Bayesian parametric estimation based on left-truncated competing risks data under bivariate Clayton copula models

Abstract

In observational/field studies, competing risks and left-truncation may co-exist, yielding ‘left-truncated competing risks’ settings. Under the assumption of independent competing risks, parametric estimation methods were developed for left-truncated competing risks data. However, competing risks may be dependent in real applications. In this paper, we propose a Bayesian estimator for both independent competing risks and copula-based dependent competing risks models under left-truncation. The simulations show that the Bayesian estimator for the copula-based dependent risks model yields the desired performance when competing risks are dependent. We also comprehensively explore the choice of the prior distributions (Gamma, Inverse-Gamma, Uniform, half Normal and half Cauchy) and hyperparameters via simulations. Finally, two real datasets are analyzed to demonstrate the proposed estimators.

1. Introduction

Competing risks often arise in observational or field studies, where survival/failure time of interest may not be ascertained due to the incidence of other events, known as competing risks. Three different models were available to analyze competing risks data; the first and the most classical one is the latent failure time model [9], the second one is the cause-specific hazard model [35], and the third one is the subdistribution hazard model [19]. The latent failure time model explicitly specifies a multivariate distribution on latent failure times under the independence model [5,27], multivariate dependence models [4,17,30] or copula models [16,48,49]. The cause-specific and subdistribution models do not explicitly specify the independence/dependence structures for competing risks. All these competing risks models are widely applied to reliability engineering [17,30,50] and medicine [1,2,11,38]. While the choice of the models depends on the goal of the research and application fields, we will focus on the latent failure time model.

Besides competing risks, ‘left-truncation’ arises in observational/field studies. Subjects who have already experienced the first milestone (e.g. onset of disease) before starting a study may be excluded from the data, which is known as left-truncation in survival analysis [25]. It is well known that ignorance of left-truncation causes a systematic bias toward favorable survival [25]. To analyze such left-truncated data in observational/field studies, Hong et al. [23] divided the data into two parts: a truncated part (e.g. times before the starting date of study) and an untruncated part (e.g. times after the starting date of study), and then introduced a combined likelihood function consisting of the two parts. Recently, this likelihood-based approach for left-truncated observational/field data (with right-censoring) received considerable attentions: see [3,12,14,24,29,32,36,43–46], and references therein.

Competing risks and left-truncation may co-exist, yielding a ‘left-truncated competing risks’ setting. Kundu et al. [27] first considered this setting under observational/field left-truncated data by extending the likelihood approach of Hong et al. [23] to introduce competing risks. Under the ‘independent’ competing risks model, Kundu et al. [27] developed their inference methods based on left-truncated competing risks data. However, competing event times may be dependent in real applications. For example, in many cancer studies focusing on time to progression (TTP) and overall survival (OS), strong correlation is observed between TTP and OS [8,15,20,28]. Therefore, the assumption of the independent risks may lead to biases due to model misspecification. Accordingly, some authors suggested specific models for left-truncated competing risks, such as the Marshall–Olkin bivariate Rayleigh model [46], and the Marshall–Olkin bivariate Weibull model [43,44]. These specific bivariate models are not the standard models, though they are good starting points for fitting dependent competing risks.

Copula-based competing risks models provide a flexible approach to incorporate the standard parametric models, such as the Weibull model. To the best of our knowledge, copula-based competing risks models with left-truncation are limited to the nonparametric models of Uña-Álvarez and Veraverbeke [10] and the parametric models of Michimae and Emura [31]. However, a problem remains: it is difficult to estimate a copula parameter. Uña-Álvarez and Veraverbeke [10] resorted to an assumed copula. Also, Michimae and Emura [31] pointed out the low estimation accuracy for parameter estimates when the copula parameter is unknown. Bayes estimators may offer a solution to this problem.

In this paper, we propose a Bayesian estimator for both the independent competing risks model and copula-based dependent competing risks models under left-truncation. There are two main contributions: First, we improve the estimation accuracy of the copula model of Michimae and Emura [31] by developing Bayesian estimators. Second, we extend the Bayesian estimator of Kundu et al. [27] that were restricted to Gamma priors and independent risks. We implement five prior distributions (Gamma, Inverse-Gamma, Uniform, half Normal and half Cauchy) and dependent risks. Our simulations show that the Bayesian estimators for the copula-based dependent risks model yield superior performance than these existing methods. Finally, two real datasets are analyzed for illustration. Thus, our paper is the comprehensive extension and improvement of the previous papers, Kundu et al. [27] and Michimae and Emura [31].

The structure of the paper is as follows. Section 2 briefly introduces the left-truncated competing risks setting. Section 3 proposes two Bayesian estimators under the independence model and copula-based model, respectively. Five different prior distributions are also suggested in this section. Section 4 presents Monte Carlo simulation results, comparing the two Bayesian estimators and the five prior distributions. Section 5 analyzes two datasets. Section 6 concludes with discussions.

2. Backgrounds

We review left-truncated competing risks data that were introduced by Kundu et al. [27].

Figure 1 shows a schematic description for an observational clinical study with follow-up. The study begins at a calendar time point, where subjects who experience disease onsets are recruited for further follow-up. During the study, a recruited subject after the starting date may experience one of two failure events, Event 1 (the event of interest: Case 1) and Event 2 (the competing risk: Case 2). Subjects who experienced the disease onset and who failed (by either Event 1 or Event 2) before the recruitment time were not observable, leading to truncation of subjects (‘Not observed’). However, we can observe the subjects if they did not fail before the recruitment time (Cases 4–6). They may experience one of two failure events, Event 1 (Case 4) and Event 2 (Case 5) during the study. As the follow-up time is often limited, the recruited subjects may be censored at the end of the study without experiencing any event (Cases 3 and 6). This leads to left-truncated competing risks data with right-censoring.

Figure 1.

Left-truncated competing risks (Events 1 and 2), and onset of disease ( $\bullet$) from a hypothetical observational clinical study. Any event occurring in the first duration (D1) is not observable. An event is observable only within the second duration (D2). T and $\tau$ indicate observed event times and truncation times.

