Exploring Causal Effects of Hormone- and Radio-Treatments in an Observational Study of Breast Cancer Using Copula-Based Semi-Competing Risks Models

Abstract

Breast cancer patients may experience relapse or death after surgery during the follow-up period, leading to dependent censoring of relapse. This phenomenon, known as semi-competing risk, imposes challenges in analyzing treatment effects on breast cancer and necessitates advanced statistical tools for unbiased analysis. Despite progress in estimation and inference within semi-competing risks regression, its application to causal inference is still in its early stages. This article aims to propose a frequentist and semi-parametric framework based on copula models that can facilitate valid causal inference, net quantity estimation and interpretation, and sensitivity analysis for unmeasured factors under right-censored semi-competing risks data. We also propose novel procedures to enhance parameter estimation and its applicability in practice. After that, we apply the proposed framework to a breast cancer study and detect the time-varying causal effects of hormone- and radio-treatments on patients’ relapse and overall survival. Moreover, extensive numerical evaluations demonstrate the method’s feasibility, highlighting minimal estimation bias and reliable statistical inference.

1 |. Introduction

1.1 |. Semi-Competing Risks Data

In clinical and biomedical studies, time-to-event endpoints often play a pivotal role, offering a tangible measure of the impact and effectiveness of medical interventions over time [1]. However, the analysis of time-to-event endpoints becomes intricate in the presence of semi-competing risks, a scenario frequently observed in various observational/clinical studies [2]. This paper focuses on studying the effects of hormone and radiotherapies in the treatment of breast cancer patients. While these two therapies are widely adopted in clinical practice to improve survival for patients with breast cancer, it remains unclear whether the combination of hormone and radiotherapies is more beneficial to patients after surgery than using hormone therapy alone [3,4]. To unbiasedly address this question while accounting for different time-to-event endpoints during breast cancer progression, we aim to investigate both relapse time (RFS, that is, the time from the initial diagnosis to the time of tumor loco-regional or distant relapse) and overall survival (OS) and develop a valid causal inference tool to assess the effects of hormone and radiotherapies on these semi-competing risks.

Formally, semi-competing risks survival times encompass both the time to a non-terminal event (e.g., RFS) and the time to a terminal event (e.g., OS) from the study entry, denoted by $T_1$ and $T_2$, respectively. The pair $\left(T_1,T_2\right)$ can be conceptualized as a bivariate failure time, where $T_2$ acts as a censoring variable for $T_1$, but the reverse is not true-$T_1$ does not censor $T_2$. Unlike the classic competing risk setting [5, 6] focusing on the first event and ignoring the information after the non-terminal event, the two endpoints in the semi-competing risk setting focus on both outcomes in a sequential manner and interplay between these outcomes [7]. In our application, we are particularly interested in the time to cancer relapse and time to death, where the latter event will truncate the former event, but the reverse is not true. Both events are highly valued by patients and their caregivers. Understanding how factors influence both events and how two events interact would inform clinical decisions and promote dynamic predictions on the terminal event [8]. The above motivation underscores the necessity for specialized statistical methods that appropriately account for the interplay between these two distinct time-to-event processes in the analysis of semi-competing risks.

1.2 |. Semi-Parametric Regression With Semi-Competing Risks

Semi-parametric regressions within the framework of semi-competing risks are of importance and can be divided into two distinct categories: Crude quantities and net quantities [7]. Concerning crude quantities, the sub-distribution approach and multi-state approach (specifically the so-called illness-death approach) are two kinds of typical means to handle survival data subject to dependently censored data. The sub-distribution approach [5, 9] evaluates the probability of each event occurring as the first one. Accordingly, it treats semi-competing risks akin to conventional competing risks. The illness-death approach [10] encompasses two cause-specific hazard models, quantifying the impact of covariates on the occurrence of either the terminal or non-terminal event, along with a Markov model that characterizes the progression from the non-terminal to the terminal event. However, both approaches focus on observable quantities rather than the marginal distributions of specific event times. According to the bounds established by Peterson [11], the marginal distribution of $T_1$ is bounded from below by the sub-distribution function of the non-terminal event and from above by the distribution of the first event occurring. Such wide bounds suggest that predicting the non-terminal event time based solely on cause-specific hazards in the illness-death model could lead to biases in the case of heavily dependent censoring. Moreover, such shared frailty models encounter challenges in effectively describing the negative correlation between the two event times [7].

The net-quantity-based approaches concern primarily the “net” effects of potential covariates on the marginal distribution of non-terminal and terminal event times. The marginal distribution in the absence of external influences, that is, hypothesizing the removal of the terminal event, may hold more direct clinical implications [7]. Marginal methodologies include joint models specified through copulas [12–14] or two separate marginal models augmented with artificial censoring techniques [15, 16]. The latter approaches concentrate on estimating regression coefficients in marginal models of $T_1$ and $T_2$. They, however, fail to provide insights into the dependence between the non-terminal and terminal events. Additionally, the marginal distribution of the non-terminal event time faces an issue of unidentifiability without making substantial additional assumptions. Therefore, copula-based models may offer a more meaningful framework, given their capacity to simultaneously explore marginal distributions and quantify the association between the two event times [17–20].

1.3 |. Causal Inference Under Semi-Competing Risks

Several efforts have been made for causal interpretation under survival data with competing risks in the literature [6, 21, 22]. The average treatment effects on the event of interest in the competing risk setting [6] is constructed based on “marginal” risk; however, it puts aside the causal effects on both events when they occur sequentially. In practice, treatment can influence both non-terminal and terminal events, and these two events are inter-connected in complex ways. Jointly modeling these dual events provides a more accurate representation of this complexity [23] However, the application of semi-competing risks to causal inference is still in its early stages. Our focus in this article is to study the treatment effect on dual survival outcomes, that is, relapse and death times. A notable challenge arises when comparing outcomes between treated and untreated groups after treatment initiation, where the studied subjects may experience death before reaching the target event, and the death rates in the treated and control groups may provide informative insights. Consequently, the conventional average causal effect becomes neither well-defined nor easily interpretable in a causal context, except under additional assumptions [23, 24]. Robins [25] showed the identifiability of average causal effect when censoring time is always observed, and without distinguishing between censoring due to death or other reasons.

The existing causal estimands tailored to semi-competing risks data can be broadly categorized into several classes. In terms of methodological approaches, bayesian methods include works such as [26, 27], while frequentist methods include [23, 28–31]. In terms of causal frameworks, mediation analysis and Survivor Average Causal Effect (SACE) are two active topics in recent years. The mediation analysis, which regards the time to relapse as a time-varying mediator and aims to estimate direct and indirect causal effects, has been studied in References [28–31]. On the other hand, SACE quantifies the causal effect within the stratum of individuals who would have survived under both treatment values [32], and has been explored in References [23, 26, 27] for semi-competing risks data. Among these, Comment et al. [26] and Xu et al. [27] focused on causal effects for time-varying populations and estimated SACE via Bayesian methods. Under fixed population partitions, Nevo and Gorfine [23] further investigated its non-parametric partial identifiability and proposed a sensitivity analysis using frailty-based illness-death models.

Despite advancements, the frailty-based illness-death approach restricts its applicability to datasets where the association between non-terminal and terminal events is strictly non-negative. Notably, flipping the sign of association patterns can lead to invalid causal inferences for semi-competing risk data. This limitation motivates the integration of copula-based models into causal inference frameworks for semi-competing risks. As outlined earlier, copula-based models offer advantages in accommodating both negative and positive associations between the two event times, estimating net quantities, and supporting clinically meaningful interpretations. In this article, we propose and evaluate a copula-based causal inference framework for semi-competing risks, an area that remains understudied in existing literature, and apply it to analyze our breast cancer data. The goal is to enable valid causal interpretation, estimate net causal effects, and perform sensitivity analyses for unmeasured confounding in the context of semi-competing risks. Compared to frailty-based illness-death models [23], the proposed method offers greater flexibility in modeling diverse associations between events, facilitates more intuitive interpretation of covariate effects, and demonstrates improved robustness to model misspecification in numerical studies.

