Expectation-Maximization Algorithm for the Weibull Proportional Hazard Model under Current Status Data

Abstract

Due to the flexibility of the Weibull distribution and the proportional hazard (PH) model, Weibull PH is widely used in survival analysis under right censored data and interval censored data but it is seldom investigated under current status data, partially because there is less information in current status data than in right censored data and interval censored data. This paper considers the Weibull PH model under the current status data and introduces the Poisson latent variables to augment the data, then uses the expectation-maximization (EM) algorithm to obtain the maximum likelihood estimators of the model parameters. The EM algorithm is compared with the Newton–Raphson (NR) algorithm from several perspectives in the simulation studies, and the results show that the proposed method has several highlights, such as computational simplicity, improved convergence stability, and overall estimator results that are either comparable or slightly better in terms of bias. Furthermore, the performance of the Weibull PH model and the semi-parametric PH model is compared under two simulation scenarios, and two standard model selection criteria are used for model selection. The results indicate that the Weibull PH model has significant advantages when failure time follows a Weibull distribution. Lastly, the Weibull PH model along with EM algorithm is applied to lung tumor data and intraocular lens (IOL) calcification data with the aim of assessing the impact of covariates, including environmental factors and gender, on event timing and risk.

1. Introduction

The proportional hazard (PH) model proposed by Cox [1] has gradually gained popularity as one of the most widely used models in survival analysis. The PH model is a semi-parametric model consisting of two parts: the parametric part assuming that the explanatory variables have exponentially multiplicative effects on the hazard function of survival time, and the non-parametric part with an unspecified baseline hazard function. The two-parameter Weibull distribution [2] can have increasing, decreasing, or constant hazard function depending on the shape parameter; thus, it is flexible to describe hazard functions with different shapes. When the baseline hazard function is specified as the Weibull distribution, we obtain the Weibull PH model, which has wide applications in survival analysis and reliability analysis. For example, Alakuş [3] estimated confidence intervals for the survival function of the Weibull PH model with censored survival time data; Gong and Fang [4] investigated the performance of the Weibull PH model, exponential PH model, and 10-piece exponential PH model under interval censored data with different underlying data distributions and censoring patterns, and they advocated the use of a parametric PH model to analyze interval censored data; Sha and Pan [5] used the Weibull PH model to analysis the step-stress accelerated life testing data in a Bayesian framework; Nemati et al. [6] used the Weibull PH model to assess the impact of different factors on the failure rate of cables; Liu and Xie [7] estimated the parameters of the Weibull PH model for right censored data. Few studies on Weibull PH modeling concerned current status data.

In survival analysis, current status censoring also refers to Case I interval censoring, which is an extreme case of interval censoring. It happens if each subject is observed only once and the survival time of interest is known only to be either smaller or greater than the observation time. Sun [8] gave a detailed introduction to the estimation theories and applications for current status data. Current status data often appear along with covariates in cross-sectional studies and tumorigenicity experiments. Under current status censoring, the event times are left censored for events prior to the examination and are otherwise right censored. The main interest in such studies includes estimating the distribution of the lifetime and evaluating the effect of covariates on the lifetime or hazard of failure. A major challenge in the analysis of current status data is that information on lifetime for an individual is limited to the status of the event under consideration at a single monitoring time.

Compared to right censored data, current status data is less informative, and current statistical techniques for right censored data cannot be directly used to analyze current status data. So, it is necessary to develop new techniques to efficiently deal with these kinds of data. Recently, McMahan et al. [9] introduced a two-stage Poisson data augmentation method for estimating a PH model and proportional odds model with current status data. This approach was characterized by its simplicity and high effectiveness; furthermore, it provided closed-form variance estimators. Subsequently, numerous scholars applied this methodology to various models and data types. For example, Wang et al. [10] extended it to a PH model with interval censored data; Zhou et al. [11] discussed the problem of fitting a PH model to interval censored data with missing covariates; Withana et al. [12] analyzed a left-truncated arbitrarily censored data under PH model; Cui and Tee [13] analyzed the Bayesian additive PH model for the current status data. Most of the literature that used this Poisson data augmentation method to study the PH model estimated the cumulative baseline hazard function using splines, such as monotone spline, I-splines, and B-spline. However, it is more reasonable to utilize the corresponding parametric model when the baseline hazard function can be approximated by a flexible parameter distribution.

In this paper, we focus on the Weibull PH model under current status data with the baseline hazard function characterized by the popular Weibull distribution, and discuss the maximum likelihood estimators (MLE) of the model parameters. Instead of directly maximizing the observed likelihood function with the Newton–Raphson (NR) method, we propose a one-step Poisson data augmentation expectation-maximization (EM) algorithm inspired by McMahan et al. [9]. This methodology introduces Poisson latent variables to augment the data; thus, it establishes a missing data structure and produces a simpler complete likelihood. The procedure simplifies the likelihood structure and improves the computation efficiency as well as the convergence speed of the algorithm. Simultaneously, the variance estimators can be provided in closed form by Louis’ method [14]. We evaluate the performance of the EM algorithm through simulation studies involving various sample sizes and censoring ratios, and compare the results with the ones from the NR method. In summary, the EM method in this study yields estimator results comparable to those of the NR method, with the advantage of simplicity and computational convenience. Furthermore, it guarantees that the scale parameter remains positive, contributing to slightly improved convergence. In addition, we compare the Weibull PH model with the I-spline-based semi-parametric PH model through parameter estimation and model selection. Finally, we apply the Weibull PH model with EM algorithm to analyze lung tumor experimental data and IOL calcification data, and obtain some interesting findings.

2. The Proposed Method

2.1. The Weibull PH Model

The PH model is one of the most popular models in survival analysis, with a multiplicative effect of its covariates on the hazard function $\lambda (t,\mathit{x})$ with

$$\lambda \left(t;\mathit{x}\right)={\lambda }_0\left(t\right)exp\left({\mathit{x}}^{\prime }\mathit{\beta }\right),$$

where ${\lambda }_0\left(t\right)$ is the baseline hazard function, $\mathit{x}={\left(x_1,\cdots ,x_p\right)}^{\prime }$, and $\mathit{\beta }={\left({\beta }_1,\cdots ,{\beta }_p\right)}^{\prime }$. When the baseline hazard function is unknown, it is often estimated by splines. However, when its distribution is known, the use of the corresponding specific parameter model makes the results more precise. The two-parameter Weibull model is a popular parametric model in survival analysis with the flexible hazard function

$${\lambda }_0\left(t\right)=\lambda \gamma {\left(\lambda t\right)}^{\gamma -1},$$

(1)

where scale parameter $\lambda >0$ and shape parameter $\gamma >0$. Obviously, this hazard function is monotone decreasing for $\gamma <1$ and increasing for $\gamma <1$, and it degenerates to an exponential distribution with a scale parameter $\lambda$ when $\gamma =1$. In this paper, we assume that the baseline hazard function comes from the two-parameter Weibull distribution which is given by (1), and discuss the Weibull PH model

$$\lambda \left(t;\mathit{x}\right)=\lambda \gamma {\left(\lambda t\right)}^{\gamma -1}exp\left({\mathit{x}}^{\prime }\mathit{\beta }\right).$$

The distribution function and the survival function of the failure time T have the following forms

$$F\left(t;\mathit{x}\right)=1-exp\left\{-{\left(\lambda t\right)}^{\gamma }exp\left({\mathit{x}}^{\prime }\mathit{\beta }\right)\right\},$$

and

$$S\left(t;\mathit{x}\right)=exp\left\{-{\left(\lambda t\right)}^{\gamma }exp\left({\mathit{x}}^{\prime }\mathit{\beta }\right)\right\}.$$

Assume that there are n independently observed samples. Let $T_i$, $C_i$ and ${\mathit{x}}_i$ denote the failure time, censoring time, and covariate for subject i, respectively. In current status data, the failure time $T_i$ is not directly observable; we only know whether $T_i$ is larger than the observation time $C_i$. Let ${\delta }_i=I\left(T_i\le C_i\right)$ denote the censoring indicator, where $I(·)$ is the indicator function. When ${\delta }_i=1$, it indicates that the failure time is smaller than the observation time, which is left censored. Conversely, it is right censored in the case ${\delta }_i=0$. Let $\mathit{D}=\left\{D_i=\left(C_i,{\delta }_i,{\mathit{x}}_i\right),i=1,\cdots ,n\right\}$ denote all observed data and $\mathit{\theta }={\left({\mathit{\beta }}^{\prime },\gamma ,\lambda \right)}^{\prime }$ denote all parameters. Then, the observed data likelihood is given by

$$\begin{array}{cc}\hfill L_{obs}\left(\mathit{\theta }|\mathit{D}\right)=& \prod _{i=1}^{n}F{\left(C_i;{\mathit{x}}_i\right)}^{{\delta }_i}{\left[1-F\left(C_i;{\mathit{x}}_i\right)\right]}^{1-{\delta }_i}\hfill \\ \hfill =& \prod _{i=1}^n{\left[1-exp\left\{-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right\}\right]}^{{\delta }_i}\hfill \\ \hfill & \times exp\left\{-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\left(1-{\delta }_i\right)\right\}.\hfill \end{array}$$

Correspondingly, the observed data log-likelihood is given by

$$log{L}_{obs}\left(\mathit{\theta }|\mathit{D}\right)=\sum _{i=1}^n\left\{{\delta }_{i}log\left[1-exp\left\{-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right\}\right]-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\left(1-{\delta }_i\right)\right\}.$$

(2)

The NR method obtains the MLEs of the parameters by directly maximizing the above equation, and the corresponding derivation procedure is shown in Appendix A. It can be seen that the method is complicated in the calculation process, the expression is tedious, and there is no closed-form solution for all three parameters.

2.2. Data Augmentation

Motivated by McMahan et al. [9] and Wang et al. [10], we introduce Poisson latent variables to augment the data and obtain the complete data likelihood by considering the relationship between the PH model and the non-homogeneous Poisson process. According to the idea of Wang et al. [10], consider a non-homogenous Poisson process $N\left(t\right)$ with mean parameter ${\Lambda }_0\left(t\right)\exp \left({\mathit{x}}^{\prime }\mathit{\beta }\right)$, where ${\Lambda }_0\left(t\right)$ is the cumulative baseline hazard function, and in the Weibull situation, ${\Lambda }_0\left(t\right)={\left(\lambda t\right)}^{\gamma }$. The lifetime T can be viewed as the time of the first jump of this process, i.e., $T=\inf \{t:N(t)>0\}$; then, the survival function of T is

$$S(t;\mathit{x})=P(T>t)=P(N\left(t\right)=0)=\exp \{-{\Lambda }_0\left(t\right)\exp \left({\mathit{x}}^{\prime }\mathit{\beta }\right)\}=1-F(t;\mathit{x});$$

thus, T follows the PH model. This equivalence between the Poisson process and PH model makes the augmentation of the likelihood feasible. In the Weibull PH model, let $N_i\left(t\right)$ denote the latent Poisson process for subject i with mean parameter ${\left(\lambda C_i\right)}^{\gamma }\exp \left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)$, and lifetime $T_i=\inf \{t:N_i\left(t\right)>0\}$. We introduce latent variable $Z_i=N_i\left(C_i\right)$, then

$$P(Z_i=0)=P(N\left(C_i\right)=0)=P(T_i>C_i)=\exp \{-{\left(\lambda C_i\right)}^{\gamma }\exp \left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\}=S(C_i;{\mathit{x}}_i),$$

and

$$P(Z_i>0)=P(N\left(C_i\right)>0)=P(T_i\le C_i)=1-\exp \{-{\left(\lambda C_i\right)}^{\gamma }\exp \left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\}=F(C_i;{\mathit{x}}_i);$$

therefore, $Z_i$ follows the Poisson distribution with mean ${\left(\lambda C_i\right)}^{\gamma }\exp \left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)$, that is, $Z_i\sim \mathrm{Poisson}\left({\left(\lambda C_i\right)}^{\gamma }\exp \left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right)$, and as a byproduct, we have ${\delta }_i=\mathrm{I}(Z_i>0)$.

