\Delta ,K,\underset{\_}{T},\underset{\_}{N}\right] \\ =\sum _{j=1}^K\Delta {\left\{\Delta N_j-\Delta {\Lambda }_j\left(Y\right)\right\}}^2+\sum _{j=1}^K\left(1-\Delta \right)\frac{{\int }_Y^{\infty }{\left\{△N_j-\Delta {\Lambda }_j\left(u\right)\right\}}^{2}dF\left(u\right)}{1-F\left(Y\right)}, \end{gathered}$$

hence we propose a least-squares based loss function

$$\begin{gathered} {\ell }_n(\Lambda ,F;𝒳)=\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^{K_i}\left[{\Delta }_i{\left\{\Delta N_{i,j}-\Delta {\Lambda }_{i,j}\left(Y_i\right)\right\}}^2\right. \\ \left.+\left(1-{\Delta }_i\right)\frac{{\int }_{Y_i}^{\infty }{\left\{\Delta N_{i,j}-\Delta {\Lambda }_{i,j}\left(u\right)\right\}}^{2}dF\left(u\right)}{1-F\left(Y_i\right)}\right], \end{gathered}$$ (2.2)

where $△N_{i,j}=N_i\left(T_{ij}\right)-N_i\left(T_{i(j-1)}\right)$ and $△{\Lambda }_{i,j}\left(u\right)=\Lambda \left(u-T_{i(j-1)}\right)-\Lambda (u-\left.T_{ij}\right)$. A natural idea is to take the minimizer of $l_n(\Lambda ,F;𝒳)$ defined in (2.2) as the estimator of the parameter. However, since the loss function involves an unknown distribution function $F$, it is difficult to estimate $\Lambda$ and $F$ simultaneously. To tackle the problem, we propose a two-stage approach. Specifically, in Stage 1, we estimate $F$ by using the Kaplan-Meier (KM) estimator ${\hat{F}}_n\left(u\right)$ Kaplan and Meier, 1958). In Stage 2, ${\hat{\Lambda }}_n$ is obtained by minimizing the loss function ${\ell }_n\left(\Lambda ,{\hat{F}}_n;𝒳\right)$ with respect to $\Lambda$. Since the observed data of $(Y,\Delta )$ are used in both Stages 1 and 2, to distinguish them, we use the notation $(\tilde{Y},\tilde{\Delta })$ to represent the data when we obtain the KM estimator in Stage 1 without any ambiguity.

We adopt the spline-based sieve estimation method to estimate the nondecreasing function $\Lambda$ because of its numerical advantage and good statistical properties (Lu, Zhang and Huang, 2007, 2009). Let $\left\{t_i:i=\right.\left.1,\cdots ,m_n+2d\right\}$ be a sequence of knots that partition $[0,\tau ]$ into $m_n+1$ subintervals, where

$$0=t_1=\cdots =t_d<t_{d+1}<\cdots <t_{m_n+d}<t_{m_n+d+1}=\cdots =t_{m_n+2d}=\tau .$$

Let $q_n=m_n+d$ and $\left\{I_l\left(s\right),l=1,\cdots ,q_n\right\}$ be the I-spline basis functions of order $d$ (Ramsay, 1988). We then define the functional space of the estimator for $\Lambda$ to be

$${\Phi }_n=\left\{\sum _{l=1}^{q_n}{\alpha }_l{I}_l\left(s\right):{\alpha }_l\ge 0,l=1,\cdots ,q_n\right\}.$$

Since ${\hat{F}}_n$ is a monotone step function, as shown in Section S1 in the online Supplementary Material, minimizing the loss function $l_n\left(\Lambda ,{\hat{F}}_n;𝒳\right)$ is a quadratic programming with the constraint that ${\alpha }_l\ge 0$ for $l=1,\cdots ,q_n$. Let $\mathbf{I}\left(s\right)={\left(I_1\left(s\right),\cdots ,I_{q_n}\left(s\right)\right)}^T,\mathit{\alpha }={\left({\alpha }_1,\cdots ,{\alpha }_{q_n}\right)}^T$, and the solution of quadratic programming be $\stackrel{\mathit{ˆ}}{\mathit{\alpha }}={\left({\hat{\alpha }}_1,\cdots ,{\hat{\alpha }}_{q_n}\right)}^T$. Then the spline estimator of $\Lambda \left(s\right)$ is ${\hat{\Lambda }}_n\left(s\right)=\mathbf{I}(s{)}^T\stackrel{\mathit{ˆ}}{\mathit{\alpha }}$.

3. Asymptotic Properties

To present the asymptotic results, we introduce some notation. Let $g^{\left(r\right)}$ be the $r$th derivative of function $g$. For $r\ge 1$, define

$$\begin{gathered} {\mathscr{H}}_r=\left\{g:\left|g^{(r-1)}\left(s\right)-g^{(r-1)}\left(t\right)\right|\le c_0|s-t|\text{for}\text{all}0\le s,t\le \tau \right\} \\ \Phi =\left\{\Lambda \in {\mathscr{H}}_r:\Lambda \right.\text{is}\text{a}\text{nondecreasing}\text{continuous}\text{function} \\ \text{on}[0,\tau ]\text{with}\Lambda \left(0\right)=0\}, \\ ℱ= \\ \left\{F:F\text{is}\text{a}\text{distribution}\text{function}\text{on}\left[0,\infty \right)\right\}. \end{gathered}$$

Denote the true value of $(\Lambda ,F)$ to be $\left({\Lambda }_0,F_0\right)\in \Phi \times ℱ$. For

$$B_1,B_2\in {\mathcal{B}}_{\left[0,\tau \right]}=:\left\{B\cap \left[0,\tau \right]:B\in \mathcal{B}\right\},$$

where $\mathcal{B}$ denotes the collection of Borel sets, set

$$\begin{gathered} {\mu }_1\left(B_1\times B_2\right)=\int \sum _{k=1}^{\infty }P(K=k\mid U=u) \\ \times \sum _{j=1}^{k}P\left(\left(u-T_j\right)\in B_1,\left(u-T_{j-1}\right)\in B_2\mid K=k,U=u\right)d{F}_0\left(u\right), \\ {\mu }_2\left(B_1\times B_2\right)=\int \sum _{k=1}^{\infty }P(K=k\mid U=u) \\ \times P\left(\left(u-T_K\right)\in B_1,u\in B_2\mid K=k,U=u\right)d{F}_0\left(u\right). \end{gathered}$$ (3.3)