Observed data from n subjects consist of $(\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })$, where $\mathbf{\text{T}}=(t_1,\dots ,t_n)$, $\mathit{\tau }=({\tau }_1,\dots ,{\tau }_n)$, $\mathit{\delta }=({\delta }_1,\dots ,{\delta }_n)$, and $\mathit{\nu }=({\nu }_1,\dots ,{\nu }_n)$ will be defined below. Let $(T_{1i},T_{2i})$ be continuous failure times (measured from the time of disease onset) for Event 1 and Event 2, respectively, for a subject $i=1,2,\dots ,n$. Also, let $C_i$ be random right-censored time, which is assumed to be independent of $(T_{1i},T_{2i})$. What we observe is an event time $t_i$ defined as a realized value of the first occurring event time $T_i=min(T_{1i},T_{2i},C_i)$. We also observe an event type ${\delta }_i=I(T_{1i}\le T_{2i},T_{1i}\le C_i)+2I(T_{2i}\le T_{1i},T_{2i}\le C_i)$, namely, ${\delta }_i=1$ for Event 1, ${\delta }_i=2$ for Event 2, or ${\delta }_i=0$ for censoring. We also observe a ‘left-truncation time’ ${\tau }_i$ (see dotted lines in Figure 1) for a subject who experienced the disease onset before starting the study. For such truncated subjects, we account for the truncation constraint $T_i\ge {\tau }_i$ since a subject has survived long enough. We observe a truncation indicator defined as

$${\nu }_i=\{\begin{array}{ll}0,& \text{if the subject is constrained by truncation\hspace{0.17em}}(T_i\ge {\tau }_i\ge 0)\\ 1,& \text{if the subject is unconstrained by truncation\hspace{0.17em}}({\tau }_i=\mathrm{undefined})\end{array}$$

3. Proposed Bayesian methods

First, the Bayesian estimators for the independent competing risks model will be proposed using different prior distributions from the independent risks model of Kundu et al. [27]. Second, the copula-based Bayesian estimators will be proposed as a general dependent case.

3.1. Bayesian estimators for independent competing risks model

Let $f_1(\cdot |{\mathit{\theta }}_1)$ be the density of $T_{1i}$ and $f_2(\cdot |{\mathit{\theta }}_2)$ be the density of $T_{2i}$; the corresponding survival functions are $S_1(\cdot |{\mathit{\theta }}_1)$ and $S_2(\cdot |{\mathit{\theta }}_2)$, where ${\mathit{\theta }}_j$ $j=0,1,2.$, is the vector of parameters. The likelihood of an individual sample $(t_i,{\tau }_i,{\delta }_i,{\nu }_i)$ is divided into the following cases (see Figure 1) according to the observed events and truncation status (Kundu et al. [27]):

$$\begin{array}{rl}\mathbf{\text{Case 1:}}& P(T_{1i}=t_i,T_{2i}>t_i)=f_1(t_i)S_2(t_i),i\in I_1,{\nu }_i=1\end{array}$$

$$\begin{array}{rl}\mathbf{\text{Case 2:}}& P(T_{2i}=t_i,T_{1i}>t_i)=f_2(t_i)S_1(t_i),i\in I_2,{\nu }_i=1\end{array}$$

$$\begin{array}{rl}\mathbf{\text{Case 3:}}& P(T_{1i}>t_i,T_{2i}>t_i)=S_1(t_i)S_2(t_i),i\in I_0,{\nu }_i=1\end{array}$$

$$\begin{array}{rl}\mathbf{\text{Case 4:}}& P(T_{1i}=t_i,T_{2i}>t_i|T_{1i}>{\tau }_i,T_{2i}>{\tau }_i)=\frac{f_1(t_i)}{S_1({\tau }_i)}\cdot \frac{S_2(t_i)}{S_2({\tau }_i)},i\in I_1,{\nu }_i=0\end{array}$$

$$\begin{array}{rl}\mathbf{\text{Case 5:}}& P(T_{2i}=t_i,T_{1i}>t_i|T_{1i}>{\tau }_i,T_{2i}>{\tau }_i)=\frac{f_2(t_i)}{S_2({\tau }_i)}\cdot \frac{S_1(t_i)}{S_1({\tau }_i)},i\in I_2,{\nu }_i=0\end{array}$$

$$\begin{array}{rl}\mathbf{\text{Case 6:}}& P(T_{1i}>t_i,T_{2i}>t_i|T_{1i}>{\tau }_i,T_{2i}>{\tau }_i)=\frac{S_1(t_i)}{S_1({\tau }_i)}\cdot \frac{S_2(t_i)}{S_2({\tau }_i)},i\in I_0,{\nu }_i=0\end{array}$$

where $I_0$ is a set of censored observations, and I₁ and I₂ are sets of failure observations due to Event 1 and Event 2, respectively. That is, $I_j=\{i;{\delta }_i=j\},j=1,2.$

Combining the likelihoods of Cases 1–6, the likelihood function is given as follows:

$$\begin{array}{rl}L({\mathit{\theta }}_1,{\mathit{\theta }}_2|\mathbf{\text{T}},\mathit{\tau },\delta ,\mathit{\nu })& =\prod _{i\in I_0}{\{S_1(t_i|{\mathit{\theta }}_1)S_2(t_i|{\mathit{\theta }}_2)\}}^{{\nu }_i}{\left\{\frac{S_1(t_i|{\mathit{\theta }}_1)}{S_1({\tau }_i|{\mathit{\theta }}_1)}\cdot \frac{S_2(t_i|{\mathit{\theta }}_2)}{S_2({\tau }_i|{\mathit{\theta }}_2)}\right\}}^{1-{\nu }_i}\\ & \times \prod _{i\in I_1}{\{f_1(t_i|{\mathit{\theta }}_1)S_2(t_i|{\mathit{\theta }}_2)\}}^{{\nu }_i}{\left\{\frac{f_1(t_i|{\mathit{\theta }}_1)}{S_1({\tau }_i|{\mathit{\theta }}_1)}\cdot \frac{S_2(t_i|{\mathit{\theta }}_2)}{S_2({\tau }_i|{\mathit{\theta }}_2)}\right\}}^{1-{\nu }_i}\\ & \times \prod _{i\in I_2}{\{f_2(t_i|{\mathit{\theta }}_2)S_1(t_i|{\mathit{\theta }}_1)\}}^{{\nu }_i}{\left\{\frac{f_2(t_i|{\mathit{\theta }}_2)}{S_2({\tau }_i|{\mathit{\theta }}_2)}\cdot \frac{S_1(t_i|{\mathit{\theta }}_1)}{S_1({\tau }_i|{\mathit{\theta }}_1)}\right\}}^{1-{\nu }_i}\end{array}$$