Based on the latent variable $\mathit{Z}={(Z_1,\cdots ,Z_n)}^{\prime }$, the augmented likelihood has the following form

$$\begin{array}{cc}\hfill L_{*}\left(\mathit{\theta }|\mathit{D}\right)=& \prod _{i=1}^n{\delta }_i^{I\left(Z_i>0\right)}{\left(1-{\delta }_i\right)}^{I\left(Z_i=0\right)}\hfill \\ \hfill & \times {\left\{{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right\}}^{Z_i}exp\left\{-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right\}/Z_i!.\hfill \end{array}$$

Its corresponding log-likelihood is

$$log{L}_{*}\left(\mathit{\theta }|\mathit{D}\right)=\sum _{i=1}^n\left\{Z_i\left[\gamma log\left(\lambda C_i\right)+{\mathit{x}}_i^{\prime }\mathit{\beta }\right]-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)-log\left(Z_i!\right)\right\}.$$

(3)

According to the augmented likelihood, the conditional distribution of $Z_i$ can be obtained as

$$P(Z_i=z_i|\mathit{D},\mathit{\theta })=\left\{\begin{array}{cc}0,\hfill & \hfill {\delta }_i=0,\\ \frac{{\left\{{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right\}}^{z_i}exp\left\{-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right\}}{z_i!},\hfill & \hfill {\delta }_i=1.\end{array}\right.$$

(4)

Then, the conditional expectation and conditional variance of $Z_i$ can be easily calculated, which are given by

$$\mathrm{E}\left(Z_i|\mathit{\theta },\mathit{D}\right)=\frac{{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right){\delta }_i}{1-exp\{-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\}},$$

(5)

and

$$\mathrm{Var}\left(Z_i|\mathit{\theta },\mathit{D}\right)=\mathrm{E}\left(Z_i|\mathit{\theta },\mathit{D}\right)-{\left(\mathrm{E}\left(Z_i|\mathit{\theta },\mathit{D}\right)\right)}^{2}exp\{-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\}.$$

(6)

2.3. EM Algorithm

The EM algorithm is an iterative algorithm used to obtain MLEs of model parameters when the model has a missing data structure [15]. The EM algorithm in this paper operates on the augmented likelihood by treating it as the complete data likelihood, where Z is viewed as missing data. The E-step is to compute the expectation of $log{L}_{*}$ with respect to the conditional distribution of Z given the current iteration values and the observed data. According to Equation (3), the expression for the Q function can be obtained as

$$\begin{array}{cc}\hfill Q\left(\mathit{\theta }|{\mathit{\theta }}^{\left(d\right)},\mathit{D}\right)=& \mathrm{E}\left(log\left(L_{*}\left(\mathit{\theta }\right)\right)|{\mathit{\theta }}^{\left(d\right)},\mathit{D}\right)\hfill \\ \hfill =& \sum _{i=1}^n\left\{u_i^{\left(d\right)}\left[\gamma log\left(\lambda C_i\right)+{\mathit{x}}_i^{\prime }\mathit{\beta }\right]-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)-\mathrm{E}\left(log\left(Z_i!\right)\right)\right\},\hfill \end{array}$$

where $u_i^{\left(d\right)}=\mathrm{E}\left(Z_i|{\mathit{\theta }}^{\left(d\right)},\mathit{D}\right)$ is obtained by (6) with $\mathit{\theta }$ replaced by ${\mathit{\theta }}^{\left(d\right)}$.

The M-step seeks to find ${\mathit{\theta }}^{(d+1)}=arg\underset{\mathit{\theta }}{max}Q\left(\mathit{\theta }|{\mathit{\theta }}^{\left(d\right)},\mathit{D}\right)$. For this purpose, we need to derive the first-order partial derivatives of $Q\left(\mathit{\theta }|{\mathit{\theta }}^{\left(d\right)},\mathit{D}\right)$ with respect to all components of $\mathit{\theta }$.

$$\begin{array}{cc}\hfill \frac{\partial Q}{\partial \lambda }& =\sum _{i=1}^n\left[u_i^{\left(d\right)}\frac{\gamma }{\lambda }-\gamma {\lambda }^{\gamma -1}{C}_i^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right],\hfill \\ \hfill \frac{\partial Q}{\partial \gamma }& =\sum _{i=1}^n\left[u_i^{\left(d\right)}log\left(\lambda C_i\right)-{\left(\lambda C_i\right)}^{\gamma }log\left(\lambda C_i\right)exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right],\hfill \\ \hfill \frac{\partial Q}{\partial \mathit{\beta }}& =\sum _{i=1}^n\left[u_i^{\left(d\right)}{\mathit{x}}_i-{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right){\mathit{x}}_i\right].\hfill \end{array}$$

Setting these derivatives equal to zero, we can obtain

$$\lambda ={\left(\frac{{\sum }_{i=1}^n{u}_i^{\left(d\right)}}{{\sum }_{i=1}^n{C}_i^{{\gamma }^{\left(d\right)}}exp\left({\mathit{x}}_i^{\prime }{\mathit{\beta }}^{\left(d\right)}\right)}\right)}^{\frac{1}{{\gamma }^{\left(d\right)}}},$$

(7)

$$\sum _{i=1}^n{u}_i^{\left(d\right)}log\left(\lambda C_i\right)=\sum _{i=1}^n{\left(\lambda C_i\right)}^{\gamma }log\left(\lambda C_i\right)exp\left({\mathit{x}}_i^{\prime }{\mathit{\beta }}^{\left(d\right)}\right),$$

(8)

$$\sum _{i=1}^n{u}_i^{\left(d\right)}{\mathit{x}}_i=\sum _{i=1}^n{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right){\mathit{x}}_i.$$

(9)

These equations are considerably simple and easy to solve compared to Appendix A. Specifically, Equation (7) provides an explicit solution for $\lambda$ and guarantees that $\lambda$ is positive, while the NR method has no explicit solution for any of the three parameters. Then, we summarize the EM algorithm as follows:

Step 1: 

Initialize ${\mathit{\theta }}^{\left(d\right)}={\left({\mathit{\beta }}^{\left(d\right)\prime },{\gamma }^{\left(d\right)},{\lambda }^{\left(d\right)}\right)}^{\prime }$ for $d=0$.

Step 2: 

Calculate ${\lambda }^{\left(d+1\right)}={\left(\frac{{\sum }_{i=1}^n{u}_i^{\left(d\right)}}{{\sum }_{i=1}^n{C}_i^{{\gamma }^{\left(d\right)}}exp\left({\mathit{x}}_i^{\prime }{\mathit{\beta }}^{\left(d\right)}\right)}\right)}^{\frac{1}{{\gamma }^{\left(d\right)}}}.$

Step 3: 

Based on the ${\lambda }^{(d+1)}$, calculate ${\gamma }^{(d+1)}$ by solving the Equation (8) via NR method.

Step 4: 

Based on ${\lambda }^{(d+1)}$, ${\gamma }^{(d+1)}$ and NR method, calculate ${\mathit{\beta }}^{\left(d+1\right)}$ according to the Equation (9), and update $d=d+1$.

Step 5: 

Repeat Steps 2–4 until convergence.

Denote $\widehat{\mathit{\theta }}$ as the final convergence value, that is, the MLE of $\mathit{\theta }$.

2.4. Asymptotic Variances and Covariance

According to the asymptotic normality of the MLE, we have $\widehat{\mathit{\theta }}\sim N\left(0,I^{-1}\left(\widehat{\mathit{\theta }}\right)\right)$, where $I\left(\mathit{\theta }\right)$ is the information matrix of the observed likelihood, and Appendix A gives the calculation of $I\left(\mathit{\theta }\right)$. Within the EM framework, Louis’ method [14] gives a closed-form expression for the observed information matrix in a simple and straightforward way, that is,

$$I\left(\mathit{\theta }\right)=-\frac{{\partial }^{2}Q\left(\mathit{\theta }\right|\widehat{\mathit{\theta }},\mathit{D})}{\partial \mathit{\theta }\partial {\mathit{\theta }}^{\prime }}-\mathrm{Cov}\left\{\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \mathit{\theta }}\right\},$$

where $\mathrm{Cov}\left\{\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \mathit{\theta }}\right\}$ represents $\mathrm{Cov}\left\{\left.\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \mathit{\theta }}\right|\mathit{D},\mathit{\theta }\right\}$. The second-order derivatives of Q-function with respect to $\mathit{\theta }$ are a symmetric matrix with upper triangle components given by

$$\begin{array}{cc}\hfill \frac{{\partial }^{2}Q}{{\partial }^2\lambda }& =-\sum _{i=1}^n\left[{\widehat{u}}_i\frac{\gamma }{{\lambda }^2}+\gamma \left(\gamma -1\right){\lambda }^{\gamma -2}{C}_i^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right],\hfill \\ \hfill \frac{{\partial }^{2}Q}{{\partial }^2\gamma }& =-\sum _{i=1}^n{\left(\lambda C_i\right)}^{\gamma }{\left(log\left(\lambda C_i\right)\right)}^{2}exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right),\hfill \\ \hfill \frac{{\partial }^{2}Q}{\partial \mathit{\beta }\partial {\mathit{\beta }}^{\prime }}& =-\sum _{i=1}^n{\left(\lambda C_i\right)}^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right){\mathit{x}}_i{\mathit{x}}_i^{\prime },\hfill \\ \hfill \frac{{\partial }^{2}Q}{\partial \lambda \partial \gamma }& =\sum _{i=1}^n\left[{\widehat{u}}_i\frac{1}{\lambda }-{\lambda }^{\gamma -1}{C}_i^{\gamma }\left(1+\gamma log\left(\lambda C_i\right)\right)exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right)\right],\hfill \\ \hfill \frac{{\partial }^{2}Q}{\partial \lambda \partial {\mathit{\beta }}^{\prime }}& =-\sum _{i=1}^n\gamma {\lambda }^{\gamma -1}{C}_i^{\gamma }exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right){\mathit{x}}_i^{\prime },\hfill \\ \hfill \frac{{\partial }^{2}Q}{\partial \gamma \partial {\mathit{\beta }}^{\prime }}& =-\sum _{i=1}^n{\left(\lambda C_i\right)}^{\gamma }log\left(\lambda C_i\right)exp\left({\mathit{x}}_i^{\prime }\mathit{\beta }\right){\mathit{x}}_i^{\prime },\hfill \end{array}$$

where ${\widehat{u}}_i=\frac{{\left(\widehat{\lambda }{C}_i\right)}^{\widehat{\gamma }}exp\left({\mathit{x}}_i^{\prime }\widehat{\mathit{\beta }}\right){\delta }_i}{1-exp\{-{\left(\widehat{\lambda }{C}_i\right)}^{\widehat{\gamma }}exp\left({\mathit{x}}_i^{\prime }\widehat{\mathit{\beta }}\right)\}}$. To this end, consider the conditional covariance matrix $\mathrm{Cov}\left\{\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \mathit{\theta }}\right\}$, whose upper triangle components are given by

$$\begin{array}{cc}\hfill & \mathrm{Var}\left(\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \lambda }\right)={\left(\frac{\gamma }{\lambda }\right)}^2\sum _{i=1}^n\mathrm{Var}\left(Z_i\right),\hfill \\ \hfill & \mathrm{Var}\left(\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \gamma }\right)=\sum _{i=1}^n\mathrm{Var}\left(Z_i\right){\left(log\left(\lambda C_i\right)\right)}^2,\hfill \\ \hfill & \mathrm{Cov}\left(\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \mathit{\beta }}\right)=\sum _{i=1}^n\mathrm{Var}\left(Z_i\right){\mathit{x}}_i{\mathit{x}}_i^{\prime },\hfill \\ \hfill & \mathrm{Cov}\left(\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \lambda },\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \gamma }\right)=\frac{\gamma }{\lambda }\sum _{i=1}^n\mathrm{Var}\left(Z_i\right)log\left(\lambda C_i\right),\hfill \\ \hfill & \mathrm{Cov}\left(\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \lambda },\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \mathit{\beta }}\right)=\frac{\gamma }{\lambda }\sum _{i=1}^n\mathrm{Var}\left(Z_i\right){\mathit{x}}_i^{\prime },\hfill \\ \hfill & \mathrm{Cov}\left(\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \gamma },\frac{\partial log{L}_{*}\left(\mathit{\theta }\right)}{\partial \mathit{\beta }}\right)=\sum _{i=1}^n\mathrm{Var}\left(Z_i\right){\mathit{x}}_i^{\prime }log\left(\lambda C_i\right),\hfill \end{array}$$

where $\mathrm{Var}\left(Z_i\right)$ denotes $\mathrm{Var}\left(Z_i|\mathit{\theta },\mathit{D}\right)$ given by (6). Apparently, the EM algorithm is simpler and has a more concise expression regarding the computation of the covariance matrix than the NR method in Appendix A.

3. Simulation Study

In this section, a series of simulations are used to evaluate the performance of the proposed method. We compare the proposed EM method with the NR method and compare the Weibull PH model with the semi-parametric PH model.

