Analyzing Interval-Censored Recurrence Event Data with Adjusting Informative Observation Times by Propensity Scores

Abstract

In this paper, we discuss the statistical inference of interval-censored recurrence event data under an informative observation process. We establish an additive semiparametric mean model for the recurrence event process. Since the observation process may contain relevant information about potential underlying recurrence event processes, which leads to confounding bias, therefore, we introduced a propensity score into the additive semiparametric mean model to adjust for confounding bias, which possibly exists. Furthermore, the estimation equations were used to estimate the parameters of the covariate effects, and the asymptotic normality of the estimator under large samples is proven. Through simulation studies, we illustrated that the proposed method works well, and it was applied to the analysis of bladder cancer data.

1. Introduction

When it comes to survival analysis, a certain survival phenomenon or event may be repeated two or more times. We often refer to such survival events as recurrence events, such as multiple asthma exacerbations or multiple recurrences of tumors or epilepsy. During practical research, we are limited to observing subjects at specific time intervals, allowing us to calculate the frequency of recurrence events between these intervals rather than the exact timing of each event. This type of data is commonly referred to as interval-censored recurrence event data or panel count data [1].

Numerous researchers have examined interval-censored recurrence event data. Kalbfleisch et al. [2] suggested a Markov model. Balakrishnan et al. [3] created nonparametric estimations, and Nielsen and Dean [4] examined an estimation equation for interval-censored recurrence event data. Two methods based on the mixed Poisson model have been proposed by Chen et al. [5]. One approach posits that various recurrence events are connected by multivariate lognormal random effects, and the alternative approach employs a marginal model strategy. However, these approaches assume that the recurrence event process is entirely independent of the observation process, or they are independent of each other given certain covariates. The process under observation frequently provides useful information about the underlying recurrence event process, known as the dependent observation process. These recurrence event data are commonly called recurrence event data with information or with dependent observation times. For cases where two processes may be related to each other, Sun et al. [6] suggested joint modeling techniques for modeling the connection between the recurrence event process and the observation process through the incorporation of random effects. Zhao et al. [7] proposed a robust estimation method for regression analysis of interval-censored recurrence event data with informative observation times, assuming that the observation process follows a more general rate function rather than a Poisson process. Wang et al. [8] developed a more general model for interval-censored recurrence event data and introduced a penalized composite quantile regression to obtain robust estimates of the model. Wang and Lin [9] presented a Bayesian semiparametric approach under a proportional mean model. Satter et al. [10] introduced an empirical likelihood method. The literature [11] showed a partial varying coefficient model for interval-censored count data with informative observation times, aimed at addressing nonlinear interaction effects among covariates. Li et al. [12] suggested a group of semiparametric transformation models suitable for analyzing multivariate interval-censored recurrence event data with informative observations. Although numerous literature works discuss the situation where two processes are correlated, few literature works consider the confounding bias that may arise from the dependent observation process. Therefore, we hope to extend the semiparametric model for a recurrence event proposed by Li et al. [12] and introduce a propensity score adjustment in the model to address potential confounding bias.

The propensity score represents the likelihood that an individual will receive a specific treatment or intervention, conditional on a set of covariates, and it is a crucial tool for controlling for confounding variables in observational research. The utilization of the propensity score is prevalent in the medical industry [13,14,15,16,17]. Inverse probability of treatment weighting (IPTW) is one of the most important methods to use the propensity score to deal with confounding bias. Xie and Liu [18] introduced a modified survival function estimator called AKME, utilizing inverse probability weighting to minimize confounding bias, and suggested a weighted log-rank test for comparing survival function disparities. Yet, it overlooked the variability in the sampling of the propensity score. As an extension of this approach, Sugihara et al. [19] studied the generalized propensity score (GPS) for multiple treatments and developed an exact formula for the IPTW log-rank test. In addition, the Cox proportional hazard model is commonly utilized for estimating the hazards ratio, and its accuracy requires the proper specification of the propensity score model. Shu et al. [20] proposed a weighted estimation method based on empirical likelihood theory to prevent the misspecification of the model. The literature [21] demonstrates through a series of extensive Monte Carlo studies that the inverse probability weighting method based on propensity scores can estimate marginal hazards ratios with minimal bias, and compared to the propensity score matching method, the treatment effect estimates derived from inverse probability weighting have a lower mean-squared error. Wang et al. [22] analyzed the impact of different treatments on pancreatic cancer survival rates using Kaplan–Meier estimates and Cox proportional hazards models with inverse probability weighting based on propensity scores. One of the most frequently utilized techniques for addressing time-varying confounding variables is also inverse probability weighting. Linden and Adems [23] describe an example of the application of the inverse probability weighting of the propensity score in longitudinal data. Despite the widespread use of the propensity score for addressing confounding bias in survival analysis, there is a lack of literature on its application in recurrence events. This paper aims to incorporate the propensity score-based inverse probability weighting method into interval-censored recurrence event data to address potential confounding bias resulting from the dependent observation process.

We will introduce some notations and assumptions in Section 2. In Section 3, we introduce the model. The recurrence event process is modeled using the additive semiparametric mean model, with the propensity score introduced to adjust for potential confounding bias caused by the dependent observation process. The covariate effect is estimated by the estimation equation. Section 4 demonstrates the parameter estimates’ asymptotic normality under large samples, and Section 5 conducts a set of numerical simulation studies to validate the method’s rationality. The proposed method is applied to bladder cancer data in Section 6. Section 7 is the summary of this paper.

2. Statistical Models

We assume that n subjects experience the recurrence event of interest. Define $V_i\left(t\right)$ to be the recurrence event process that represents the cumulative count of such events observed up to time t and only observed at specific discrete points $0<t_{i,1}<\cdots <t_{i,m_i}$; here, $m_i$ represents the total observation number for subject i. Let $H_i\left(t\right)={\displaystyle \sum _{j=1}^{m_i}}I(t_{ij}<t)$ be the cumulative counting process that denotes the number of observations of the event on subject i by time t, called a potential observation process. We assume $X_i\left(t\right)$ is a covariate process, $C_i$ denotes the follow-up or censoring time for subject i, and indicator function $I_i\left(t\right)=I(t\le C_i)$ shows whether subject i is censored or not at time t. In practice, $V_i\left(t\right)$ is observed at $t_{i,j}$ only if $t_{i,j}<C_i<\tau$, where $\tau$ denotes the longest follow-up time. Define $H_i^{*}\left(t\right)=H_i(t\wedge C_i)$ as the actual observation process for subject i and $m_i^{*}\left(t\right)=H_i\left(C_i\right)=H_i^{*}\left(C_i\right)$ as the total number of actual observations on subject i, then we have $H_i^{*}\left(t\right)={\int }_0^t{I}_i\left(s\right)dH\left(s\right)$.

Following [6], we can assume that the observation process is a nonhomogeneous Poisson process with

$$E\left[d{H}_i\left(t\right)\mid X_i\left(t\right)\right]=e^{{\delta }^{\prime }{X}_i\left(t\right)}d{\varphi }_0\left(t\right),$$

(1)

$i=1,\dots ,n$, where $\delta$ is a vector of a regression parameters and ${\varphi }_0\left(t\right)$ denotes the mean of the cumulative number of observations up to time t.

For subject i, Let ${\mathcal{F}}_{it}=\left\{H_i\left(s\right),0\le s<t\right\}$ denote the history or filtration of the observation process; we define the function $h\left({\mathcal{F}}_{it}\right)=H_i(t-)$ or $h\left({\mathcal{F}}_{it}\right)=H_i(t-)-H_i(t-\rho ),\rho >0$. The choice of $h\left({\mathcal{F}}_{it}\right)$ depends on the relationship between the observation process and the recurrence event process. The former choice assumes that the recurrence event is influenced by the entire observation process prior to time t, while the latter choice assumes that the recurrence event is influenced by the observation process during the time interval $[t-\rho ,t)$. The former scenarios might occur where patients, feeling worse than usual regardless of receiving treatment, could visit their doctors or clinics more frequently. The latter typically assumes that the recent observations may carry relevant information about the recurrence rate [1]. In order to reduce the problem of confounding bias arising from the dependent observation, we introduce the propensity score:

$$E_{h\left({\mathcal{F}}_{it}\right)}=E\left\{h\left({\mathcal{F}}_{it}\right)\mid X_i\left(s\right),\delta ,{\varphi }_0,0\le s<t\right\}$$

(2)

defined by the conditional expectation of $h\left({\mathcal{F}}_{it}\right)$. Furthermore, Given ${\mathcal{F}}_{it}$ and $X_i\left(t\right)$, we extend the semiparametric model proposed by Li et al. [12], assuming that $V_i\left(t\right)$ follows the semiparametric mean model with an inverse probability weighting by propensity score:

$$E\left\{V_i\left(t\right)\mid X_i\left(t\right),{\mathcal{F}}_{it}\right\}={\psi }_0\left(t\right)+{\gamma }^{\prime }{X}_i\left(t\right)+{\alpha }^{\prime }\frac{h\left({\mathcal{F}}_{it}\right)}{E_{h\left({\mathcal{F}}_{it}\right)}}$$

(3)

where ${\psi }_0\left(t\right)$ is an unspecified fundamental mean function and $\gamma$ and $\alpha$ are vectors of regression parameters of interest to be estimated. We assume that, given $X_i\left(t\right)$, $C_i\left(t\right)$ is independent of $H_i\left(t\right)$ and $C_i\left(t\right)$ is also independent of $V_i\left(t\right)$. Also, we assume that, given $X_i\left(t\right)$ and ${\mathcal{F}}_i\left(t\right)$, $H_i\left(t\right)$ and $V_i\left(t\right)$ are mutually independent.

3. Inference Procedures

In this section, we estimate the parameters in the proposed models. Let ${\gamma }_0$, ${\alpha }_0$, and ${\delta }_0$ denote the true values of $\gamma$, $\alpha$, and $\delta$, respectively. Define $B_i(t;\delta )={\left(X_i^{\prime }\left(t\right),\frac{h{\left({\mathcal{F}}_{it}\right)}^{\prime }}{E_{h\left({\mathcal{F}}_{it}\right)}}\right)}^{\prime }$$\theta ={\left({\gamma }^{\prime },{\alpha }^{\prime }\right)}^{\prime }$, ${\theta }_0={\left({\gamma }_0^{\prime },{\alpha }_0^{\prime }\right)}^{\prime }$.