The rest of this article is organized as follows. Section 2 introduces basic notations for semi-competing risks data. Section 3 introduces the causal framework. Section 4 summarizes estimation methods and provides a sensitivity analysis for unmeasured factors. Section 5 describes the analysis for breast cancer data. Section 6 displays extensive simulation studies to illustrate the proposed causal framework. Section 7 provides discussions and future extensions.

2 |. Copula-Based Models for Bivariate Survival Function

In line with Sklar’s theorem [33], we establish a functional relationship between the bivariate survival function of $\left(T_1,T_2\right)$ and their respective marginal distributions as follows:

$$\mathrm{Pr}\left(T_1\ge t_1,T_2\ge t_2\mid \mathbf{Z}\right)=C\left\{S_1\left(t_1\mid \mathbf{Z}\right),S_2\left(t_2\mid \mathbf{Z}\right);\alpha \right\},0\le t_1\le t_2$$ (1)

where $S_k(t\mid \mathbf{Z})$ is the survival functions of $T_k$ conditional on the covariate vector $\mathbf{Z}$, for $k=1,2$, and $C\{u,v;\alpha \}$ denotes a specified copula function modeling inter-dependence between two conditional survival distributions with an unknown association parameter $\alpha$. This copula-based joint distribution is defined within the upper wedge where $T_1\le T_2$. In contrast to models based solely on observable quantities (i.e., whichever event occurs first), the marginal models for non-terminal and terminal events-detailed in later sections-allow for the evaluation of net covariate effects. Specifically, they estimate the effect on the non-terminal event time $T_1$, independent of or adjusted for the influence of the terminal event $T_2$. The “net” covariate effects hold notable clinical implications from the causal perspective in biomedical studies [16, 34]. In this article, we consider the Archimedean copula, which can be expressed as

$$C\{u,v;\alpha \}={\varphi }_{\alpha }^{-1}\left\{{\varphi }_{\alpha }(u)+{\varphi }_{\alpha }(v)\right\},0\le u,v\le 1$$ (2)

where the generator function ${\varphi }_{\alpha }:[0,1]\to [0,\infty )$ is continuous and strictly decreasing from ${\varphi }_{\alpha }(0)>0$ to ${\varphi }_{\alpha }(1)=0$. Consequently, Kendall’s $\tau$, a widely recognized metric for assessing the association between the two survival times, can be expressed as [35]

$$\tau =4{\int }_0^1{\varphi }_{\alpha }(u)/{\varphi }_{\alpha }^{\prime }(u)du+1$$ (3)

Upon the above models (1–3), we consider a sample of $n$ individuals. Let $\left\{T_{i1},T_{i2},C_i,{\mathbf{Z}}_i\right\}$, be $n$ independent copies of $\left\{T_1,T_2,C,\mathbf{Z}\right\}$, where $C_i$ is censoring time such that $T_{i1}$ and $T_{i2}$ are observed as $\left(X_i,Y_i,{\delta }_{i1},{\delta }_{i2}\right)$. Here, $X_i=\min \left(T_{i1},T_{i2},C_i\right)$ and $Y_i=\min \left(T_{i2},C_i\right)$, ${\delta }_{i1}=I\left(X_i=T_{i1}\right)$ and ${\delta }_{i2}=I\left(Y_i=T_{i2}\right)$.

The implementation of copula-based estimation techniques poses challenges due to their intricate nature. These approaches are often perceived as sophisticated and not extensively explored, particularly in the context of semi-competing risk causal inference. The remainder of this article aims to fill the gap and investigate the feasibility and potential advantages of applying the net-quantity-based semi-parametric framework to the field of causal inference. We begin by delving into the causal parameter specific to the context of semi-competing risks data. The estimation of regression parameters is elaborated upon in Section 4.

3 |. Treatment Effects

3.1 |. Observed Data and Potential Outcomes

In clinical and biomedical research, investigators are often interested in evaluating the causal impact of treatments on both slowing the advancement of disease and extending overall survival following an intermediate event. To address these questions, we employ the potential outcomes framework [36, 37] to define causal effects. The data is then distinguished as observed data and potential outcomes.

Observed data.

In addition to $\left(T_{i1},T_{i2}\right)$ and their observations $\left(X_i,Y_i,{\delta }_{i1},{\delta }_{i2}\right)$ defined in the previous section, there exists a treatment variable $A_i$, indicating whether individual $i$ receive a treatment or not. It is assumed that ${\mathbf{Z}}_i$, are pre-treatment covariates unaffected by the treatment assignment. Consequently, the observed data is denoted as ${\mathcal{O}}_i=\left\{X_i,Y_i,{\delta }_{i1},{\delta }_{i2},A_i,{\mathbf{Z}}_i\right\}$, $i=1,\dots ,n$.

Potential outcomes.

Before defining the potential outcomes, we first take into account the stable unit treatment value assumption (SUTVA) [38]:

Assumption 1.

There is no interference between patients, and each treatment value corresponds to a single and consistent outcome, without the existence of multiple variations.

Let $\left(T_{i1}(a),T_{i2}(a)\right)$ be the potential time-to-event outcomes when the individual $i$ is subjected to treatment $A=a$ and $a=0$ or $1$. This establishes pairs of potential outcomes $\left\{\left(T_{i1}(0),T_{i2}(0)\right),\left(T_{i1}(1),T_{i2}(1)\right)\right\}$ for each individual. The causal effect pertains to the comparison between potential outcomes for either $T_1$ or $T_2$ within the same group of subjects, each under the influence of two competing treatments. We notice here that, within the framework of causal inference, at most one of the potential outcomes can be observed for each individual, while the others remain unobserved. The SUTVA suggests that the observed event times $T_1$ and $T_2$ can be represented as linear combinations of the potential outcomes under different treatments, specifically $T_1=T_1(1)A+T_1(0)(1-A)$ and $T_2=T_2(1)A+T_2(0)(1-A)$. Similarly, in the presence of censoring, denoted by $C$, the underlying potential outcomes $\left(T_{i1}(a),T_{i2}(a)\right)$ could be observed in the form of $\left(X_i(a),Y_i(a)\right)$ accompanied by the corresponding censoring indicators $\left({\delta }_{i1}(a),{\delta }_{i2}(a)\right)$. For further defining causal estimands, we adopt the no-unmeasured confounding assumption, which is often imposed in the literature:

Assumption 2.

(i) (Conditional randomization) $A\perp \left(T_1(a)\right.$, $\left.T_2(a),C\right)\mid \mathbf{Z}$ and (ii) (Non-informative censoring) $C\perp \left(T_1(a)\right.,\left.T_2(a)\right)\mid \mathbf{Z}$.

We notice here that Assumption 2ii is less restrictive than it initially appears. Given the fact that the non-terminal event time is subject to either terminal event time or random censoring $C$, as a result, informative censoring is applied to the non-terminal event, while non-informative censoring is applied to the terminal event.

3.2 |. Causal Estimands

An appropriate causal estimand should involve a valid causal interpretation, ensuring comparable individual profiles between treated and control groups throughout the time course [23]. One popular approach to achieving this is by employing the concept of principal stratification [32, 39–41]. Specifically, to account for censoring, the stratum-specific survivor causal effect (SCE) of treatment on the terminal event time is defined by the comparison of potential outcomes of $T_2$ among individuals who would have survived regardless of treatment exposure [42]. However, in the semi-competing risks setting, the presence of a non-terminal event and the sequential ordering of the two events complicate causal inference. The treatment may delay the occurrence of the non-terminal event, which in turn postpones the terminal event. Therefore, it is more appropriate to evaluate causal effects on $T_2$ while accounting for the non-terminal event time. Additionally, $E\left(T_1(1)-T_1(0)\right)$ is not well-defined in the semi-competing risks framework, as some individuals may die without experiencing the non-terminal event. To account for dependence between the two event times, as described by Nevo and Gorfine [23], the entire population can be categorized into four strata: (1) the “always-diseased” stratum, where individuals experience cancer relapse before death regardless of treatment $\left(T_1(0)\le T_2(0),T_1(1)\le T_2(1)\right)$; (2) the “never-diseased” stratum, where individuals never experience cancer relapse under either treatment condition $\left(T_1(0)>\right.\left.T_2(0),T_1(1)>T_2(1)\right)$; (3) the “harmed”, where treatment $(a=1)$ accelerates cancer relapse $\left(T_1(0)>T_2(0),T_1(1)\le T_2(1)\right)$; and (4) the “protected”, where the treatment $(a=1)$ prevents cancer relapse $\left(T_1(0)\le T_2(0),T_1(1)>T_2(1)\right)$. The opposite effects in the latter two strata complicate result interpretation and may not align with the clinical relevance of treatment decisions. Therefore, excluding the “harmed” and “protected” strata renders the remaining causal estimands identifiable, interpretable, and applicable to real-world decision-making [23]. Given the considerable interest in the survival probability of non-terminal events in clinical studies, we primarily focus on the following causal estimand:

$$\begin{gathered} \text{AD-SC}{\text{E}}_1(t)=\mathrm{Pr}\left(T_1(1)\ge t\mid T_1(0)\le T_2(0),T_1(1)\le T_2(1)\right) \\ -\mathrm{Pr}\left(T_1(0)\ge t\mid T_1(0)\le T_2(0),T_1(1)\le T_2(1)\right) \end{gathered}$$ (4)