3.1. Simulation Study I: Compare the Proposed EM Method with the NR Method

This study considers that the failure time $T_i$ is generated from the following Weibull PH model

$$F\left(t;{\mathit{x}}_i\right)=1-exp\left\{-{\left(\lambda t\right)}^{\gamma }exp\left(x_{i1}{\beta }_1+x_{i2}{\beta }_2\right)\right\},$$

where ${\mathit{x}}_i={\left(x_{i1},x_{i2}\right)}^{\prime }$, $x_{i1}\sim \mathrm{Bernoulli}\left(0.5\right)$, $x_{i2}\sim \mathrm{N}\left(0,0.5^2\right)$, $i=1,\cdots ,n$, $\lambda =3$, $\gamma =2$.

In the simulation, the sample sizes are set to be $n=30,60,$ 100, 200 and 400. Both ${\beta }_1$ and ${\beta }_2$ can be taken as $-0.5$, 0 and $0.5$, totaling nine parameter combinations, and the left censoring ratio P is set as $0.3$, $0.4$, and $0.5$, respectively. The failure time $T_i$ can be obtained by solving $F\left(t;{\mathit{x}}_i\right)=u_i$, where $u_i\sim \mathrm{U}\left(0,1\right)$, and the censoring time $C_i$ obeys $\mathrm{U}\left(0,a\right)$, where a is calculated according to the censoring ratio.

Each simulation is repeated 1000 times, with an initial value ${\mathit{\theta }}^{\left(0\right)}={\left({\mathit{\beta }}^{{\left(0\right)}^{\prime }},{\lambda }^{\left(0\right)},{\gamma }^{\left(0\right)}\right)}^{\prime }={\left({{\mathbf{1}}_{\mathbf{2}}}^{\prime },1,1\right)}^{\prime }$, where ${\mathbf{1}}_{\mathbf{2}}={(1,1)}^{\prime }$. Then, the algorithm is recognized to have reached convergence when $max|{\mathit{\theta }}^{\left(d+1\right)}-{\mathit{\theta }}^{\left(d\right)}|<0.005$. The bias (Bias), the estimated standard errors (ESE), the standard deviation (SD), and the coverage probabilities (CP) are used as indicators of the goodness of the estimators, where the Bias is the absolute value of the difference between the estimated value and the true value, the ESE is the average of the estimated standard errors obtained by squaring the diagonal elements of the asymptotic covariance matrix computed by Louis’ method, the SD is the standard deviation of the 1000 estimates of $\mathit{\beta }$, and the CP is the proportion of the true value that falls into the $95\%$ confidence interval. Here, we compare the results of the proposed EM method with the ones based on the NR algorithm. All of the above simulations are implemented in R [16]. Table 1, Table 2, Table 3, Table 4 and Table 5 are the estimate results, from which we can draw the following conclusion:

Table 1. Comparison of the EM method and the NR method with different censoring ratios at $n=30$.

			EM				NR
			Bias	ESE	SD	CP	Bias	ESE	SD	CP
P = 0.3	${\beta }_1$	−0.5	0.0830	1.8440	2.2309	0.959	0.0836	1.9107	2.4515	0.968
	${\beta }_2$	−0.5	0.0484	1.0089	1.0913	0.969	0.0328	1.2368	1.2633	0.970
	${\beta }_1$	−0.5	0.2092	1.9040	2.3507	0.961	0.2147	1.9659	2.5252	0.969
	${\beta }_2$	0	0.0206	0.9289	1.0585	0.962	0.0262	1.0142	1.1342	0.971
	${\beta }_1$	−0.5	0.1791	1.8637	2.2418	0.975	0.2132	1.9850	2.5868	0.981
	${\beta }_2$	0.5	0.1805	1.0694	1.1345	0.966	0.2178	1.5543	1.3574	0.970
	${\beta }_1$	0	0.0429	1.7708	2.1292	0.966	0.0392	1.8299	2.3199	0.973
	${\beta }_2$	−0.5	0.0800	0.9298	1.0756	0.968	0.0849	0.9939	1.1996	0.972
	${\beta }_1$	0	0.0028	1.8364	2.2490	0.961	0.0280	1.9182	2.5170	0.973
	${\beta }_2$	0	0.0007	0.8648	0.9720	0.963	0.0011	0.8916	1.0309	0.971
	${\beta }_1$	0	0.0102	1.8739	2.2762	0.970	0.0048	1.9260	2.4167	0.975
	${\beta }_2$	0.5	0.2084	0.9739	1.0988	0.966	0.2233	1.0044	1.1707	0.971
	${\beta }_1$	0.5	0.2492	1.8562	2.2518	0.966	0.2433	1.9033	2.3463	0.972
	${\beta }_2$	−0.5	0.1610	0.9248	1.0719	0.961	0.1584	0.9628	1.1288	0.970
	${\beta }_1$	0.5	0.1354	1.8649	2.3146	0.954	0.1446	1.9187	2.4616	0.964
	${\beta }_2$	0	0.0122	0.8776	0.9651	0.959	0.0095	0.9147	1.1997	0.962
	${\beta }_1$	0.5	0.2410	1.9257	2.3118	0.962	0.2821	2.0136	2.6029	0.969
	${\beta }_2$	0.5	0.2649	0.9864	1.0996	0.974	0.2984	1.0783	1.2327	0.981
P = 0.4	${\beta }_1$	−0.5	0.1729	1.7020	2.0879	0.963	0.2270	1.8282	2.5046	0.977
	${\beta }_2$	−0.5	0.1813	0.8450	0.9767	0.956	0.2266	1.0621	1.3028	0.972
	${\beta }_1$	−0.5	0.2832	1.7186	2.0628	0.964	0.3393	1.8398	2.3169	0.973
	${\beta }_2$	0	0.0239	0.8061	0.9461	0.954	0.0420	0.8696	1.0821	0.959
	${\beta }_1$	−0.5	0.1104	1.7200	2.0367	0.960	0.1262	1.8137	2.2480	0.973
	${\beta }_2$	0.5	0.1507	0.8318	0.9381	0.970	0.1857	0.8719	1.0466	0.977
	${\beta }_1$	0	0.1023	1.6709	1.9692	0.957	0.1306	1.7874	2.2489	0.973
	${\beta }_2$	−0.5	0.1277	0.8259	0.9891	0.956	0.1544	0.9196	1.1636	0.970
	${\beta }_1$	0	0.0474	1.6652	2.0079	0.954	0.0963	1.8191	2.5094	0.971
	${\beta }_2$	0	0.0393	0.7970	0.9040	0.969	0.0522	0.8661	1.0923	0.976
	${\beta }_1$	0	0.0390	1.6845	1.9040	0.968	0.0403	1.8056	2.2031	0.978
	${\beta }_2$	0.5	0.1673	0.8339	0.9646	0.957	0.2160	0.8931	1.1492	0.966
	${\beta }_1$	0.5	0.1590	1.6926	1.9928	0.962	0.1831	1.7901	2.2137	0.979
	${\beta }_2$	−0.5	0.1838	0.8250	0.9279	0.963	0.1990	0.8734	1.0495	0.975
	${\beta }_1$	0.5	0.2011	1.6773	1.9646	0.966	0.2285	1.7751	2.1901	0.976
	${\beta }_2$	0	0.0177	0.7909	0.8990	0.971	0.0346	0.8421	1.0441	0.977
	${\beta }_1$	0.5	0.1129	1.7145	2.1417	0.960	0.1533	1.8410	2.5154	0.971
	${\beta }_2$	0.5	0.1499	0.8264	0.9115	0.972	0.2068	0.8921	1.1331	0.983
P = 0.5	${\beta }_1$	−0.5	0.1404	1.6381	1.9355	0.956	0.1861	1.9337	2.9388	0.983
	${\beta }_2$	−0.5	0.1871	0.7933	0.8984	0.964	0.2680	0.9546	1.4432	0.980
	${\beta }_1$	−0.5	0.2128	1.6384	1.8534	0.962	0.2979	1.9294	2.6214	0.982
	${\beta }_2$	0	0.0253	0.7582	0.8298	0.959	0.0106	0.8936	1.2374	0.977
	${\beta }_1$	−0.5	0.0929	1.6298	1.7967	0.966	0.1344	1.8299	2.2020	0.979
	${\beta }_2$	0.5	0.0989	0.8058	0.8608	0.966	0.1651	0.9246	1.1044	0.980
	${\beta }_1$	0	0.0444	1.5998	1.8181	0.965	0.0435	1.8125	2.2458	0.983
	${\beta }_2$	−0.5	0.1861	0.7997	0.8878	0.969	0.2333	0.9375	1.2308	0.983
	${\beta }_1$	0	0.0304	1.6357	1.8762	0.956	0.0364	1.9037	2.5094	0.977
	${\beta }_2$	0	0.0535	0.7591	0.8195	0.953	0.0039	1.0509	1.1162	0.972
	${\beta }_1$	0	0.0308	1.6139	1.7789	0.956	0.0331	1.8368	2.3188	0.979
	${\beta }_2$	0.5	0.0903	0.8036	0.8984	0.970	0.1933	0.9571	1.2448	0.982
	${\beta }_1$	0.5	0.1448	1.6689	1.9246	0.964	0.2251	1.9309	2.7892	0.983
	${\beta }_2$	−0.5	0.1694	0.8016	0.8754	0.964	0.1987	0.9200	1.1835	0.977
	${\beta }_1$	0.5	0.1584	1.6408	1.8934	0.962	0.2116	1.8851	2.4036	0.979
	${\beta }_2$	0	0.0426	0.7553	0.8390	0.968	0.0198	0.8727	1.1649	0.978
	${\beta }_1$	0.5	0.1499	1.6196	1.8070	0.970	0.1770	1.8445	2.2799	0.984
	${\beta }_2$	0.5	0.0900	0.8012	0.8667	0.962	0.1762	0.9118	1.1601	0.977

Table 2. Comparison of the EM method and the NR method with different censoring ratios at $n=60$.