Under the Weibull survival function $S_i(t;{\alpha }_i,{\lambda }_i)=\exp (-{\lambda }_i{t}^{{\alpha }_i}),i=1,2$, Kundu et al. [27] used the following likelihood function for estimating the parameters:

$$\begin{array}{rl}L({\lambda }_1,{\alpha }_1,{\lambda }_2,{\alpha }_2|\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })& =\prod _{i\in l_0}{\{\exp (-{\lambda }_1{t}_i^{{\alpha }_1})\exp (-{\lambda }_2{t}_i^{{\alpha }_2})\}}^{{\nu }_i}\\ & \times {\left\{\frac{\exp (-{\lambda }_1{t}_i^{{\alpha }_1})\exp (-{\lambda }_2{t}_i^{{\alpha }_2})}{\exp (-{\lambda }_1{\tau }_i^{{\alpha }_1})\exp (-{\lambda }_2{\tau }_i^{{\alpha }_2})}\right\}}^{1-{\nu }_i}\\ & \times \prod _{i\in l_1}{\{{\alpha }_1{\lambda }_1{t}_i^{{\alpha }_1-1}\exp (-{\lambda }_1{t}_i^{{\alpha }_1})\exp (-{\lambda }_2{t}_i^{{\alpha }_2})\}}^{{\nu }_i}\\ & \times {\left\{\frac{{\alpha }_1{\lambda }_1{t}_i^{{\alpha }_1-1}\exp (-{\lambda }_1{t}_i^{{\alpha }_1})\exp (-{\lambda }_2{t}_i^{{\alpha }_2})}{\exp (-{\lambda }_1{\tau }_i^{{\alpha }_1})\exp (-{\lambda }_2{\tau }_i^{{\alpha }_2})}\right\}}^{1-{\nu }_i}\\ & \times \prod _{i\in l_2}{\{{\alpha }_2{\lambda }_2{t}_i^{{\alpha }_2-1}\exp (-{\lambda }_2{t}_i^{{\alpha }_2})\exp (-{\lambda }_1{t}_i^{{\alpha }_1})\}}^{{\nu }_i}\\ & \times {\left\{\frac{{\alpha }_2{\lambda }_2{t}_i^{{\alpha }_2-1}\exp (-{\lambda }_2{t}_i^{{\alpha }_2})\exp (-{\lambda }_1{t}_i^{{\alpha }_1})}{\exp (-{\lambda }_1{\tau }_i^{{\alpha }_1})\exp (-{\lambda }_2{\tau }_i^{{\alpha }_2})}\right\}}^{1-{\nu }_i}\end{array}$$

Kundu et al. [27] suggested using Gamma prior distributions for the two Weibull parameters; ${\lambda }_1\sim G(a_1,b_1)$, ${\alpha }_1\sim G(c_1,d_1)$, ${\lambda }_2\sim G(a_2,b_2)$ and ${\alpha }_2\sim G(c_2,d_2)$, where all the Gamma parameters are positive. Under the Gamma priors, the joint prior is $f({\lambda }_1,{\alpha }_1,{\lambda }_2,{\alpha }_2)=f({\lambda }_1)f({\alpha }_1)f({\lambda }_2)f({\alpha }_2)$.

Since the choice of the gamma priors is somewhat ad-hoc, we also considered different priors not considered in Kundu et al. [27]. Specifically, we considered the Inverse-Gamma prior, the Uniform prior, half Normal prior and half Cauchy prior in addition to the Gamma prior. In the five priors, hyperparameters were chosen for the prior distributions to be uninformative. Table S1 gives the details of the prior specifications, including the hyperparameter values of the prior distributions.

Given the observed data and prior distributions, the joint posterior density becomes

$$f({\lambda }_1,{\alpha }_1,{\lambda }_2,{\alpha }_2|\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })\propto L({\lambda }_1,{\alpha }_1,{\lambda }_2,{\alpha }_2|\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })f({\lambda }_1)f({\alpha }_1)f({\lambda }_2)f({\alpha }_2)$$

Given the observed data, one obtains a Bayesian estimator $\hat{\mathit{\theta }}=({\hat{\lambda }}_1,{\hat{\alpha }}_1,{\hat{\lambda }}_2,{\hat{\alpha }}_2)$ defined as the posterior median under a given prior. We will call the estimator described above as the independent Bayesian estimator.

3.2. Bayesian estimators for copula-based dependent competing risks model

A copula gives a dependence structure in a bivariate distribution. Let $F_1(\cdot )$ and $F_2(\cdot )$ be cumulative distribution functions (CDFs). Then, a bivariate CDF can be defined as $F(x_1,x_2)=C(F_1(x_1),F_2(x_2))$, where $C:[0,1{]}^2\to [0,1]$ is a copula [34,13].

In this paper, we consider bivariate latent failure times $(T_1,T_2)$ whose marginal survival functions are $S_1(t)=P(T_1>t)$ and $S_2(t)=P(T_2>t)$. Instead of the bivariate CDF, we adopt a bivariate survival function for dependent competing risks [16] as follows;

$$P(T_1>t_1,T_2>t_2)=C(S_1(t_1),S_2(t_2))$$

As in Escarela and Carriere [16], we separately parametrize the copula and marginal survival functions. For the copula, we adopt the bivariate Clayton copula due to its simplicity and popularity, defined by

$$C_{\varphi }(u,v|\varphi )=(u^{-\varphi }+v^{-\varphi }-1{)}^{-\frac{1}{\varphi }},\varphi \in [-1,\mathrm{\infty })\mathrm{\setminus }\{0\}$$

where $\varphi$ is a dependence parameter.