Then ${\mu }_1$ and ${\mu }_2$ are two measures on $\left([0,\tau {]}^2,{\mathcal{B}}_{[0,\tau ]}^2\right)$. For any functions ${\Lambda }_1,{\Lambda }_2\in \Phi$, we define the metric as

$$\begin{gathered} d_1{\left({\Lambda }_1,{\Lambda }_2\right)}^2={∥\Delta {\Lambda }_1\left(s_1,s_2\right)-\Delta {\Lambda }_2\left(s_1,s_2\right)∥}_{L_2\left({\mu }_1\right)}^2 \\ =E\left[\sum _{j=1}^K{\left(\Delta {\Lambda }_{1,j}\left(U\right)-\Delta {\Lambda }_{2,j}\left(U\right)\right)}^2\right] \\ =E\left[\sum _{j=1}^K\left\{\Delta {\left(\Delta {\Lambda }_{1,j}\left(Y\right)-\Delta {\Lambda }_{2,j}\left(Y\right)\right)}^2\right.\right. \\ +\left.\left.(1-\Delta )\frac{{\int }_Y^{\infty }{\left(\Delta {\Lambda }_{1,j}\left(u\right)-\Delta {\Lambda }_{2,j}\left(u\right)\right)}^{2}d{F}_0\left(u\right)}{1-F_0\left(Y\right)}\right\}\right], \end{gathered}$$

where $△\Lambda \left(s_1,s_2\right)=\Lambda \left(s_2\right)-\Lambda \left(s_1\right)$. For any $F_1,F_2\in ℱ$, we define the metric as

$$d_2\left(F_1,F_2\right)={∥F_1-F_2∥}_{\infty },$$

where $\parallel \cdot {\parallel }_{\infty }$ represents the $L_{\infty }$ norm. Denote the $\delta$-neighbourhood of $F_0$ by ${ℱ}_{\delta }=\left\{F\in ℱ:d_2\left(F,F_0\right)\le \delta \right\}$ for any small $\delta >0$.

To establish the asymptotic properties of the proposed estimator, we assume:

• (C1) $0<{\Lambda }_0\left(\tau \right)<\infty .$

• (C2) $0<F_0\left(\tau \right)<1.$ $F_0$ is absolutely continuous with respect to Lebesgue measure. Moreover, the density function $f_0\left(s\right)$ has a uniform positive lower bound for all $s\in \left[M_1,\tau \right]$, where $M_1$ is a constant representing the minimum value of the support of $F_0$.

• (C3) $E\left[{\sum }_{j=1}^K{\left\{△N_j-△{\Lambda }_{0,j}\left(U\right)\right\}}^2\right]<\infty$.

• (C4) The probability of censoring $\varrho =P(Y<U)$ satisfies that $0<\varrho <$ 1.

• (C5) $m_n=O\left(n^{\nu }\right)$ for $0<\nu <1/2$. Moreover, we suppose that

  $$\underset{d+1\le i\le m_n+d+1}{\text{max}}\left|t_i-t_{i-1}\right|=O\left(n^{-\nu }\right),$$

  and there is a constant $M_2>0$ such that

  $$\frac{{\text{max}}_{d+1\le i\le m_n+d+1}\left|t_i-t_{i-1}\right|}{{\text{min}}_{d+1\le i\le m_n+d+1}\left|t_i-t_{i-1}\right|}\le M_2$$

  uniformly for $n$.

• (C6) There is a constant $M_3>0$ such that $P\left(K\le M_3\right)=1$.

• (C7) $P\left(T_j-T_{j-1}\ge M_4\right.$ for all $\left.j=1,\cdots ,K\right)=1$ with some constant $M_4>0$.

Remark 1. Condition (C1) is standard in the literature of nonparametric estimation. Condition (C2) generally holds for the cumulative distribution function of a continuous random variable in view of applications. Condition (C3) requires that ${\sum }_{j=1}^K△N_j$ has finite second order central moment. Condition (C4) ensures the censoring rate between 0 and 1, which is commonly assumed in practice of survival data analysis. Condition (C5) is a regularity condition for monotone spline estimation, see Lu, Zhang, and Huang (2007, 2009). Condition (C6) is similar to Condition (C2) in Wellner and Zhang (2007), which indicates that the number of observations is bounded. Condition (C7) requires that the adjacent observation times are separable (Wellner and Zhang, 2007), which is generally the case in applications.

Theorem 1 (Consistency for the Two-Stage Estimator). Suppose that Conditions (C1)-(C7) hold. Then, for every $0\le b_1\le b_2\le \tau$ satisfying ${\mu }_2\left(\left[0,b_1\right]\times \left[b_2,\tau \right]\right)>0$, we have

$${∥△{\hat{\Lambda }}_n\left(s_1,s_2\right)1_{\left\{\left(s_1,s_2\right)\in \left[b_1,b_2\right]\times \left[b_1,b_2\right]\right\}}-{\Lambda }_0\left(s_1,s_2\right)1_{\left\{\left(s_1,s_2\right)\in \left[b_1,b_2\right]\times \left[b_1,b_2\right]\right\}}∥}_{L_2\left({\mu }_1\right)}^2=o_p\left(1\right)$$

In particular, if ${\mu }_2\left(\right\{0\}\times \{\tau \left\}\right)>0$, then $d_1\left({\hat{\Lambda }}_n,{\Lambda }_0\right)=o_p\left(1\right)$.

To establish the rate of convergence and the asymptotic normality, we further assume:

• (C8) ${\mu }_1$ is absolutely continuous with respect to Lebesgue measure with a derivative ${\dot{\mu }}_1$, and ${\dot{\mu }}_1$ has a uniform positive lower bound.

• (C9) There is a constant $0<{\lambda }_a<\infty$ such that $1/{\lambda }_a<{\Lambda }_0^{\prime }\left(s\right)<{\lambda }_a$ for all $s\in \left[{\tau }^{\prime },\tau \right]$ with $0<{\tau }^{\prime }\le \tau$ such that ${\Lambda }_0\left({\tau }^{\prime }\right)>0$.

• (C10) $P(U\ge \tau )={\omega }_1>0$ and $P(C\ge \tau )={\omega }_2>0$.

• (C11) $E\left(e^{cN\left(t\right)}\right)$ is uniformly bounded for $t\in [0,\tau ]$ and some constant $c$.