First, for the estimation of $\delta$ and ${\varphi }_0$, we define

$$\begin{array}{c}{M}_i^1\left(t\right)={\int }_0^t{I}_i\left(s\right)\left[d{H}_i\left(s\right)-e^{{\delta }^{\prime }{X}_i\left(s\right)}d{\varphi }_0\left(s\right)\right],\\ U_1\left(\delta \right)=\sum _{i=1}^n{\int }_0^{\tau }{X}_i\left(t\right)d{M}_i^1(t;\delta ).\end{array}$$

It is evident that $M_i^1\left(t\right)$ is a stochastic process with a mean of zero and $E\left\{U_1\left({\delta }_0\right)\right\}=0$. We also define

$$L_k(t;\delta )=n^{-1}\sum _{i=1}^n{I}_i\left(t\right)X_i^k\left(t\right)e^{{\delta }^{\prime }{X}_i\left(t\right)},$$

k = 0, 1. Then, we can obtain estimates of $\delta$ and ${\varphi }_0$ by

$$\sum _{i=1}^{n}d{M}_i^1(t;\delta )=0$$

and

$$U_1\left(\delta \right)=0.$$

Thus, estimate $\widehat{\delta }$ has the closed form, which can be obtained by [24,25]

$$\sum _{i=1}^n{\int }_0^{\tau }\left(X_i\left(t\right)-{\overline{X}}_i\left(t\right)\right)I_i\left(t\right)d{H}_i\left(t\right)=0,$$

where $\overline{X}=\frac{L_1(t;\delta )}{L_0(t;\delta )}$. Furthermore, we have

$${\widehat{\varphi }}_0(t;\delta )={\displaystyle \sum _{i=1}^n}{\int }_0^t\frac{I_i\left(s\right)d{H}_i\left(s\right)}{{\displaystyle \sum _{j=1}^n}{I}_j\left(s\right)e^{{\delta }^{\prime }{X}_j\left(s\right)}}={\displaystyle \sum _{i=1}^n}{\int }_0^t\frac{I_i\left(s\right)d{H}_i\left(s\right)}{n{L}_0\left(s\right)}.$$

(4)

By substituting ${\widehat{\varphi }}_0(t;\widehat{\delta })$ and $\widehat{\delta }$ into model (2), the corresponding estimate of $E_{h\left({\mathcal{F}}_{it}\right)}$ can be obtained by

$${\widehat{E}}_{h\left({\mathcal{F}}_{it}\right)}={\int }_0^{t}h\left(e^{{\widehat{\delta }}^{\prime }{X}_i\left(s\right)}d{\widehat{\varphi }}_0(s;\widehat{\delta })\right).$$

Next, we hope to obtain the estimate of parameters $\theta ={\left({\gamma }^{\prime },{\alpha }^{\prime }\right)}^{\prime }$ and ${\psi }_0\left(t\right)$ in model (1), and we assume $W\left(t\right)$ is a possibly data-dependent weight function, then we define

$$\begin{array}{c}{M}_i^2\left(t\right)={\int }_0^t{I}_i\left(s\right)\left[V_i\left(s\right)d{H}_i\left(s\right)-\left[{\psi }_0\left(s\right)+{\theta }^{\prime }{B}_i\left(s\right)\right]e^{{\delta }^{\prime }{X}_i\left(s\right)}d{\varphi }_0\left(s\right)\right].\\ U_2(\theta ,\delta )=\sum _{i=1}^n{\int }_0^{\tau }W\left(t\right)B_i\left(t\right)d{M}_i^2\left(t\right),\end{array}$$

Under models (1) and (3), we have

$$\begin{array}{cc}\hfill E\left\{V_i\left(s\right)I_i\left(s\right)d{H}_i\left(s\right)\right\}& =E\left\{E\left\{V_i\left(s\right)I_i\left(s\right)d{H}_i\left(s\right)\mid X_i\left(s\right),{\mathcal{F}}_{is}\right\}\right\}\hfill \\ \hfill & =E\left\{E\left\{I_i\left(s\right)\mid X_i\left(s\right)\right\}E\left\{V_i\left(s\right)\mid X_i\left(s\right),{\mathcal{F}}_{is}\right\}E\left\{d{H}_i\left(s\right)\mid X_i\left(s\right)\right\}\right\}\hfill \\ \hfill & =E\left\{E\left\{I_i\left(s\right)·\left\{{\psi }_0\left(s\right)+{\theta }^{\prime }{B}_i\left(s\right)\right\}{e}^{{\delta }^{\prime }{X}_i\left(s\right)}d{\varphi }_0\left(s\right)\mid X_i\left(s\right),{\mathcal{F}}_{is}\right\}\right\}\hfill \\ \hfill & =E\left\{I_i\left(s\right)\left\{{\psi }_0\left(s\right)+{\theta }^{\prime }{B}_i\left(s\right)\right\}{e}^{{\delta }^{\prime }{X}_i\left(s\right)}d{\varphi }_0\left(s\right)\right\},\hfill \end{array}$$

that is $E\left\{M_i^2(t;{\theta }_0,{\delta }_0)\right\}=0$, which means $M_i^2(t;{\theta }_0,{\delta }_0)$ is a zero-mean stochastic process. Thus, the parameter ${\psi }_0\left(t\right)$ can be achieved through the solution to

$$\sum _{i=1}^{n}d{M}_i^2\left(t,\widehat{\delta },{\widehat{\varphi }}_0\right)=0.$$

In addition, we can readily prove that $E\left\{U_2({\theta }_0;{\delta }_0)\right\}=0$. Therefore, according to the generalized estimating equation method in the literature [26], the parameter $\theta$ can be consistently estimated by the estimating equation as

$$U_2\left(\theta ,\widehat{\delta },{\widehat{\varphi }}_0\right)=0.$$

Thus, the estimates of $\theta$ and ${\psi }_0\left(t\right)$ can be provided by

$$\begin{array}{c}\widehat{\theta }=\frac{1}{n}{\widehat{\phi }}_{\theta }^{-1}\sum _{i=1}^n{\int }_0^{\tau }W\left(t\right)I_i\left(t\right)\left\{\widehat{B_i}\left(t\right)-{\widehat{\overline{B}}}_i(t;\widehat{\delta })\right\}{V}_i\left(t\right)d{H}_i\left(t\right),\\ {\widehat{\psi }}_0(t;\widehat{\theta },\widehat{\delta })=\frac{{\displaystyle \sum _{i=1}^n}{I}_i\left(t\right)V_i\left(t\right)d{H}_i\left(t\right)}{{\displaystyle \sum _{i=1}^n}{I}_i\left(t\right)e^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}d{\widehat{\varphi }}_0\left(t\right)}-{\widehat{\theta }}^{\prime }\widehat{\overline{B}}(t;\widehat{\delta }).\end{array}$$

$${\widehat{\phi }}_{\theta }=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\int }_0^{\tau }W\left(t\right)I_i\left(t\right){\widehat{B}}_i\left(t\right){\left\{{\widehat{B}}_i\left(t\right)-\widehat{\overline{B}}(t;\widehat{\delta })\right\}}^{\prime }{e}^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}d{\widehat{\varphi }}_0(t;\widehat{\delta }),$$

$${\widehat{\overline{B}}}_i(t;\widehat{\delta })=\frac{{\sum }_{i=1}^n{I}_i\left(t\right){\widehat{B}}_i(t;\widehat{\delta })e^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}}{{\sum }_{i=1}^n{I}_i\left(t\right)e^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}}.$$

4. Asymptotic Properties of Parameter Estimation

In this section, we will prove the asymptotic normality of the parameters under large samples. We provide some conditions necessary for proving the theorem:

(a)

$\{C_i,I_i,X_i\}$ are independent and identically distributed.

(b)

$h(·)$, $X_i\left(t\right)$, and $W\left(t\right)$ are bounded for all $t\in [0,\tau ]$.

(c)

The function $W\left(t\right)$ converges almost surely to a function $w\left(t\right)$ for all $t\in [0,\tau ]$.

(d)

There exists the longest follow-up time $\tau >0$ and $P(\tau \le C_i)>0$.

(e)

$Q\left({\theta }_0\right)=E\left[{\int }_0^{\tau }w\left(t\right)I_i\left(t\right){\left\{B_i\left(t\right)-\overline{B}\left(t\right)\right\}}^{\otimes 2}{e}^{{\delta }_0^{\prime }{X}_i\left(t\right)}d{\varphi }_0\left(t\right)\right]$ is positive definite.

We define

$${\widehat{E}}_{h\left({\mathcal{F}}_{it}\right)}^{*}={\int }_0^{\tau }h\left(e^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}d{\widehat{\varphi }}_0(t;\widehat{\delta })\right)+{\int }_0^{\tau }h\left(\frac{e^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}}{n{L}_0(t;\widehat{\delta })}\sum _{i=1}^{n}d{\widehat{M}}_i^1\left(t\right)\right),$$

$${\widehat{B}}_i^{*}\left(t\right)={\left(X_i{\left(t\right)}^{\prime },\frac{h{\left({\mathcal{F}}_{it}\right)}^{\prime }}{{\widehat{E}}_{h\left({\mathcal{F}}_{it}\right)}^{*}}\right)}^{\prime },$$

$${\widehat{\overline{B}}}^{*}(t;\widehat{\delta })=\frac{{\sum }_{i=1}^n{I}_i\left(t\right){\widehat{B}}_i^{*}(t;\widehat{\delta })e^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}}{{\sum }_{i=1}^n{I}_i\left(t\right)e^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}},$$

$${\widehat{R}}_1(t;\widehat{\theta },\widehat{\delta })=n^{-1}\sum _{i=1}^n\left\{{\widehat{B}}_i^{*}(t;\widehat{\delta })-{\widehat{\overline{B}}}^{*}(t;\widehat{\delta })\right\}{I}_i\left(t\right)\left\{{\widehat{\theta }}^{\prime }{\widehat{B}}_i(t;\widehat{\delta })+{\widehat{\psi }}_0\left(t\right)\right\}{e}^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)},$$

$${\widehat{R}}_2(t;\widehat{\theta },\widehat{\delta })=n^{-1}\sum _{i=1}^n\left\{{\widehat{B}}_i^{*}(t;\widehat{\delta })-{\widehat{\overline{B}}}^{*}(t;\widehat{\delta })\right\}{I}_i\left(t\right)e^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}{\widehat{\alpha }}^{\prime }\frac{h\left({\mathcal{F}}_{it}\right)}{{\widehat{E}}_{h\left({\mathcal{F}}_{it}\right)}{\widehat{E}}_{h\left({\mathcal{F}}_{it}\right)}^{*}},$$

$$\begin{array}{c}\hfill {\widehat{P}}_{\delta }=n^{-1}\sum _{i=1}^n\left[{\int }_0^{\tau }W\left(t\right)I_i\left(t\right){\widehat{B}}_i\left(t\right){\widehat{\theta }}^{\prime }{\left.\frac{\partial {\widehat{B}}_i(t;\delta )}{\partial {\delta }^{\prime }}\right|}_{\delta =\widehat{\delta }}{e}^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}d{\widehat{\varphi }}_0\left(t\right)-{\int }_0^{\tau }W\left(t\right)I_i\left(t\right){\left.\frac{\partial {\widehat{B}}_i(t;\delta )}{\partial {\delta }^{\prime }}\right|}_{\delta =\widehat{\delta }}d{\widehat{M}}_i^2\left(t\right)\right.\\ \hfill \left.+{\int }_0^{\tau }W\left(t\right)I_i\left(t\right)e^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}\left\{{\widehat{\psi }}_0\left(t\right)+{\widehat{\theta }}^{\prime }{\widehat{B}}_i\left(t\right)\right\}\left\{{\widehat{B}}_i(t;\widehat{\delta })-\widehat{\overline{B}}(t;\widehat{\delta })\right\}\left\{X_i\left(t\right)-\overline{X}(t;\widehat{\delta })\right\}d{\widehat{\varphi }}_0\left(t\right)\right],\end{array}$$

$${\widehat{D}}_{\delta }=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }{\left\{X_i\left(t\right)-\overline{X}(t;\widehat{\delta })\right\}}^{\otimes 2}{I}_i\left(t\right)d{H}_i\left(t\right).$$

Theorem 1. 