Herein, we study the time-varying SCE estimands in Equation (4) rather than its integration version (or restricted survival mean) defined in Nevo and Gorfine [23] because estimators of survival means may lead to inevitably inaccurate and unsatisfactory results caused by point predictions in many finite-sample numerical examples [43]. The goal of this article is to explore varying (dynamic) treatment effects over time. In addition, since the terminal event could occur after or before the non-terminal event, we consider two types of stratum-specific SCE for the terminal event:

$$\begin{gathered} \mathrm{AD}-{\text{SCE}}_2(t)=\mathrm{Pr}\left(T_2(1)\ge t\mid T_1(0)\le T_2(0),T_1(1)\le T_2(1)\right) \\ -\mathrm{Pr}\left(T_2(0)\ge t\mid T_1(0)\le T_2(0),T_1(1)\le T_2(1)\right) \end{gathered}$$ (5)

$$\begin{gathered} \mathrm{ND}-{\text{SCE}}_2(t)=\mathrm{Pr}\left(T_2(1)\ge t\mid T_1(0)>T_2(0),T_1(1)>T_2(1)\right) \\ -\mathrm{Pr}\left(T_2(0)\ge t\mid T_1(0)>T_2(0),T_1(1)>T_2(1)\right) \end{gathered}$$ (6)

We remark here that the estimands above are subject to the joint distribution of event times from both worlds, that is, $a=0,1$, which cannot be observed simultaneously. To facilitate non-parametric identification, we introduce a combined covariate vector $\mathbf{Z}$ to describe the alterations in the occurrences of both event times, and impose the following assumption.

Assumption 3.

The cross-world dependence is captured by pre-treatment covariates $\mathbf{Z}$ in the sense that $\left(T_1(0),T_2(0)\right)\perp \left(T_1(1),T_2(1)\right)\mid \mathbf{Z}$.

We notice here that the cross-world conditional independence in Assumption 3 is strong and untestable in practice [44–46]. More discussion about cross-world unmeasured factors and a weaker assumption can be found in Section 4.2. Under Assumptions 1-3, we establish the identification of causal estimands of our interest in Proposition 1, with proof provided in the Supplementary Section 1. The above estimands (4–6) can alternatively be defined across observable strata, such as $\{X(0)\le Y(0),X(1)\le Y(1)\}$ instead of the “AD” stratum, and $\{X(0)\ge Y(0),X(1)\ge Y(1)\}$ instead of the “ND” stratum, which can be verified that these two types of definitions are equivalent under non-informative censoring in Assumption 2.

Proposition 1 (Non-parametric identification).

Under Assumptions 1 - 3, the stratum-specific survivor average causal effects in Equations (4–6) are identified by

$$\begin{gathered} {\mathbb{E}}_{\mathbf{Z}}\left[\left(S_{k\mid T_1\le T_2,A=1,\mathbf{Z}}(t)\right.\right. \\ \mathrm{AD}-{\text{SCE}}_k(t)=\frac{\left.\left.-S_{k\mid T_1\le T_2,A=0,\mathbf{Z}}(t)\right){\Pi }_{A=0,\mathbf{Z}}{\Pi }_{A=1,\mathbf{Z}}\right]}{{\mathbb{E}}_{\mathbf{Z}}\left[{\Pi }_{A=0,\mathbf{Z}}{\Pi }_{A=1,\mathbf{Z}}\right]},k=1,2 \end{gathered}$$ (7)

and

$$\begin{gathered} {\mathbb{E}}_{\mathbf{Z}}\left[\left(S_{2\mid T_1\ge T_2,A=1,\mathbf{Z}}(t)\right.\right. \\ \text{ND}-\text{SC}{\text{E}}_2(t)=\frac{\left.\left.-S_{2\mid T_1\ge T_2,A=0,\mathbf{Z}}(t)\right){\widetilde{\text{Π}}}_{A=0,\mathbf{Z}}{\widetilde{\text{Π}}}_{A=1,\mathbf{Z}}\right]}{{\mathbb{E}}_{\mathbf{Z}}\left[{\widetilde{\text{Π}}}_{A=0,\mathbf{Z}}{\widetilde{\text{Π}}}_{A=1,\mathbf{Z}}\right]},t>0 \end{gathered}$$ (8)

where $S_{k\mid \mathcal{Q}}(t)=\mathrm{Pr}\left(T_k\ge t\mid \mathcal{Q}\right)$, $\text{Π}(\mathcal{Q})=\mathrm{Pr}\left(T_1\le T_2\mid \mathcal{Q}\right)$, $\widetilde{\text{Π}}(\mathcal{Q})=\mathrm{Pr}\left(T_1\ge T_2\mid \mathcal{Q}\right)$ for any event $\mathcal{Q}$.

4 |. Identification and Estimation of Principal Causal Effects

Proposition 1 justifies the validity and identifiability of the proposed estimands in general. In this section, we detail how identifiability is maintained when applying copula-based techniques. The estimation of copula-based semicompeting risks regression has been extensively explored by existing literature. Nevertheless, adapting this framework to address causal inference involves significant adjustments and remains inadequately studied, where the literature either exhibits bias in cases of heavily dependent censoring or encounters challenges in effectively portraying the negative correlation between the two event times [7,11]. Herein, we provide an improved procedure to estimate parameters in the copula-based causal model.

4.1 |. Identification Based on Stringent Assumption

We first consider a stringent assumption as follows:

Assumption 4.

Conditional on covariates $\mathbf{Z}$, the joint distribution function of $\left(T_1(a),T_2(a)\right)$ follows a copula model

$$\begin{gathered} D\left(t_1,t_2\mid a,\mathbf{Z}\right)=\mathrm{Pr}\left(T_1\ge t_1,T_2\ge t_2\mid A=a,\mathbf{Z}\right) \\ =C^{(a)}\left\{S_1\left(t_1\mid a,\mathbf{Z}\right),S_2\left(t_2\mid a,\mathbf{Z}\right);{\alpha }^{(a)}\right\} \end{gathered}$$ (9)

where $0\le t_1\le t_2,S_k(t\mid a,\mathbf{Z})$ is the survival functions of $T_k(a)(k=1,2)$ given treatment and covariates $\mathbf{Z}$; $C^{(a)}\{u,v;\alpha \}$ is a pre-specified copula function with ${\alpha }^{(a)}$ being a constant parameter describing extra association between event times under the treatment status $a$.

It is noteworthy that Assumption 4 permits the utilization of various copula structures corresponding to different treatment assignments. Consequently, the proposed model exhibits greater flexibility compared to parametric frailty models found in the existing literature [23]. A weaker substitution of Assumption 4 will be discussed in Section 4.2.