			EM				NR
			Bias	ESE	SD	CP	Bias	ESE	SD	CP
P = 0.3	${\beta }_1$	−0.5	0.1074	1.1225	1.2900	0.944	0.1125	1.1261	1.3128	0.944
	${\beta }_2$	−0.5	0.0405	0.5654	0.5989	0.950	0.0446	0.5657	0.5983	0.952
	${\beta }_1$	−0.5	0.0499	1.1132	1.1935	0.953	0.0515	1.1145	1.2006	0.954
	${\beta }_2$	0	0.0155	0.5302	0.5485	0.966	0.0123	0.5304	0.5491	0.966
	${\beta }_1$	−0.5	0.0050	1.1140	1.2353	0.941	0.0056	1.1153	1.2443	0.941
	${\beta }_2$	0.5	0.0564	0.5686	0.6086	0.951	0.0548	0.5693	0.6139	0.951
	${\beta }_1$	0	0.0375	1.1005	1.1955	0.964	0.0375	1.1017	1.2020	0.963
	${\beta }_2$	−0.5	0.0888	0.5657	0.5999	0.948	0.0931	0.5662	0.6003	0.947
	${\beta }_1$	0	0.0507	1.0971	1.1601	0.957	0.0515	1.0978	1.1646	0.955
	${\beta }_2$	0	0.0171	0.5331	0.5844	0.948	0.0135	0.5331	0.5839	0.950
	${\beta }_1$	0	0.0093	1.1012	1.2218	0.956	0.0088	1.1025	1.2313	0.956
	${\beta }_2$	0.5	0.0819	0.5655	0.6033	0.951	0.0810	0.5666	0.6111	0.949
	${\beta }_1$	0.5	0.0867	1.1226	1.2370	0.951	0.0877	1.1249	1.2510	0.951
	${\beta }_2$	−0.5	0.0422	0.5666	0.5999	0.945	0.0460	0.5670	0.6010	0.946
	${\beta }_1$	0.5	0.0016	1.1134	1.2137	0.948	0.0006	1.1142	1.2196	0.948
	${\beta }_2$	0	0.0168	0.5332	0.5942	0.944	0.0132	0.5332	0.5934	0.946
	${\beta }_1$	0.5	0.0852	1.1258	1.2744	0.944	0.0889	1.1289	1.2905	0.943
	${\beta }_2$	0.5	0.0787	0.5677	0.6177	0.957	0.0770	0.5687	0.6211	0.953
P = 0.4	${\beta }_1$	−0.5	0.1149	1.0102	1.1372	0.951	0.1204	1.0180	1.1682	0.950
	${\beta }_2$	−0.5	0.0569	0.5098	0.5621	0.949	0.0598	0.5132	0.5738	0.951
	${\beta }_1$	−0.5	0.0105	1.0175	1.1324	0.947	0.0131	1.0227	1.1519	0.947
	${\beta }_2$	0	0.0035	0.4781	0.5207	0.943	0.0034	0.4801	0.5284	0.944
	${\beta }_1$	−0.5	0.0945	1.0070	1.1029	0.944	0.0977	1.0110	1.1191	0.943
	${\beta }_2$	0.5	0.0455	0.5031	0.5380	0.959	0.0469	0.5048	0.5479	0.959
	${\beta }_1$	0	0.0298	1.0027	1.1316	0.947	0.0315	1.0089	1.1553	0.949
	${\beta }_2$	−0.5	0.0193	0.5074	0.5474	0.957	0.0220	0.5101	0.5605	0.958
	${\beta }_1$	0	0.0019	0.9944	1.1156	0.952	0.0022	1.0007	1.1410	0.951
	${\beta }_2$	0	0.0102	0.4780	0.5253	0.946	0.0111	0.4799	0.5321	0.944
	${\beta }_1$	0	0.0098	1.0039	1.1104	0.949	0.0098	1.0094	1.1322	0.951
	${\beta }_2$	0.5	0.0718	0.5100	0.5616	0.946	0.0730	0.5120	0.5716	0.942
	${\beta }_1$	0.5	0.0679	1.0202	1.2119	0.937	0.0702	1.0254	1.2306	0.936
	${\beta }_2$	−0.5	0.0792	0.5076	0.5621	0.948	0.0812	0.5098	0.5682	0.947
	${\beta }_1$	0.5	0.0610	1.0068	1.1049	0.944	0.0635	1.0113	1.1221	0.944
	${\beta }_2$	0	0.0165	0.4758	0.5096	0.944	0.0162	0.4769	0.5130	0.945
	${\beta }_1$	0.5	0.0678	1.0126	1.1113	0.948	0.0693	1.0184	1.1322	0.948
	${\beta }_2$	0.5	0.0865	0.5088	0.5526	0.953	0.0892	0.5116	0.5673	0.949
P = 0.5	${\beta }_1$	−0.5	0.0513	0.9776	1.0617	0.956	0.0602	0.9957	1.1085	0.957
	${\beta }_2$	−0.5	0.0913	0.4900	0.5125	0.964	0.0871	0.4995	0.5419	0.963
	${\beta }_1$	−0.5	0.1144	0.9747	0.9752	0.964	0.1270	0.9908	1.0187	0.962
	${\beta }_2$	0	0.0224	0.4615	0.4980	0.953	0.0161	0.4703	0.5389	0.956
	${\beta }_1$	−0.5	0.0834	0.9733	1.0536	0.947	0.0979	0.9964	1.1363	0.949
	${\beta }_2$	0.5	0.0552	0.4915	0.5375	0.953	0.0822	0.5041	0.5997	0.949
	${\beta }_1$	0	0.0237	0.9685	1.0653	0.952	0.0242	0.9860	1.1128	0.951
	${\beta }_2$	−0.5	0.0763	0.4870	0.5234	0.958	0.0722	0.4965	0.5581	0.960
	${\beta }_1$	0	0.0472	0.9686	1.0561	0.949	0.0452	0.9913	1.1351	0.946
	${\beta }_2$	0	0.0096	0.4679	0.5137	0.937	0.0054	0.4775	0.5502	0.934
	${\beta }_1$	0	0.0181	0.9681	1.0731	0.954	0.0189	0.9899	1.1438	0.957
	${\beta }_2$	0.5	0.0525	0.4906	0.5291	0.950	0.0766	0.5018	0.5889	0.949
	${\beta }_1$	0.5	0.1028	0.9805	1.0581	0.959	0.1111	1.0013	1.1321	0.960
	${\beta }_2$	−0.5	0.0701	0.4887	0.5296	0.946	0.0696	0.5002	0.5764	0.949
	${\beta }_1$	0.5	0.0539	0.9740	1.0550	0.954	0.0616	0.9952	1.1130	0.954
	${\beta }_2$	0	0.0447	0.4655	0.5116	0.944	0.0325	0.4763	0.5572	0.945
	${\beta }_1$	0.5	0.0757	0.9826	1.0414	0.956	0.0832	1.0036	1.1060	0.957
	${\beta }_2$	0.5	0.0586	0.4884	0.5083	0.964	0.0830	0.4989	0.5545	0.961

Table 3. Comparison of the EM method and the NR method with different censoring ratios at $n=100$.

			EM				NR
			Bias	ESE	SD	CP	Bias	ESE	SD	CP
P = 0.3	${\beta }_1$	−0.5	0.0506	0.8134	0.8641	0.942	0.0522	0.8132	0.8661	0.941
	${\beta }_2$	−0.5	0.0483	0.4208	0.4457	0.949	0.0523	0.4207	0.4453	0.950
	${\beta }_1$	−0.5	0.0867	0.8140	0.8513	0.954	0.0878	0.8137	0.8533	0.954
	${\beta }_2$	0	0.0061	0.3944	0.4099	0.945	0.0024	0.3942	0.4094	0.945
	${\beta }_1$	−0.5	0.0382	0.8190	0.8794	0.946	0.0392	0.8186	0.8815	0.946
	${\beta }_2$	0.5	0.0548	0.4191	0.4344	0.956	0.0512	0.4190	0.4347	0.956
	${\beta }_1$	0	0.0154	0.8024	0.8590	0.946	0.0152	0.8023	0.8612	0.946
	${\beta }_2$	−0.5	0.0147	0.4177	0.4349	0.947	0.0188	0.4175	0.4342	0.947
	${\beta }_1$	0	0.0028	0.8053	0.8456	0.953	0.0028	0.8049	0.8472	0.952
	${\beta }_2$	0	0.0193	0.3965	0.4129	0.953	0.0153	0.3963	0.4121	0.953
	${\beta }_1$	0	0.0258	0.8061	0.8418	0.946	0.0260	0.8056	0.8436	0.945
	${\beta }_2$	0.5	0.0350	0.4180	0.4458	0.941	0.0310	0.4179	0.4450	0.941
	${\beta }_1$	0.5	0.0352	0.8157	0.8365	0.953	0.0367	0.8156	0.8393	0.952
	${\beta }_2$	−0.5	0.0232	0.4188	0.4344	0.952	0.0271	0.4186	0.4339	0.953
	${\beta }_1$	0.5	0.0561	0.8169	0.8579	0.959	0.0572	0.8166	0.8599	0.958
	${\beta }_2$	0	0.0028	0.3950	0.4089	0.951	0.0010	0.3948	0.4083	0.951
	${\beta }_1$	0.5	0.0429	0.8208	0.8814	0.941	0.0439	0.8204	0.8838	0.941
	${\beta }_2$	0.5	0.0551	0.4190	0.4311	0.948	0.0515	0.4189	0.4310	0.948
P = 0.4	${\beta }_1$	−0.5	0.0033	0.7281	0.7429	0.959	0.0052	0.7290	0.7497	0.957
	${\beta }_2$	−0.5	0.0168	0.3725	0.3818	0.953	0.0201	0.3730	0.3823	0.954
	${\beta }_1$	−0.5	0.0907	0.7382	0.7621	0.954	0.0929	0.7388	0.7678	0.952
	${\beta }_2$	0	0.0196	0.3545	0.3669	0.950	0.0161	0.3546	0.3674	0.951
	${\beta }_1$	−0.5	0.0772	0.7384	0.7902	0.942	0.0794	0.7387	0.7962	0.942
	${\beta }_2$	0.5	0.0524	0.3736	0.3938	0.943	0.0487	0.3738	0.3958	0.942
	${\beta }_1$	0	0.0115	0.7229	0.7605	0.948	0.0118	0.7239	0.7680	0.947
	${\beta }_2$	−0.5	0.0366	0.3725	0.3880	0.949	0.0396	0.3728	0.3891	0.950
	${\beta }_1$	0	0.0152	0.7260	0.7776	0.945	0.0157	0.7267	0.7848	0.942
	${\beta }_2$	0	0.0204	0.3533	0.3725	0.934	0.0243	0.3535	0.3733	0.935
	${\beta }_1$	0	0.0044	0.7292	0.7800	0.939	0.0049	0.7295	0.7862	0.938
	${\beta }_2$	0.5	0.0310	0.3726	0.3736	0.958	0.0270	0.3728	0.3758	0.956
	${\beta }_1$	0.5	0.0686	0.7347	0.8041	0.944	0.0711	0.7356	0.8109	0.941
	${\beta }_2$	−0.5	0.0185	0.3718	0.3893	0.956	0.0215	0.3721	0.3894	0.955
	${\beta }_1$	0.5	0.0215	0.7390	0.7908	0.949	0.0234	0.7397	0.7970	0.947
	${\beta }_2$	0	0.0057	0.3555	0.3735	0.956	0.0017	0.3557	0.3739	0.953
	${\beta }_1$	0.5	0.0266	0.7370	0.8072	0.931	0.0289	0.7378	0.8167	0.931
	${\beta }_2$	0.5	0.0383	0.3733	0.3984	0.940	0.0352	0.3736	0.4015	0.939
P = 0.5	${\beta }_1$	−0.5	0.0095	0.7045	0.7172	0.955	0.0065	0.7091	0.7345	0.955
	${\beta }_2$	−0.5	0.0410	0.3555	0.3732	0.949	0.0352	0.3577	0.3807	0.946
	${\beta }_1$	−0.5	0.0552	0.7090	0.7134	0.963	0.0592	0.7138	0.7315	0.962
	${\beta }_2$	0	0.0003	0.3407	0.3488	0.957	0.0047	0.3424	0.3559	0.951
	${\beta }_1$	−0.5	0.0606	0.7058	0.7553	0.939	0.0662	0.7109	0.7783	0.936
	${\beta }_2$	0.5	0.0321	0.3564	0.3700	0.950	0.0410	0.3582	0.3851	0.942
	${\beta }_1$	0	0.0154	0.6972	0.7254	0.949	0.0153	0.7014	0.7414	0.947
	${\beta }_2$	−0.5	0.0563	0.3558	0.3697	0.953	0.0516	0.3578	0.3757	0.953
	${\beta }_1$	0	0.0004	0.6969	0.7125	0.967	0.0003	0.7014	0.7308	0.966
	${\beta }_2$	0	0.0008	0.3402	0.3669	0.931	0.0039	0.3417	0.3747	0.928
	${\beta }_1$	0	0.0220	0.7012	0.7403	0.950	0.0221	0.7059	0.7606	0.946
	${\beta }_2$	0.5	0.0008	0.3560	0.3761	0.946	0.0058	0.3575	0.3889	0.944
	${\beta }_1$	0.5	0.0198	0.7076	0.7766	0.944	0.0251	0.7128	0.7970	0.944
	${\beta }_2$	−0.5	0.0525	0.3571	0.3679	0.955	0.0478	0.3593	0.3753	0.954
	${\beta }_1$	0.5	0.0341	0.7077	0.7485	0.950	0.0380	0.7128	0.7702	0.948
	${\beta }_2$	0	0.0100	0.3404	0.3424	0.957	0.0054	0.3418	0.3494	0.955
	${\beta }_1$	0.5	0.0163	0.7048	0.7260	0.950	0.0208	0.7095	0.7481	0.948
	${\beta }_2$	0.5	0.0468	0.3558	0.3727	0.949	0.0555	0.3577	0.3903	0.943

Table 4. Comparison of the EM method and the NR method with different censoring ratios at $n=200$.