Remark 2. Condition (C8) implies that the metric ${\mu }_1$ defined in (3.3) has a strictly positive intensity. Condition (C9) requires the true conditional mean function be absolutely continuous with bounded intensity function, which is reasonable as explained in Wellner and Zhang (2007). (C10) is used as a technical condition in the proof of the uniform weak convergence rate of the KM estimator according to Kong et al. (2018). Condition (C11) holds when $N\left(t\right)$ is from a Poisson-type process or is uniformly bounded conditional on terminal event time, which is also generally true in view of applications.

Theorem 2 (Rate of Convergence). Suppose that Conditions (C1)-(C11) hold and ${\mu }_2\left(\right\{0\}\times \{\tau \left\}\right)>0$. Taking $\nu =1/(1+2r)$, we have $d_1\left({\hat{\Lambda }}_n,{\Lambda }_0\right)=O_p\left(n^{-r/(1+2r)}\right)$.

To present the asymptotic normality, we introduce more notation. Let $𝒫$ and ${\mathbb{P}}_n$ denote probability measure and empirical measure with $𝒫f=\int fd𝒫$ and ${\mathbb{P}}_{n}f=n^{-1}{\sum }_{i=1}^{n}f\left(X_i\right)$, respectively. Write ${\ell }_n(\Lambda ,F;𝒳)={\mathbb{P}}_{n}m(\Lambda ,F;X)$. Let $l^{\infty }\left({\mathscr{H}}_r\right)$ be the space of bounded functionals on ${\mathscr{H}}_r$ under the $L_{\infty }$ norm. For $h\in {\mathscr{H}}_r$, define a sequence of maps $Q_n$ of a neighborhood of $\left({\Lambda }_0,F_0\right)$, denoted by $𝒰$, in the parameter space for $(\Lambda ,F)$ into $l^{\infty }\left({\mathscr{H}}_r\right)$ as the derivative of ${\ell }_n(\Lambda ,F;𝒳)$ with respect to $\Lambda$ in the direction $h$:

$$\begin{gathered} Q_n(\Lambda ,F)\left[h\right]=\underset{\epsilon \to 0}{\text{lim}}\frac{{\ell }_n(\Lambda +\epsilon h,F;𝒳)-l_n(\Lambda ,F;𝒳)}{\epsilon } \\ ={\mathbb{P}}_n\underset{\epsilon \to 0}{\text{lim}}\frac{m(\Lambda +\epsilon h,F;X)-m(\Lambda ,F;X)}{\epsilon } \\ ={\mathbb{P}}_n\psi (\Lambda ,F;X)\left[h\right] \end{gathered}$$

and define $Q(\Lambda ,F)\left[h\right]=𝒫\psi (\Lambda ,F;X)\left[h\right]$, where

$$\begin{gathered} \psi (\Lambda ,F;X)\left[h\right]=\sum _{j=1}^K\left[\Delta \left\{\Delta N_j-\Delta {\Lambda }_j\left(Y\right)\right\}△h_j\left(Y\right)\right. \\ \left.+(1-\Delta )\frac{{\int }_Y^{\infty }\left\{\Delta N_j-△{\Lambda }_j\left(u\right)\right\}△h_j\left(u\right)dF\left(u\right)}{1-F\left(Y\right)}\right] \end{gathered}$$

with $△h_j\left(u\right)=h\left(u-T_{j-1}\right)-h\left(u-T_j\right),j=1,\dots ,K$. Furthermore, define

$$\begin{gathered} {\dot{Q}}_{{\Lambda }_0,{\hat{F}}_n}^{\left(1\right)}\left({\hat{\Lambda }}_n-{\Lambda }_0\right)\left[h\right]=\underset{\epsilon \to 0}{\text{lim}}\frac{Q\left({\Lambda }_0+\epsilon \left({\hat{\Lambda }}_n-{\Lambda }_0\right),{\hat{F}}_n\right)\left[h\right]-Q\left({\Lambda }_0,{\hat{F}}_n\right)\left[h\right]}{\epsilon } \\ =-{𝒫}_{\varsigma }\left({\hat{\Lambda }}_n,{\hat{F}}_n;X\right)\left[h\right] \end{gathered}$$

with

$$\begin{gathered} \varsigma (\Lambda ,F;X)\left[h\right]=\sum _{j=1}^K\left[\Delta \left\{△{\Lambda }_j\left(Y\right)-△{\Lambda }_{0,j}\left(Y\right)\right\}△h_j\left(Y\right)\right. \\ \left.+(1-\Delta )\frac{{\int }_Y^{\infty }\left\{△{\Lambda }_j\left(u\right)-△{\Lambda }_{0,j}\left(u\right)\right\}△h_j\left(u\right)dF\left(u\right)}{1-F\left(Y\right)}\right], \end{gathered}$$

and

$$\begin{gathered} {\dot{Q}}_{{\Lambda }_0,F_0}^{\left(2\right)}\left({\hat{F}}_n-F_0\right)\left[h\right]=\underset{\epsilon \to 0}{\text{lim}}\frac{Q\left({\Lambda }_0,F_0+\epsilon \left({\hat{F}}_n-F_0\right)\right)\left[h\right]-Q\left({\Lambda }_0,F_0\right)\left[h\right]}{\epsilon } \\ =𝒫\left[{\int }_Y^{\infty }{\bar{\phi }}_{{\Lambda }_0,F_0}(u;X)\left[h\right]d\left({\hat{F}}_n-F_0\right)\left(u\right)\right], \end{gathered}$$

with

$$\begin{gathered} {\bar{\phi }}_{\Lambda ,F}(u;X)\left[h\right]=\frac{1-\Delta }{1-F\left(Y\right)}\sum _{j=1}^K\left\{\left(\Delta N_j-\Delta {\Lambda }_j\left(u\right)\right)\cdot \Delta h_j\left(u\right)\right. \\ \left.-\frac{{\int }_Y^{\infty }\left(\Delta N_j-\Delta {\Lambda }_j\left(s\right)\right)\cdot \Delta h_j\left(s\right)dF\left(s\right)}{1-F\left(Y\right)}\right\}. \end{gathered}$$

Theorem 3 (Asymptotic Normality). Suppose that Conditions (C1)-(C11) hold and ${\mu }_2\left(\right\{0\}\times \{\tau \left\}\right)>0$.