We will now present the identification formulas within the framework of copulas. To ease presentation, we define $C_1^{(a)}(u,v,\alpha )=\partial C^{(a)}(u,v;\alpha )/\partial u$, $C_2^{(a)}(u,v,\alpha )=\partial C^{(a)}(u,v;\alpha )/\partial v$, $D\{s,t\mid a,\mathbf{Z}\}=C^{(a)}\left\{S_1(s\mid a,\mathbf{Z}),S_2(t\mid a,\mathbf{Z});{\alpha }^{(a)}\right\}$, $D_k\{s,t\mid a,\mathbf{Z}\}=C_k^{(a)}\left\{S_1(s\mid a,\mathbf{Z}),S_2(t\mid a,\mathbf{Z});{\alpha }^{(a)}\right\}$, $k=1,2$. Then, under Assumptions 1-4, by simple derivation, the quantities in Proposition 1 can be rewritten as

$$S_{1\mid T_1\le T_2,A=a,\mathbf{Z}}(t)=\frac{{\int }_t^{\infty }{D}_1(s,s\mid a,\mathbf{Z})d{S}_1(s\mid a,\mathbf{Z})}{{\int }_0^{\infty }{D}_1(s,s\mid a,\mathbf{Z})d{S}_1(s\mid a,\mathbf{Z})}$$ (10)

$$S_{2\mid T_1\le T_2,A=a,\mathbf{Z}}(t)=\frac{S_2(t\mid a,\mathbf{Z})+{\int }_t^{\infty }{D}_2(s,s\mid a,\mathbf{Z})d{S}_2(s\mid a,\mathbf{Z})}{1+{\int }_0^{\infty }{D}_2(s,s\mid a,\mathbf{Z})d{S}_2(s\mid a,\mathbf{Z})}$$ (11)

$$S_{2\mid T_1>T_2,A=a,\mathbf{Z}}(t)=\frac{{\int }_t^{\infty }{D}_2(s,s\mid a,\mathbf{Z})d{S}_2(s\mid a,\mathbf{Z})}{{\int }_0^{\infty }{D}_2(s,s\mid a,\mathbf{Z})d{S}_2(s\mid a,\mathbf{Z})}$$ (12)

The denominators in Equations (10–12) are the explicit expressions of ${\text{Π}}_{\mathcal{Q}}$ and ${\widetilde{\text{Π}}}_{\mathcal{Q}}$ for event $\mathcal{Q}$. Quantities (10–12) use observable data in the whole upper wedge of $\left(T_1,T_2\right)$ and thus can be estimable.

Plugging estimates of ${\alpha }^{(a)}$, $S_1(\cdot \mid a,\mathbf{Z})$, and $S_2(\cdot \mid a,\mathbf{Z})$ assisted by methods in Peng and Fine [12] and Zhu et al. [47], estimates of causal effects (4–6) can be obtained from Equations (7, 8), (10–12). To facilitate the estimation, we limit $C^{(a)}\{\cdot \}$ to the Archimedean copula class, and for illustration, we use (but not limited to) the proportional hazards models as in Equation (13) to marginally fit $T_1$ and $T_2$, respectively

$${\lambda }_k(t\mid \mathbf{Z},A=a)={\lambda }_{0k}^{(a)}(t)\exp \left({\mathit{\beta }}_k^{(a)T}\mathbf{Z}\right)$$ (13)

where ${\lambda }_k(t\mid Z,A=a)$ is the conditional hazard function of $T_k(a)$, $k=1,2$, given the covariates $\mathbf{Z}$ and treatment $A=a$; ${\mathit{\beta }}_k^{(a)}$ is a vector of unknown regression coefficients, and ${\lambda }_{0k}^{(a)}$ is an unspecified baseline hazard function. Let ${\text{Λ}}_{0k}^{(a)}(t)={\int }_0^t{\lambda }_{0k}^{(a)}(s)ds$ be the baseline cumulative hazard function. And the conditional survival functions are in the forms of $S_k(t\mid \mathbf{Z},A)=\exp \left[-{\text{Λ}}_{0k}^{(a)}(t)\exp \left({\beta }_k^{(a)T}\mathbf{Z}\right)\right]$. We denote $C_{21}(u,v,\alpha )=C_{12}(u,v,\alpha )={\partial }^{2}C(u,v;\alpha )/\partial u\partial v$ and $D_{jk}\left\{X_i,Y_i;{\alpha }^{(a)}\right\}=C_{jk}\left\{S_1\left(X_i\mid {\mathbf{Z}}_i,a,{\gamma }_i\right),S_2\left(Y_i\mid {\mathbf{Z}}_i,a,{\gamma }_i\right);{\alpha }^{(a)}\right\}$, $j$, $k=1,2$. Under models (9) and (13), the log-likelihood function concerning the parameters to be estimated given observed data $\mathcal{O}=\left\{X_i,Y_i,{\delta }_{i1},{\delta }_{i2},A_i,{\mathbf{Z}}_i:i=1,\dots ,n\right\}$ can be written as

$$\begin{array}{l}l\left({\alpha }^{(a)},{\text{Λ}}_{01}^{(a)},{\text{Λ}}_{02}^{(a)},{\mathit{\beta }}_1^{(a)},{\mathit{\beta }}_2^{(a)}\right)\\ =\sum _{i=1}^n\left(1-{\delta }_{i1}\right)\left(1-{\delta }_{i2}\right)I\left(A_i=a\right)\log \left[D\left\{X_i,Y_i;{\alpha }^{\left(A_i\right)}\right\}\right]\\ +{\delta }_{i1}{\delta }_{i2}I\left(A_i=a\right)\log \left[D_{12}\left\{X_i,Y_i;{\alpha }^{\left(A_i\right)}\right\}\right]\\ +{\delta }_{i1}\left(1-{\delta }_{i2}\right)I\left(A_i=a\right)\log \left[D_1\left\{X_i,Y_i;{\alpha }^{\left(A_i\right)}\right\}\right]\\ +\left(1-{\delta }_{i1}\right){\delta }_{i2}I\left(A_i=a\right)\log \left[D_2\left\{X_i,Y_i;{\alpha }^{\left(A_i\right)}\right\}\right]\\ +{\delta }_{i1}I\left(A_i=a\right)\log \left[S_1\left(X_i\mid {\mathbf{Z}}_i,A_i\right){\lambda }_{01}^{\left(A_i\right)}\left(X_i\right)\exp \left({\mathit{\beta }}_1^{\left(A_i\right)T}{\mathbf{Z}}_i\right)\right]\\ +{\delta }_{i2}I\left(A_i=a\right)\log \left[S_2\left(Y_i\mid {\mathbf{Z}}_i,A_i\right){\lambda }_{02}^{\left(A_i\right)}\left(Y_i\right)\exp \left({\mathit{\beta }}_2^{\left(A_i\right)T}{\mathbf{Z}}_i\right)\right]\end{array}$$

These parameters can be estimated by adapting the non-parametric maximum likelihood estimation (NPMLE) [14, 48]. However, the objective function of the aforementioned NPMLE method is only locally convex in a small neighborhood of the truth and thus sensitive to initial estimates. Poor initialization will lead to biases in the final estimation or even algorithm convergence issues. Accordingly, we improve the above NPMLE method by substituting initial values with estimates derived from Peng et al. [14] and Zhu et al. [47], which have proven effective in simulation studies. Such adaptation and integration can effectively reduce the number of iterations and computation time as well as estimation biases. It is worth noting that the proposed estimation is not limited to the time-independent Cox proportional hazards model aforementioned. We can easily extend it to other model forms, including the proportional odds model and varying-coefficient models [14, 47].

4.2 |. Sensitivity Analysis

While Assumptions 3 and 4 could be valid in some applications, they may not hold—and are generally not testable—in broader contexts, as cross-world data cannot be simultaneously observed for any individual. In practice, we recommend incorporating expert knowledge to guide the identification of crucial covariates while concurrently assessing the sensitivity of estimates to unmeasured factors [23, 24]. To this end, we propose the copula-based sensitivity analysis. To capture the remaining dependency attributed to unmeasured factors shared across two worlds, we introduce a frailty $\gamma$ and a weaker restriction than Assumptions 3 and 4.

Assumption 5.

There exists an unmeasured time-unvarying frailty variable $\gamma >0$ such that the cross-world dependence is captured by pre-treatment covariates $\mathbf{Z}$ together with $\gamma$ in the sense that $\left(T_1(0),T_2(0)\right)\perp \left(T_1(1),T_2(1)\right)\mid (\gamma ,\mathbf{Z})$.

Assumption 6.