			EM				NR
			Bias	ESE	SD	CP	Bias	ESE	SD	CP
P = 0.3	${\beta }_1$	−0.5	0.0171	0.5542	0.5636	0.947	0.0159	0.5538	0.5644	0.947
	${\beta }_2$	−0.5	0.0171	0.2881	0.2979	0.949	0.0210	0.2879	0.2977	0.950
	${\beta }_1$	−0.5	0.0377	0.5560	0.5725	0.943	0.0386	0.5556	0.5729	0.943
	${\beta }_2$	0	0.0105	0.2721	0.2862	0.951	0.0071	0.2720	0.2856	0.951
	${\beta }_1$	−0.5	0.0407	0.5549	0.5784	0.939	0.0414	0.5544	0.5788	0.940
	${\beta }_2$	0.5	0.0294	0.2875	0.2938	0.952	0.0258	0.2874	0.2932	0.951
	${\beta }_1$	0	0.0277	0.5474	0.5568	0.947	0.0278	0.5471	0.5575	0.947
	${\beta }_2$	−0.5	0.0141	0.2876	0.2920	0.957	0.0180	0.2875	0.2918	0.958
	${\beta }_1$	0	0.0262	0.5498	0.5774	0.950	0.0262	0.5494	0.5782	0.950
	${\beta }_2$	0	0.0069	0.2728	0.2800	0.943	0.0035	0.2726	0.2796	0.945
	${\beta }_1$	0	0.0097	0.5481	0.5652	0.942	0.0097	0.5476	0.5656	0.942
	${\beta }_2$	0.5	0.0350	0.2871	0.3066	0.941	0.0311	0.2869	0.3059	0.944
	${\beta }_1$	0.5	0.0216	0.5541	0.5484	0.957	0.0229	0.5537	0.5491	0.956
	${\beta }_2$	−0.5	0.0095	0.2866	0.2873	0.956	0.0134	0.2864	0.2871	0.956
	${\beta }_1$	0.5	0.0159	0.5543	0.5892	0.941	0.0150	0.5539	0.5898	0.941
	${\beta }_2$	0	0.0036	0.2712	0.2792	0.944	0.0002	0.2711	0.2788	0.944
	${\beta }_1$	0.5	0.0226	0.5556	0.5878	0.942	0.0234	0.5551	0.5882	0.942
	${\beta }_2$	0.5	0.0209	0.2878	0.3060	0.933	0.0171	0.2877	0.3053	0.935
P = 0.4	${\beta }_1$	−0.5	0.0231	0.4987	0.5155	0.955	0.0251	0.4988	0.5186	0.953
	${\beta }_2$	−0.5	0.0257	0.2560	0.2611	0.951	0.0300	0.2560	0.2608	0.950
	${\beta }_1$	−0.5	0.0211	0.4958	0.4925	0.955	0.0226	0.4956	0.4950	0.954
	${\beta }_2$	0	0.0112	0.2430	0.2543	0.941	0.0065	0.2429	0.2541	0.939
	${\beta }_1$	−0.5	0.0265	0.4994	0.5045	0.950	0.0275	0.4990	0.5067	0.948
	${\beta }_2$	0.5	0.0227	0.2553	0.2637	0.949	0.0169	0.2553	0.2629	0.950
	${\beta }_1$	0	0.0153	0.4898	0.5004	0.957	0.0155	0.4898	0.5033	0.956
	${\beta }_2$	−0.5	0.0236	0.2565	0.2649	0.945	0.0278	0.2565	0.2644	0.945
	${\beta }_1$	0	0.0145	0.4904	0.4968	0.951	0.0147	0.4902	0.4993	0.950
	${\beta }_2$	0	0.0043	0.2422	0.2467	0.956	0.0090	0.2421	0.2462	0.955
	${\beta }_1$	0	0.0269	0.4928	0.5021	0.953	0.0271	0.4925	0.5042	0.953
	${\beta }_2$	0.5	0.0190	0.2561	0.2672	0.947	0.0133	0.2560	0.2667	0.948
	${\beta }_1$	0.5	0.0275	0.4981	0.5009	0.947	0.0297	0.4982	0.5042	0.946
	${\beta }_2$	−0.5	0.0149	0.2558	0.2654	0.948	0.0191	0.2558	0.2651	0.948
	${\beta }_1$	0.5	0.0428	0.4996	0.5153	0.947	0.0447	0.4994	0.5183	0.946
	${\beta }_2$	0	0.0113	0.2426	0.2510	0.937	0.0066	0.2425	0.2507	0.937
	${\beta }_1$	0.5	0.0086	0.4985	0.5189	0.945	0.0079	0.4981	0.5212	0.942
	${\beta }_2$	0.5	0.0442	0.2559	0.2641	0.947	0.0383	0.2558	0.2634	0.944
P = 0.5	${\beta }_1$	−0.5	0.0027	0.4753	0.4787	0.960	0.0005	0.4767	0.4868	0.958
	${\beta }_2$	−0.5	0.0249	0.2427	0.2563	0.935	0.0213	0.2434	0.2577	0.933
	${\beta }_1$	−0.5	0.0508	0.4757	0.4795	0.958	0.0552	0.4770	0.4883	0.956
	${\beta }_2$	0	0.0021	0.2319	0.2333	0.955	0.0021	0.2323	0.2356	0.956
	${\beta }_1$	−0.5	0.0232	0.4747	0.4631	0.964	0.0263	0.4757	0.4718	0.962
	${\beta }_2$	0.5	0.0153	0.2424	0.2492	0.942	0.0149	0.2427	0.2542	0.939
	${\beta }_1$	0	0.0063	0.4699	0.4734	0.953	0.0063	0.4713	0.4812	0.949
	${\beta }_2$	−0.5	0.0256	0.2423	0.2434	0.954	0.0230	0.2429	0.2440	0.953
	${\beta }_1$	0	0.0195	0.4710	0.4612	0.964	0.0197	0.4722	0.4700	0.959
	${\beta }_2$	0	0.0098	0.2322	0.2453	0.936	0.0099	0.2327	0.2484	0.935
	${\beta }_1$	0	0.0181	0.4716	0.4720	0.954	0.0182	0.4726	0.4809	0.950
	${\beta }_2$	0.5	0.0179	0.2429	0.2558	0.947	0.0175	0.2432	0.2620	0.944
	${\beta }_1$	0.5	0.0171	0.4744	0.4878	0.950	0.0207	0.4758	0.4953	0.948
	${\beta }_2$	−0.5	0.0174	0.2422	0.2465	0.951	0.0146	0.2429	0.2475	0.949
	${\beta }_1$	0.5	0.0429	0.4766	0.4820	0.960	0.0470	0.4779	0.4911	0.956
	${\beta }_2$	0	0.0122	0.2318	0.2408	0.932	0.0123	0.2322	0.2433	0.930
	${\beta }_1$	0.5	0.0257	0.4752	0.4959	0.953	0.0283	0.4762	0.5055	0.947
	${\beta }_2$	0.5	0.0267	0.2427	0.2397	0.950	0.0261	0.2430	0.2446	0.946

Table 5. Comparison of the EM method and the NR method with different censoring ratios at $n=400$.

			EM				NR
			Bias	ESE	SD	CP	Bias	ESE	SD	CP
P = 0.3	${\beta }_1$	−0.5	0.0120	0.3864	0.3899	0.950	0.0131	0.3861	0.3903	0.949
	${\beta }_2$	−0.5	0.0027	0.2008	0.1964	0.954	0.0064	0.2006	0.1963	0.954
	${\beta }_1$	−0.5	0.0042	0.3881	0.4012	0.951	0.0049	0.3878	0.4015	0.951
	${\beta }_2$	0	0.0002	0.1899	0.1810	0.966	0.0033	0.1898	0.1808	0.965
	${\beta }_1$	−0.5	0.0223	0.3877	0.3857	0.954	0.0229	0.3873	0.3857	0.955
	${\beta }_2$	0.5	0.0195	0.2012	0.2049	0.937	0.0161	0.2012	0.2046	0.936
	${\beta }_1$	0	0.0078	0.3812	0.3774	0.949	0.0078	0.3809	0.3777	0.949
	${\beta }_2$	−0.5	0.0063	0.2006	0.2055	0.939	0.0100	0.2004	0.2054	0.938
	${\beta }_1$	0	0.0063	0.3800	0.3849	0.951	0.0063	0.3798	0.3851	0.951
	${\beta }_2$	0	0.0101	0.1897	0.1983	0.943	0.0069	0.1896	0.1980	0.942
	${\beta }_1$	0	0.0133	0.3809	0.3715	0.959	0.0133	0.3805	0.3717	0.960
	${\beta }_2$	0.5	0.0147	0.2003	0.2079	0.943	0.0112	0.2003	0.2074	0.946
	${\beta }_1$	0.5	0.0203	0.3860	0.3842	0.948	0.0213	0.3858	0.3847	0.948
	${\beta }_2$	−0.5	0.0007	0.2007	0.1969	0.956	0.0029	0.2005	0.1968	0.958
	${\beta }_1$	0.5	0.0010	0.3872	0.4031	0.936	0.0017	0.3869	0.4034	0.936
	${\beta }_2$	0	0.0004	0.1899	0.1929	0.944	0.0036	0.1898	0.1926	0.941
	${\beta }_1$	0.5	0.0171	0.3873	0.4030	0.943	0.0178	0.3869	0.4032	0.943
	${\beta }_2$	0.5	0.0204	0.2004	0.2038	0.942	0.0170	0.2004	0.2034	0.942
P = 0.4	${\beta }_1$	−0.5	0.0229	0.3475	0.3633	0.945	0.0250	0.3474	0.3651	0.945
	${\beta }_2$	−0.5	0.0028	0.1785	0.1787	0.954	0.0075	0.1785	0.1784	0.954
	${\beta }_1$	−0.5	0.0050	0.3447	0.3499	0.949	0.0062	0.3445	0.3513	0.947
	${\beta }_2$	0	0.0025	0.1694	0.1748	0.948	0.0072	0.1693	0.1744	0.950
	${\beta }_1$	−0.5	0.0039	0.3450	0.3449	0.948	0.0044	0.3447	0.3457	0.947
	${\beta }_2$	0.5	0.0140	0.1782	0.1757	0.951	0.0078	0.1782	0.1746	0.955
	${\beta }_1$	0	0.0053	0.3405	0.3411	0.959	0.0052	0.3403	0.3428	0.957
	${\beta }_2$	−0.5	0.0079	0.1786	0.1825	0.954	0.0127	0.1786	0.1822	0.954
	${\beta }_1$	0	0.0032	0.3412	0.3425	0.951	0.0034	0.3410	0.3438	0.951
	${\beta }_2$	0	0.0013	0.1697	0.1725	0.938	0.0034	0.1697	0.1722	0.935
	${\beta }_1$	0	0.0070	0.3414	0.3452	0.959	0.0069	0.3410	0.3459	0.957
	${\beta }_2$	0.5	0.0174	0.1784	0.1795	0.956	0.0112	0.1784	0.1784	0.957
	${\beta }_1$	0.5	0.0105	0.3460	0.3415	0.960	0.0124	0.3459	0.3432	0.960
	${\beta }_2$	−0.5	0.0054	0.1787	0.1776	0.949	0.0007	0.1787	0.1773	0.952
	${\beta }_1$	0.5	0.0022	0.3458	0.3406	0.962	0.0009	0.3456	0.3419	0.962
	${\beta }_2$	0	0.0096	0.1697	0.1785	0.939	0.0048	0.1696	0.1782	0.938
	${\beta }_1$	0.5	0.0053	0.3461	0.3528	0.953	0.0059	0.3457	0.3536	0.952
	${\beta }_2$	0.5	0.0211	0.1787	0.1794	0.947	0.0149	0.1787	0.1782	0.949
P = 0.5	${\beta }_1$	−0.5	0.0043	0.3280	0.3215	0.958	0.0076	0.3287	0.3261	0.954
	${\beta }_2$	−0.5	0.0122	0.1685	0.1762	0.945	0.0100	0.1689	0.1760	0.942
	${\beta }_1$	−0.5	0.0197	0.3286	0.3305	0.953	0.0233	0.3291	0.3361	0.950
	${\beta }_2$	0	0.0066	0.1612	0.1631	0.949	0.0041	0.1613	0.1637	0.948
	${\beta }_1$	−0.5	0.0103	0.3288	0.3458	0.943	0.0127	0.3291	0.3512	0.941
	${\beta }_2$	0.5	0.0183	0.1688	0.1694	0.952	0.0148	0.1689	0.1720	0.944
	${\beta }_1$	0	0.0009	0.3253	0.3371	0.956	0.0010	0.3259	0.3419	0.954
	${\beta }_2$	−0.5	0.0074	0.1687	0.1736	0.945	0.0051	0.1691	0.1735	0.947
	${\beta }_1$	0	0.0095	0.3255	0.3245	0.958	0.0096	0.3259	0.3298	0.955
	${\beta }_2$	0	0.0071	0.1613	0.1611	0.949	0.0046	0.1615	0.1619	0.949
	${\beta }_1$	0	0.0055	0.3259	0.3167	0.965	0.0056	0.3261	0.3218	0.960
	${\beta }_2$	0.5	0.0099	0.1688	0.1634	0.956	0.0059	0.1689	0.1652	0.956
	${\beta }_1$	0.5	0.0019	0.3298	0.3357	0.951	0.0051	0.3305	0.3404	0.945
	${\beta }_2$	−0.5	0.0093	0.1688	0.1718	0.946	0.0071	0.1692	0.1718	0.945
	${\beta }_1$	0.5	0.0004	0.3293	0.3208	0.951	0.0031	0.3298	0.3263	0.948
	${\beta }_2$	0	0.0069	0.1614	0.1709	0.935	0.0045	0.1615	0.1716	0.936
	${\beta }_1$	0.5	0.0133	0.3295	0.3433	0.938	0.0158	0.3298	0.3487	0.936
	${\beta }_2$	0.5	0.0147	0.1689	0.1738	0.950	0.0112	0.1691	0.1767	0.946

1.