As in the independent competing risks model, the likelihood of an individual sample $(t_i,{\tau }_i,{\delta }_i,{\nu }_i)$ is divided into the following cases (see Figure 1) according to the observed events and truncation status:

$$\begin{array}{rl}\mathbf{\text{Case 1:}}& P(T_{1i}=t_i,T_{2i}>t_i)=-\frac{\mathrm{\partial }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{\mathrm{\partial }{u}_i}\frac{d{u}_i}{d{t}_i},i\in I_1,{\nu }_i=1\end{array}$$

$$\begin{array}{rl}\mathbf{\text{Case 2:}}& P(T_{2i}=t_i,T_{1i}>t_i)=-\frac{\mathrm{\partial }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{\mathrm{\partial }{v}_i}\frac{d{v}_i}{d{t}_i},i\in I_2,{\nu }_i=1\end{array}$$

$$\begin{array}{rl}\mathbf{\text{Case 3:}}& P(T_{1i}>t_i,T_{2i}>t_i)=(u_i^{-\varphi }+v_i^{-\varphi }-1{)}^{-\frac{1}{\varphi }},i\in I_0,{\nu }_i=1\end{array}$$

$$\begin{array}{rl}\mathbf{\text{Case 4:}}& P(T_{1i}=t_i,T_{2i}>t_i|T_{1i}>{\tau }_i,T_{2i}>{\tau }_i)\\ & =-\frac{1}{{(x_i^{-\varphi }+y_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}\frac{\mathrm{\partial }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{\mathrm{\partial }{u}_i}\frac{d{u}_i}{d{t}_i},i\in I_1,{\nu }_i=0\end{array}$$

$$\begin{array}{rl}\mathbf{\text{Case 5:}}& P(T_{2i}=t_i,T_{1i}>t_i|T_{1i}>{\tau }_i,T_{2i}>{\tau }_i)\\ & =-\frac{1}{{(x_i^{-\varphi }+y_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}\frac{\mathrm{\partial }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{\mathrm{\partial }{v}_i}\frac{d{v}_i}{d{t}_i},i\in I_2,{\nu }_i=0\end{array}$$

$$\begin{array}{rl}\mathbf{\text{Case 6:}}& P(T_{1i}>t_i,T_{2i}>t_i|T_{1i}>{\tau }_i,T_{2i}>{\tau }_i)=\frac{{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{{(x_i^{-\varphi }+y_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}},i\in I_0,{\nu }_i=0\end{array}$$

Here, $u_i=\exp (-{\lambda }_1{t}_i^{{\alpha }_1})$, $v_i=\exp (-{\lambda }_2{t}_i^{{\alpha }_2})$, $x_i=\exp (-{\lambda }_1{\tau }_i^{{\alpha }_1})$ and $y_i=\exp (-{\lambda }_2{\tau }_i^{{\alpha }_2})$ for the case of the Weibull model.

Combining the likelihoods of Cases 1–6, the copula-based likelihood function is given as follows:

$$\begin{array}{rl}L({\mathit{\theta }}_1,{\mathit{\theta }}_2,\varphi |\mathbf{\text{T}},\mathit{\tau },\delta ,\mathit{\nu })& =\prod _{i\in I_0}{\left\{{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}\right\}}^{{\nu }_i}{\left\{\frac{{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{{(x_i^{-\varphi }+y_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}\right\}}^{1-{\nu }_i}\\ & \times \prod _{i\in I_1}{\left\{-\frac{\mathrm{\partial }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{\mathrm{\partial }{u}_i}\frac{d{u}_i}{d{t}_i}\right\}}^{{\nu }_i}\\ & \times {\left\{-\frac{1}{{(x_i^{-\varphi }+y_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}\frac{\mathrm{\partial }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{\mathrm{\partial }{u}_i}\frac{d{u}_i}{d{t}_i}\right\}}^{1-{\nu }_i}\\ & \times \prod _{i\in I_2}{\left\{-\frac{\mathrm{\partial }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{\mathrm{\partial }{v}_i}\frac{d{v}_i}{d{t}_i}\right\}}^{{\nu }_i}\\ & \times {\left\{-\frac{1}{{(x_i^{-\varphi }+y_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}\frac{\mathrm{\partial }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{\mathrm{\partial }{v}_i}\frac{d{v}_i}{d{t}_i}\right\}}^{1-{\nu }_i}\end{array}$$

For the marginal survival functions, we set Weibull distributions $S_i(t)=\exp (-{\lambda }_i{t}^{{\alpha }_i}),i=1,2$. The Weibull distribution is perhaps the most standard and convenient parametric model for analyzing biomedical survival data [37] and reliability engineering data [17,47]. Nonetheless, the Weibull distribution can be flexibly changed to other distributions under the copula model. Following Michimae and Emura [31] who dealt with left-truncation and copula-based competing risks, the corresponding likelihood function is

$$\begin{array}{rl}L({\lambda }_1,{\alpha }_1,{\lambda }_2,{\alpha }_2,\varphi |\mathbf{\text{T}},\mathit{\tau },\delta ,\mathit{\nu })& =\prod _{i\in I_0}{\left\{{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}\right\}}^{{\nu }_i}{\left\{\frac{{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}{{(x_i^{-\varphi }+y_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}\right\}}^{1-{\nu }_i}\\ & \times \prod _{i\in I_1}{\left\{{\alpha }_1{\lambda }_1{t}_i^{{\alpha }_1-1}{u}_i^{-\varphi }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\left(1+\frac{1}{\varphi }\right)}\right\}}^{{\nu }_i}\\ & \times {\left\{\frac{{\alpha }_1{\lambda }_1{t}_i^{{\alpha }_1-1}{u}_i^{-\varphi }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\left(1+\frac{1}{\varphi }\right)}}{{(x_i^{-\varphi }+y_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}\right\}}^{1-{\nu }_i}\\ & \times \prod _{i\in I_2}{\left\{{\alpha }_2{\lambda }_2{t}_i^{{\alpha }_2-1}{v}_i^{-\varphi }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\left(1+\frac{1}{\varphi }\right)}\right\}}^{{\nu }_i}\\ & \times {\left\{\frac{{\alpha }_2{\lambda }_2{t}_i^{{\alpha }_2-1}{v}_i^{-\varphi }{(u_i^{-\varphi }+v_i^{-\varphi }-1)}^{-\left(1+\frac{1}{\varphi }\right)}}{{(x_i^{-\varphi }+y_i^{-\varphi }-1)}^{-\frac{1}{\varphi }}}\right\}}^{1-{\nu }_i}\end{array}$$