The time-unvarying frailty variable $\gamma >0$, from a distribution with mean 1 and variance $\sigma$, satisfies:

1. The frailty variable operates multiplicatively on conditional cumulative hazard function of $T_k(a)$, that is, ${\text{Λ}}_k(t\mid A=a,\gamma ,\mathbf{Z})=\gamma {\widetilde{\text{Λ}}}_k(t\mid A=a,\mathbf{Z})$, where ${\text{Λ}}_k(t\mid A=a,\gamma ,\mathbf{Z})$ is the conditional cumulative hazard function of $T_k(a)$ given treatment $a$ and $(\gamma ,\mathbf{Z})$, and ${\widetilde{\text{Λ}}}_k(t\mid A=a,\mathbf{Z})$ is irrelevant to $\gamma$.

2. The conditional joint distribution function of $\left(T_1(a),T_2(a)\right)$, given $(\gamma ,\mathbf{Z})$, follows a copula model $D\left(t_1,t_2\mid a,\gamma ,\mathbf{Z}\right)=C^{(a)}\left\{S_1\left(t_1\mid a,\gamma ,\mathbf{Z}\right),S_2\left(t_2\mid a,\gamma ,\mathbf{Z}\right);{\alpha }^{(a)}\right\}$ where $S_k(t\mid a,\gamma ,\mathbf{Z})=\exp \left[-{\text{Λ}}_k(t\mid a,\gamma ,\mathbf{Z})\right]$ is the survival functions of $T_k(a)(k=1,2)$ given treatment and $(\gamma ,\mathbf{Z})$, the definitions of ${\alpha }^{(a)}$ and $C^{(a)}\{u,v;\alpha \}$ are provided in Assumption 4.

The frailty variable in Assumption 5 is commonly used in the survival analysis literature to explain unobserved heterogeneity due to unmeasured covariates or other sources of natural variation. It is also widely applied in many causal inference literature to account for the cross-world dependence [24, 49, 50], a plausible remedy when the cross-world conditional independence in Assumption 3 is violated. This assumption also implies their marginal hazard ratios at time t on treatment $A=1$ vs. control $A=0$ for the same patient population with covariates $\mathbf{Z}$ and unmeasured factor $\gamma$ who would survive that time, no matter what treatment, which is related to the so-called conditional causal hazard ratio [50]. We stress in Assumption 6ii that the unmeasured factor has impacts on marginal distributions except for the global association between two event times under each counterfactual arm. We refer readers to the discussion section for more general identification assumptions and possible extensions.

Based on the weaker Assumptions 5 and 6, and coupled with Assumptions 1 and 2, the causal estimands in Equations (4–6) are identifiable with detailed formulas in Supplementary Section 2. It is important to note that the distribution of frailty $\gamma$ may not be estimable in general. As a result, we evaluate the effect of unmeasured factors by pre-specifying values of the unidentifiable parameter $\sigma$. The expectation with respect to frailty as Equations (A.2-A.4) in the Supporting Information can then be numerically calculated by Monte Carlo integration techniques. Note that incorporating a frailty, even with a pre-specified value of $\sigma$, is non-trivial and necessitates more intricate estimation, thus warranting further methodological development. Suppose that the marginal regressions of $T_k(a)$ on $\mathbf{Z}$, $k=1,2$, follow the shared-frailty Cox proportional hazards models

$${\lambda }_k(t\mid \mathbf{Z},A=a,\gamma )=\gamma {\lambda }_{0k}^{(a)}(t)\exp \left({\mathit{\beta }}_k^{(a)T}\mathbf{Z}\right)$$ (14)

where ${\lambda }_k(t\mid Z,A=a,\gamma )$ is the conditional hazard function of $T_k(a)$ given the covariates $\mathbf{Z}$ and treatment a; the definitions of ${\mathit{\beta }}_k^{(a)}$ and ${\lambda }_{0k}^{(a)}$ are same as model (13); and $\gamma$ is the frailty defined in Assumption 6. The log-likelihood function regarding to the parameters $\left({\alpha }^{(a)},{\text{Λ}}_{01}^{(a)},{\text{Λ}}_{02}^{(a)},{\mathit{\beta }}_1^{(a)},{\mathit{\beta }}_2^{(a)}\right)$ given observed data $\mathcal{O}$ can be expressed by

$$\begin{array}{l}{l}_o\left({\alpha }^{(a)},{\text{Λ}}_{01}^{(a)},{\text{Λ}}_{02}^{(a)},{\mathit{\beta }}_1^{(a)},{\mathit{\beta }}_2^{(a)};\sigma \right)\\ =\sum _{i=1}^n\log \left\{\int \exp \left[l_i\left({\alpha }^{(a)},{\text{Λ}}_{01}^{(a)},{\text{Λ}}_{02}^{(a)},{\mathit{\beta }}_1^{(a)},{\mathit{\beta }}_2^{(a)}\right)\right]f_{\gamma }\left({\gamma }_i\mid \sigma \right)d{\gamma }_i\right\}\end{array}$$ (15)

where $f_{\gamma }(\gamma \mid \sigma )$ is the density function of $\gamma$ with prespecified variance parameter $\sigma$, and

$$\begin{array}{l}{l}_i\left({\alpha }^{(a)},{\text{Λ}}_{01}^{(a)},{\text{Λ}}_{02}^{(a)},{\mathit{\beta }}_1^{(a)},{\mathit{\beta }}_2^{(a)}\right)\\ =\left(1-{\delta }_{i1}\right)\left(1-{\delta }_{i2}\right)\log \left[D\left\{X_i,Y_i;{\alpha }^{\left(A_i\right)}\right\}\right]\\ +{\delta }_{i1}{\delta }_{i2}\log \left[D_{12}\left\{X_i,Y_i;{\alpha }^{\left(A_i\right)}\right\}{S}_1\left(X_i\mid {\mathbf{Z}}_i,A_i,{\gamma }_i\right)\right.\\ \left.\times {\lambda }_{01}^{\left(A_i\right)}\left(X_i\right){\gamma }_i\exp \left({\mathit{\beta }}_1^{\left(A_i\right)T}{\mathbf{Z}}_i\right)\right]\\ +{\delta }_{i1}{\delta }_{i2}\log \left[S_2\left(Y_i\mid {\mathbf{Z}}_i,A_i,{\gamma }_i\right){\lambda }_{02}^{\left(A_i\right)}\left(Y_i\right){\gamma }_i\exp \left({\mathit{\beta }}_2^{\left(A_i\right)T}{\mathbf{Z}}_i\right)\right]\\ +{\delta }_{i1}\left(1-{\delta }_{i2}\right)\log \left[D_1\left\{X_i,Y_i;{\alpha }^{\left(A_i\right)}\right\}{S}_1\left(X_i\mid {\mathbf{Z}}_i,A_i,{\gamma }_i\right)\right.\\ \left.\times {\lambda }_{01}^{\left(A_i\right)}\left(X_i\right){\gamma }_i\exp \left({\mathit{\beta }}_1^{\left(A_i\right)T}{\mathbf{Z}}_i\right)\right]\\ +\left(1-{\delta }_{i1}\right){\delta }_{i2}\log \left[D_2\left\{X_i,Y_i;{\alpha }^{\left(A_i\right)}\right\}{S}_2\left(Y_i\mid {\mathbf{Z}}_i,A_i,{\gamma }_i\right)\right.\\ \left.\times {\lambda }_{02}^{\left(A_i\right)}\left(Y_i\right){\gamma }_i\exp \left({\mathit{\beta }}_2^{\left(A_i\right)T}{\mathbf{Z}}_i\right)\right]\end{array}$$

The integral in Equation (15) poses analytical intractability, making a direct evaluation of the likelihood function challenging. A prevalent approach involves approximating it and subsequently utilizing the approximated likelihood to conduct inference about the model parameters. We modify the MCEM algorithm [14] to accommodate our causal framework. To reduce computation complexity, we start from a small MCMC sample size and increase it as computation progresses to approximate the likelihood above to derive estimates of $\left({\alpha }^{(a)},{\text{Λ}}_{01}^{(a)},{\text{Λ}}_{02}^{(a)},{\mathit{\beta }}_1^{(a)},{\mathit{\beta }}_2^{(a)}\right)$. We refer readers to Section 3 of the Supporting Information for more implementation details. Unlike Peng et al. [14]’s regression model, frailty variance is not estimable and should be prespecified in our potential outcomes framework, given that the two-world outcomes cannot be simultaneously observed. Nevertheless, it is noteworthy that their theoretical results remain applicable to our setting. In addition, from Equation (15), the proposed method is based upon (but not limited to) gamma-frailty for illustration. The proposed method can be extended to other general survival models and frailty distributions as well.

