Non‐Parametric Mediation Analysis of Non‐Markov Illness‐Death Model

ABSTRACT

The illness‐death model is widely used to characterize disease progression over time. Previous work focuses either on estimation without causal interpretation or on causal interpretation under a strong Markov assumption where the terminal event depends on the status but not the timing of the intermediate event. To bridge the research gap, we propose a new definition of counterfactual hazard that relaxes the Markov assumption by considering the entire history of the intermediate event. We derive an identification formula that involves an integral with respect to the probability density function of the intermediate event time. Direct and indirect effects refer to the influence of an exposure on the terminal event not mediated by, and mediated through, the intermediate event, respectively. We propose non‐parametric kernel estimators for the two effects and study their asymptotic properties. We conduct numerical simulations to examine the proposed estimators' finite‐sample performance. Applying the method to a hepatitis study where the Markov assumption is violated, we show that the effect of hepatitis C on mortality is not mediated through septicemia during the first 15 years of follow‐up.

1. Introduction

Illness‐death models, which describe transitions between different states, are widely used to characterize disease progression in clinical studies [1, 2]. A motivating example is provided by the REVEAL study [3] on the disease progression of hepatitis C, which can be formulated as an illness‐death model to assess how hepatitis C influences the risk of death through the intermediate development of septicemia. The illness‐death model consists of three states: a healthy state (state 0), an intermediate illness state (state 1; septicemia), and an absorbing terminal state (state 2; death). Individuals may transition either indirectly from the healthy state to the illness state and subsequently to the terminal state or directly from the healthy state to the terminal state.

Modeling transitions in the illness‐death framework using transition hazard functions is a well‐established approach [4, 5, 6]. Specifically, the transition intensities characterize the rates at which individuals move across states:

$$\begin{array}{ll}\hfill {\lambda }_{01}(t)& =\underset{\mathrm{\Delta }\to 0}{\lim }P\left(T_1\in [t,t+\mathrm{\Delta })|T_1\ge t,T_2\ge t\right)/\mathrm{\Delta },\end{array}$$ (1)

$$\begin{array}{ll}\hfill {\lambda }_{02}(t)& =\underset{\mathrm{\Delta }\to 0}{\lim }P\left(T_2\in [t,t+\mathrm{\Delta })|T_1\ge t,T_2\ge t\right)/\mathrm{\Delta },\end{array}$$ (2)

$$\begin{array}{ll}\hfill {\lambda }_{012}(t|t_1)& =\underset{\mathrm{\Delta }\to 0}{\lim }P\left(T_2\in [t,t+\mathrm{\Delta })|T_1=t_1,T_2\ge t\right)/\mathrm{\Delta },\end{array}$$ (3)

where $T_1$ and $T_2$ denote the intermediate event and terminal event times, respectively.

A commonly made assumption in illness‐death models is that the occurrence of the terminal event (e.g., death) depends only on whether the intermediate event (e.g., disease) occurs, but not on when it occurs, the so‐called Markov assumption. Formally, ${\lambda }_{012}(t|t_1)={\lambda }_{012}(t)={\lim }_{\mathrm{\Delta }\to 0}P\left(T_2\in [t,t+\mathrm{\Delta })|T_1<t,T_2\ge t\right)/\mathrm{\Delta }$. In practice, the Markov assumption does not always hold, despite its analytical convenience in the analysis of illness‐death data. In particular, formal tests for the Markov assumption in illness‐death models have been developed [7, 8, 9]. When violated, classical estimators of transition probabilities‐ such as the Aalen‐Johansen estimator [4], an extension of the Kaplan‐Meier estimator [10]‐ may yield biased estimates of transition rates [6]. To address this issue, a series of papers has proposed non‐parametric methods to assess transition probabilities without the Markov condition [5, 6, 11], many of which were based on Kaplan‐Meier or Pepe‐type estimators [12, 13]. Although these approaches successfully address the issue of the Markov condition, their estimands, such as transition rates, lack causal interpretations.

Additionally, another way to relax the Markov assumption is to allow the illness‐to‐death transition hazard to depend on the sojourn time, or gap time, since the intermediate event. That is, instead of assuming ${\lambda }_{012}(t|t_1)={\lambda }_{012}(t)$, one may specify ${\lambda }_{012}(t|t_1)={\lambda }_{012}(t-t_1)$ [14], or, more generally, ${\lambda }_{012}(t|t_1)={\lambda }_{012}(t,t-t_1)$, where $t-t_1$ represents the time elapsed since the occurrence of the intermediate event [15, 16]. Nevertheless, such sojourn‐time‐based approaches still require specifying how the elapsed time since the intermediate event enters the illness‐to‐death transition hazard. In particular, a model depending only on the gap time implicitly assumes that, conditional on the same elapsed time since the intermediate event, the illness‐to‐death hazard is homogeneous with respect to the timing of the intermediate event. Thus, although these approaches relax the Markov assumption, they still rely on specific transition‐hazard modeling assumptions.