For small sample sizes, such as $n=30$, 60 and 100, both the Bias and ESE of the EM method are generally smaller than those of the NR method. For example, Table 2 presents the estimation results for $n=60$. Among 54 cases, the EM method exhibits smaller biases in 35 cases, and a majority of the ESEs are also smaller than those of the NR method. Similar trends are observed in Table 1 and Table 3.

2.

When the sample size is medium or large, such as $n=200$ and 400, the EM method still maintains an advantage in terms of the Bias, with over half of the 54 cases exhibiting smaller Bias. However, when considering ESEs, the EM method performs unsatisfactorily. For instance, when $n=200$, the EM method produces $53.7\%$ smaller Bias and only $37.0\%$ smaller ESE.

3.

With the increase in sample size, the Bias and ESE from both methods diminish towards zero, indicating that the performance of both methods improves with larger sample sizes, and the estimates can converge to the true values.

4.

The estimated standard errors and sample standard deviations decrease and approach each other as the sample size increases, suggesting that the asymptotic covariance matrix obtained by Louis’ method performs well for finite sample sizes.

In summary, the estimate results of the EM method are comparable to those of the NR method, and even have an advantage when the sample size is small. In terms of bias, the EM method consistently outperforms the NR method; in terms of SD and ESE, the EM method performs better when the sample size is small, and the NR method slightly outperforms when the sample size is large; in terms of CP, both methods perform well, essentially around $95\%$, and the censoring ratio has no impact on the performance of the two methods. With the increase in sample size, the estimation performance of both methods improves.

Remark 1. 

It should be noted that when the small sample size is small, both the EM algorithm and the NR algorithm can sometimes fail to converge, which is similar to the case in Balakrishnan and Mitra [17], and in this case, the simulation data are excluded from the experiment. The EM algorithm fails to converge much less frequently than the NR approach. This is partially due to the fact that the scale parameter λ in the EM algorithm has an analytical solution and is guaranteed to be positive, whereas the NR method relies on numerical computation to solve for all three parameters. When the sample size is large, none of the methods fail to converge.

3.2. Simulation Study II: Compare the Weibull PH Model with the Semi-Parametric PH Model

In this simulation, we compare the Weibull PH model with a semi-parametric PH model proposed by McMahan et al. [9], who utilize a spline-based EM method for estimating the PH model under the current status data. This semi-parametric PH model can be implemented using the R package ICsurv [18]. To ensure a comprehensive comparison, we consider two scenarios:

1.

The failure time $T_i$ follows a Weibull distribution, with

$$F\left(t;{\mathit{x}}_i\right)=1-exp\left\{-{\left(\lambda t\right)}^{\gamma }exp\left(x_{i1}{\beta }_1+x_{i2}{\beta }_2\right)\right\},$$

where $\lambda =1$, $\gamma =2$.

2.

Referring to the simulation setting in McMahan et al. [9], the failure time $T_i$ follows a non-Weibull distribution, with

$$F\left(t;{\mathit{x}}_i\right)=1-exp\left\{-{\Lambda }_0\left(t\right)exp\left(x_{i1}{\beta }_1+x_{i2}{\beta }_2\right)\right\},$$

where ${\Lambda }_0\left(t\right)=log(1+t)+t^{3/2}$.

In the above equation, ${\mathit{x}}_i={\left(x_{i1},x_{i2}\right)}^{\prime }$, $x_{i1}\sim \mathrm{Bernoulli}\left(0.5\right)$, $x_{i2}\sim \mathrm{N}\left(0,0.5^2\right)$, $i=1,\cdots ,n$. Regression coefficients ${\beta }_1$ and ${\beta }_2$ can be taken as $-0.5$, 0 and $0.5$, totaling nine parameter combinations, and the sample size n is specified as 200. The failure time $T_i$ can be obtained by solving $F\left(t;{\mathit{x}}_i\right)=u_i$, where $u_i\sim \mathrm{U}\left(0,1\right)$, and the censoring time $C_i$ obeys a truncated exponential distribution $\mathrm{Exp}\left(1\right)$ with support $(0,2)$.

Each simulation is repeated 500 times, with an initial value of ${\mathit{\theta }}^{\left(0\right)}={\left({\mathit{\beta }}^{{\left(0\right)}^{\prime }},{\lambda }^{\left(0\right)},{\gamma }^{\left(0\right)}\right)}^{\prime }$$={\left({{\mathbf{1}}_{\mathbf{2}}}^{\prime },1,1\right)}^{\prime }$ for the Weibull PH model, and ${\mathit{\theta }}^{\left(0\right)}={\left({\mathit{\beta }}^{{\left(0\right)}^{\prime }},{\gamma }^{{\left(0\right)}^{\prime }}\right)}^{\prime }={\left({{\mathbf{1}}_{\mathbf{2}}}^{\prime },{{\mathbf{1}}_{\mathbf{6}}}^{\prime }\right)}^{\prime }$ for the semi-parametric PH model. In McMahan et al. [9], the I-spline with suitable degree is used to control the smoothness of the hazard function. For comparison, we consider the degree to be 1, 2, and 3, corresponding to the linear, quadratic, and cubic basis function, respectively, and the number of knots is set as 5. The algorithm is recognized to have reached convergence when $max|{\mathit{\theta }}^{\left(d+1\right)}-{\mathit{\theta }}^{\left(d\right)}|<0.005$. All evaluation criteria are consistent with the Simulation study I. Table 6 and Table 7 show the results.

Table 6. Estimates for the Weibull PH model and semi-parametric PH model when the failure time comes from the Weibull distribution.

			Weibull PH Model				Semi-Parametric PH Model
			Bias	ESE	SD	CP	Bias	ESE	SD	CP
degree = 1	${\beta }_1$	−0.5	0.0022	0.5756	0.5697	0.962	0.0354	0.6093	0.6149	0.958
	${\beta }_2$	−0.5	0.0137	0.2844	0.2796	0.948	0.0496	0.3090	0.2987	0.950
	${\beta }_1$	−0.5	0.0096	0.5559	0.5643	0.958	0.0481	0.5923	0.6142	0.952
	${\beta }_2$	0	0.0032	0.2707	0.2700	0.956	0.0075	0.2933	0.2830	0.964
	${\beta }_1$	−0.5	0.0435	0.5471	0.5661	0.952	0.0866	0.5852	0.6243	0.944
	${\beta }_2$	0.5	0.0155	0.2724	0.2656	0.954	0.0440	0.2937	0.2878	0.962
	${\beta }_1$	0	0.0203	0.5696	0.5649	0.960	0.0086	0.6031	0.6104	0.952
	${\beta }_2$	−0.5	0.0114	0.2837	0.2929	0.944	0.0428	0.3123	0.3122	0.940
	${\beta }_1$	0	0.0056	0.5511	0.5436	0.958	0.0004	0.5863	0.5916	0.948
	${\beta }_2$	0	0.0123	0.2700	0.2753	0.952	0.0049	0.2936	0.2961	0.960
	${\beta }_1$	0	0.0096	0.5436	0.5245	0.960	0.0084	0.5806	0.5714	0.962
	${\beta }_2$	0.5	0.0242	0.2728	0.2812	0.964	0.0573	0.2956	0.3097	0.958
	${\beta }_1$	0.5	0.0383	0.5731	0.6035	0.936	0.0086	0.6096	0.6400	0.944
	${\beta }_2$	−0.5	0.0121	0.2831	0.2844	0.966	0.0221	0.3074	0.3050	0.960
	${\beta }_1$	0.5	0.0006	0.5506	0.5546	0.950	0.0269	0.5877	0.6049	0.944
	${\beta }_2$	0	0.0085	0.2708	0.2638	0.958	0.0026	0.2925	0.2775	0.968
	${\beta }_1$	0.5	0.0218	0.5512	0.5378	0.954	0.0724	0.5904	0.5991	0.948
	${\beta }_2$	0.5	0.0219	0.2735	0.2760	0.956	0.0513	0.2966	0.3040	0.954
degree = 2	${\beta }_1$	−0.5	0.0484	0.5732	0.5917	0.952	0.0735	0.6211	0.6411	0.946
	${\beta }_2$	−0.5	0.0386	0.2848	0.2827	0.954	0.0759	0.3399	0.3085	0.942
	${\beta }_1$	−0.5	0.0283	0.5526	0.5676	0.956	0.0695	0.5998	0.6258	0.946
	${\beta }_2$	0	0.0050	0.2715	0.2757	0.948	0.0126	0.3158	0.2964	0.944
	${\beta }_1$	−0.5	0.0248	0.5468	0.5494	0.970	0.0650	0.5994	0.6057	0.976
	${\beta }_2$	0.5	0.0225	0.2728	0.2620	0.974	0.0497	0.3200	0.2876	0.974
	${\beta }_1$	0	0.0127	0.5669	0.5739	0.960	0.0120	0.6178	0.6121	0.954
	${\beta }_2$	−0.5	0.0139	0.2813	0.2847	0.950	0.0245	0.3336	0.3087	0.944
	${\beta }_1$	0	0.0285	0.5458	0.5416	0.964	0.0313	0.5902	0.5855	0.966
	${\beta }_2$	0	0.0026	0.2702	0.2815	0.954	0.0098	0.3162	0.3003	0.956
	${\beta }_1$	0	0.0206	0.5450	0.5471	0.944	0.0176	0.5884	0.6037	0.946
	${\beta }_2$	0.5	0.0293	0.2729	0.2717	0.952	0.0592	0.3317	0.2906	0.962
	${\beta }_1$	0.5	0.0566	0.5789	0.5918	0.956	0.0960	0.6406	0.6362	0.956
	${\beta }_2$	−0.5	0.0035	0.2852	0.3038	0.942	0.0402	0.3485	0.3233	0.952
	${\beta }_1$	0.5	0.0121	0.5535	0.5423	0.964	0.0491	0.5996	0.5908	0.958
	${\beta }_2$	0	0.0016	0.2706	0.2913	0.936	0.0100	0.3174	0.3096	0.936
	${\beta }_1$	0.5	0.0691	0.5478	0.5645	0.956	0.1089	0.6086	0.6183	0.952
	${\beta }_2$	0.5	0.0305	0.2725	0.2785	0.952	0.0583	0.3370	0.3095	0.960
degree = 3	${\beta }_1$	−0.5	0.0005	0.5744	0.5983	0.942	0.0275	0.6254	0.6327	0.948
	${\beta }_2$	−0.5	0.0281	0.2843	0.3043	0.938	0.0645	0.3296	0.3249	0.864
	${\beta }_1$	−0.5	0.0243	0.5497	0.5632	0.948	0.0650	0.5998	0.6210	0.940
	${\beta }_2$	0	0.0013	0.2714	0.2802	0.948	0.0298	0.3381	0.3099	0.914
	${\beta }_1$	−0.5	0.0356	0.5584	0.5848	0.942	0.0744	0.6160	0.6435	0.948
	${\beta }_2$	0.5	0.0107	0.2725	0.2775	0.934	0.0164	0.3191	0.2932	0.884
	${\beta }_1$	0	0.0341	0.5746	0.6104	0.932	0.0670	0.6332	0.6644	0.926
	${\beta }_2$	−0.5	0.0340	0.2854	0.3001	0.942	0.0692	0.3334	0.3262	0.876
	${\beta }_1$	0	0.0468	0.5475	0.5622	0.962	0.0893	0.6012	0.6137	0.956
	${\beta }_2$	0	0.0091	0.2713	0.2693	0.962	0.0189	0.3355	0.3009	0.918
	${\beta }_1$	0	0.0208	0.5576	0.5458	0.962	0.0596	0.6079	0.5873	0.960
	${\beta }_2$	0.5	0.0092	0.2725	0.2668	0.958	0.0198	0.3094	0.2809	0.914
	${\beta }_1$	0.5	0.0143	0.5727	0.5730	0.960	0.0178	0.6301	0.6142	0.964
	${\beta }_2$	−0.5	0.0027	0.2831	0.2879	0.946	0.0374	0.3252	0.3105	0.894
	${\beta }_1$	0.5	0.0165	0.5474	0.5586	0.956	0.0202	0.5923	0.6116	0.964
	${\beta }_2$	0	0.0021	0.2721	0.2896	0.936	0.0269	0.3156	0.3190	0.878
	${\beta }_1$	0.5	0.0234	0.5750	0.6228	0.942	0.0243	0.6327	0.6693	0.940
	${\beta }_2$	0.5	0.0360	0.2851	0.3033	0.944	0.0711	0.3465	0.3184	0.906

Table 7. Estimates for the the Weibull PH model and semi-parametric PH model when the failure time comes from the non-Weibull distribution.