where $u_i=\exp (-{\lambda }_1{t}_i^{{\alpha }_1})$, $v_i=\exp (-{\lambda }_2{t}_i^{{\alpha }_2})$, $x_i=\exp (-{\lambda }_1{\tau }_i^{{\alpha }_1})$ and $y_i=\exp (-{\lambda }_2{\tau }_i^{{\alpha }_2})$.

Similar to the independent risks model, we propose five priors (Gamma, Inverse-Gamma, Uniform, half Normal and half Cauchy) as shown in Table S1. Under the copula-based dependent risks model, the joint posterior density is

$$f({\lambda }_1,{\alpha }_1,{\lambda }_2,{\alpha }_2,\varphi |\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })\propto L({\lambda }_1,{\alpha }_1,{\lambda }_2,{\alpha }_2,\varphi |\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })f({\lambda }_1)f({\alpha }_1)f({\lambda }_2)f({\alpha }_2)f(\varphi )$$

We set the improper Uniform prior distribution $U(-1,\mathrm{\infty })$ for the copula parameter $\varphi$. Given the observed data, one obtains a Bayesian estimator $\hat{\mathit{\theta }}=({\hat{\lambda }}_1,{\hat{\alpha }}_1,{\hat{\lambda }}_2,{\hat{\alpha }}_2,\hat{\varphi })$ as the posterior median with respect to the posterior density. We will call it the copula-based Bayesian estimator.

3.3. Computation of proposed Bayesian estimators

Let $f(\mathit{\theta })$ be a prior density function for unknown parameters $\mathit{\theta }$. Irrespective of the independence or dependence model, the posterior density is

$$f(\mathit{\theta }|\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })\propto L(\mathit{\theta }|\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })f(\mathit{\theta })$$

The posterior median $\hat{\mathit{\theta }}=\mathrm{Median}(\mathit{\theta }|\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })$ is the Bayes estimator, where $\mathit{\theta }=({\lambda }_1,{\alpha }_1,{\lambda }_2,{\alpha }_2)$ for the independent risks model while $\mathit{\theta }=({\lambda }_1,{\alpha }_1,{\lambda }_2,{\alpha }_2,\varphi )$ for the copula-based dependent risks model. The forms of $L(\mathit{\theta }|\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })$ are as stated above. Here, the two main issues are the choice of the prior density $f(\mathit{\theta })$ and computation methods for $\hat{\mathit{\theta }}$.

For computing with complex posterior distributions such as $f(\mathit{\theta }|\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })$, analytical formulas for the posterior medians through conjugacy may not exist. A simulation-based approach to compute the posterior medians through MCMC (Markov Chain Monte Carlo) algorithms is valid even in such a case. The Stan software [41,40] uses the No-U-Turn Sampler [22], an improved type of MCMC algorithm known as the Hamiltonian Monte Carlo (HMC), which can efficiently sample from complex posterior distributions with correlated parameters. We adopt the posterior median as the point estimate, and the posterior 2.5% and 97.5% points as the 95% credible interval (95% CI).

We adopted the Stan software to compute the posterior distribution by inputting the formulas of $f(\mathit{\theta })$ and $L(\mathit{\theta }|\mathbf{\text{T}},\mathit{\tau },\mathit{\delta },\mathit{\nu })$. We set the number of Markov chains to 4, the number of iterations for each chain to 4000, and the number of worm-up iterations per chain to 2000 for both the simulation (Section 4) and data analysis (Section 5). We checked ‘Rhat’ criterion in Stan, known as the Gelman-Rubin convergence diagnostic [21] for the successful convergence of the MCMC algorithm. As a rule of thumb, we employed $R\le 1.10$ for all parameters, which indicates that four chains have converged.

The issues of choosing the prior densities will be explored in the following sections. Some conjugate gamma prior densities were explored in the Bayesian estimator by Kundu et al. [27] under the independent risks model with the ‘common shape parameter’. However, their explanations for different shape parameters (uncommon parameters) were not sufficient and unclear. Since we do not assume the common shape parameter, we need to explore this issue under the independent risks model.

4. Simulation study

We conducted simulation studies to assess the performance of the proposed estimators (the independent Bayesian estimator and copula-based Bayesian estimator). The main purpose is to compare the independent Bayesian estimator and copula-based Bayesian estimator when the true model has independent competing risks or dependent competing risks. Another purpose is to see how different priors (Gamma, Inverse-Gamma, Uniform, half Normal and half Cauchy) influence the performance of the Bayesian estimators. Supplementary Materials report another simulation study comparing the Bayes estimator with the maximum likelihood estimator, which confirm the superiority of the Bayes estimators.

4.1. Simulation designs

Table S2 shows eight scenarios (Scenarios 1–8) with different configurations of parameters.

For Scenarios 1–4, we set the first duration $D1=6$ and the second duration $D2=1$ (see Figure 1 for the definitions of $D1$ and $D2$). For Scenarios 5–8, we set $D1=3$ and $D2=1$. In all the scenarios, we set the sample size n = 100 or 500. The truncation percentage was set to be 50% (50 truncated subjects occur in $D1$, and 50 untruncated subjects occur in $D2$ for n = 100). We sampled a birth time $B_i=U(0,D1)$ for a truncated sample $({\nu }_i=0)$; we also sampled a birth time $B_i=U(D1,D1+D2)$ for an untruncated sample $({\nu }_i=1)$. For ${\nu }_i=0$, the truncation time ${\tau }_i$ was set by ${\tau }_i=D1-B_i$. For ${\nu }_i=1$, we do not define ${\tau }_i$. Censoring time was defined as $C_i=D1+D2-B_i$.