4.3 |. Statistical Inference for Causal Parameters

Utilizing the estimators derived from Sections 4.1 and 4.2, we proceed to estimate the causal effects via principal stratification in the context of semi-competing risks data. Specifically, the forms of estimators for baseline cumulative hazard functions ${\text{Λ}}_{01}^{(a)}$ and ${\text{Λ}}_{02}^{(a)}$) are provided in Equations (A.5-A.8) of the Supporting Information. When the conditional expectations of frailty in Equations (A.5) and (A.6) are excluded, these estimators reduce to estimators for ${\text{Λ}}_{01}^{(a)}$ and ${\text{Λ}}_{02}^{(a)}$ in the semiparametric model without frailty. Estimators of association parameter ${\alpha }^{(a)}$) and regression coefficients $\left({\mathit{\beta }}_1^{(a)},{\mathit{\beta }}_2^{(a)}\right)(a=0,1)$ are obtained by maximizing the log-likelihood functions referred in Sections 4.1 and 4.2. It can be seen that the estimations for regression parameters in the two worlds are separate under the semiparametric model without frailty, whereas these parameters are jointly estimated under the model with frailty. If there is no unmeasured confounding, plugging estimators for $\left({\alpha }^{(a)},{\text{Λ}}_{01}^{(a)},{\text{Λ}}_{02}^{(a)},{\mathit{\beta }}_1^{(a)},{\mathit{\beta }}_2^{(a)}\right)$, $a=0,1$, obtained from Section 4.1, which are consistent and asymptotically normal [12, 14, 47], into (10–12) and replacing $S_{k\mid T_1\le T_2,A=a,\mathbf{Z}}$ and $S_{2\mid T_1\ge T_2,A=a,\mathbf{Z}}$ in Equations (4–6) with its empirical counterpart yields consistent and asymptotically normal estimators for the three SCEs including ND-SCE₂(t), AD-SCE₁(t), and AD-SCE₂(t) with each time point $t$. The asymptotic properties for the causal parameters follow from the continuous mapping theorem and the delta method. The same asymptotic properties are applied to the proposed method for sensitivity analysis. These estimators share the same asymptotic properties as those discussed in the literature [14], though their asymptotic variance can be more complex and lacks an explicit form. Thus, the classical bootstrap method based on resampling with replacement is more suggested to be used in practice to construct confidence intervals for both regression and causal parameters. Our simulation studies have further demonstrated the utility and effectiveness of the bootstrap method.

5 |. Application to a Breast Cancer Study

Hormone therapy or neoadjuvant endocrine therapy is often used in clinical practice for breast cancer patients as an adjuvant treatment following surgery. Despite its effectiveness, there is considerable debate surrounding the decision of whether to combine it with chemotherapy or radiotherapy or to pursue separate therapy after surgery [3]. Our objective was to apply our developed methods to evaluate the impact of hormone therapy and compare it with the combination of hormone therapy and radiotherapy in breast cancer patients who have undergone mastectomy surgery.

Data.

We applied our methods to analyze the data from the Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) cohort [51, 52]. The response variable of our interest was the pair of time to relapse (RFS, $T_1$) and overall survival (OS, $T_2$) outcomes measured in months (i.e., semi-competing risks). Due to loss to follow-up and dependent censoring, the overall censoring rates for RFS and OS were 55% and 34%, respectively. Covariates used in our analysis included age at diagnosis (21–96 years old), ER status (ER, 1=positive, 0=negative), PR status (PR, 1=positive, 0=negative), HER2 status (HER2,1=positive, 0=negative), inferred menopausal state (MENO, 1=post-menopausal, 0=pre-menopausal), whether the number of lymph nodes examined positive bigger than 0 or not (NODE, 1= yes, 0=no), Nottingham prognostic index (NPI), and tumor size (SIZE). After data processing, there were 1114 patients taking mastectomies with complete records for covariates, RFS, and OS. In the following analyses, we used the Frank copula to describe the joint distribution of RFS and OS.

Primary analysis.

We conducted two primary analyses: (1) We first quantified the causal effect of hormone treatment alone $\left(a=1\right)$ in comparison to no treatment $\left(a=0\right)$ using the proposed causal framework. To eliminate the influence and interaction of other treatments, we specifically focused on a subset of patients who did not receive chemotherapy or radiotherapy, leaving 611 patients (59.4% patients with hormone treatment alone vs. 40.6% without any treatment) in this analysis. The estimation of regression coefficients is summarized in Table 1. We dropped the variables PR and HER2 in this analysis, considering that the estimated coefficients for PR and HER2 status were very small in this limited sample. Figure 1 displays the curves of causal parameters varying over time under two principal strata (AD, ND). (2) We continued to study the causal effect of the combination of hormone and radio treatment $\left(a=1\right)$ in comparison to the use of hormone treatment alone $\left(a=0\right)$. Thus, there were 588 patients in this subgroup (38.3% patients with hormone and radiotherapy, vs. 61.7% with hormone therapy alone). Supplementary Table S.4 summarizes the effects of covariates on the marginal survival distribution for each outcome. Figure 2 displays causal estimands under the AD or ND stratum. Similar to the analysis above, we dropped the variable ER status due to very small estimated coefficients (Supplementary Table S.4).

TABLE 1 |.

Estimation of regression coefficients and Kendall’s tau ${\tau }^{(a)}$ for the breast cancer study under different specified frailty variance $\sigma$, where $a=0$ and $1$ represent no treatment and hormone treatment, respectively.

		$\text{RFS}(T_1)$						$\text{OS}(T_2)$
$\sigma$	${\tau }^{(a)}$	Age	ER	Meno	Node	NPI	size	Age	ER	Meno	Node	NPI	Size
a=0 (no treatment)
0	0.574**	0.005	0.457	−0.057	0.613	0.228 *	0.008	0.056**	−0.021	−0.732**	0.909**	0.175 *	0.011
0.5	0.566**	0.000	0.403	0.033	0.696 *	0.248 *	0.006	0.055	−0.003	−0.705**	0.981**	0.253**	0.013
1	0.438**	−0.002	0.388	−0.025	0.762	0.319**	0.005	0.054	−0.083	−0.730*	1.090**	0.283**	0.012
1.5	0.548**	0.000	0.334	−0.034	0.678*	0.266*	0.008	0.054	−0.012	−0.710**	0.927**	0.213*	0.011
10	0.498**	−0.005	0.369	−0.037	0.743	0.252*	0.005	0.051	−0.011	−0.749*	0.964*	0.197	0.011
a=1 (hormone treatment)
0	0.675**	0.016	0.008	−0.081	0.500**	0.179*	0.021**	0.044**	−0.071	−0.027	0.270	0.188**	0.017**
0.5	0.655**	0.010	−0.037	−0.134	0.515**	0.281*	0.025*	0.045	−0.097	−0.005	0.337*	0.236**	0.017**
1	0.505**	0.004	−0.082	−0.124	0.559*	0.289**	0.024**	0.042	−0.104	0.093	0.312	0.239	0.019**
1.5	0.652**	0.007	−0.082	−0.152	0.558**	0.273*	0.025	0.044	−0.129	0.008	0.355*	0.249**	0.019**
10	0.584**	0.004	−0.056	−0.151	0.563**	0.270**	0.024	0.041	−0.111	0.000	0.329*	0.226**	0.016**

Note:

^{* and **}indicate significance computed by 100 Bootstrap replicates at levels 0.1 and 0.05, respectively.

FIGURE 1 |.

Causal effects of hormone treatment alone compared to no treatment use in breast cancer. The three plots in the top row present the estimated ND-SCE₂, AD-SCE1, and AD-SCE₂ with different specified frailty variance $\sigma$ (sensitivity analysis). The three plots in the bottom row present the estimated causal parameters along with their 90% confidence interval under pre-specified $\sigma =0$ (i.e., no unmeasured confounding).