			Weibull PH Model				Semi-Parametric PH Model
			Bias	ESE	SD	CP	Bias	ESE	SD	CP
degree = 1	${\beta }_1$	−0.5	0.0451	0.5006	0.4966	0.960	0.0183	0.5508	0.5653	0.954
	${\beta }_2$	−0.5	0.0323	0.2484	0.2578	0.928	0.0334	0.2794	0.2922	0.942
	${\beta }_1$	−0.5	0.0462	0.4860	0.4362	0.974	0.0348	0.5362	0.5162	0.962
	${\beta }_2$	0	0.0019	0.2397	0.2291	0.966	0.0027	0.2647	0.2625	0.952
	${\beta }_1$	−0.5	0.0462	0.4812	0.4395	0.966	0.0314	0.5310	0.5094	0.970
	${\beta }_2$	0.5	0.0427	0.2399	0.2146	0.972	0.0389	0.2679	0.2571	0.964
	${\beta }_1$	0	0.0207	0.4985	0.4739	0.954	0.0271	0.5483	0.5487	0.952
	${\beta }_2$	−0.5	0.0302	0.2473	0.2559	0.936	0.0434	0.2776	0.2967	0.942
	${\beta }_1$	0	0.0265	0.4878	0.5002	0.960	0.0273	0.5409	0.5906	0.946
	${\beta }_2$	0	0.0251	0.2403	0.2409	0.956	0.0299	0.2697	0.2824	0.952
	${\beta }_1$	0	0.0172	0.4800	0.4538	0.970	0.0283	0.5294	0.5377	0.952
	${\beta }_2$	0.5	0.0649	0.2404	0.2216	0.962	0.0133	0.2683	0.2609	0.954
	${\beta }_1$	0.5	0.0535	0.5007	0.5003	0.954	0.0159	0.5519	0.5806	0.948
	${\beta }_2$	−0.5	0.0363	0.2486	0.2494	0.946	0.0402	0.2811	0.2853	0.948
	${\beta }_1$	0.5	0.0468	0.4902	0.4690	0.966	0.0324	0.5436	0.5550	0.960
	${\beta }_2$	0	0.0017	0.2403	0.2385	0.956	0.0061	0.2655	0.2672	0.958
	${\beta }_1$	0.5	0.0675	0.4826	0.4342	0.970	0.0027	0.5301	0.4968	0.966
	${\beta }_2$	0.5	0.0514	0.2404	0.2176	0.968	0.0247	0.2680	0.2574	0.962
degree = 2	${\beta }_1$	−0.5	0.0086	0.5037	0.5100	0.956	0.0616	0.5580	0.5918	0.950
	${\beta }_2$	−0.5	0.0397	0.2489	0.2369	0.962	0.0324	0.2847	0.2748	0.970
	${\beta }_1$	−0.5	0.0715	0.4850	0.4107	0.984	0.0077	0.5370	0.4818	0.974
	${\beta }_2$	0	0.0023	0.2395	0.2349	0.942	0.0018	0.2683	0.2708	0.942
	${\beta }_1$	−0.5	0.0872	0.4841	0.4895	0.950	0.0129	0.5369	0.5775	0.946
	${\beta }_2$	0.5	0.0628	0.2411	0.2208	0.958	0.0166	0.2722	0.2673	0.956
	${\beta }_1$	0	0.0030	0.4977	0.5126	0.946	0.0026	0.5517	0.5945	0.938
	${\beta }_2$	−0.5	0.0223	0.2486	0.2436	0.952	0.0500	0.2859	0.2845	0.952
	${\beta }_1$	0	0.0311	0.4885	0.4641	0.968	0.0385	0.5417	0.5457	0.954
	${\beta }_2$	0	0.0005	0.2408	0.2351	0.958	0.0010	0.2696	0.2697	0.956
	${\beta }_1$	0	0.0076	0.4813	0.4343	0.974	0.0154	0.5318	0.5157	0.962
	${\beta }_2$	0.5	0.0671	0.2411	0.2297	0.944	0.0122	0.2715	0.2733	0.944
	${\beta }_1$	0.5	0.0177	0.5027	0.4798	0.974	0.0696	0.5572	0.5592	0.964
	${\beta }_2$	−0.5	0.0485	0.2483	0.2382	0.966	0.0260	0.2848	0.2733	0.964
	${\beta }_1$	0.5	0.0369	0.4912	0.4734	0.960	0.0481	0.5459	0.5538	0.958
	${\beta }_2$	0	0.0156	0.2405	0.2404	0.960	0.0135	0.2689	0.2727	0.956
	${\beta }_1$	0.5	0.0353	0.4841	0.4587	0.950	0.0467	0.5390	0.5336	0.948
	${\beta }_2$	0.5	0.0527	0.2414	0.2294	0.950	0.0295	0.2742	0.2743	0.958
degree = 3	${\beta }_1$	−0.5	0.0045	0.5026	0.4690	0.974	0.0697	0.5629	0.5497	0.960
	${\beta }_2$	−0.5	0.0227	0.2486	0.2355	0.964	0.0485	0.2920	0.2750	0.976
	${\beta }_1$	−0.5	0.0234	0.4889	0.4452	0.968	0.0674	0.5461	0.5250	0.954
	${\beta }_2$	0	0.0063	0.2401	0.2314	0.958	0.0123	0.2723	0.2673	0.956
	${\beta }_1$	−0.5	0.0416	0.4822	0.4445	0.976	0.0284	0.5341	0.5132	0.974
	${\beta }_2$	0.5	0.0690	0.2399	0.2276	0.948	0.0079	0.2736	0.2692	0.962
	${\beta }_1$	0	0.0061	0.4965	0.4888	0.966	0.0093	0.5557	0.5721	0.956
	${\beta }_2$	−0.5	0.0292	0.2479	0.2471	0.950	0.0438	0.2931	0.2825	0.958
	${\beta }_1$	0	0.0063	0.4861	0.4607	0.964	0.0090	0.5430	0.5525	0.956
	${\beta }_2$	0	0.0032	0.2400	0.2282	0.962	0.0059	0.2747	0.2662	0.958
	${\beta }_1$	0	0.0254	0.4795	0.4485	0.970	0.0324	0.5313	0.5234	0.968
	${\beta }_2$	0.5	0.0530	0.2405	0.2332	0.946	0.0269	0.2751	0.2757	0.968
	${\beta }_1$	0.5	0.0030	0.5040	0.4761	0.982	0.0702	0.5648	0.5647	0.974
	${\beta }_2$	−0.5	0.0297	0.2494	0.2517	0.956	0.0464	0.2917	0.2920	0.954
	${\beta }_1$	0.5	0.0579	0.4879	0.4614	0.972	0.0163	0.5414	0.5352	0.960
	${\beta }_2$	0	0.0186	0.2396	0.2390	0.962	0.0201	0.2713	0.2684	0.962
	${\beta }_1$	0.5	0.0741	0.4833	0.4801	0.956	0.0000	0.5361	0.5590	0.950
	${\beta }_2$	0.5	0.0494	0.2409	0.2098	0.962	0.0277	0.2778	0.2562	0.974

Table 6 presents the results when the failure times follow the Weibull distribution. In this case, the Weibull PH model performs overwhelmingly superior to the semi-parametric model, with smaller Bias and ESE in the majority of the 54 cases. For instance, when the degree is 2, the proportion with smaller biases from the Weibull PH model is $88.9\%$, and almost all ESEs are smaller than those from the semi-parametric PH model. Meanwhile, in the non-Weibull situation, Table 7 indicates the superiority of the semi-parametric PH model. This finding is natural and consistent with expectations.

Simultaneously, we utilize Akaike’s information criterion (AIC) and the Bayesian information criterion (BIC) for model selection and the results are shown in Table 8 and Table 9. Clearly, the Weibull PH model produces smaller AIC and BIC values in all situations, indicating that the Weibull PH model is superior to the semi-parametric PH model in some sense. This may be explained by the fact that the AIC and BIC values prefer a concise model with fewer parameters, while the semi-parametric PH model introduces several splines to describe the smoothness of the unspecified baseline hazard function, which increases the complexity of the model.

Table 8. The AIC and BIC of the Weibull PH model and the semi-parametric PH model when the failure time comes from a Weibull distribution.

		Weibull PH Model		Semi-Parametric PH Model
		AIC	BIC	AIC	BIC
degree = 1	${\mathit{\beta }}^{\prime }=(-0.5,-0.5)$	158.0671	171.2604	162.5652	188.9517
	${\mathit{\beta }}^{\prime }=(-0.5,0)$	159.3115	172.5048	164.0589	190.4455
	${\mathit{\beta }}^{\prime }=(-0.5,0.5)$	153.2519	166.4451	158.3429	184.7294
	${\mathit{\beta }}^{\prime }=(0,-0.5)$	158.5114	171.7047	163.0422	189.4287
	${\mathit{\beta }}^{\prime }=(0,0)$	159.2377	172.4310	164.0106	190.3972
	${\mathit{\beta }}^{\prime }=(0,0.5)$	153.1766	166.3699	158.3069	184.6934
	${\mathit{\beta }}^{\prime }=(0.5,-0.5)$	159.3043	172.4976	164.0052	190.3918
	${\mathit{\beta }}^{\prime }=(0.5,0)$	158.5679	171.7612	163.3746	189.7611
	${\mathit{\beta }}^{\prime }=(0.5,0.5)$	153.1342	166.3275	158.1237	184.5102
degree = 2	${\mathit{\beta }}^{\prime }=(-0.5,-0.5)$	157.5231	170.7164	163.7349	193.4198
	${\mathit{\beta }}^{\prime }=(-0.5,0)$	158.5837	171.7769	165.0200	194.7048
	${\mathit{\beta }}^{\prime }=(-0.5,0.5)$	153.4395	166.6327	160.0308	189.7156
	${\mathit{\beta }}^{\prime }=(0,-0.5)$	159.7065	172.8997	165.9552	195.6400
	${\mathit{\beta }}^{\prime }=(0,0)$	158.5474	171.7407	164.8709	194.5557
	${\mathit{\beta }}^{\prime }=(0,0.5)$	153.4417	166.6350	159.9141	189.5989
	${\mathit{\beta }}^{\prime }=(0.5,-0.5)$	157.3260	170.5192	163.4532	193.1381
	${\mathit{\beta }}^{\prime }=(0.5,0)$	159.0321	172.2254	165.3613	195.0462
	${\mathit{\beta }}^{\prime }=(0.5,0.5)$	153.7928	166.9861	160.1865	189.8714
degree = 3	${\mathit{\beta }}^{\prime }=(-0.5,-0.5)$	158.0656	171.2589	166.2319	199.2151
	${\mathit{\beta }}^{\prime }=(-0.5,0)$	155.7645	168.9577	164.1663	197.1495
	${\mathit{\beta }}^{\prime }=(-0.5,0.5)$	157.2065	170.3998	165.3615	198.3447
	${\mathit{\beta }}^{\prime }=(0,-0.5)$	157.1589	170.3521	165.2879	198.2711
	${\mathit{\beta }}^{\prime }=(0,0)$	154.8845	168.0777	163.3521	196.3353
	${\mathit{\beta }}^{\prime }=(0,0.5)$	157.3314	170.5247	165.5741	198.5573
	${\mathit{\beta }}^{\prime }=(0.5,-0.5)$	158.9620	172.1552	167.2433	200.2265
	${\mathit{\beta }}^{\prime }=(0.5,0)$	153.8931	167.0864	162.4024	195.3856
	${\mathit{\beta }}^{\prime }=(0.5,0.5)$	157.6660	170.8592	165.8563	198.8395

Table 9. The AIC and BIC of the Weibull PH model and the semi-parametric PH model when the failure time does not follow a Weibull distribution.