In the case of independent risks, we sampled two independent latent failure times, $T_{1i}$ and $T_{2i}$, under Weibull distributions with the shape parameter ${\alpha }_j$ and scale parameter ${\lambda }_j$. In the case of dependent competing risks, we sampled two dependent Weibull failure times, $T_{1i}$ and $T_{2i}$, form the Clayton copula model with parameter $\varphi =2$ (Kendall’s tau = 0.5). We considered the common parameters across events: $({\lambda }_1,{\alpha }_1)=({\lambda }_2,{\alpha }_2)=(1.00,1.00)$ for Scenarios 1–2, $(1.00,0.50)$ for Scenarios 3–4, and $(1.00,1.50)$ for Scenarios 5–6, corresponding to constant, decreasing and increasing hazards, respectively. We also considered the case of the varying parameters across events: $({\lambda }_1,{\alpha }_1)=(2.00,1.50)$ and $({\lambda }_2,{\alpha }_2)=(1.00,0.50)$ for Scenarios 7 and 8.

Following the left-truncated competing risks setting, we used the first occurring event time $T_i=min(T_{1i},T_{2i},C_i)$, the event type ${\delta }_i=I(T_{1i}\le T_{2i},T_{1i}\le C_i)+2I(T_{2i}\le T_{1i},T_{2i}\le C_i)$, and the truncation indicator ${\nu }_i$ as observations to calculate the Weibull parameter estimators $({\hat{\lambda }}_1,{\hat{\alpha }}_1,{\hat{\lambda }}_2,{\hat{\alpha }}_2)$ as well as the copula parameter estimator (for the case of the copula model) in 10 different ways: five independent Bayesian (five priors) and five copula-based Bayesian (five priors) estimators with the 95% CIs. Also, we calculated the estimators of the survival probabilities, ${\hat{S}}_i(t)=\exp (-{\hat{\lambda }}_i{t}^{{\hat{\alpha }}_i}),i=1,2$, for $t=0.5,1,2$.

1000 Monte Carlo repetitions (for n = 100) were performed to evaluate the performance of the estimators based on the median absolute errors (MAEs), and coverage probabilities of the 95%CI. For n = 100, the computing time ranged from 10,683 s (Scenario 4 for the independent model using the uniform prior) to 253,396 s (Scenario 1 for the copula-based model using the inverse-Gamma prior). For n = 500, we reduced the number of Monte Carlo repetitions from 1000 to 100; the computing time ranged from 4942 s (Scenario 2 for the independent model using the uniform prior) to 151,641 s (Scenario 2 for the copula-based model using the uniform prior).

4.2. Simulation results

In all the scenarios (Scenarios 1–8), the MAEs of all the estimators tend to decrease when the sample sizes increase (Tables S3–S10) and when the model is correctly specified. However, the comparative performance in terms of the MAEs depend largely on the true models (independent vs. dependent) and priors (Gamma, Inverse-Gamma, Uniform, half Normal and half Cauchy).

4.2.1. Weibull parameters and survival probabilities

When the true model is the independent risks model (Scenarios 1, 3, 5 and 7), the independent Bayesian estimator showed all very small MAEs for the Weibull parameters (Tables S3–S6) and the survival probabilities (Tables S7–S10). This is a natural conclusion since the true model is correctly specified. Also, the MAEs are similar across different prior densities (Tables S3–S10). In summary, the independent Bayesian estimator gives very accurate estimates irrespective of the prior choices as long as the independent risks model is correct. However, this independent estimator exhibits unacceptable biases and inflated MAEs when the true model is the copula-based dependent risks model.

When the true model is the copula-based dependent risks model (Scenarios 2, 4, 6 and 8), the independent Bayesian estimator yielded large MAEs due to biases in estimation. In contrast, the copula-based Bayesian estimator shows smaller MAEs compared to the independent Bayesian estimator. Among the different priors (Gamma, Inverse-Gamma, Uniform, half Normal and half Cauchy), the Uniform, the half Normal and the half Cauchy priors showed better performance than the Gamma and the Inverse-Gamma priors, especially in Scenarios 2, 4 and 6. However, in Scenario 8, there was no best estimator in terms of MAEs (Tables S6 and S10). Therefore, we conclude that the Bayes estimator with the improper Uniform, the half Normal or the half Cauchy prior provides the most accurate estimate for unknown Weibull parameters and survival probabilities.

4.2.2. Copula parameters

When the true model is the independent risks model (Scenarios 1, 3, 5 and 7), the Gamma and the Inverse-Gamma priors showed better performance than the Uniform, the half Normal and the half Cauchy priors, especially in Scenarios 1, 3 and 5. However, in the large sample size of Scenario 7, there was no better estimator in terms of MAEs (Table S6).

Contrary to the results, when the true model is the copula-based dependent risks model (Scenarios 2, 4, 6 and 8), the Uniform, the half Normal and the half Cauchy priors showed better performance than the Gamma and the Inverse-Gamma priors, in Scenarios 2, 4 and 6. However, in Scenario 8, the result was reversed. Therefore, we conclude that the best choice of the prior depends on the situations and cannot be uniquely determined for unknown copula parameters.

The conclusions from our simulations are clear-cut except for estimating copula parameters. When the true model has dependent risks, the copula-based Bayesian estimator with the Uniform, the half Normal and the half Cauchy priors showed better performance. When the true model has independent risks, the independent Bayes estimator showed similar performance. Therefore, we suggest the copula-based Bayesian estimator with the Uniform, the half Normal and the half Cauchy priors when the true competing risks are dependent. The independent estimators are recommended only when one can assure the assumption of the independent competing risks.

5. Data analysis

Two real data were analyzed to illustrate the proposed methods.