1. Then for any bounded function $h\in {\mathscr{H}}_r$, we have

   $$-\sqrt{n}{\dot{Q}}_{{\Lambda }_0,{\hat{F}}_n}^{\left(1\right)}\left({\hat{\Lambda }}_n-{\Lambda }_0\right)\left[h\right]=\sqrt{n}{\dot{Q}}_{{\Lambda }_0,F_0}^{\left(2\right)}\left({\hat{F}}_n-F_0\right)\left[h\right]+\sqrt{n}{Q}_n\left({\Lambda }_0,F_0\right)\left[h\right]+o_p\left(1\right).$$
2. Moreover, $\sqrt{n}{𝒫}_{\varsigma }\left({\hat{\Lambda }}_n,{\hat{F}}_n;X\right)\left[h\right]\stackrel{d}{\to }N\left(0,{\sigma }_0^2\right)$, where

   $${\sigma }_0^2=E\left[{\left\{𝒫\phi \left({\Lambda }_0,F_0;X;\tilde{Y},\tilde{\Delta }\right)\left[h\right]+\psi \left({\Lambda }_0,F_0;X\right)\left[h\right]\right\}}^2\right],$$

   and $\phi (\Lambda ,F;X;\tilde{Y},\tilde{\Delta })\left[h\right]$ is defined in the online Supplementary Material.

4. Two-Sample Test

Suppose that $n$ subjects are from two groups with corresponding sample sizes $n_1$ and $n_2$, where $n_1+n_2=n$. Denote the observed data of the $l$th group as $\left\{X_i^{\left(l\right)}:i=1,\cdots ,n_l\right\}=\left\{\left(Y_i^{\left(l\right)},{\Delta }_i^{\left(l\right)},K_i^{\left(l\right)},\underset{\_}{T_i^{\left(l\right)}},\underset{\_}{N_i^{\left(l\right)}}\right):i=\right.\left.1,\cdots ,n_l\right\}$, for $l=1,2$. Given the terminal event time $U^{\left(l\right)}=u$, the conditional mean function of ${\tilde{N}}^{\left(l\right)}\left(t\right)$ is ${\Lambda }_l(u-t)$.

4.1. Terminal Events with an Identical Distribution

In this subsection, we assume that terminal event times share the same distribution function $F_0$ for all the subjects so that its estimator ${\hat{F}}_n$ can be obtained based on the pooled data. We investigate two-sample test with the null hypothesis $H_0:{\Lambda }_1={\Lambda }_2={\Lambda }_0$. Denote ${\hat{\Lambda }}_l$ and ${\hat{\Lambda }}_n$ as the estimates of ${\Lambda }_l$ and ${\Lambda }_0$ based on the data set of group $l$ and the pooled data, respectively.

Theorem 4 (Two-Sample Test with an Identical Distribution of Terminal Events). In addition to the conditions in Theorem 3, we suppose that $h_n(\cdot )$ is a bounded weight process, and there is a bounded function $h\in {\mathscr{H}}_r$ such that

$$d_1^2\left(h_n,h\right)=E\left[\sum _{j=1}^K{\left\{△h_{n,j}\left(U\right)-△h_j\left(U\right)\right\}}^2\right]=o_p\left(n^{-1/(1+2r)}\right).$$

Assume that $n_1/n\to p$ as $n\to \infty$, where $0<p<1$. Then under $H_0:{\Lambda }_1={\Lambda }_2,U_n=\sqrt{n}{\mathbb{P}}_n\left(\varsigma \left({\hat{\Lambda }}_1,{\hat{F}}_n;X\right)\left[h_n\right]-\varsigma \left({\hat{\Lambda }}_2,{\hat{F}}_n;X\right)\left[h_n\right]\right)$ converges in distribution to $N\left(0,(1/p+1/(1-p\left)\right){\sigma }_0^2\right)$, where ${\sigma }_0^2=E\left[{\psi }^2\left({\Lambda }_0,F_0;X\right)\left[h\right]\right]$. Moreover, ${\sigma }_0^2$ can be consistently estimated by ${\hat{\sigma }}_n^2={\mathbb{P}}_n\left[{\psi }^2\left({\hat{\Lambda }}_n,{\hat{F}}_n;X\right)\left[h_n\right]\right]$.

Remark 5. Theorem 4 states the asymptotic normality of the new statistic $U_n$ and gives a consistent estimator of its asymptotic variance. Then the $h_n$-specific standardized statistic, $T_n\left(h_n\right)=U_n{\left\{{\hat{\sigma }}_n\sqrt{\frac{n}{n_1}+\frac{n}{n_2}}\right\}}^{-1}$ can be applied to conduct the two-sample hypothesis test. A natural choice is to take $h_n={\hat{\Lambda }}_n$. Other possible choices of the weight processes are explored and evaluated in the simulation studies. It is worth noting that the weight process is only required to be bounded, which is more flexible compared to the monotone condition in Zhang (2006) and Balakrishnan and Zhao (2009).

4.2. Terminal Events with Different Distributions

In this subsection, we assume the distribution and density functions of $U^{\left(l\right)},l=\mathrm{1,2}$ to be $F_l$ and $f_l$, which may be different for the two groups. Let ${\hat{F}}_l$ and ${\hat{\Lambda }}_l$ be the estimator of $F_l$ and ${\Lambda }_l$, respectively. Given a partition $0=t_0^{\left(l\right)}<t_1^{\left(l\right)}<\cdots <t_{{\nu }_{n_l}}^{\left(l\right)}=\tau$, we define the histogram-type estimators of $f_l$ as ${\hat{f}}_l\left(u\right)=\left({\hat{F}}_l\left(t_{i_l}^{\left(l\right)}\right)-{\hat{F}}_l\left(t_{i_l-1}^{\left(l\right)}\right)\right)/\left(t_i^{\left(l\right)}-t_{i-1}^{\left(l\right)}\right)$ for $t_{i_l-1}^{\left(l\right)}\le u<t_{i_l}^{\left(l\right)}$ following Földes, Rejtő, and Winter (1981). Set $f_T\left(u\right)=p_1{f}_1\left(u\right)+p_2{f}_2\left(u\right)$,

$$\begin{gathered} w_l\left(u-t_1\right)-w_l\left(u-t_2\right)=\left(h\left(u-t_1\right)-h\left(u-t_2\right)\right)f_T\left(u\right)/f_l\left(u\right), \\ {w}_n^{\left(l\right)}\left(u-t_1\right)-w_n^{\left(l\right)}\left(u-t_2\right)=\left(h_n\left(u-t_1\right)-h_n\left(u-t_2\right)\right)\left(\frac{n_l}{n}+\frac{n_r}{n}\frac{{\hat{f}}_r\left(u\right)}{{\hat{f}}_l\left(u\right)}\right), \end{gathered}$$

where $p_l={\text{lim}}_{n\to \infty }{n}_l/n$ for $l,r=\mathrm{1,2},l\ne r$.