		Weibull PH Model		Semi-Parametric PH Model
		AIC	BIC	AIC	BIC
degree = 1	${\mathit{\beta }}^{\prime }=(-0.5,-0.5)$	178.4425	191.6358	180.6047	206.9913
	${\mathit{\beta }}^{\prime }=(-0.5,0)$	176.8683	190.0616	179.8233	206.2098
	${\mathit{\beta }}^{\prime }=(-0.5,0.5)$	171.7429	184.9362	174.7805	201.1671
	${\mathit{\beta }}^{\prime }=(0,-0.5)$	180.2393	193.4325	182.3858	208.7723
	${\mathit{\beta }}^{\prime }=(0,0)$	176.2032	189.3964	180.6287	210.3136
	${\mathit{\beta }}^{\prime }=(0,0.5)$	171.2697	184.4630	174.6974	201.0839
	${\mathit{\beta }}^{\prime }=(0.5,-0.5)$	180.0552	193.2485	182.0399	208.4264
	${\mathit{\beta }}^{\prime }=(0.5,0)$	176.5421	189.7354	179.3505	205.7371
	${\mathit{\beta }}^{\prime }=(0.5,0.5)$	170.9358	184.1291	174.3874	200.7740
degree = 2	${\mathit{\beta }}^{\prime }=(-0.5,-0.5)$	177.6850	190.8783	181.5645	211.2493
	${\mathit{\beta }}^{\prime }=(-0.5,0)$	177.4778	190.6711	181.9235	211.6084
	${\mathit{\beta }}^{\prime }=(-0.5,0.5)$	170.4353	183.6285	175.1660	204.8509
	${\mathit{\beta }}^{\prime }=(0,-0.5)$	178.6891	191.8824	182.5802	212.2651
	${\mathit{\beta }}^{\prime }=(0,0)$	175.4079	188.6011	179.8539	209.5387
	${\mathit{\beta }}^{\prime }=(0,0.5)$	169.7078	182.9011	174.7290	204.4138
	${\mathit{\beta }}^{\prime }=(0.5,-0.5)$	179.2848	192.4781	182.8814	212.5662
	${\mathit{\beta }}^{\prime }=(0.5,0)$	175.9501	189.1434	180.4436	210.1285
	${\mathit{\beta }}^{\prime }=(0.5,0.5)$	169.5194	182.7126	174.4124	204.0973
degree = 3	${\mathit{\beta }}^{\prime }=(-0.5,-0.5)$	179.3540	192.5473	184.6973	217.6805
	${\mathit{\beta }}^{\prime }=(-0.5,0)$	176.9312	190.1245	182.8466	215.8298
	${\mathit{\beta }}^{\prime }=(-0.5,0.5)$	171.5355	184.7287	178.6081	211.5913
	${\mathit{\beta }}^{\prime }=(0,-0.5)$	179.2683	192.4616	184.9653	217.9485
	${\mathit{\beta }}^{\prime }=(0,0)$	177.3079	190.5011	183.3511	216.3343
	${\mathit{\beta }}^{\prime }=(0,0.5)$	170.1874	183.3807	177.2666	210.2498
	${\mathit{\beta }}^{\prime }=(0.5,-0.5)$	178.5908	191.7841	183.9859	216.9691
	${\mathit{\beta }}^{\prime }=(0.5,0)$	177.4439	190.6372	183.7068	216.6900
	${\mathbf{\beta }}^{\prime }=(0.5,0.5)$	170.5001	183.6934	177.4353	210.4184

Remark 2. 

The function provided by the R package ICsurv [18] sometimes converges too slowly; therefore, in this paper, we add a restriction of 1000 iterations to the simulations using this function.

4. Real Data Analysis

In this section, we use the Weibull PH model to fit two real data, and use the proposed EM method to estimate all unknown parameters.

4.1. Lung Tumor Data

Lung tumor data were obtained by Hoel and Walberg [19] in 1972 from 144 male RFM mice that were examined in a tumorigenicity experiment on lung tumors. The tumorigenicity experiment focused on whether drugs or the environment accelerate the time to tumor onset. In this experiment, the time to tumor onset $T_i$ was of interest, but lung tumors were primarily non-lethal and insidious, meaning that the onset of the tumor did not affect the survival of the mice, so the time to tumor onset was not directly observable. Instead, we were only able to observe the time of death and the presence or absence of lung tumors when the mice died. The experiment recorded the death time $C_i$ of mice in days, and used indicator variable ${\delta }_i=I\left(T_i\le C_i\right)$ to note the presence (1) or absence (0) of lung tumors for each subject at the death time.

The experiment placed the mice in two different environments, a conventional environment (CE) and a germ-free environment (GE). There were 96 rats in the CE, and 27 had lung tumors at the time of death. Similarly, there were 48 rats in GE and 35 with lung tumors. The right censoring rate of the tumor onset was approximately $43.1\%$. Detailed variable descriptions are shown in Table 10.

Table 10. Variable description of lung tumor data.

Variable	Description
$x_i$	Environment, GE = 1, CE = 0
${\delta }_i$	Presence of tumor at death time, yes = 1, no = 0
$C_i$	The time of death

In order to estimate the impact of environmental factors on tumor development, we consider the Weibull PH model and use the EM algorithm to estimate the parameters:

$$\widehat{\lambda }=0.001,\widehat{\gamma }=1.988,\widehat{\beta }=0.803.$$

Based on Louis’ method, we can easily calculate that the estimated standard deviation of $\widehat{\beta }$ is 0.247. This results in a p-value of $0.0012$ for the comparison of the two groups, meaning that the incidence rates of lung tumors in the two different environments differ significantly. The corresponding hazard function and survival function are

$$\lambda \left(t,x\right)=0.001988{\left(t/100\right)}^{0.988}exp\left(0.803x\right),$$

and

$$S\left(t,x\right)=exp\left\{-{\left(t/100\right)}^{1.988}exp\left(0.803x\right)\right\}.$$

The baseline hazard function ${\lambda }_0\left(t\right)=\lambda \gamma {\left(\lambda t\right)}^{\gamma -1}$ increases with time since $\widehat{\gamma }=1.988$, indicating that the risk of cancer in mice increases over time. Furthermore, the risk ratio of lung tumors in mice under GE and CE is

$$\frac{\lambda \left(t,x=1\right)}{\lambda \left(t,x=0\right)}=exp\left(0.803\right).$$

This suggests that the risk of tumor growth in mice in the GE is $exp\left(0.803\right)\approx 2.232$ times higher than that in the CE. Sun [8] discussed the non-parametric maximum likelihood estimation (NPMLE) of the survival function based on current status data. Figure 1 presents the estimators of the survival functions for the EM method and the NPMLE method.

Figure 1. Estimates of survival functions of time for lung onset tumor by EM method and NPMLE method.

It is noteworthy that the estimator of the survival function from the Weibull PH model is very close to that from the NPMLE. Both the EM method and NPMLE method suggest that lung tumors tend to develop earlier in the CE compared to the GE. However, overall, mice in the GE appear to have a higher risk of developing lung tumors than mice in the CE. At the same time, this indicates that the Weibull PH model is appropriate to fit these data. In addition, the EM algorithm can effectively estimate model parameters in the Weibull PH model, which provides a continuous expression for the survival function, enabling the estimation of survival probabilities at any given time. In contrast, the NPMLE method is limited to approximating survival probabilities only at the endpoints of the observation interval.

Furthermore, we compare the Weibull PH model with the semi-parametric PH model, and the results are presented in Table 11. There are slight differences in the regression coefficient estimates from the two methods. Moreover, the Weibull PH model demonstrates smaller variances, and smaller AIC and BIC values compared to the semi-parametric PH model. These findings suggest that the Weibull PH model may be more suitable for modeling the lung tumor data.

Table 11. The estimation results of the Weibull PH model and the semi-parametric PH model for lung tumor data, where the value in parentheses represents the degree of the spline basis function in the semi-parametric PH model and the $\widehat{\mathrm{Var}}$ denotes the estimated variance of $\widehat{\beta }$.

	$\widehat{\mathit{\beta }}$	$\widehat{\mathbf{Var}}$	AIC	BIC
Weibull PH model	0.803	0.061	167	176
semi-parametric PH model (1)	0.805	0.144	174	195
semi-parametric PH model (2)	0.843	0.148	177	200
semi-parametric PH model (3)	0.818	0.157	179	205

4.2. The Data of IOL Calcification

The calcification of hydrogel intraocular lenses (IOL) [20,21] is a rare complication of cataract treatment. At examination time $T_i$ for each subject i, an ophthalmologist classified the severity of a patient’s IOL calcification into four grades. We simply consider mild or serious calcification as calcified $({\delta }_i=1)$, and no or little calcification as not calcified $({\delta }_i=0)$. There are 379 samples, including 142 males and 237 females. The variables are described in Table 12.

Table 12. Variable description of IOL calcification data.

Variable	Description
$x_i$	Gender, male = 1, female = 0
${\delta }_i$	Calcified or not, calcified = 1, uncalcified = 0
$C_i$	Examination time

We aim to assess the effect of the gender factor on IOL calcification. Assuming that the failure time $T_i$ follows the Weibull PH model, we use the proposed EM algorithm to obtain the corresponding parameter estimates as follows:

$$\widehat{\lambda }=0.005,\widehat{\gamma }=0.642,\widehat{\beta }=-0.269.$$

Meanwhile, the estimated standard error of $\widehat{\beta }$ is $0.312$, calculated by Louis’ method. This results in a p-value of $0.388$ for testing $\beta =0$ according to the asymptotic normality of the MLE, indicating that there is no significant difference between males and females in terms of the time to calcification. The estimator $\widehat{\gamma }=0.642$ suggests that the risk of IOL calcification decreases with time. Although the gender is not significant, it appears that the risk of IOL calcification in males is roughly $exp\left(-0.269\right)\approx 0.76$ times that of females based on the risk ratio. Figure 2 shows the estimators of the survival functions for the IOL calcification data by the EM method and the NPMLE method [8], respectively. Figure 2 suggests that calcification tends to occur earlier in males, but overall, the risk of calcification is slightly higher in females than in males. In addition, the estimators of the survival function from the Weibull PH model for the IOL data are close to those from the NPMLE, suggesting the suitability of the Weibull PH model for describing the IOL data.

Figure 2. Estimates of survival functions of time for IOL calcification by EM method and NPMLE method.

Table 13 summarizes the comparison results of the Weibull PH model and semi-parametric PH models with varying degrees. The estimates of $\beta$ of these models are relatively close, but the Weibull PH model provides smaller variances, and smaller AIC and BIC values, indicating that it is better suited for this dataset.

Table 13. The estimation results of the Weibull PH model and the semi-parametric PH model for the IOL data, where the value in parentheses represents the degree of the spline basis function in the semi-parametric PH model and the $\widehat{\mathrm{Var}}$ denotes the estimated variance of $\widehat{\beta }$.

	$\widehat{\mathit{\beta }}$	$\widehat{\mathbf{Var}}$	AIC	BIC
Weibull PH model	−0.269	0.097	287	299
semi-parametric PH model (1)	−0.251	0.130	295	323
semi-parametric PH model (2)	−0.254	0.108	297	328
semi-parametric PH model (3)	−0.256	0.210	299	334

5. Conclusions

The Weibull PH model is a commonly used parametric model in fields like reliability analysis and survival analysis. In this paper, we consider the Weibull PH model under current status data. A Poisson-based EM algorithm is developed to estimate the parameters, where the M-step of the EM algorithm is realized by the NR method. Louis’ method is used to estimate the asymptotic variance–covariance matrix, whose diagonal elements are estimators of the variance. Then, the proposed EM method is compared with the NR method through simulations. On the whole, the estimator results of the two methods are close; the bias of the EM algorithm is consistently smaller than that of the NR method, especially when the sample size is small, while the variance of EM is smaller in small samples but larger in larger sample sizes. In addition, the proportion that the true parameter value falls in the $95\%$ confidence intervals is close for both methods. Moreover, the EM algorithm offers several advantages. It simplifies the process of parameter estimation and covariance matrix estimation. Additionally, the EM algorithm ensures that the critical scale parameter remains positive throughout the estimator process, which is a crucial requirement in the Weibull PH model. The comparison between the Weibull PH model and the semi-parametric PH model reveals that the Weibull PH model demonstrates significant advantages when the failure time follows a Weibull distribution. This is also supported by the smaller values of AIC and BIC for the Weibull PH model. The method and model of this paper are applied to two practical examples. In the lung tumor experiment, the results indicate that the risk of lung tumor increases over time, and the environment significantly influences tumor growth. Specifically, mice in the GE exhibit a higher risk of developing lung tumors compared to those in the CE. In the IOL calcification data, the risk of calcification decreases over time, and gender has no impact on the risk of IOL calcification.

The discussion in this paper is based on current status data; the EM algorithm of the Weibull PH model can be extended to analysis of general interval censored data. In addition, our future research plans include exploring Bayesian analysis for the Weibull PH model.