Given that conditions (a)–(e) are met, $\widehat{\theta }$ is a consistent estimator of ${\theta }_0$ and $\sqrt{n}(\widehat{\theta }-{\theta }_0)$ asymptotically converges to a normal distribution with mean 0 and the covariance matrix estimated by ${\widehat{Q}}^{-1}\widehat{\Sigma }{\widehat{Q}}^{-1}$, where

$$\widehat{Q}=n^{-1}\sum _{i=1}^n{\int }_0^{\tau }W\left(t\right)I_i\left(t\right){\left\{{\widehat{B}}_i(t;\widehat{\delta })-{\widehat{\overline{B}}}_i(t;\widehat{\delta })\right\}}^{\otimes 2}{e}^{{\widehat{\delta }}^{\prime }{X}_i\left(t\right)}d{\widehat{\varphi }}_0(t;\widehat{\delta }),$$

$$A_i^1={\int }_0^{\tau }W\left(t\right)\left({\widehat{B}}_i^{*}(t;\widehat{\delta })-{\widehat{\overline{B}}}^{*}(t;\widehat{\delta })\right)d{\widehat{M}}_i^2\left(t\right),$$

$$A_i^2={\int }_0^{\tau }\frac{W\left(t\right){\widehat{R}}_1(t;\widehat{\theta },\widehat{\delta })}{L_0(t;\widehat{\delta })}d{\widehat{M}}_i^1\left(t\right),$$

$$A_i^3={\int }_0^{\tau }W\left(t\right){\widehat{R}}_2{\int }_0^{t}h\left(\frac{e^{{\widehat{\delta }}^{\prime }{X}_i\left(s\right)}}{L_0(s;\widehat{\delta })}d{\widehat{M}}_i^1\left(s\right)\right)\left\{d{\widehat{\varphi }}_0(t;\widehat{\delta })+n^{-1}\sum _{i=1}^n\frac{d{\widehat{M}}_i^1\left(t\right)}{L_0(t;\widehat{\delta })}\right\}$$

$$A_i^4={\widehat{P}}_{\delta }{\widehat{D}}_{\delta }^{-1}{\int }_0^{\tau }\left\{X_i\left(t\right)-{\overline{X}}_i(t;\widehat{\delta })\right\}d{\widehat{M}}_i^1\left(t\right),$$

and

$$\widehat{\Sigma }=n^{-1}\sum _{i=1}^n{\left[A_i^1-A_i^2+A_i^3-A_i^4\right]}^{\otimes 2},$$

Proof. 

It is easy to show that $\theta$ always exist and are unique and consistent, then we prove the asymptotic normality of $\theta$; define

$$M_i^1\left(t\right)={\int }_0^t{I}_i\left(s\right)\left[d{H}_i\left(s\right)-e^{{\delta }_0^{\prime }{X}_i\left(s\right)}d{\varphi }_0\left(s\right)\right],$$

$$M_i^2\left(t\right)={\int }_0^t{I}_i\left(s\right)\left[V_i\left(s\right)d{H}_i\left(s\right)-\left[{\psi }_0\left(s\right)+{\theta }_0^{\prime }{B}_i\left(s\right)\right]e^{{\delta }_0^{\prime }{X}_i\left(s\right)}d{\varphi }_0\left(s\right)\right],$$

$${\widehat{U}}_2({\theta }_0,{\delta }_0)=\sum _{i=1}^n{\int }_0^{\tau }W\left(t\right){\widehat{B}}_i\left(t\right)I_i\left(t\right)\left\{V_i\left(t\right)d{H}_i\left(t\right)-\left[{\widehat{\psi }}_0(t;{\theta }_0,{\delta }_0)+{\theta }_0^{\prime }{\widehat{B}}_i\left(t\right)\right]e^{{\delta }_0^{\prime }{X}_i\left(t\right)}d{\widehat{\varphi }}_0\left(t\right)\right\}.$$

Obviously, ${\widehat{\psi }}_0(t;\theta ,\delta )$ satisfies

$$\sum _{i=1}^n{I}_i\left(t\right)\left[V_i\left(t\right)d{H}_i\left(t\right)-\left\{{\widehat{\psi }}_0\left(t\right)+{\theta }^{\prime }{\widehat{B}}_i\left(t\right)\right\}{e}^{{\delta }^{\prime }{X}_i\left(t\right)}d{\widehat{\varphi }}_0\left(t\right)\right]=0.$$

(5)

Let $\theta ={\theta }_0$, $\delta ={\delta }_0$; Equation (5) yields

$$\left\{{\widehat{\psi }}_0\left(t;{\theta }_0,{\delta }_0\right)-{\psi }_0\left(t\right)\right\}d{\widehat{\varphi }}_0\left(t\right)=\frac{{\sum }_{i=1}^n{I}_i\left(t\right)\left[V_i\left(t\right)d{H}_i\left(t\right)-\left\{{\psi }_0\left(t\right)+{\theta }_0^{\prime }{\widehat{B}}_i\left(t;{\delta }_0\right)\right\}{e}^{{\delta }_0^{\prime }{X}_i\left(t\right)}d{\widehat{\varphi }}_0\left(t\right)\right]}{{\sum }_{i=1}^n{I}_i\left(t\right)e^{{\delta }_0^{\prime }{X}_i\left(t\right)}}.$$

(6)

The expansion of ${\widehat{U}}_2(\theta ,\delta )$ at $\theta ={\theta }_0$ and $\delta ={\delta }_0$ yields

$$\begin{array}{cc}\hfill {\widehat{U}}_2\left({\theta }_0,{\delta }_0\right)& =\sum _{i=1}^n{\int }_0^{\tau }W\left(t\right){\widehat{B}}_i\left(t;{\delta }_0\right)I_i\left(t\right)\{V_i\left(t\right)d{H}_i\left(t\right)\hfill \\ \hfill & -e^{{\delta }_0^{\prime }{X}_i\left(t\right)}\left[{\widehat{\psi }}_0\left(t;{\theta }_0,{\delta }_0\right)-{\psi }_0\left(t\right)+{\theta }_0^{\prime }{\widehat{B}}_i\left(t;{\delta }_0\right)+{\psi }_0\left(t\right)\right]d{\widehat{\varphi }}_0\left(t\right)\}\hfill \\ \hfill & =\sum _{i=1}^n{\int }_0^{\tau }W\left(t\right){\widehat{B}}_i\left(t;{\delta }_0\right)I_i\left(t\right)\left\{V_i\left(t\right)d{H}_i\left(t\right)-e^{{\delta }_0^{\prime }{X}_i\left(t\right)}\left[{\widehat{\psi }}_0\left(t;{\theta }_0,{\delta }_0\right)-{\psi }_0\left(t\right)\right]d{\widehat{\varphi }}_0\left(t\right)\right.\hfill \\ \hfill & \left.-e^{{\delta }_0^{\prime }{X}_i\left(t\right)}\left[{\theta }_0^{\prime }{\widehat{B}}_i\left(t;{\delta }_0\right)+{\psi }_0\left(t\right)\right]d{\widehat{\varphi }}_0\left(t\right)\right\}.\hfill \end{array}$$

(7)

Then, insert (6) into (7); we have

$$\begin{array}{cc}\hfill {\widehat{U}}_2\left({\theta }_0,{\delta }_0\right)& =\sum _{i=1}^n{\int }_0^{\tau }W\left(t\right)\left\{{\widehat{B}}_i\left(t;{\delta }_0\right)-\widehat{\overline{B}}\left(t;{\delta }_0\right)\right\}·\left[d{M}_i^2\left(t\right)\right.\hfill \\ & -I_i\left(t\right)\left\{{\theta }_0^{\prime }{B}_i\left(t;{\delta }_0\right)+{\psi }_0\left(t\right)\right\}{e}^{{\delta }_0^{\prime }{X}_i\left(t\right)}\left\{d{\widehat{\varphi }}_0\left(t;{\delta }_0\right)-d{\varphi }_0\left(t\right)\right\}\hfill \\ & \left.-I_i\left(t\right)\left\{{\theta }_0^{\prime }{\widehat{B}}_i\left(t;{\delta }_0\right)-{\theta }_0^{\prime }{B}_i\left(t;{\delta }_0\right)\right\}{e}^{{\delta }_0^{\prime }{X}_i\left(t\right)}d{\widehat{\varphi }}_0\left(t;{\delta }_0\right)\right].\hfill \end{array}$$

(8)

Following from (4), it is easy to obtain

$$d{\widehat{\varphi }}_0\left(t;{\delta }_0\right)-d{\varphi }_0\left(t\right)=\frac{1}{n}\sum _{i=1}^n\frac{d{M}_i^1\left(t\right)}{L_0\left(t\right)}.$$

(9)

Define

$$B_i^{*}\left(t\right)={\left(X_i{\left(t\right)}^{\prime },\frac{h{\left({\mathcal{F}}_{it}\right)}^{\prime }}{E_{h\left({\mathcal{F}}_{it}\right)}^{*}}\right)}^{\prime },$$

(10)

where

$$E_{h\left({\mathcal{F}}_{it}\right)}^{*}={\int }_0^{t}h\left(e^{{\delta }_0^{\prime }{X}_i\left(s\right)}d{\widehat{\varphi }}_0\left(s;{\delta }_0\right)\right)={\int }_0^{t}h\left(e^{{\delta }_0^{\prime }{X}_i\left(s\right)}d{\varphi }_0\left(s;{\delta }_0\right)\right)+{\int }_0^{t}h\left(\frac{e^{{\delta }_0^{\prime }{X}_i\left(s\right)}}{n{L}_0\left(s;{\delta }_0\right)}\sum _{i=1}^{n}d{M}_i^1\left(s\right)\right);$$

(11)

also, define

$$\begin{array}{c}\hfill R_1\left(t;{\theta }_0,{\delta }_0\right)=n^{-1}\sum _{i=1}^n\left\{B_i^{*}\left(t;{\delta }_0\right)-{\overline{B}}^{*}\left(t;{\delta }_0\right)\right\}{I}_i\left(t\right)\left\{{\theta }_0^{\prime }{B}_i\left(t;{\delta }_0\right)+{\psi }_0\left(t\right)\right\}{e}^{{\delta }_0^{\prime }{X}_i\left(t\right)},\end{array}$$

(12)

$$\begin{array}{c}\hfill R_2\left(t;{\theta }_0,{\delta }_0\right)=n^{-1}\sum _{i=1}^n\left\{B_i^{*}\left(t;{\delta }_0\right)-{\overline{B}}^{*}\left(t;{\delta }_0\right)\right\}{I}_i\left(t\right)e^{{\delta }_0^{\prime }{X}_i\left(t\right)}{\alpha }_0^{\prime }\frac{h\left({\mathcal{F}}_{it}\right)}{E_{h\left({\mathcal{F}}_{it}\right)}{E}_{h\left({\mathcal{F}}_{it}\right)}^{*}}.\end{array}$$