FIGURE 2 |.

Causal effects of hormone- and radio-treatment compared to hormone therapy alone in breast cancer. The three plots in the top row present the estimated ND-SCE₂, AD-SCE₁, and AD-SCE₂ with different specified frailty variance $\sigma$ (sensitivity analysis). The three plots in the bottom row present the estimated causal parameters along with their 90% confidence interval under pre-specified $\sigma =0$ (i.e., no unmeasured confounding).

Sensitivity analysis.

To more comprehensively evaluate the robustness of our findings to unmeasured confounding, we conducted a sensitivity analysis by following the strategy in Sections 4.2, 4.3, and varying the pre-specified $\sigma$ values. We also compared our method to the Kaplan-Meier analysis based upon the samples matched by propensity scores (PS) using baseline variables as covariates in the PS model [53], a widely used method in clinical studies. Figure 3 summarizes the results.

FIGURE 3 |.

Kaplan-Meier survival curves obtained using propensity score matching method. (a) and (c) are KM survival curves of RFS by ignoring dependent censoring. (b) and (d) are KM survival curves of OS for breast cancer patients.

Result interpretation.

1. Results from the Copula regression model (Table 1, Supplementary Table S.4) indicated that there was a strongly positive association between RFS and OS, even after adjusting for covariates. This highlights the importance of modeling dependence between semi-parametric risks. These tables also indicated that the impacts of covariates (such as Her2, Node, Size) on RFS $\left(T_1\right)$ and OS $\left(T_2\right)$ varied between the two scenarios (i.e., $a=1$ vs. $a=0$). However, sensitivity analyses, involving variations in the pre-specified $\sigma$ values, suggested that the frailty variance may primarily affect the association between $T_1$ and $T_2$ and has a lesser influence on the significance of covariates. Specifically, in the first primary analysis, only the NPI variable consistently showed a positive association with $T_1$ for patients with $a=0$, while the variables Node, NPI, and Size were positively associated with $T_1$ for patients with $a=1$. In the second primary analysis, the variables Node, NPI, and Size were positively associated with $T_1$ for patients with $a=0$, while HER2, Size, and Meno had an impact for patients with $a=1$, where the first two showed a positive association and the last one showed a negative association.

2. Results from causal estimand estimation: By assuming no unmeasured confounding (i.e., $\sigma =0$), using Hormone therapy alone compared to no treatment had a positive impact and was significantly beneficial to those patients in the “ND” stratum at some duration after surgery. But it may not have significant effects on both RFS and OS for patients in the “AD” stratum with long-time treatment (Figure 1). These imply that using Hormone therapy alone could be good for patients’ overall survival in the “ND” stratum, compared to no treatment use. Sensitivity analysis with varied $\sigma$ also supported the above conclusion (Figure 1). On the other hand, we detected a weak effect by comparing combined therapy with Hormone therapy alone (Figure 2) in the “AD” stratum at some duration after surgery. This implies that using the combined therapy might be even better for patients’ overall survival in the “AD” stratum, compared to using Hormone therapy alone. More discussions about AD/ND stratum are provided in Section 7.

3. In the Kaplan-Meier analysis under PS-matched cohorts (Figure 3), we observed that not undergoing any treatment resulted in a significantly higher survival rate compared to hormone treatment, and the combined treatment did not exhibit a clear effect on OS, contrary to expectations in clinical practice. This finding supports the necessity of the principal stratification analysis and demonstrates that the proposed causal framework for terminal event times provides more conservative and arguably more reliable results than conventional survival analysis. As for the KM survival curves of RFS, the combined treatment exhibited a significant effect on postponing the occurrence of relapse. This discrepancy could be attributed to the Kaplan-Meier analysis, even after PS matching, ignoring the dependence structure between RFS and OS, and failing to account for potential differences over principal strata. Consequently, we place more emphasis on and advocate for the use of our proposed methods, which offer a more valid causal comparison and yield more reasonable final results.

We also applied the semi-parametric estimator from the Nevo-Gorfine illness-death model [23] to our breast cancer dataset, with results presented in Figure S.9 in the Supporting Information. Their method suggested that hormone treatment alone reduced the survival rate for patients in the “ND” (never-diseased) stratum at certain durations post-surgery compared to no treatment. In the “AD” (always-diseased) stratum, hormone therapy showed a significantly positive impact on delaying relapse and death. Furthermore, combined treatment showed a positive effect on OS in the “ND” stratum but negative effects on both RFS and OS in the “AD” stratum. However, these findings do not align well with well-recognized clinical knowledge and practice. Notably, radiotherapy, a more aggressive treatment, is typically reserved for high-risk patients, while hormone therapy, being gentler, is preferred for patients with lower relapse risk [4, 54]. Generally speaking, for patients expected to die without relapse, hormone therapy alone is often sufficient, avoiding the need for aggressive radiotherapy; for patients at high risk of relapse, radiotherapy is recommended to eliminate residual tumors and prevent cancer progression. However, the results produced using the illness-death model deviate from these clinical guidelines, suggesting treatment strategies that contradict standard practice. In contrast, our proposed estimator produces results that align more closely with clinical expectations. These also highlight the robustness of our method in real-world practice.

6 |. Simulation Studies

The data generation mechanisms are classified into two categories: The “without frailty” class (where the cross-world dependence is exclusively captured by all observed covariates) and the “with frailty” class (where the cross-world dependence is not only attributed to observed covariates but also to latent and unobservable frailty variables). We evaluated the proposed methods based on four simulation settings (Ex1-Ex4 cases), where case Ex2 contains more covariates than case Ex1; case Ex3 is used to examine the robustness of our causal framework when treatment allocation is affected by baseline covariates as indicated by Assumption 2; and case Ex4 is to evaluate our method in the presence of a frailty describing two-world dependency. Due to the page limit, we refer readers to Supplementary Section 4 for more technical details.

Under each scenario, we generated 200 Monte Carlo replicates with sample size n = 500 or 1000. In all scenarios, we first estimated regression coefficients, baseline cumulative hazard functions, and the association between non-terminal and terminal times for each arm. We provided a comprehensive evaluation of the estimation results of regression coefficients and association parameters in terms of mean biases, Monte Carlo standard deviations (MCSD), and asymptotic standard errors (ASE) across 200 simulation runs. We only present the results from case Ex1. The other three cases showed similar patterns, thus omitted in this manuscript. We then estimated stratum-specific survivor causal effects (4–6) and evaluated their performance under cases Ex1-Ex4 varying with time over a grid of 30 time points which is determined by the sample order statistics of observed $T_1$. We used the bootstrap method with 100 replicates to estimate the standard errors of the estimators for both causal parameters and regression coefficients.

Under Scenario Ex1 with low/high censoring setting, the biases and MCSDs of estimated regression coefficients appear to be negligible and diminish as the sample size increases for both Kendall’s tau values (0.3 and 0.6) (Supplementary Table S.2). The estimated standard errors are close to the empirical standard deviation of point estimates of regression coefficients. The estimated baseline cumulative hazard functions (depicted in red) for the first 50 simulated samples surround the truth in blue (Supplementary Figure S.1). In Figures 4 and 5, the mean estimated causal parameters (red curves) plus/minus 1.96 times empirical standard deviation (pink vertical bars) encapsulate the truth (depicted in black dashed curves), and discrepancies between ASE and MCSD at 20 time points seem to be negligible and fluctuate around zero. In Table 2, the biases of estimated causal parameters at specific time points are small and their coverage probabilities (CP) are reasonable but fluctuate slightly around the nominal level 0.95, which is expected regarding the complex nature of causal estimand estimation.

FIGURE 4 |.

Estimation of causal parameters under the case Ex1 with $\tau =0.3$ and n = 1000. The three plots in the top row present the mean estimated causal parameters (represented by red curves) plus/minus 1.96 times the empirical standard deviation, encapsulating the true values (depicted in black dashed curves). The three plots in the bottom row present the difference between the Bootstrap-based ASE and the MCSD, fluctuating around zero. From left to right, these plots correspond to ND-SCE₂, AD-SCE₁, and AD-SCE₂, respectively.