5.1. Analysis of electric power transformers data

We used the left-truncated competing risks dataset available in Appendix of Kundu et al. [27]. For convenience, the dataset is given in Supplementary Materials (reproduced from Appendix of Kundu et al. [27]). The dataset consists of n = 100 samples, which mimics 100 electric power transformers assumed to be installed from 1960 to 2008. The starting year of the study was assumed to be at 1980, and therefore the transformers installed between 1960 and 1980 were regarded as left-truncated ones. Accordingly, the 30 transformers were left-truncated $({\nu }_i=0)$, and other 70 transformers were untruncated $({\nu }_i=1)$. The 14 transformers were assumed to experience Event 1 $({\delta }_i=1)$, 33 transformers Event 2 $({\delta }_i=2)$, 53 transformers censoring $({\delta }_i=0)$. The dataset contains event times $\mathbf{\text{T}}=(t_1,\dots ,t_{100})$, truncation times $\mathit{\tau }=({\tau }_1,\dots ,{\tau }_{100})$, censoring indicators $\mathit{\delta }=({\delta }_1,\dots ,{\delta }_{100})$, and truncation indicators $\mathit{\nu }=({\nu }_1,\dots ,{\nu }_{100})$. As in Kundu et al. [27], we divided event times (in years) by 100 to scale the parameter estimates.

Our goal is to see how the proposed estimators produce different results from the benchmark estimates reported by Kundu et al. [27] who analyzed the same dataset under the Weibull distribution with independent competing risks. They calculated the independent Bayes estimates under the gamma prior. The last row of Table 1 quotes their estimates as the benchmark.

Table 1.

Estimated parameters (95%CI in parenthesis).

	Event 1		Event 2		Copula
Fitted method	λ1 (95%CI)	α1 (95%CI)	λ2 (95%CI)	α2 (95%CI)	ϕ (95%CI)
indep. Bayes	6.18	2.74	14.96	2.75	–
(Gamma prior)	(1.58, 27.60)	(1.71, 4.01)	(6.07, 39.02)	(2.04, 3.55)
indep. Bayes	6.13	2.74	15.02	2.76	–
(Inverse-Gamma prior)	(1.51, 29.50)	(1.67, 4.05)	(5.91, 40.72)	(2.04, 3.58)
indep. Bayes	12.86	3.33	20.33	2.99	–
(Uniform prior)	(3.10, 60.50)	(2.18, 4.69)	(7.88, 56.19)	(2.24, 3.85)
indep. Bayes	12.67	3.31	20.03	2.98	–
(half Normal prior)	(2.79, 60.80)	(2.12, 4.68)	(7.71, 54.16)	(2.23, 3.82)
indep. Bayes	6.06	2.74	12.85	2.64	–
(half Cauchy prior)	(1.93, 21.15)	(1.84, 3.81)	(5.29, 33.06)	(1.96, 3.40)
copula. Bayes	16.51	3.16	18.00	2.81	1.03
(Gamma prior)	(1.50, 80.45)	(1.71, 4.47)	(6.46, 46.43)	(2.09, 3.60)	(−0.58, 19.78)
copula. Bayes	16.45	3.16	17.69	2.81	0.97
(Inverse-Gamma prior)	(1.49, 81.67)	(1.73, 4.48)	(6.09, 48.74)	(2.07, 3.63)	(−0.60, 21.39)
copula. Bayes	49.02	3.88	31.58	3.19	2.93
(Uniform prior)	(9.06, 167.42)	(2.82, 5.14)	(11.66, 79.2)	(2.46, 3.98)	(0.11, 27.47)
copula. Bayes	48.26	3.86	30.28	3.18	2.89
(half Normal prior)	(7.65, 156.51)	(2.70, 5.09)	(12.33, 75.18)	(2.48, 3.97)	(0.10, 23.19)
copula. Bayes	9.11	2.91	13.96	2.66	0.48
(half Cauchy prior)	(1.86, 44.72)	(1.88, 4.05)	(5.58, 35.64)	(2.00, 3.44)	(−0.60, 12.92)
indep. Bayes	8.53	2.78	17.08	2.77	–
quoted from Kundu et al. [27]*	(0.58, 22.79)	(1.65, 4.01)	(4.08, 34.85)	(1.99, 3.52)

*Highest posterior density credible interval is shown.

Table 1 shows the Bayesian estimates for the Weibull parameters $({\lambda }_1,{\alpha }_1,{\lambda }_2,{\alpha }_2)$ and the copula parameter $\varphi$. The estimates of the independent Bayesian estimates by the Gamma and the inverse-Gamma priors are somewhat close to the same estimates reported by Kundu et al. [27]. However, the independent Bayes estimates under the uniform and the half Normal priors, and the copula-based estimates, especially under the uniform and the half normal priors, shows remarkably different results from those under the Gamma prior (Table 1 and Figure 2). This indicates the possibility of having different estimates by different model assumptions.

Figure 2.

Weibull parameter estimates (95% CI in parenthesis) for Event 1 (E1) and Event 2 (E2) and the estimates of the hazard functions (95% CI in shaded regions) using the left-truncated competing risks dataset provided by Kundu et al. [27].

A particularly notable result is that the estimates of copula parameter under the uniform and the half-normal priors and its lower limit of the 95% CI are greater than 0, which may indicate the positive correlation between Event 1 and Event 2.

5.2. Advanced lung cancer and idiopathic interstitial pneumonias data

We analyzed an advanced lung cancer and idiopathic interstitial pneumonias dataset, which was originally reported by Miyamoto et al. [33]. The study focused on the survival of patients with advanced-stage lung cancer, who received chemotherapy. The primary endpoint of interest is time at the occurrence of acute exacerbation (AE) of idiopathic interstitial pneumonia (IIP). The data was collected from consecutive patients from January 2012 to December 2013.

We set event times by:

• T₁: the time from the 1st line treatment to the date of death.

• T₂: time from the 1st line treatment to the occurrence date of the AE of IIP.