(13)

Then, combining Equations (8)–(13), it follows that

$$\begin{array}{cc}\hfill U_2\left({\theta }_0,{\delta }_0\right)& =\sum _{i=1}^n{\int }_0^{\tau }W\left(t\right)\left\{B_i^{*}\left(t;{\delta }_0\right)-{\overline{B}}^{*}\left(t;{\delta }_0\right)\right\}d{M}_i^2\left(t\right)-\sum _{i=1}^n{\int }_0^{\tau }\frac{W\left(t\right)R_1\left(t;{\theta }_0,{\delta }_0\right)}{L_0\left(t;{\delta }_0\right)}d{M}_i^1\left(t\right)\hfill \\ \hfill & -{\int }_0^{\tau }W\left(t\right)R_2\left(t;{\theta }_0,{\delta }_0\right){\int }_0^{t}h\left(\sum _{i=1}^n\frac{e^{{\delta }_0^{\prime }{X}_i\left(s\right)}}{L_0\left(s\right)}d{M}_i^1\left(s\right)\right)\left\{d{\varphi }_0\left(t;{\delta }_0\right)+n^{-1}\sum _{i=1}^n\frac{d{M}_i^1\left(t\right)}{L_0\left(t\right)}\right\}.\hfill \end{array}$$

(14)

In this section, let $l_0\left(t\right)$, $l_1\left(t\right)$, $\overline{b}\left(t\right)$, ${\overline{b}}^{*}\left(t\right)$, $r_1\left(t\right)$, and $r_2\left(t\right)$ denote the limit of $L_0\left(t;{\delta }_0\right)$, $L_1\left(t;{\delta }_0\right)$, $\overline{B}\left(t;{\theta }_0,{\delta }_0\right)$, ${\overline{B}}^{*}\left(t;{\theta }_0,{\delta }_0\right),$ $R_1\left(t;{\theta }_0,{\delta }_0\right)$, $R_2\left(t;{\theta }_0,{\delta }_0\right)$, respectively, when $n\to \infty$. Also, let $\overline{\mathit{x}}\left(t\right)=l_1\left(t\right)/l_0\left(t\right)$. Then, Equation (8) converges to

$$\begin{array}{c}\hfill U_2\left({\theta }_0,{\delta }_0\right)=\sum _{i=1}^n{\int }_0^{\tau }w\left(t\right)\left\{B_i^{*}\left(t\right)-{\overline{b}}^{*}\left(t\right)\right\}d{M}_i^2\left(t\right)-\sum _{i=1}^n{\int }_0^{\tau }\frac{w\left(t\right)r_1\left(t\right)}{l_0\left(t\right)}d{M}_i^1\left(t\right)\\ \hfill -{\int }_0^{\tau }w\left(t\right)r_2\left(t\right){\int }_0^{t}h\left(\sum _{i=1}^n\frac{e^{{\delta }_0^{\prime }{X}_i\left(s\right)}}{l_0\left(s\right)}d{M}_i^1\left(s\right)\right)\left\{d{\varphi }_0\left(t;{\delta }_0\right)+n^{-1}\sum _{i=1}^n\frac{d{M}_i^1\left(t\right)}{l_0\left(t\right)}\right\}+o_p\left(n^{1/2}\right).\end{array}$$

(15)

Next, It is easy to know that $\widehat{\delta }$ is a consistent estimate of ${\delta }_0$, then let the Taylor expansion of $U_1\left({\theta }_0,\widehat{\delta }\right)$ at $\left({\theta }_0,{\delta }_0\right)$ by following [27]; we have

$$\begin{array}{c}\hfill \sqrt{n}\left(\widehat{\delta }-{\delta }_0\right)=-n^{-1/2}{D}_{\delta }\sum _{i=1}^n{\int }_0^{\tau }{\left\{X_i\left(t\right)-\overline{x}\left(t;{\delta }_0\right)\right\}}^{\otimes 2}d{M}_i^1\left(t\right)+o_p\left(1\right),\end{array}$$

(16)

where

$$D_{\delta }=E\left[{\int }_0^{\tau }{\left\{X_i\left(t\right)-\overline{x}\left(t\right)\right\}}^{\otimes 2}{I}_i\left(t\right)d{H}_i\left(t\right)\right].$$

Let $P^{*}(\theta ,\delta )=-n^{-1}\partial U_2(\theta ,\delta )/\partial {\delta }^{\prime }$, then, following from law of large numbers, one can obtain that $P^{*}(\theta ,\delta )$ converges almost surely to a deterministic function $P(\theta ,\delta )$ and $P\left({\theta }_0,{\delta }_0\right)$ is denoted by $P_{\delta }$:

$$\begin{array}{cc}\hfill P_{\delta }=& E\left[{\int }_0^{\tau }w\left(t\right)I_i\left(t\right)e^{{\delta }_0^{\prime }{X}_i\left(t\right)}\left\{{\psi }_0+{\theta }^{\prime }{B}_i\left(t\right)\right\}\left\{B_i\left(t;{\delta }_0\right)-\overline{b}\left(t;{\delta }_0\right)\right\}\left\{X_i\left(t\right)-\overline{x}\left(t\right)\right\}d{\varphi }_0\left(t\right)\right.\hfill \\ \hfill & +{\int }_0^{\tau }w\left(t\right)I_i\left(t\right)B_i\left(t\right){\theta }_0^{\prime }\frac{\partial B_i\left(t;{\delta }_0\right)}{\partial {\delta }_0^{\prime }}{e}^{{\delta }_0^{\prime }{X}_i\left(t\right)}d{\varphi }_0(t;{\delta }_0)\hfill \\ \hfill & -{\int }_0^{\tau }w\left(t\right)I_i\left(t\right)\frac{\partial B_i\left(t;{\delta }_0\right)}{\partial {\delta }_0^{\prime }}\left[V_i\left(t\right)d{H}_i\left(t\right)-\left\{{\psi }_0\left(t\right)+{\theta }_0^{\prime }{B}_i\left(t\right)\right\}{e}^{{\delta }_0^{\prime }{X}_i\left(t\right)}d{\varphi }_0\left(t\right)\right].\hfill \end{array}$$

(17)

Hence, combining (16) and (17), the Taylor expansion of $U_2\left({\theta }_0,\widehat{\delta }\right)$ at $\left({\theta }_0,{\delta }_0\right)$ yields

$$U_2\left({\theta }_0,\widehat{\delta }\right)=U_2\left({\theta }_0,{\delta }_0\right)-P_{\delta }{D}_{\delta }^{-1}\sum _{i=1}^n{\int }_0^{\tau }\left\{X_i\left(t\right)-\overline{x}\left(t\right)\right\}d{M}_i^1\left(t\right)+o_p\left(n^{1/2}\right).$$

(18)

Thus, following from (15) and (18) and the multivariate central limit theorem, we can conclude that $n^{-1/2}{U}_2\left({\theta }_0;\widehat{\delta }\right)$ converges in distribution to a normal distribution with mean 0 and the following covariance:

$$\begin{array}{cc}\hfill \Sigma & =E\left[{\int }_0^{\tau }w\left(t\right)\left\{B_i^{*}\left(t\right)-{\overline{b}}^{*}\left(t\right)\right\}d{M}_i^2\left(t\right)-{\int }_0^{\tau }\frac{w\left(t\right)r_1\left(t\right)}{l_0\left(t\right)}d{M}_i^1\left(t\right)\right]\hfill \\ & +{\int }_0^{\tau }w\left(t\right)r_2\left(t\right){\int }_0^{t}h\left(\frac{e^{{\delta }_0^{\prime }{X}_i\left(s\right)}}{l_0\left(s\right)}d{M}_i^1\left(s\right)\right)\left\{d{\varphi }_0\left(t;{\delta }_0\right)+n^{-1}\sum _{i=1}^n\frac{d{M}_i^1\left(t\right)}{l_0\left(t\right)}\right\}\hfill \\ & {\left.-P_{\delta }{D}_{\delta }^{-1}\sum _{i=1}^n{\int }_0^{\tau }\left\{X_i\left(t\right)-\overline{x}\left(t\right)\right\}d{M}_i^1\left(t\right)\right]}^{\otimes 2}.\hfill \end{array}$$

(19)

Let $Q^{*}(\theta ,\widehat{\delta })=-n^{-1}\partial U_2(\theta ,\widehat{\delta })/\partial {\theta }^{\prime }$; according to the law of large numbers, $Q^{*}(\theta ,\widehat{\delta })$ almost surely converges to the nonrandom function $Q\left(\theta \right)$, and

$$Q\left({\theta }_0\right)=E\left[{\int }_0^{\tau }w\left(t\right)I_i\left(t\right){\left\{B_i\left(t\right)-\overline{b}\left(t\right)\right\}}^{\otimes 2}{e}^{{\delta }_0^{\prime }{X}_i\left(t\right)}d{\varphi }_0\left(t\right)\right].$$

Based on the Taylor expansion of $U_2(\widehat{\theta },\widehat{\delta })$ at $\left({\theta }_0,\widehat{\delta }\right)$, we can obtain

$$\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)=Q{\left({\theta }_0\right)}^{-1}{n}^{-1/2}{U}_2\left({\theta }_0,\widehat{\delta }\right)+o_p\left(1\right).$$

(20)

Therefore, combining (19) and (20), $\sqrt{n}\left(\widehat{\theta }-{\theta }_0\right)$ asymptotically converges to a normal distribution with mean 0 and the covariance matrix estimated by ${\widehat{Q}}^{-1}\widehat{\Sigma }{\widehat{Q}}^{-1}$. □

5. Simulation Studies

We will evaluate the effectiveness of the methods outlined in Section 2 through some simulation studies. In our research, we hypothesized that $X_i\left(t\right)$ was a random variable following a Bernoulli distribution with a probability of success of 0.5, and the censoring time $C_i$ was generated from a uniform distribution over $(\tau /2,\tau )$. Assume that $H_i\left(t\right)$ is a Poisson process generated by the ratio function ${\lambda }_0\left(t\right)=c/\tau$, where c is a constant. Thus, the number of observations $m_i$ was produced by the Poisson distribution with mean

$$\varphi \left(C_i\mid X_i\right)={\varphi }_0\left(C_i\right)e^{{\delta }^{\prime }{X}_i}=\frac{c{C}_i{e}^{{\delta }^{\prime }{X}_i}}{\tau }.$$

We assumed the time points $T=\left(T_1,T_2,\dots ,T_{80}\right)$ are the order statistics of a random sample from the uniform distribution $U(0.2,\tau )$ and then supposed that $\left(t_{i,1},\dots ,t_{i,m_i}\right)$ are the order statistics of a random sample of size $m_i$ from a discrete uniform distribution over T that is less than or equal to $C_i$.