To gain a causal insight, we formulate the analysis of illness‐death models under the framework of mediation analysis [17]. The mediation framework follows the classic counterfactual definition proposed by Pearl [18] and Robins [19], in which the effect of an exposure on an outcome is decomposed into a natural indirect effect operating through a mediator and a natural direct effect that affects the outcome not through the mediator. What differs from the classic setting is that the mediator and the outcome in our setting are both time‐to‐event variables that are subject to right‐censoring. Furthermore, the outcome may also censor the mediator, which is the unique feature and challenge in illness‐death models and semi‐competing risks. Causal mediation analysis of illness‐death models or semi‐competing risks has gained increasing attention in recent years [20, 21, 22, 23]. Huang [21] identified a counterfactual hazard function based on a weighted sum of ${\lambda }_{01}(·)$ and ${\lambda }_{012}(·)$ to assess causal direct and indirect effects of semicompeting risks. Nevo and Gorfine [22] proposed a principal stratification approach, stratifying the population according to potential outcomes of the two events, and under their proposed assumptions, they identified the effects on both outcomes. Valeri et al. [23] characterized the effect of stochastic interventions for the intermediate time‐to‐event on the terminal outcome. Breum et al. [20] quantified the effect of an exposure on the terminal time‐to‐event outcome mediated by the time‐to‐intermediate event under the separable effect framework [24].

Existing causal mediation analyses of illness‐death models still make the Markov or semi‐Markov assumption. For example, the approach of Huang [21] is based on the Markov assumption that the risk of hepatitis B‐induced mortality does not depend on the time of occurrence of liver cancer, but only on the state of liver cancer. Stensrud et al. [25] and Fulcher et al. [26] have argued that such a Markov assumption is overly restrictive and may not adequately capture realistic disease progression. The methods proposed by Breum et al. [20] and Valeri et al. [23] address this issue by relaxing the Markov assumption through model assumptions, that is, the semi‐Markov condition. Specifically, Valeri et al. [23] provided the partial relaxation of the Markov condition based on another model assumption regarding how the mediator's timing affects the risk of the outcome in a pre‐specified parametric form:${\lambda }_{012}(t|t_1,Z)={\lambda }_{012}^0(t)e^{{\alpha }_{1}Z+{\alpha }_2{t}_1+{\alpha }_3{t}_{1}Z}$, where ${\lambda }_{012}^0(t)$ is the baseline hazard and $Z$ is the exposure. Model misspecification of the relationship may still violate the Markov assumption. To ensure valid causal inference in more general settings, it is important to develop an approach that does not rely on either the Markov or semi‐Markov assumption.

To bridge this research gap, we develop a causal mediation analysis framework that does not rely on the Markov assumption. Specifically, we explicitly model the dependence of the outcome on the timing of the mediator through non‐parametric kernel estimators. Moreover, we rigorously specify the set of assumptions required for identifying mediation effects‐extending from the existing sequential ignorability [27]‐thereby providing a transparent foundation for valid causal inference. Our method accommodates general non‐Markov illness‐death structures, while previous approaches impose the Markovian restriction. Importantly, under the Markov assumption, our framework reduces to the existing approaches by Huang [21] and Valeri et al. [23], establishing comparability and generality.

This article is structured as follows. Section 2 introduces the counterfactual hazard framework and notation, formalizes the causal estimands of interest under a non‐Markov multi‐state process, and describes the proposed non‐parametric estimation procedure. Section 3 presents theoretical results, including unbiasedness, uniform consistency, and the asymptotic distribution of the estimators. Section 4 reports the results from simulation studies conducted under Markov and non‐Markov scenarios, as well as a real‐data application of the REVEAL study that compares the performance of the proposed method with that of Huang's and Valeri's estimators. Finally, Section 5 concludes with a discussion of the main findings and potential directions for future research. Technical details and proofs associated with Section 3 are provided in the Supporting Information.

2. Counterfactual Hazard Function and Causal Assumptions

2.1. Counterfactual Hazard

Let $T_1$ and $T_2$ denote the intermediate event time and primary event time, respectively, with the corresponding underlying counting processes for counting the disease and death events defined as ${Ñ}_1(t)=I(T_1\le t)$ and ${Ñ}_2(t)=I(T_2\le t)$. Moreover, let $Z$ be a binary exposure factor and $W$ be a $p$‐dimensional vector of confounders.