FIGURE 5 |.

Estimation of causal parameters under the case Ex1 with $\tau =0.6$ and n = 1000. The three plots in the top row present the mean estimated causal parameters (represented by red curves) plus/minus 1.96 times the empirical standard deviation, encapsulating the true values (depicted in black dashed curves). The three plots in the bottom row present the difference between the Bootstrap-based ASE and the MCSD, fluctuating around zero. From left to right, these plots correspond to ND-SCE₂, AD-SCE1, and AD-SCE₂, respectively.

TABLE 2 |.

Estimation of causal parameters (ND-SCE₂, AD-SCE₁ and AD-SCE₂) at time points t = 3,6 in the case Ex1 with low censoring. The true values of causal parameters, the mean biases, and the coverage probabilities (CP) of their estimates are shown in this table.

	ND-SCE2			AD-SCE1			AD-SCE2
Time	Truth	Bias	95%CP	Truth	Bias	95%CP	Truth	Bias	95%CP
$\tau =0.3$, $n=500$
3	0.32	−0.002	0.96	0.38	0.003	1.00	0.26	−0.004	0.94
6	0.26	0.013	0.96	0.17	0.009	0.94	0.27	−0.005	0.96
$\tau =0.3$, $n=1000$
3	0.32	0.005	0.94	0.38	−0.008	0.92	0.26	−0.004	1.00
6	0.26	0.004	0.92	0.17	0.002	0.90	0.27	−0.006	0.94
$\tau =0.6$, $n=500$
3	0.30	0.018	0.98	0.37	−0.026	0.98	0.28	−0.010	0.98
6	0.31	−0.055	0.94	0.18	−0.017	1.00	0.25	0.028	0.86
$\tau =0.6$, $n=1000$
3	0.30	0.020	0.96	0.37	−0.031	0.98	0.28	−0.011	1.00
6	0.31	−0.015	0.94	0.18	−0.010	0.98	0.25	0.016	0.92

In the presence of more covariates as in case Ex2, estimators of causal parameters still perform well in terms of negligible estimation bias (Supplementary Figure S.2). Under case Ex3, it can be seen from Figure S.3 in the Supporting Information that the proposed estimator shows satisfactory performance except for a slight deviation shown in the ND stratum. This anomaly is understandable in light of the high-censoring environment, where fewer than 8% of subjects belong to the ND stratum and are observed to have the $T_2$ event. In case Ex4 with a frailty describing two-world dependence, the proposed MCEM algorithm in Section 4 yields asymptotically unbiased estimation for regression coefficients (Supplementary Table S.3) as well as causal parameters (Figure 6). The results presented in the manuscript are based on n = 1000, Kendall’s $\tau =0.6$, and $\sigma =0.2$ or 0.4. More simulation evaluations based upon different combinations of sample size $n$, Kendall’s $\tau$, and $\sigma$ can be found in Supplementary Figures S.4-S.6. Similar patterns are observed over all the settings, demonstrating the validity of our new method.

FIGURE 6 |.

Estimation of causal parameters (ND-SCE₂, AD-SCE₁, and AD-SCE₂) in the case Ex4 (evaluating the method for sensitivity analysis) with n = 1000, $\tau =0.6$, and $\sigma =0.2/0.4$.

We further evaluated the sensitivity of our method to copula misspecification and compared its performance with the semiparametric approach proposed by [23] under various data-generating scenarios. Specifically, when the data were generated using the Clayton copula, the gamma-frailty illness-death model by [23] was correctly specified, as the Clayton copula structure aligns to some extent with a gamma frailty model [10, 55]. In contrast, data generated from the Frank copula led to misspecification in the frailty model. We conducted simulations under Case Ex4 using either the Clayton or Frank copula, with $\sigma =0.2$ governing the cross-world dependency, to assess the sensitivity of both the proposed method and the Nevo-Gorfine method under model misspecification. Results are presented in Figures S.7 and S.8 of the Supporting Information. When both models are correctly specified (Figure S.7a), the two methods yield comparable results. When the Nevo-Gorfine method is correctly specified but the proposed method is misspecified (Figure S.7b), the proposed method shows slightly inferior yet acceptable performance. However, when the proposed method is correctly specified and the Nevo-Gorfine method is misspecified (Figure S.8a), the Nevo-Gorfine method appears more sensitive to model misspecification compared to the earlier case. Finally, when both models are misspecified (Figure S.8b), the proposed method demonstrates greater robustness-particularly in estimating the terminal event time in the “ND” stratum.

7 |. Discussion

This article establishes a frequentist framework using copula models for causal interpretation, net quantity estimation, and sensitivity analysis for unmeasured factors under right-censored semi-competing risks data. We have demonstrated the non-parametric identification and justified the feasibility and utility of the proposed estimation. The advantage of the proposed method over existing methods, including Nevo and Gorfine [23], is its capacity to assess net covariate effects and dependency between non-terminal and terminal event times under each counterfactual arm. Additionally, by leveraging the copula structure and a cross-world frailty variable, the proposed causal framework offers enhanced flexibility for practical applications and demonstrates more robust performance against model misspecification in numerical studies, compared to the illness-death based approach [23].

In the observational study of breast cancer, by applying our method, we detected that hormone therapy could be beneficial to patients with a low risk of relapse compared to no treatment use, and that the combined treatment will be beneficial to those with a higher risk of relapse, compared to hormone therapy alone. These findings imply a more tailored intervention strategy and suggest intermediate intervention after assessing the risk of relapse could be much more beneficial to patients’ overall survival, which to some extent aligns with previous clinical research [56] and provides more insight into personalized medicine for breast cancer patients to improve their post-surgery outcomes.

For future method development, we introduce various extensions and pose open questions for readers to explore and enhance the current framework: (1) Copula model selection. Hsieh et al. [57] provided the hypothesis test in terms of whether a copula model fits the data. Specifically, for each covariate group, the function $G_a\left(t_1,t_2\right)=\mathrm{Pr}\left(T_1\ge t_1,T_2\ge t_2\mid {\delta }_1=1,{\delta }_2=1,A=a\right)$ is identifiable non-parametrically in the upper wedge as

$$\begin{gathered} {\tilde{G}}_a\left(t_1,t_2\right)=\sum _{i=1}^{n}I\left(X_i\ge t_1,Y_i\ge t_2,{\delta }_{i1}=1,{\delta }_{i2}=1,A_i=a\right)/ \\ \sum _{i=1}^{n}I\left({\delta }_{i1}=1,{\delta }_{i2}=1,A_i=a\right) \end{gathered}$$

The goodness-of-fit test can be thus constructed by comparing ${\tilde{G}}_a\left(t_1,t_2\right)$ and ${\widehat{G}}_a\left(t_1,t_2\right)$ based on two copula models based on some distance measure [57]. (2) This paper considers a simple copula model with homogeneous association for illustration. The extensions of such a model include time-dependent association [12] and covariate-dependent association [58]. (3) Another extension is to study multiple treatments involving considering the effects of various interventions on survival outcomes subject to semi-competing risks, while addressing potential confounding factors. This extension often requires methodologies like the generalized propensity score, instrumental variables, or causal mediation analysis, adapted to handle multiple treatment groups. (4) Recurrent event data offers a valuable opportunity for methodological extension. One potential direction is to replace the non-terminal event time in models (1) and (9) of the manuscript with the recurrence gap time. By adopting population partitions such as “always recurrent” and “never recurrent,” analogous to those used in this study, the focus shifts to paired outcomes-namely, the time to first recurrence and time to death-thereby reducing the problem to a semi-competing risks framework. (5) How to appropriately identify the latent AD and ND sub-populations is unclear in practice. This motivates our future work on defining a proper discrimination measure under the semi-competing risks setting. Such an approach would serve as a useful tool for clinical audiences. All of the above merits further investigation in future studies.

Supplementary Material

Supporting Information

Additional supporting information can be found online in the Supporting Information section. Data S1. The Supporting Information contains technical proofs of propositions referenced in Section 3, the maximum likelihood estimation algorithm referenced in Section 4, and additional numerical results referenced in Sections 5 and 6.

Data Availability Statement

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.