To generate interval-censored recurrence event data $V_i\left(t_{i,j}\right)$, we assumed

$$V_i\left(t_{i,j}\right)=V_i\left(t_{i,1}\right)+V_i\left(t_{i,2}-t_{i,1}\right)+\cdots +V_i\left(t_{i,j}-t_{i,j-1}\right),$$

with $t_{i,0}=0$. We considered two cases, For the first case, we assumed $V_i\left(t_{i,1}\right)$ and $V_i\left(t_{i,j}-t_{i,j-1}\right)$ followed a Poisson distribution such that the mean function with propensity scores are given by

$$E\left\{V_i\left(t_{i,1}\right)\mid X_i,{\mathcal{F}}_{i,1}\right\}={\psi }_0\left(t_{i,1}\right)+{\gamma }^{\prime }{X}_i+{\alpha }^{\prime }\frac{h\left({\mathcal{F}}_{i{t}_{i,1}}\right)}{E_{h\left({\mathcal{F}}_{i{t}_{i,1}}\right)}},$$

and

$$\begin{array}{cc}\hfill E\left\{V_i\left(t_{i,j}-t_{i,j-1}\right)\mid X_i,{\mathcal{F}}_{i,t_{i,j}}\right\}& =\left({\psi }_0\left(t_{i,j}\right)+{\gamma }^{\prime }{X}_i+{\alpha }^{\prime }\frac{h\left({\mathcal{F}}_{i{t}_{i,j}}\right)}{E_{h\left({\mathcal{F}}_{i{t}_{i,j}}\right)}}\right)\hfill \\ \hfill & -\left({\psi }_0\left(t_{i,j-1}\right)+{\gamma }^{\prime }{X}_i+{\alpha }^{\prime }\frac{h\left({\mathcal{F}}_{i{t}_{i,j-1}}\right)}{\left.E_{h\left({\mathcal{F}}_{i{t}_{i,j}-1}\right)}\right)}\right),\hfill \end{array}$$

respectively. In the second case, without considering the propensity score, it was hypothesized that $V_i\left(t_{i,1}\right)$ and $V_i\left(t_{i,j}-t_{i,j-1}\right)$ follow a Poisson distribution with the mean function modeled by

$$E\left\{V_i\left(t_{i,1}\right)\mid X_i,{\mathcal{F}}_{i,t_{i,1}}\right\}={\psi }_0\left(t_{i,1}\right)+{\gamma }^{\prime }{X}_i+{\alpha }^{\prime }h\left({\mathcal{F}}_{i{t}_{i,1}}\right),$$

and

$$E\left\{V_i\left(t_{i,j}-t_{i,j-1}\right)\mid X_i,{\mathcal{F}}_{i,t_{i,j}}\right\}=\left({\psi }_0\left(t_{i,j}\right)+{\gamma }^{\prime }{X}_i+{\alpha }^{\prime }h\left({\mathcal{F}}_{i{t}_{i,j}}\right)\right)-\left({\psi }_0\left(t_{i,j-1}\right)+{\gamma }^{\prime }{X}_i+{\alpha }^{\prime }h\left({\mathcal{F}}_{i{t}_{i,j-1}}\right)\right),$$

respectively, $j=1,\dots ,m_i$, $i=1,\dots ,n$. For all simulations, we let $h\left({\mathcal{F}}_{it}\right)=H_i(t-)$, $W\left(t\right)=1$, and $n=100$ or $n=300$. The results presented in the tables are based on 1000 replications, and we chose four different sets of the true value of $(\gamma ,\alpha )$: (0, 0), (0.1, 0), (0, 0.1), and (0.1, 0.1). The results presented in Table 1 and Table 2 are based on the simulated data that were produced by the Poisson distribution with the propensity score and were estimated by the semiparametric model with and without the propensity score, respectively. In addition, Table 3 and Table 4 were obtained under the same settings as in Table 1 and Table 2, respectively, except that the simulated data were produced by a Poisson distribution without the propensity score. The results include the estimated biases (BIAS) obtained by the mean deviations of the point estimates $\widehat{\gamma }$ and $\widehat{\alpha }$ from their true values, the empirical standard deviation of $\widehat{\gamma }$ and $\widehat{\alpha }$ (ESD), the average estimated standard errors $\widehat{\gamma }$ and $\widehat{\alpha }$ (ASE), and the $95\%$ empirical coverage probabilities (CPs) for both $\gamma$ and $\alpha$.

Table 1. Estimation of $\gamma$ and $\alpha$ estimated by the model with the propensity score, where simulated data were produced by a Poisson process with the propensity score and $Z\sim B(1,0.5)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	0.0086	0.0098	0.0047	0.0132	0.0019	0.0094	−0.0011	0.0109
ESD	0.1499	0.0572	0.1998	0.0780	0.0866	0.0334	0.1179	0.0447
ASE	0.1447	0.0546	0.1949	0.0781	0.0854	0.0325	0.1149	0.0451
CP	0.9414	0.9588	0.9435	0.9653	0.9480	0.9541	0.9495	0.9538
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0012	0.0081	0.0083	0.0101	−0.0027	0.0067	0.0061	0.0089
ESD	0.1588	0.0562	0.2141	0.0724	0.0915	0.0317	0.1195	0.0418
ASE	0.1514	0.0507	0.2034	0.0714	0.0901	0.0301	0.1203	0.0409
CP	0.9360	0.9602	0.9320	0.9662	0.9491	0.9480	0.9521	0.9565
$\gamma =0$, $\alpha =0.1$
BIAS	0.0068	−0.0164	−0.0012	0.0079	0.0011	−0.0109	−0.0002	0.0056
ESD	0.1479	0.0921	0.1963	0.1227	0.0874	0.0559	0.1127	0.0707
ASE	0.1461	0.0915	0.1894	0.1229	0.0863	0.0554	0.1111	0.0729
CP	0.9487	0.9536	0.9440	0.9427	0.9513	0.9324	0.9456	0.9552
$\gamma =0.1$, $\alpha =0.1$
BIAS	0.0017	−0.0186	0.0038	0.0032	0.0006	−0.0123	0.0026	0.0018
ESD	0.1619	0.0912	0.2005	0.1235	0.0911	0.0565	0.1165	0.0707
ASE	0.1563	0.0927	0.1973	0.1205	0.0919	0.0560	0.1164	0.0709
CP	0.9380	0.9356	0.9446	0.9412	0.9521	0.9472	0.9516	0.9447

Table 2. Estimation of $\gamma$ and $\alpha$ estimated by the model without the propensity score, where simulated data were produced by a Poisson process with the propensity score and $Z\sim B(1,0.5)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	0.0021	0.0101	−0.0015	0.0145	−0.0005	0.0138	−0.0017	0.0185
ESD	0.1496	0.0810	0.2032	0.1236	0.0872	0.0472	0.1167	0.0713
ASE	0.1437	0.0957	0.1943	0.1534	0.0851	0.0532	0.1157	0.0852
CP	0.9358	0.9792	0.9381	0.9851	0.9408	0.9742	0.9495	0.9538
$\gamma =0.1$, $\alpha =0$
BIAS	0.0077	0.0111	0.0032	0.0171	−0.0059	0.0149	0.0081	0.0236
ESD	0.1592	0.1003	0.2106	0.1543	0.0907	0.0580	0.1199	0.0864
ASE	0.1515	0.1169	0.2030	0.1837	0.0897	0.0646	0.1201	0.1014
CP	0.9356	0.9734	0.9370	0.9808	0.9460	0.9718	0.9456	0.9778
$\gamma =0$, $\alpha =0.1$
BIAS	−0.0030	−0.0299	0.0078	−0.0223	−0.0008	−0.0285	−0.0009	−0.0141
ESD	0.1589	0.0966	0.2109	0.1363	0.0928	0.0550	0.1220	0.0790
ASE	0.1539	0.1067	0.2029	0.1642	0.0911	0.0607	0.1202	0.0929
CP	0.9422	0.9674	0.9387	0.9845	0.9444	0.9314	0.9470	0.9756
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0120	−0.0263	−0.0105	−0.0191	−0.0090	−0.0235	−0.0044	−0.0221
ESD	0.1686	0.0995	0.2149	0.1409	0.1068	0.0579	0.1247	0.0815
ASE	0.1621	0.1119	0.2091	0.1698	0.0936	0.0618	0.1277	0.0916
CP	0.9334	0.9708	0.9420	0.9854	0.9466	0.9420	0.9440	0.9773

Table 3. Estimation of $\gamma$ and $\alpha$ estimated by the model with the propensity score, where simulated data were produced by a Poisson process without the propensity score and $Z\sim B(1,0.5)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	−0.0007	0.0095	−0.0016	0.0118	0.0032	0.0069	−0.0013	0.0124
ESD	0.1482	0.0562	0.2035	0.0796	0.0862	0.0328	0.1174	0.0456
ASE	0.1447	0.0546	0.1952	0.0779	0.0854	0.0324	0.1148	0.0451
CP	0.9475	0.9550	0.9403	0.9612	0.9465	0.9550	0.9450	0.9536
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0047	0.0076	0.0069	0.0109	−0.0021	0.0083	0.0057	0.0095
ESD	0.1602	0.0539	0.2174	0.0732	0.0918	0.0328	0.1226	0.0411
ASE	0.1522	0.0509	0.2037	0.0713	0.0897	0.0301	0.1203	0.0412
CP	0.9318	0.9630	0.9300	0.9610	0.9415	0.9483	0.9412	0.9590
$\gamma =0$, $\alpha =0.1$
BIAS	0.0047	0.0019	0.0057	0.0037	−0.0019	0.0007	0.0042	0.0012
ESD	0.1737	0.0831	0.2159	0.1048	0.0989	0.0489	0.1239	0.0596
ASE	0.1639	0.0757	0.2095	0.0987	0.0971	0.0469	0.1237	0.0586
CP	0.9305	0.9367	0.9370	0.9350	0.9440	0.9399	0.9466	0.9440
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0071	−0.0031	−0.0046	−0.0043	−0.0041	−0.0028	−0.0033	0.0003
ESD	0.1788	0.0863	0.2176	0.1073	0.1028	0.0510	0.1305	0.0592
ASE	0.1709	0.0781	0.2151	0.0995	0.1015	0.0478	0.1275	0.0596
CP	0.9333	0.9493	0.9406	0.9356	0.9480	0.9390	0.9526	0.9468