Theorem 5 (Two-Sample Test with Different Distributions of Terminal Events). Suppose that the conditions in Theorem 4 hold for each group and $f_l$ ‘s are Lipschitz continuous. Then for ${\Lambda }_0\in {\mathscr{H}}_r,r\ge 2$, under the null hypothesis $H_0:{\Lambda }_1={\Lambda }_2={\Lambda }_0$ we have

1. ${\tilde{U}}_n=\frac{1}{\sqrt{n}}{\sum }_{l=1}^2{\sum }_{i=1}^{n_l}\left(\varsigma \left({\hat{\Lambda }}_1,{\hat{F}}_l;X_i^{\left(l\right)}\right)\left[h_n\right]-\varsigma \left({\hat{\Lambda }}_2,{\hat{F}}_l;X_i^{\left(l\right)}\right)\left[h_n\right]\right)$ converges in distribution to $N\left(0,\left({\sigma }_1^2/p_1+{\sigma }_2^2/p_2\right)\right)$, where

   $${\sigma }_l^2=E\left[{\left\{𝒫{\phi }_l\left({\Lambda }_0,F_l;X^{\left(l\right)};{\tilde{Y}}^{\left(l\right)},{\tilde{\Delta }}^{\left(l\right)}\right)\left[w_l\right]+\psi \left({\Lambda }_0,F_l;X^{\left(l\right)}\right)\left[w_l\right]\right\}}^2\right]$$

   with ${\phi }_l(\Lambda ,F;X;\tilde{Y},\tilde{\Delta })\left[w\right]$ defined in the online Supplementary Material.

2. In addition, suppose the knots of partition satisfy

   $$\underset{i=1,\cdots ,{{\nu }_{n_l}}^{\left(l\right)}}{\text{max}}\left\{\left|t_i^{\left(l\right)}-t_{i-1}^{\left(l\right)}\right|\right\}\to 0\text{and}{\left(\frac{n}{\text{log}n}\right)}^{1/4}\underset{i=1,\cdots ,{{\nu }_{n_l}}^{\left(l\right)}}{\text{min}}\left\{\left|t_i^{\left(l\right)}-t_{i-1}^{\left(l\right)}\right|\right\}\to \infty$$
   as $n\to \infty$. Then ${\sigma }_l^2$ can be consistently estimated by ${\hat{\sigma }}_l^2$, and the asymptotic variance of ${\tilde{U}}_n$ can be consistently estimated by ${\tilde{\sigma }}_n^2=n\left({\hat{\sigma }}_1^2/n_1+{\hat{\sigma }}_2^2/n_2\right)$, where

   $${\hat{\sigma }}_l^2={\mathbb{P}}_{n_l}\left[{\left\{{\mathbb{P}}_{n_l}{\phi }_n^{\left(l\right)}\left({\hat{\Lambda }}_l,{\hat{F}}_l;X^{\left(l\right)};{\tilde{Y}}^{\left(l\right)},{\tilde{\Delta }}^{\left(l\right)}\right)\left[w_n^{\left(l\right)}\right]+\psi \left({\hat{\Lambda }}_l,{\hat{F}}_l;X^{\left(l\right)}\right)\left[w_n^{\left(l\right)}\right]\right\}}^2\right]$$

   with ${\phi }_n^{\left(l\right)}(\Lambda ,F;X;\tilde{Y},\tilde{\Delta })\left[w\right]$ defined in the online Supplementary Material.

According to Theorem 5, we can apply the statistics ${\tilde{T}}_n\left(h_n\right)={\tilde{U}}_n/{\tilde{\sigma }}_n$ to test the equality of the mean function of counting processes under the scenario of different terminal event time distributions between the two samples.

5. Simulation Studies

5.1. Two-Stage Estimation for the Mean Function

We conducted simulation studies to evaluate the finite-sample performance of the two-stage estimator ${\hat{\Lambda }}_n$ with sample size $n=100$ and 200. For subject $i$, the observation $X_i=\left(Y_i,{\Delta }_i,K_i,\underset{\_}{T_i},\underset{\_}{N_i}\right)$ was generated as below. The latent terminal event time $U_i$ was from 6 + exp(1), and the censoring time $C_i$ was from $6+\kappa \text{exp}\left(3\right)$. The study ending time was set at $\tau =10$. The values of $\kappa$ were chosen to be 1.375 and 0.500 such that the terminal events were censored by 20% and 40%, respectively, in the study. Let $Y_i=U_i\wedge C_i$ and ${\Delta }_i=1_{\left\{U_i\le C_i\right\}}$. The number of observation $K_i$ was generated from the discrete uniform distribution in $\{1,\dots ,6\}$. Given the censored terminal event time $Y_i$, observation time $\underset{\_}{T_i}=\left(T_{i1},T_{i2},\dots ,T_{i{K}_i}\right)$ was an ordered sample from uniform distribution $\text{Unif}\left(0,Y_i\right)$. Let $N_i$ be a Poisson process with $E\left(N_i\left(t\right)\mid U=u\right)={\Lambda }_0\left(u\right)-{\Lambda }_0(u-t)$ and ${\Lambda }_0\left(s\right)=s$, which means that $N_i\left(T_{i1}\right)$ was from Poisson distribution with mean $T_{i1}$, and $N_i\left(T_{ij}\right)-N_i\left(T_{i(j-1)}\right)$ was from Poisson distribution with mean $T_{ij}-T_{i(j-1)}$. For the knots of spline, took $d=m_n=3$ and let $t_{d+1},t_{d+2},t_{d+3}$ be quartiles of $\left\{Y_i-T_{ij}:j=1,\dots ,K_i,i=1,\dots ,n\right\}$. All simulation studies were based on 1000 replications.