The starting date of the study was set at January 2013, and therefore the patients started the 1st line of therapy before the staring time have truncation times (truncated samples). However, the patients experienced any failure events before the starting date were completely missed out of the sampling protocol. Some of the patients were censored at December 2013. In short, the 572 patients were considered in this analysis, of which the 205 were truncated samples $({\nu }_i=0)$ and the 367 were untruncated samples $({\nu }_i=1)$. Also, the 175 patients died $({\delta }_i=1)$, the 61 patients experienced the AE of IIP $({\delta }_i=2)$, and the 336 patients were censored $({\delta }_i=0)$. We scaled the failure times (in days) by dividing them by 1000 to avoid many decimal places in parameter estimates.

Table 2 compares the results of the estimates. The estimates by the independent Bayes estimators showed values close to each other (Table 2 and Figure 3). However, the estimates by the copula-based estimators take somewhat different values to each other. Especially, the copula-based Bayesian estimates of copula parameter by both the Uniform and half Normal priors considerably differ from other corresponding copula-based estimates (Table S8 and Figure 3), although both of the lower limits of the 95% CI are less than 0. It is likely that time to death and time to acute exacerbation are positively correlated because acute exacerbation in patients with IIP is recognized as a lethal event. Therefore, chemotherapy-related acute exacerbation may be a direct mortality cause in these patients [39,26]. Thus, it would be advantageous to adopt the copula-based Bayesian estimator, considering the superior performance of simulation results under dependent competing risk events.

Table 2.

Estimated parameters (95%CI in parenthesis).

	Death		AE*
Fitted method	λ1 (95%CI)	α1 (95%CI)	λ2 (95%CI)	α2 (95%CI)	Copula ϕ (95%CI)
indep. Bayes	2.58	1.32	0.80	0.51	–
(Gamma prior)	(2.05, 3.21)	(1.16, 1.48)	(0.64, 1.00)	(0.36, 0.71)
indep. Bayes	2.58	1.32	0.80	0.51	–
(Inverse-Gamma prior)	(2.05, 3.21)	(1.16, 1.48)	(0.64, 1.00)	(0.36, 0.71)
indep. Bayes	2.62	1.33	0.83	0.53	–
(Uniform prior)	(2.10, 3.31)	(1.17, 1.51)	(0.66, 1.03)	(0.38, 0.74)
indep. Bayes	2.62	1.33	0.83	0.53	–
(half Normal prior)	(2.10, 3.31)	(1.17, 1.51)	(0.66, 1.03)	(0.38, 0.74)
indep. Bayes	2.62	1.33	0.83	0.53	–
(half Cauchy prior)	(2.08, 3.26)	(1.17, 1.49)	(0.65, 1.02)	(0.38, 0.73)
copula. Bayes	2.82	1.34	0.65	0.89	0.57
(Gamma prior)	(2.07, 3.61)	(1.18, 1.51)	(0.36, 1.31)	(0.65, 1.18)	(−0.48, 2.48)
copula. Bayes	2.86	1.33	0.69	0.92	0.71
(Inverse-Gamma prior)	(2.05, 3.65)	(1.13, 1.51)	(0.37, 3.02)	(0.66, 1.29)	(−0.49, 54.58)
copula. Bayes	3.02	1.32	0.98	1.05	1.49
(Uniform prior)	(2.29, 3.74)	(1.11, 1.51)	(0.45, 3.45)	(0.74, 1.37)	(−0.17, 103.71)
copula. Bayes	2.99	1.32	0.96	1.05	1.41
(half Normal prior)	(2.26, 3.73)	(1.11, 1.51)	(0.44, 3.31)	(0.74, 1.35)	(−0.23, 71.79)
copula. Bayes	2.97	1.34	0.83	0.99	1.09
(half Cauchy prior)	(2.25, 3.74)	(1.13, 1.51)	(0.43, 3.17)	(0.71, 1.33)	(−0.21, 55.66)

*Acute exacerbation of idiopathic interstitial pneumonia (IIP).

Figure 3.

Weibull parameter estimates (95%CI in parenthesis) for death and AE (Acute Exacerbation) using the lung cancer data.

6. Discussion

This paper proposes a copula-based Bayesian estimator for dependent competing risks data subject to left-truncation. With two dependent competing event times, the simulation results confirm that the proposed estimators were highly reliable and better than the independent Bayesian estimator (Tables S3–S10). Among the five non-informative priors (Gamma, Inverse-Gamma, Uniform, half Normal and half Cauchy), the Uniform, the half Normal and the half Cauchy priors give more accurate results in Weibull parameter estimates than the Gamma and Inverse-Gamma prior. However, this is not the case for the copula parameter estimates. On the other hand, the performance of the independent risks model does not depend on priors. However, the independence is a quite strong assumption since competing risks are often positively dependent in real applications (Section 5.2). Therefore, we generally recommend the copula-based Bayesian estimator.

We formulate the likelihood functions separately for the independent risks (Section 3.1) and dependent risks by the Clayton copula (Section 3.2). There are two reasons for this separate formulation. First, the Clayton copula does not strictly include the independence copula as a special case. The Clayton copula approaches to the independence copula when the parameter is close to zero, but it does not exactly equal to the independence copula. Second, the treatment of the unknown parameter in any parametric copula is crucial in the present paper. Hence, the Clayton copula model differs from the independence copula since the former produces an unknown parameter in the likelihood function.

We take up a few important issues to be addressed as future works. First, the present failure time distribution is restricted to the Weibull model. It is possible to extend the Weibull model to other parametric models, such as the log-normal, gamma, Lomax, and Gompertz models [5,14,31,32,36]. Second, it is interesting to estimate the effects of covariates on times to failures [23]. Third, the joint distribution of latent failure times is constructed by the Clayton copula only: many other copulas [34,13] could be tried, and a data-driven copula selection criterion could be considered. Finally, our implicit assumption of independent left-truncation could be modeled by copulas if the assumption is rejected by the quasi-independence tests [6,7,18,42]. The resultant new model makes the likelihood more complex by two copulas: one is for competing risks and the other for dependent truncation.

Funding Statement

This work was financially supported by JSPS KAKENHI [JP21K12127, JP22K11948, 22K11948].

Disclosure statement

No potential conflict of interest was reported by the author(s).