Table 4. Estimation of $\gamma$ and $\alpha$ estimated by the model without the propensity score, where simulated data were produced by a Poisson process without the propensity score and $Z\sim B(1,0.5)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	−0.0008	0.0129	0.0035	0.0170	0.0006	0.0121	0.0027	0.0204
ESD	0.1479	0.0801	0.2019	0.1228	0.0878	0.0469	0.1145	0.0719
ASE	0.1446	0.0957	0.1943	0.1538	0.0852	0.0531	0.1147	0.0850
CP	0.9430	0.9812	0.9359	0.9811	0.9438	0.9736	0.9450	0.9791
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0065	0.0078	0.0085	0.0175	−0.0063	0.0145	0.0081	0.0213
ESD	0.1587	0.1010	0.2136	0.1535	0.0917	0.0574	0.1231	0.0866
ASE	0.1517	0.1164	0.2041	0.1855	0.0896	0.0647	0.1199	0.1010
CP	0.9334	0.9792	0.9387	0.9787	0.9427	0.9710	0.9426	0.9790
$\gamma =0$, $\alpha =0.1$
BIAS	0.0011	0.0099	0.0066	0.0159	−0.0012	0.0139	0.0009	0.0183
ESD	0.1682	0.0947	0.2197	0.1258	0.0993	0.0562	0.1243	0.0739
ASE	0.1620	0.1077	0.2083	0.1589	0.0959	0.0621	0.1224	0.0899
CP	0.9404	0.9778	0.9382	0.9872	0.9433	0.9741	0.9448	0.9871
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0084	0.0104	−0.0057	0.0146	−0.0068	0.0145	−0.0033	0.0166
ESD	0.1748	0.0994	0.2177	0.1464	0.1018	0.0578	0.1212	0.0786
ASE	0.1684	0.1133	0.2133	0.1659	0.1001	0.0657	0.1258	0.0937
CP	0.9388	0.9788	0.9383	0.9840	0.9495	0.9745	0.9513	0.9786

Table 1 and Table 3 suggest that the point estimates seem to be unbiased and the ASE and ESD are quite close to each other and decreased as the sample size increased; what is more, the $\mathrm{CP}$ value is around 95%. In Table 2, compared to Table 1, the CP value deviates further from $95\%$, and the estimated biases is larger. Analogously, we can obtain a similar conclusion by comparing Table 4 to Table 3. Therefore, the results indicate that the estimation effect of the semiparametric model with the propensity score performed well and better than the semiparametric model without the propensity score.

In addition, more general settings were considered. Assuming all other configurations are identical to those in Table 1, Table 2, Table 3 and Table 4, we hypothesized that $X_i\left(t\right)$ were sampled from a Gaussian distribution with a mean of 0 and a variance of 0.25, and the outcomes are presented in Table 5, Table 6, Table 7 and Table 8, respectively.

Table 5. Estimation of $\gamma$ and $\alpha$ estimated by the model with the propensity score, where simulated data were produced by a Poisson process with the propensity score and $Z\sim N(0,0.25)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	−0.0026	0.0087	−0.0016	0.0106	−0.0043	0.0107	−0.0021	0.0136
ESD	0.1513	0.0593	0.2033	0.0778	0.0873	0.0339	0.1174	0.0456
ASE	0.1410	0.0646	0.1917	0.0934	0.0851	0.0359	0.1139	0.0524
CP	0.9308	0.9664	0.9338	0.9816	0.9455	0.9693	0.9415	0.9768
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0077	0.0096	−0.0044	0.0092	−0.0047	0.0074	0.0046	0.0133
ESD	0.1526	0.0587	0.2067	0.0804	0.0865	0.0336	0.1187	0.0449
ASE	0.1409	0.0642	0.1919	0.0949	0.0850	0.0357	0.1149	0.0520
CP	0.9342	0.9721	0.9294	0.9783	0.9415	0.9715	0.9467	0.9790
$\gamma =0$, $\alpha =0.1$
BIAS	0.0009	−0.0172	0.0007	0.0050	0.0003	−0.0156	−0.0011	0.0025
ESD	0.1526	0.0974	0.1998	0.1219	0.0877	0.0547	0.1115	0.0704
ASE	0.1453	0.1044	0.1876	0.1473	0.0859	0.0605	0.1112	0.0851
CP	0.9386	0.9542	0.9437	0.9755	0.9454	0.9494	0.9492	0.9662
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0012	−0.0164	0.0013	0.0020	−0.0014	−0.0161	0.0003	0.0049
ESD	0.1544	0.0976	0.1969	0.1228	0.0889	0.0552	0.1137	0.0701
ASE	0.1447	0.1042	0.1881	0.1478	0.0858	0.0608	0.1109	0.0851
CP	0.9316	0.9548	0.9384	0.9822	0.9414	0.9472	0.9460	0.9731

Table 6. Estimation of $\gamma$ and $\alpha$ estimated by the model without the propensity score, where simulated data were produced by a Poisson process with the propensity score and $Z\sim N(0,0.25)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	0.0009	0.0108	0.0032	0.0135	−0.0005	0.0129	0.0020	0.0208
ESD	0.1495	0.0741	0.2031	0.1088	0.0867	0.0412	0.1164	0.0601
ASE	0.1410	0.0981	0.1803	0.1555	0.0841	0.0496	0.1124	0.0814
CP	0.9394	0.9862	0.9366	0.9852	0.9394	0.9780	0.9427	0.9838
$\gamma =0.1$, $\alpha =0$
BIAS	0.0099	0.0088	−0.0009	0.0159	−0.0064	0.0138	−0.0007	0.0218
ESD	0.1534	0.0742	0.2019	0.1102	0.0866	0.0438	0.1135	0.0646
ASE	0.1422	0.0986	0.1913	0.1593	0.0848	0.0539	0.1132	0.0863
CP	0.9323	0.9875	0.9383	0.9837	0.9441	0.9788	0.9492	0.9825
$\gamma =0$, $\alpha =0.1$
BIAS	0.0050	−0.0471	−0.0047	−0.0283	0.0023	−0.0333	−0.0044	−0.0222
ESD	0.1627	0.0847	0.2108	0.1228	0.0921	0.0505	0.1212	0.0715
ASE	0.1513	0.1097	0.1998	0.1705	0.0906	0.0603	0.1181	0.0936
CP	0.9315	0.9795	0.9412	0.9820	0.9496	0.9412	0.9446	0.9835
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0096	−0.0378	−0.0070	−0.0258	−0.0081	−0.0331	−0.0068	−0.0235
ESD	0.1616	0.0847	0.2102	0.1233	0.0916	0.0517	0.1177	0.0922
ASE	0.1519	0.1091	0.1974	0.1704	0.0904	0.0598	0.1228	0.0708
CP	0.9348	0.9798	0.9343	0.9876	0.9468	0.9418	0.9391	0.9842

Table 7. Estimation of $\gamma$ and $\alpha$ estimated by the model with the propensity score, where simulated data were produced by a Poisson process without the propensity score and $Z\sim N(0,0.25)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	0.0012	0.0064	0.0006	0.0062	−0.0014	0.0073	0.0003	0.0095
ESD	0.1532	0.0492	0.2033	0.0674	0.0875	0.0285	0.1187	0.0385
ASE	0.1387	0.0554	0.1919	0.0855	0.0839	0.0299	0.1148	0.0438
CP	0.9368	0.9763	0.9476	0.9673	0.9416	0.9722	0.9432	0.9639
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0076	0.0084	−0.0021	0.0106	−0.0045	0.0071	0.0013	0.0122
ESD	0.1539	0.0548	0.2059	0.0767	0.0866	0.0319	0.1168	0.0419
ASE	0.1419	0.0605	0.1922	0.0919	0.0847	0.0329	0.1151	0.0483
CP	0.9334	0.9756	0.9350	0.9705	0.9478	0.9680	0.9462	0.9782
$\gamma =0$, $\alpha =0.1$
BIAS	0.0014	−0.0021	−0.0023	0.0018	−0.0022	0.0015	−0.0043	0.0017
ESD	0.1755	0.0843	0.2171	0.1018	0.0997	0.0494	0.1251	0.0595
ASE	0.1624	0.0869	0.2044	0.1179	0.0959	0.0504	0.1223	0.0668
CP	0.9318	0.9596	0.9306	0.9778	0.9447	0.9522	0.9490	0.9751
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0098	0.0031	−0.0074	0.0006	−0.0045	0.0014	−0.0063	0.0044
ESD	0.1726	0.0834	0.2029	0.1021	0.0974	0.0488	0.1258	0.0585
ASE	0.1610	0.0873	0.2063	0.1185	0.0962	0.0506	0.1227	0.0668
CP	0.9274	0.9662	0.9320	0.9789	0.9490	0.9586	0.9537	0.9770

Table 8. Estimation of $\gamma$ and $\alpha$ estimated by the model without the propensity score, where simulated data were produced by a Poisson process without the propensity score and $Z\sim N(0,0.25)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	−0.0088	0.0105	−0.0025	0.0134	0.0003	0.0138	−0.0012	0.0153
ESD	0.1465	0.0848	0.1987	0.1290	0.0869	0.0484	0.1199	0.0691
ASE	0.1417	0.1085	0.1904	0.1740	0.0846	0.0579	0.1141	0.0942
CP	0.9330	0.9830	0.9440	0.9850	0.9427	0.9782	0.9310	0.9820
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0017	0.0101	−0.0031	0.0174	−0.0026	0.0105	−0.0016	0.0145
ESD	0.1548	0.0876	0.2032	0.1345	0.0873	0.0515	0.1207	0.0719
ASE	0.1418	0.1180	0.1926	0.1924	0.0864	0.0638	0.1148	0.1031
CP	0.9345	0.9879	0.9478	0.9876	0.9480	0.9270	0.9360	0.9890
$\gamma =0$, $\alpha =0.1$
BIAS	0.0025	0.0056	−0.0027	0.0139	0.0014	0.0138	0.0015	0.0184
ESD	0.1684	0.0942	0.2181	0.1286	0.0951	0.0664	0.1230	0.0981
ASE	0.1593	0.1187	0.2051	0.1768	0.0963	0.0548	0.1259	0.0981
CP	0.9354	0.9822	0.9328	0.9823	0.9433	0.9835	0.9456	0.9821
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0092	0.0048	−0.0167	0.0155	−0.0023	0.0157	−0.0087	0.0175
ESD	0.1716	0.0921	0.2194	0.1292	0.0988	0.0556	0.1229	0.0729
ASE	0.1583	0.1183	0.2053	0.1767	0.0950	0.0665	0.1213	0.0934
CP	0.9291	0.9837	0.9351	0.9827	0.9382	0.9781	0.9478	0.9813

Interval-censored recurrence event data with an abundance of zeros relative to the Poisson distribution are frequently observed in numerous medical scenarios. In order to test the robustness of our suggested approach with this type of data, we created data using the zero-inflated Poisson (ZIP) regression model outlined in [28].

$$P\left(V_i\left(t_{i,j}-t_{i,j-1}\right)=y\right)=\left\{\begin{array}{c}{w}_{ij}+\left(1-w_{ij}\right)exp\left(-{\lambda }_{ij}\right),y=0\hfill \\ \left(1-w_{ij}\right)exp\left(-{\lambda }_{ij}\right){\lambda }_{ij}^y/y!,y>0\hfill \end{array},\right.$$

$$log\left(\frac{w_{ij}}{1-w_{ij}}\right)={\xi }_0+{\xi }_1{u}_{ij}+v_i,$$

for $j=1,\dots ,m_i$, $i=1,\dots ,n$. We set ${\xi }_0=-1$, ${\xi }_1=0.5$. The covariate $u_{ij}$ was drawn from the uniform distribution $U(0,1)$, while the random effect $v_i$ followed a normal distribution with a mean of 0 and a standard deviation of 0.1. Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15 and Table 16 present the simulation results generated by the ZIP model with the other settings as in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7 and Table 8, respectively. From Table 9 and Table 11, it can be observed that the bias values tend toward 0, further indicating that the estimators are unbiased. The values of the ESD and ASE are close to each other, and as the sample size increases, both the ASE and ESD values decrease correspondingly. The CP values are also around 95%. However, compared to Table 9, Table 10 shows larger overall biases, and some CP values deviate more from 95%. Comparing Table 12 with Table 11, a similar conclusion can also be drawn. They all reach a similar conclusion as Table 1, Table 2, Table 3 and Table 4. This indicates that the method proposed in this article is also robust for zero-inflated data. Therefore, the simulation results of this section suggest that the proposed inference methods performed well and better than the estimation without adjusting for confounding bias.