Figure 1 shows the results of two-stage estimators for the conditional mean functions. The solid line represents the true mean function, the dashed line represents the average of the estimated mean function, and the dotted-dash lines represent the 2.5% and 97.5% pointwise percentiles of the functional estimator based on 1000 replications. From (a)-(d) in Figure 1, we can conclude that the sample mean of the estimated mean functions is very close to the true one, the degree of the variability of the functional estimator decreases as the sample increases indicating the asymptotic unbiasedness of the proposed method, and the sampling distribution of the estimator does not appear to be much impacted by the censoring rate.

Figure 1:

Estimates of the mean function.

5.2. Two-Sample Test with an Identical Distribution of Terminal Event Time

We conducted the two-sample test under the scenario where the two groups have a same distribution of terminal event time. We generated two groups of independent and identically distributed sample $\left\{X_i^{\left(l\right)}:i=1,\cdots ,n_l\right\}$ by the same way as in Subsection 5.1 for $l=\mathrm{1,2}$, with sample size $n_1=n_2=100$, 150 or 200. Let $N^{\left(l\right)}$ be a Poisson process with $E\left(N^{\left(l\right)}\left(t\right)\mid U^{\left(l\right)}=u\right)={\Lambda }_l\left(u\right)-{\Lambda }_l(u-t)$ for the $l$th group. We considered the following two cases:

$$\text{Case}1:{\Lambda }_1\left(s\right)=s,{\Lambda }_2\left(s\right)=\beta s;$$

$$\text{Case}2:{\Lambda }_1\left(s\right)=s,{\Lambda }_2\left(s\right)=\frac{10s}{s+\beta }.$$

We took $\beta =\mathrm{1,1.1,1.2,1.3}$ in Case 1 and $\beta =\mathrm{1,3},5$ in Case 2.

Figure 2 plots the true mean functions for the two cases with different values of $\beta$. It can be seen that the conditional mean functions of two groups in Case 1 do not overlap but they cross over in Case 2. The weight processes $h_n^{\left(j\right)}\left(t\right),j=\mathrm{1,2},3$ in Theorem 4 were chosen to be

$$h_n^{\left(1\right)}\left(t\right)=t,h_n^{\left(2\right)}\left(t\right)=\frac{1}{n}\sum _{i=1}^n{1}_{\left(t\ge T_{i{K}_i}\right)},h_n^{\left(3\right)}\left(t\right)={\hat{\Lambda }}_n\left(t\right).$$

The results of the simulation study were evaluated with 1000 replications.

Figure 2:

Discrepancy of the mean functions

Figure 3 presents the quantile plots of the test statistics with the three weight processes against the standard normal distribution for Case 1 with $n_1=n_2=200,\beta =1$ and censoring rate 20%. They demonstrate that the normality of the test statistics is generally satisfactory with sample size of 200. Similar results were obtained for other situations and hence were not presented here.

Figure 3:

Q-Q plots for $n_1=n_2=200,\beta =1$, and censoring rate = 20% when the distribution functions of the terminal event time are identical for the two groups.

Tables 1 and 2 report the sizes and powers of the proposed test statistics $T_n\left(h_n^{\left(j\right)}\right)$ with the three weight processes at significance level 0.05 for different values of $\beta$ for Cases 1 and 2, respectively, where $T_n\left(h_n^{\left(j\right)}\right)$ represents the test statistic with the $j$th weight process for $j=\mathrm{1,2},3$. The simulation results show that the proposed test possesses good properties: (i) the estimated sizes are all around the significance level of 0.05; (ii) for any fixed value of $\beta$ in both cases, the power increases when sample size increases; (iii) the power increases to 1 when the discrepancy of the mean functions becomes more evident implying the consistency of the proposed test; (iv) the test powers also appear not to be much impacted by the censoring rate.

Table 1:

Simulation results of the two-sample tests with different weights for Case 1 when the distribution functions of the terminal event time are identical for the two groups.

	Censoring rate 20%			Censoring rate 40%
$\beta$	$T_n\left(h_n^{(1)}\right)$	$T_n\left(h_n^{(2)}\right)$	$T_n\left(h_n^{(3)}\right)$	$T_n\left(h_n^{(1)}\right)$	$T_n\left(h_n^{(2)}\right)$	$T_n\left(h_n^{(3)}\right)$
	n1=n2=100
1	0.051	0.047	0.052	0.052	0.048	0.051
1.1	0.242	0.229	0.243	0.258	0.240	0.255
1.2	0.728	0.703	0.728	0.715	0.690	0.717
1.3	0.967	0.956	0.967	0.962	0.956	0.960
	n1=n2=150
1	0.048	0.047	0.048	0.049	0.048	0.048
1.1	0.359	0.344	0.361	0.355	0.338	0.354
1.2	0.873	0.851	0.873	0.877	0.859	0.878
1.3	0.996	0.989	0.996	0.995	0.991	0.995
	n1=n2=200
1	0.042	0.044	0.043	0.043	0.051	0.043
1.1	0.438	0.398	0.443	0.467	0.433	0.467
1.2	0.944	0.924	0.943	0.954	0.939	0.954
1.3	1.000	1.000	1.000	0.999	0.998	0.999

Table 2:

Simulation results of the two-sample tests with different weights for Case 2 when the distribution functions of the terminal event time are identical for the two groups.

	Censoring rate 20%			Censoring rate 40%
$\beta$	$T_n\left(h_n^{(1)}\right)$	$T_n\left(h_n^{(2)}\right)$	$T_n\left(h_n^{(3)}\right)$	$T_n\left(h_n^{(1)}\right)$	$T_n\left(h_n^{(2)}\right)$	$T_n\left(h_n^{(3)}\right)$
	n1=n2=100
1	0.999	1.000	0.752	1.000	1.000	0.868
3	0.982	0.971	0.859	0.987	1.000	0.892
5	0.998	1.000	0.991	0.999	1.000	0.991
	n1=n2=150
1	1.000	1.000	0.900	1.000	1.000	0.960
3	0.998	1.000	0.949	0.998	1.000	0.970
5	1.000	1.000	0.998	1.000	1.000	0.999
	n1=n2=200
1	1.000	1.000	0.941	1.000	1.000	0.977
3	1.000	1.000	0.986	1.000	1.000	0.989
5	1.000	1.000	1.000	1.000	1.000	1.000