Table 9. Estimation of $\gamma$ and $\alpha$ estimated by the model with the propensity score, where simulated data were produced by the ZIP model with the propensity score and $Z\sim B(1,0.5)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	0.0012	0.0071	−0.0011	0.0062	0.0010	0.0069	0.0017	0.0081
ESD	0.1249	0.0477	0.1765	0.0647	0.0745	0.0278	0.1011	0.0379
ASE	0.1231	0.0440	0.1703	0.0629	0.0732	0.0264	0.1007	0.0366
CP	0.9458	0.9520	0.9391	0.9584	0.9470	0.9493	0.9491	0.9467
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0181	0.0092	−0.0025	0.0092	−0.0146	0.0085	−0.0021	0.0110
ESD	0.1351	0.0550	0.1820	0.0717	0.0781	0.0327	0.1055	0.0419
ASE	0.1305	0.0507	0.1748	0.0688	0.0775	0.0306	0.1034	0.0404
CP	0.9315	0.9483	0.9342	0.9598	0.9480	0.9320	0.9430	0.9443
$\gamma =0$, $\alpha =0.1$
BIAS	0.0027	−0.0191	−0.0017	0.0011	−0.0006	−0.0187	0.0003	0.0071
ESD	0.1243	0.1095	0.1581	0.1373	0.0709	0.0636	0.0905	0.0771
ASE	0.1194	0.1025	0.1538	0.1322	0.0703	0.0627	0.0906	0.0782
CP	0.9435	0.9275	0.9392	0.9394	0.9484	0.9136	0.9542	0.9503
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0321	−0.0159	−0.0320	0.0021	−0.0292	−0.0138	−0.0298	0.0037
ESD	0.1295	0.1115	0.1682	0.1364	0.0742	0.0649	0.0939	0.0806
ASE	0.1266	0.1077	0.1598	0.1368	0.0743	0.0642	0.0937	0.0811
CP	0.9332	0.9294	0.9356	0.9493	0.9265	0.9220	0.9326	0.9547

Table 10. Estimation of $\gamma$ and $\alpha$ estimated by the model without the propensity score, where simulated data were produced by the ZIP model with the propensity score and $Z\sim B(1,0.5)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	0.0013	0.0068	0.0042	0.0127	0.0008	0.0093	0.0012	0.0152
ESD	0.1277	0.0661	0.1772	0.1015	0.0748	0.0387	0.1019	0.0587
ASE	0.1230	0.0770	0.1772	0.1236	0.0730	0.0425	0.1005	0.0668
CP	0.9456	0.9798	0.9390	0.9770	0.9456	0.9755	0.9488	0.9770
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0181	0.0074	−0.0050	0.0134	−0.0164	0.0103	−0.0043	0.0151
ESD	0.1363	0.0659	0.1835	0.0955	0.0774	0.0378	0.1064	0.0540
ASE	0.1324	0.0737	0.1778	0.1129	0.0776	0.0410	0.1041	0.0625
CP	0.9377	0.9787	0.9330	0.9772	0.9404	0.9716	0.9452	0.9720
$\gamma =0$, $\alpha =0.1$
BIAS	0.0006	−0.0417	−0.0005	−0.0361	−0.0001	−0.0368	−0.0014	−0.0274
ESD	0.1364	0.0956	0.1851	0.1348	0.0796	0.0548	0.1111	0.0791
ASE	0.1296	0.1009	0.1788	0.1555	0.0779	0.0578	0.1060	0.0867
CP	0.9434	0.9166	0.9431	0.9837	0.9490	0.8910	0.9356	0.9586
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0244	−0.0456	0.0019	−0.0386	−0.0236	−0.0411	−0.0297	0.0006
ESD	0.1465	0.0932	0.1941	0.1265	0.0838	0.0577	0.1093	0.0742
ASE	0.1396	0.0988	0.1859	0.1473	0.0827	0.0570	0.1099	0.0826
CP	0.9354	0.9334	0.9355	0.9830	0.9360	0.8875	0.9495	0.9585

Table 11. Estimation of $\gamma$ and $\alpha$ estimated by the model with the propensity score, where simulated data were produced by the ZIP model without the propensity score and $Z\sim B(1,0.5)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	0.0017	0.0068	0.0016	0.0098	0.0002	0.0052	−0.0002	0.0076
ESD	0.1297	0.0419	0.1752	0.0662	0.0742	0.0278	0.1023	0.0374
ASE	0.1231	0.0440	0.1706	0.0628	0.0732	0.0264	0.1006	0.0367
CP	0.9352	0.9556	0.9424	0.9618	0.9438	0.9480	0.9470	0.9550
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0158	0.0071	0.0065	0.0080	−0.0133	0.0076	0.0061	0.0091
ESD	0.1378	0.0523	0.1694	0.0669	0.0795	0.0295	0.1058	0.0386
ASE	0.1310	0.0470	0.1759	0.0645	0.0778	0.0226	0.1047	0.0378
CP	0.9342	0.9511	0.9380	0.9600	0.9402	0.9526	0.9476	0.9554
$\gamma =0$, $\alpha =0.1$
BIAS	0.0019	0.0049	0.0029	0.0051	0.0001	0.0059	−0.0010	0.0071
ESD	0.1502	0.0820	0.1872	0.0989	0.0866	0.0478	0.1069	0.0578
ASE	0.1425	0.0743	0.1818	0.0942	0.0845	0.0458	0.1071	0.0564
CP	0.9354	0.9473	0.9400	0.9420	0.9486	0.9510	0.9486	0.9510
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0084	0.0024	−0.0035	0.0066	−0.0051	0.0061	−0.0027	0.0051
ESD	0.1559	0.0853	0.1945	0.1004	0.0884	0.0498	0.1106	0.0591
ASE	0.1492	0.0767	0.1871	0.0959	0.0883	0.0476	0.1107	0.0575
CP	0.9323	0.9589	0.9406	0.9372	0.9479	0.9374	0.9468	0.9410

Table 12. Estimation of $\gamma$ and $\alpha$ estimated by the model without the propensity score, where simulated data were produced by the ZIP model without the propensity score and $Z\sim B(1,0.5)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	−0.0018	0.0086	0.0019	0.0106	0.0004	0.0059	−0.0002	0.0139
ESD	0.1285	0.0667	0.1742	0.0988	0.0751	0.0392	0.1006	0.0576
ASE	0.1226	0.0769	0.1698	0.1215	0.0731	0.0425	0.1005	0.0667
CP	0.9301	0.9778	0.9510	0.9810	0.9440	0.9726	0.9532	0.9745
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0176	0.0073	0.0071	0.0156	−0.0143	0.0107	0.0027	0.0137
ESD	0.1371	0.0701	0.1819	0.1043	0.0789	0.0416	0.1076	0.0591
ASE	0.1317	0.0864	0.1779	0.1248	0.0779	0.0452	0.1049	0.0688
CP	0.9467	0.9751	0.9422	0.9814	0.9353	0.9691	0.9418	0.9761
$\gamma =0$, $\alpha =0.1$
BIAS	0.0006	−0.0205	−0.0038	−0.0169	0.0007	0.0192	−0.0032	−0.0137
ESD	0.1476	0.0617	0.1883	0.0814	0.0849	0.0345	0.1088	0.0481
ASE	0.1414	0.0719	0.1806	0.1019	0.0838	0.0412	0.1070	0.0579
CP	0.9336	0.9652	0.9396	0.9830	0.9433	0.9466	0.9444	0.9760
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0336	−0.0213	−0.0319	−0.0189	−0.0317	−0.0180	−0.0286	−0.0147
ESD	0.1520	0.0655	0.1895	0.0834	0.0729	0.0380	0.1104	0.0483
ASE	0.1492	0.0750	0.1881	0.1047	0.0879	0.0431	0.1105	0.0591
CP	0.9378	0.9610	0.9431	0.9837	0.9240	0.9408	0.9384	0.9770

Table 13. Estimation of $\gamma$ and $\alpha$ estimated by the model with the propensity score, where simulated data were produced by the ZIP model with the propensity score and $Z\sim N(0,0.25)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	0.0022	0.0066	−0.0129	0.0082	0.0010	0.0063	−0.0002	0.0079
ESD	0.1313	0.0492	0.1789	0.0631	0.0720	0.0285	0.1035	0.0364
ASE	0.1213	0.0521	0.1659	0.0755	0.0744	0.0278	0.0996	0.0411
CP	0.9378	0.9728	0.9423	0.9861	0.9452	0.9714	0.9556	0.9727
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0173	0.0076	−0.0005	0.0078	−0.0153	0.0068	0.0007	0.0087
ESD	0.1371	0.0484	0.1758	0.0662	0.0788	0.0281	0.1045	0.0368
ASE	0.1241	0.0514	0.1701	0.0746	0.0752	0.0287	0.1020	0.0407
CP	0.9413	0.9752	0.9296	0.9753	0.9218	0.9678	0.9443	0.9767
$\gamma =0$, $\alpha =0.1$
BIAS	0.0027	−0.0249	0.0048	−0.0048	0.0003	−0.0218	0.0005	−0.0037
ESD	0.1310	0.1002	0.1719	0.1280	0.0753	0.0592	0.0954	0.0742
ASE	0.1224	0.1054	0.1582	0.1424	0.0726	0.0624	0.0938	0.0831
CP	0.9335	0.9381	0.9507	0.9691	0.9501	0.9226	0.9404	0.9666
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0313	−0.0245	−0.0295	−0.0024	−0.0288	−0.0221	−0.0314	−0.0012
ESD	0.1252	0.1039	0.1625	0.1287	0.0713	0.0601	0.0929	0.0758
ASE	0.1185	0.1094	0.1537	0.1519	0.0707	0.0645	0.0908	0.0881
CP	0.9217	0.9435	0.9268	0.9780	0.9526	0.9366	0.9511	0.9754

Table 14. Estimation of $\gamma$ and $\alpha$ estimated by the model without the propensity score, where simulated data were produced by the ZIP model with the propensity score and $Z\sim N(0,0.25)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	0.0046	0.0017	−0.0040	0.0104	−0.0009	0.0086	0.0024	0.0141
ESD	0.1269	0.0674	0.1745	0.0982	0.0755	0.0388	0.1009	0.0589
ASE	0.1206	0.0857	0.1663	0.1372	0.0713	0.0453	0.0987	0.0734
CP	0.9350	0.9826	0.9317	0.9832	0.9501	0.9789	0.9475	0.9823
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0143	0.0071	0.0013	0.0075	−0.0139	0.0100	−0.0004	0.0161
ESD	0.1321	0.0671	0.1846	0.0977	0.0786	0.0383	0.1014	0.0575
ASE	0.1252	0.0839	0.1741	0.1341	0.0753	0.0447	0.1031	0.0715
CP	0.9378	0.9782	0.9258	0.9839	0.9211	0.9796	0.9523	0.9809
$\gamma =0$, $\alpha =0.1$
BIAS	−0.0016	−0.0417	−0.0052	−0.0355	−0.0008	−0.0381	0.0041	−0.0243
ESD	0.1335	0.0961	0.1878	0.1361	0.0783	0.0568	0.1073	0.0866
ASE	0.1268	0.1163	0.1748	0.1829	0.0772	0.0633	0.1046	0.0961
CP	0.9333	0.9847	0.9431	0.9843	0.9465	0.9152	0.9436	0.9731
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0356	−0.0421	0.0041	−0.0364	−0.0275	−0.0372	−0.0035	−0.0238
ESD	0.1398	0.0966	0.1903	0.1301	0.0796	0.0567	0.1116	0.0753
ASE	0.1325	0.1156	0.1791	0.1799	0.0788	0.0618	0.1076	0.0947
CP	0.9163	0.9785	0.9271	0.9863	0.9206	0.9172	0.9628	0.9816

Table 15. Estimation of $\gamma$ and $\alpha$ estimated by the model with the propensity score, where simulated data were produced by the ZIP model without the propensity score and $Z\sim N(0,0.25)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	−0.0033	0.0068	0.0047	0.0077	−0.0008	0.0063	0.0004	0.0095
ESD	0.1311	0.0477	0.1790	0.0637	0.0748	0.0279	0.1029	0.0468
ASE	0.1202	0.0514	0.1673	0.0754	0.0721	0.0286	0.1023	0.0415
CP	0.9330	0.9683	0.9312	0.9862	0.9498	0.9637	0.9468	0.9732
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0146	0.0056	0.0056	0.0087	−0.0124	0.0053	0.0085	0.0079
ESD	0.1348	0.0458	0.1845	0.0607	0.0786	0.0261	0.1054	0.0344
ASE	0.1232	0.0481	0.1708	0.0705	0.0748	0.0265	0.1022	0.0382
CP	0.9208	0.9813	0.9485	0.9722	0.9217	0.9685	0.9512	0.9728
$\gamma =0$, $\alpha =0.1$
BIAS	−0.0035	0.0059	−0.0021	0.0077	−0.0016	0.0049	0.0008	0.0089
ESD	0.1889	0.0981	0.1510	0.0821	0.0865	0.0482	0.1083	0.0577
ASE	0.1785	0.1105	0.1402	0.0847	0.0839	0.0496	0.1064	0.0637
CP	0.9375	0.9615	0.9392	0.9616	0.9472	0.9571	0.9476	0.9682
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0053	−0.0077	−0.0023	−0.0036	−0.0006	−0.0037	−0.0007	−0.0008
ESD	0.1546	0.0796	0.1944	0.0965	0.0888	0.0459	0.1117	0.0553
ASE	0.1409	0.0788	0.1819	0.1046	0.0856	0.0467	0.1085	0.0603
CP	0.9352	0.9474	0.9312	0.9665	0.9428	0.9533	0.9437	0.9616

Table 16. Estimation of $\gamma$ and $\alpha$ estimated by the model without the propensity score, where simulated data were produced by the ZIP model without the propensity score and $Z\sim N(0,0.25)$.

	n = 100		n = 100		n = 300		n = 300
	$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$		$\mathit{\tau }=\mathbf{1}$		$\mathit{\tau }=\mathbf{2}$
	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\gamma }}$	$\widehat{\mathit{\alpha }}$
$\gamma =0$, $\alpha =0$
BIAS	−0.0018	0.0089	0.0017	0.0108	−0.0007	0.0059	0.0023	0.0148
ESD	0.1289	0.0634	0.1794	0.0987	0.0753	0.0388	0.1027	0.0583
ASE	0.1201	0.0853	0.1667	0.1363	0.0721	0.0457	0.0997	0.0733
CP	0.9321	0.9863	0.9387	0.9815	0.9461	0.9782	0.9475	0.9813
$\gamma =0.1$, $\alpha =0$
BIAS	−0.0184	0.0068	0.0032	0.0097	−0.0178	0.0096	0.0006	0.0043
ESD	0.1355	0.0753	0.1855	0.1088	0.0781	0.0417	0.1053	0.0621
ASE	0.1249	0.0923	0.1721	0.1476	0.0747	0.0493	0.1029	0.0785
CP	0.9257	0.9841	0.9323	0.9842	0.9277	0.9836	0.9516	0.9847
$\gamma =0$, $\alpha =0.1$
BIAS	−0.0023	−0.0188	0.0017	−0.0137	0.0006	−0.0163	0.0019	−0.0135
ESD	0.1519	0.0551	0.1869	0.0733	0.0844	0.0324	0.1085	0.0406
ASE	0.1422	0.0695	0.1779	0.0778	0.0847	0.0397	0.1058	0.0548
CP	0.9351	0.9747	0.9303	0.9835	0.9507	0.9633	0.9451	0.9754
$\gamma =0.1$, $\alpha =0.1$
BIAS	−0.0304	−0.0208	−0.0285	−0.0182	−0.0273	−0.0185	−0.0254	−0.0142
ESD	0.1506	0.0641	0.1942	0.0867	0.0877	0.0386	0.1081	0.0513
ASE	0.1431	0.0783	0.1839	0.1141	0.0854	0.0453	0.1091	0.0638
CP	0.9427	0.9731	0.9267	0.9867	0.9254	0.9471	0.9383	0.9789

6. An Application

A notable instance of interval-censored recurrence event data comes from a study on bladder cancer carried out by the Veterans Administration Cooperative Urological Research Group [1,29]. Patients included in the study had superficial bladder tumors upon entry and were randomly assigned to one of three treatment groups: placebo, thiotepa, or pyridoxine. As pyridoxine treatment has proven ineffective in preventing bladder tumor recurrence, our attention will be directed towards the 47 patients in the placebo group and the 38 patients in the thiotepa-treated group.

We applied the methodology presented in this paper to the bladder cancer experiment. A sequence of clinical visit times and the number of tumors occurring between visits are included in the observed information for every patient. Bladder tumors were removed at the start of each visit. For the analysis, we define the covariate $X_{i1}=0$ for the subjects in the placebo group while $X_{i1}=1$ in the thiotepa-treated group, with $X_{i2}$ representing the number of initial tumors. We aimed to assess how thiotepa treatment, initial tumor number, and observation history impact the recurrence process.

The estimated regression parameters from the models with and without the propensity score are displayed in Table 17 and Table 18, respectively. The results in both tables indicate that thiotepa significantly reduced the recurrence rate of bladder tumors, and the recurrence rate was also significantly positively correlated with the initial number of bladder tumors. $\widehat{\alpha }$ represents the degree of influence of observation history on the recurrence process in these two tables, and the estimates of $\alpha$ with and without adjustment for the propensity score are 0.2855 and 0.0143, respectively. The 95% confidence intervals are (−0.0546, 0.6257) and (−0.0446, 0.0733), respectively. And the corresponding p-values are 0.0999 and 0.6358, respectively. This indicate that the estimation of $\alpha$ in Table 17 is more significant than that in Table 18, which is due to proposed model with the propensity score reducing the confounding bias caused by the observation process, thus improving the significance of the coefficient estimation.

Table 17. Estimation of parameters with adjusted by propensity score in the bladder tumor study.

$\widehat{{\gamma }_1}$	$\widehat{{\gamma }_2}$	$\widehat{\alpha }$
$SE\left(\widehat{{\gamma }_1}\right)$	$SE\left(\widehat{{\gamma }_2}\right)$	$SE\left(\widehat{\alpha }\right)$
$95\%\mathrm{CI}\left({\gamma }_1\right)$	$95\%\mathrm{CI}\left({\gamma }_2\right)$	$95\%\mathrm{CI}\left(\alpha \right)$
$-0.7959$	$0.1981$	$0.2855$
$0.1700$	$0.0536$	$0.1735$
$(-1.1291,-0.4627)$	$(0.0930,0.3032)$	$(-0.0546,0.6257)$

Table 18. Estimation of parameters without adjusted by propensity score in the bladder tumor study.

$\widehat{{\gamma }_1}$	$\widehat{{\gamma }_2}$	$\widehat{\alpha }$
$SE\left(\widehat{{\gamma }_1}\right)$	$SE\left(\widehat{{\gamma }_2}\right)$	$SE\left(\widehat{\alpha }\right)$
$95\%\mathrm{CI}\left({\gamma }_1\right)$	$95\%\mathrm{CI}\left({\gamma }_2\right)$	$95\%\mathrm{CI}\left(\alpha \right)$
−0.8333	0.1941	0.0143
0.1572	0.0552	0.0301
(−1.1414, −0.5252)	(0.0858, 0.3023)	(−0.0446, 0.0733)

7. Conclusions

This article explored the use of regression analysis for interval-censored recurrence event data, in which the observation process may contain relevant information about the underlying recurrence event. An additive semiparametric mean model of recurrence events was established. Note that this model has been discussed by many authors in the literature, but it seems no one has considered the confounding bias. For the problem, we introduced a propensity score to the additive semiparametric mean model to adjust the confounding bias. Furthermore, the estimation equations were used to estimate the parameters of the covariate effects, and the asymptotic normality of the resulting estimate under a large sample was established. Through simulation studies, we showed that our proposed model performs well and better than the model without adjusting for confounding bias. Finally, the method was applied to a study on bladder cancer.

Propensity score methods are being widely used to adjust confounding, and inverse probability of treatment weighting (IPTW) is one of the most important ways of using the propensity score to reduce confounding. We introduced the IPTW technique in this article to the semiparametric model in order to minimize confounding. In addition, the additive model proposed in this paper may be preferable to a multiplicative model since an additive model gives direct estimation of absolute differences.

In future research, we can introduce terminal events into the model, which often relate to recurrence events. The major difference between terminal events and typical right censoring is that once a terminal event occurs, recurrence events will cease permanently. Therefore, we can further consider statistical inference for interval-censored recurrence event data with informative observation times adjusted by propensity scores under the informative terminal event. In this paper, a semiparametric mean model was used for recurrence events, and in the future, we can further explore statistical inference for confounding bias adjustment using nonparametric methods. What is more, we assumed $H_i\left(t\right)$ is a nonhomogeneous Poisson process, but this assumption may not hold in some more complicated scenarios. As suggested by the literature [7], it is assumed that the observation process follows a general rate function. Therefore, further studies could consider conducting inference and simulation under the assumption that the observation process follows some general settings. In addition, for the weight function $W\left(t\right)$, it will be helpful if we can develop some methods that select the appropriate weight function for a given data set.