5.3. Two-Sample Test with Different Distributions of Terminal Event Times

We conducted a parallel simulation study to evaluate the proposed two-sample test for two groups with different distributions of terminal event time. The data sets were generated in a similar mechanism as in Subsection 5.2 except that the terminal event times $U_i^{\left(1\right)}$ and $U_i^{\left(2\right)}$ were from 6 + exp(1) and 6 + exp(2), and the censoring times $C_i^{\left(1\right)}$ and $C_i^{\left(2\right)}$ were from $6+{\kappa }_1\text{exp}\left(3\right)$ and $6+{\kappa }_2\text{exp}\left(3\right)$, where $\left({\kappa }_1,{\kappa }_2\right)=\left(\mathrm{1.375,5.769}\right)$ and (0.500, 1.010) to result in 20% and 40% censoring rate, respectively. The three weight processes $h_n^{\left(j\right)}\left(t\right),j=1,\cdots ,3$ were similarly chosen as

$$h_n^{\left(1\right)}\left(t\right)=t,h_n^{\left(2\right)}\left(t\right)=\frac{1}{n}\sum _{i=1}^n{1}_{\left(t\ge T_{i{K}_i}\right)},h_n^{\left(3\right)}\left(t\right)=\frac{{\hat{\Lambda }}_1\left(t\right)+{\hat{\Lambda }}_2\left(t\right)}{2}.$$

To calculate the histogram-type estimators of $f_l$, we evenly divided $[6,\tau ]$ into 5 subintervals. The results of the simulation study were again evaluated with 1000 replications.

Figure 4 presents the quantile plots of the test statistics with the three weight processes against the standard normal distribution with $n_1=n_2=200,\beta =1$ and censoring rate 20% for Case 1. The normality of the test statistics does appear to be justified when sample size is 200. The plots for other situations are similar and hence not presented here. Tables 3 and 4 summarize the sizes and powers of proposed statistics ${\tilde{T}}_n\left(h_n^{\left(j\right)}\right),j=\mathrm{1,2},3$ at significance level 0.05 for Cases 1 and 2, respectively, which results in the same conclusions as those in Subsection 5.2.

Figure 4:

Q-Q plots for $n_1=n_2=200,\beta =1$, and censoring rate = 20% when the distribution functions of the terminal event time are different for the two groups.

Table 3:

Simulation results of the two-sample tests with different weights for Case 1 when the distribution functions of the terminal event time are different for the two groups.

	Censoring rate 20%			Censoring rate 40%
$\beta$	${\tilde{T}}_n\left(h_n^{(1)}\right)$	${\tilde{T}}_n\left(h_n^{(2)}\right)$	${\tilde{T}}_n\left(h_n^{(3)}\right)$	${\tilde{T}}_n\left(h_n^{(1)}\right)$	${\tilde{T}}_n\left(h_n^{(2)}\right)$	${\tilde{T}}_n\left(h_n^{(3)}\right)$
	n1=n2=100
1	0.056	0.061	0.061	0.068	0.056	0.072
1.1	0.231	0.221	0.231	0.260	0.241	0.260
1.2	0.688	0.664	0.693	0.721	0.687	0.717
1.3	0.960	0.956	0.960	0.956	0.951	0.955
	n1=n2=150
1	0.043	0.049	0.041	0.056	0.052	0.054
1.1	0.338	0.311	0.341	0.352	0.330	0.348
1.2	0.863	0.835	0.865	0.872	0.851	0.869
1.3	0.993	0.992	0.994	0.994	0.985	0.992
	n1=n2=200
1	0.046	0.048	0.045	0.058	0.057	0.060
1.1	0.447	0.427	0.450	0.469	0.451	0.469
1.2	0.951	0.933	0.953	0.952	0.941	0.954
1.3	1.000	0.998	1.000	0.999	0.999	0.998

Table 4:

Simulation results of the two-sample tests with different weights for Case 2 when the distribution functions of the terminal event time are different for the two groups.

	Censoring rate 20%			Censoring rate 40%
$\beta$	${\tilde{T}}_n\left(h_n^{(1)}\right)$	${\tilde{T}}_n\left(h_n^{(2)}\right)$	${\tilde{T}}_n\left(h_n^{(3)}\right)$	${\tilde{T}}_n\left(h_n^{(1)}\right)$	${\tilde{T}}_n\left(h_n^{(2)}\right)$	${\tilde{T}}_n\left(h_n^{(3)}\right)$
	n1=n2=100
1	1.000	1.000	0.834	1.000	1.000	0.926
3	0.986	1.000	0.884	0.999	1.000	0.947
5	0.997	1.000	0.987	1.000	1.000	0.996
	n1=n2=150
1	1.000	1.000	0.940	1.000	1.000	0.985
3	0.998	1.000	0.976	0.999	1.000	1.000
5	1.000	1.000	0.999	1.000	1.000	0.999
	n1=n2=200
1	1.000	1.000	0.972	1.000	1.000	0.990
3	1.000	1.000	0.994	1.000	1.000	0.995
5	1.000	1.000	1.000	1.000	1.000	1.000

6. Real Data Analysis

In this section, we applied our method to analyzing the dataset from Chinese Longitudinal Healthy Longevity Survey (CLHLS) during the period from 1998 to 2014 (Zeng et al., 2017). This survey was conducted by the Center for Healthy Aging and Development Studies (CHADS) of National School of Development at Peking University and Chinese Center for Disease Control and Prevention (CDC) to study the determinants of healthy human longevity and the later-life mortality. The data consist of survey information in the seven waves at 1998, 2000, 2002, 2005, 2008, 2011 and 2014 by interviewing 9093 respondents who were 77 years or older in the 1998 baseline survey. In each wave, the respondents were asked to provide information regarding their health, socioeconomic status, family, lifestyle and demographic profile on a date randomly chosen in the interview year.

For this dataset, we analyzed panel count data of serious illness before death, for which the serious illness is defined as the illness making people bedridden, and compared the incidences of serious illness among elderly people living in different areas during the whole study period. The incidences of serious illness were counted repeatedly if the respondents were confined to bed again after recovering from the previous serious illness. We chose to analyze data for those subjects who had at least one follow-up observation after 1998 to allow for a meaningful analysis of panel count data. Therefore, we included 4362 elders for analysis by removing 3368 individuals, among which 2,005 dies in 2000 or between 1998 and 2000; 894 lost to follow-up in 2000, and 469 had missing or mistakenly documented records. Among the 4362 individuals, 1489 and 1561 lived in the urban and rural areas, respectively, all the time, and 1312 lived in both areas during this period. For the $i$th individual, we defined $T_{ij}$ as the time in month between the $j$ survey and the baseline survey, $N_i\left(t\right)$ as the accumulated number of serious illness that the individual suffered up to time $t$, and $K_i$ as the total number of follow-up surveys in the study. Death was the terminal event and loss to follow-up was treated as censoring event to death. In sum, the longest follow-up time was $\tau =200$ (months) and the censoring rate was 25.72%.

We studied the mean function of the serious illness process using model (2.1). We estimated the mean function using I-spline with order $d=3$ and $m_n=3$ to divide $[0,\tau ]$ into 4 subintervals with the internal knots of splines selected at $t_{d+1}=\tau /4,t_{d+2}=\tau /2,t_{d+3}=3\tau /4$. Figure 5 plots the estimated mean functions for the combined sample (solid line), urban sample (dash line), rural sample (dotted-dash line), and duo-area sample (longdash line), respectively. It reveals that the urban-area residents tended to experience more serious illnesses than the rural-area and duo-area residents, which may be the outcome that the rural-area and duo-area residents had limited access to advanced medical system compared to their counterparts in the urban area.

Figure 5:

The two-stage estimates of the mean function for the CLHLS data.

We then used the same three weight functions $h_n^{\left(j\right)},j=\mathrm{1,2},3$ as those in Subsection 5.2 to conduct the two-sample test for each pair of two mean functions for the urban sample, rural sample, and duo-area sample. That is the null hypotheses were $H_0^{\left(1\right)}:{\Lambda }_U\left(s\right)={\Lambda }_R\left(s\right),H_0^{\left(2\right)}:{\Lambda }_U\left(s\right)={\Lambda }_D\left(s\right)$, and $H_0^{\left(3\right)}:{\Lambda }_R\left(s\right)={\Lambda }_D\left(s\right)$, where ${\Lambda }_U,{\Lambda }_R$, and ${\Lambda }_D$ are the corresponding mean functions for the urban sample, rural sample, and duo-area sample, respectively. To conduct the test, we first concluded by the log-rank test that the survival function of death for the urban-area residents was significantly different from the rural-area residents and the duo-area residents with $p$-values $3\times {10}^{-6}$ and 0.006, respectively. Furthermore, the log-rank test revealed that the survival functions of death for the rural-area residents were not significantly different from the duo-area residents with $p$-value =0.09. Figure 6 displays the KM estimates for the survival functions of death for elders in the three groups. Therefore, we used the test statistic in Theorem 4 to compare the mean functions for the rural-area and duo-area residents. For the tests $H_0^{\left(1\right)}:{\Lambda }_U\left(s\right)={\Lambda }_R\left(s\right)$ and $H_0^{\left(2\right)}:{\Lambda }_U\left(s\right)={\Lambda }_D\left(s\right)$, we divided $[0,\tau ]$ evenly into 8 subintervals to obtain the histogram-type estimators, and then applied the test statistic in Theorem 5. The test results are summarized in Table 5 suggesting a significant difference in later-life serious illness experience at level 0.01 between the urban-area and the rural-area residents, and between the urban-area and the duo-area residents. Though there is no significant difference in the mean function between the duo-area and rural-area residents from a statistical perspective (p-value >0.05), the duoarea residents seemed to have fewer incidences of severe illness compared to rural-area residents. It may be due to the facts that: (i) many rural-area residents were transitioned to urban along with the urbanization movement in China in recent years; (ii) the elder duo-area residents tend to be in a better family socioeconomic status with better health status compared to other rural-area residents. The conclusion was consistent regardless of the choice of the weight processes.

Figure 6:

The KM estimates of the survival function for the CLHLS data.

Table 5:

Two-sample test results with the three weights for the CLHLS data.

	$h_n^{(1)}$	$h_n^{(2)}$	$h_n^{(3)}$
	$H_0^{(1)}:{\Lambda }_U(s)={\Lambda }_R(s)$
${\tilde{U}}_n$	287.904	3.968	4.325
${\tilde{\sigma }}_n$	77.849	1.138	1.182
${\tilde{T}}_n$	3.698	3.488	3.659
p-value	< 0.001***	< 0.001***	< 0.001***
	$H_0^{(2)}:{\Lambda }_U(s)={\Lambda }_D(s)$
${\tilde{U}}_n$	368.945	4.132	5.225
${\tilde{\sigma }}_n$	79.298	1.193	1.205
${\tilde{T}}_n$	4.653	3.465	4.335
p-value	< 0.001***	< 0.001***	< 0.001***
	$H_0^{(3)}:{\Lambda }_R(s)={\Lambda }_D(s)$
$U_n$	84.872	0.384	0.932
${\sigma }_n$	66.322	0.957	0.833
$T_n$	1.280	0.401	1.119
p-value	0.201	0.688	0.263

$T_n\left(h\right)$ (or $\left.{\tilde{T}}_n\left(h\right)\right)$: the observed value of the test statistic with different weight functions;

^***represents significance level of 0.01

7. Discussion

Considering the potential association between recurrent event and terminal event processes that often occur in life sciences, we have developed a conditional nonparametric mean functional model to study the explicit effects of the terminal events on the recurrent event process with panel count data. This approach is particularly useful in ascertaining the recurrent event behavior prior to the terminal event. However, the approach has challenges in both numerical computation and statistical inference due to the nature of panel count data subject to informative termination. We have designed a two-stage spline-based estimation procedure to ease the computation burden, and studied the asymptotic behavior of the proposed nonparametric estimator using modern empirical process theories. In particular, the asymptotic normality can be readily expanded to address multi-sample hypothesis testing problems. Our extensive simulation studies have demonstrated the good performance of the proposed method in finite-sample settings.

The proposed method can be extended to consider a conditional semiparametric mean function model:

$$E\left(\tilde{N}\left(t;U\right)∣U=u,Z\right)=\text{exp}\left({\mathit{\beta }}^T\mathbf{Z}\right)\Lambda \left(u-t\right),0\le t\le u,$$

where $\tilde{N}(t;u)=N\left(u\right)-N(u-t)$ and $N\left(t\right)$ is the number of recurrent events up to time $t,U$ is the terminal event time, $\mathbf{Z}$ is a covariate vector, $\mathit{\beta }$ is an unknown parameter, and $\Lambda (\cdot )$ is an unknown nondecreasing baseline function. This model has a broader application that not only captures the effect of the terminal event but also allows to assess external risk factors on recurrent event process, and is currently under our investigation.