In this article, we establish the inference of causal validity within the counterfactual outcome framework. Specifically, ${Ñ}_2(t;z,n_1({\underset{\_}{t}}^{-}))$ represents the counterfactual process of ${Ñ}_2(·)$ at time $t$, under the setting $(Z,{Ñ}_1({\underset{\_}{t}}^{-}))=(z,n_1({\underset{\_}{t}}^{-}))$, where $n_1({\underset{\_}{t}}^{-})$ is the realization of ${Ñ}_1({\underset{\_}{t}}^{-})$, with ${\underset{\_}{t}}^{-}=\{x|0\le x<t\}$ denoting the history of the time to the intermediate event before time $t$. Additionally, $n_1({\underset{\_}{t}}^{-})$ tracks the history of the underlying process up to time $t$. Furthermore, ${Ñ}_1(t;z)$ denotes the counterfactual process of ${Ñ}_1(·)$ at time $t$, had the exposure factor $Z$ been set to $z$.

Therefore, we can define the jump of counterfactual cumulative hazard at time $t$ using the counterfactual counting processes described above, as follows:

$$\begin{array}{cc}\hfill \mathrm{d}\mathrm{\Lambda }(t;z_a,z_b)& :=P\left(\mathrm{d}{Ñ}_2\left(t;z_a,{Ñ}_1({\underset{\_}{t}}^{-};z_b)\right)\right.\hfill \\ \hfill & \left.=1|{Ñ}_2\left(t^{-};z_a,{Ñ}_1({\underset{\_}{t}}^{-};z_b)\right)=0,W\right).\hfill \end{array}$$ (4)

Precisely, $\mathrm{d}N(t)$ represents the index function capturing the instantaneous jump of the arbitrary counting process $N(·)$ in the interval $[t,t+\mathrm{d}t)$. The major difference between Equation (4) and the counterfactual hazard function mentioned in Huang [21] is that the cumulative hazard of the primary event depends on the entire history of the intermediate event up to $t^{-}$, ${Ñ}_1$, rather than only the status at time $t^{-}$.

Building on the ideas proposed by Pearl [18] and Robins [19], we define the natural direct effect (DE) and natural indirect effect (IE) on the scale of cumulative hazard at time $t$ as follows:

$$\begin{array}{ll}\hfill {\mathrm{\Lambda }}_{DE}(t)& =\mathrm{\Lambda }(t;z_a=1,z_b=0)-\mathrm{\Lambda }(t;z_a=0,z_b=0)\\ \hfill & =\mathrm{\Lambda }(t;1,0)-\mathrm{\Lambda }(t;0,0),\end{array}$$ (5)

and

$$\begin{array}{ll}\hfill {\mathrm{\Lambda }}_{IE}(t)& =\mathrm{\Lambda }(t;z_a=1,z_b=1)-\mathrm{\Lambda }(t;z_a=1,z_b=0)\\ \hfill & =\mathrm{\Lambda }(t;1,1)-\mathrm{\Lambda }(t;1,0).\end{array}$$ (6)

We also define two survival‐scale causal effect measures. The first is the survival difference: ${\mathrm{\Delta }}_{DE}(t)=e^{-\mathrm{\Lambda }(t;1,0)}-e^{-\mathrm{\Lambda }(t;0,0)}$ and ${\mathrm{\Delta }}_{IE}(t)=e^{-\mathrm{\Lambda }(t;1,1)}-e^{-\mathrm{\Lambda }(t;1,0)}$. The second is the survival ratio: ${\rho }_{DE}(t)=e^{-{\mathrm{\Lambda }}_{DE}(t)}$ and ${\rho }_{IE}(t)=e^{-{\mathrm{\Lambda }}_{IE}(t)}$.

2.2. Causal Assumptions

We introduce the causal assumptions used in this article to identify the proposed cumulative counterfactual hazard as follows:

• B1.Ignorability: $\left(\mathrm{d}{Ñ}_1(t;z_b),\mathrm{d}{Ñ}_2(t;z_a,n_1({\underset{\_}{t}}^{-})),0\le t\le \zeta \right)\perp Z|W$.
• B2.
  Sequential Ignorability I:

  $$\begin{array}{ll}\hfill & P\left(\mathrm{d}{Ñ}_2\left(t;z_a,n_1({\underset{\_}{t}}^{-})\right)=1|Z=z_a,{Ñ}_1({\underset{\_}{t}}^{-};z_a)\right.\\ \hfill & \left.=n_1({\underset{\_}{t}}^{-}),{Ñ}_2\left(t^{-};z_a,n_1({\underset{\_}{t}}^{-})\right)=0,W\right)\\ \hfill & =P\left(\mathrm{d}{Ñ}_2\left(t;z_a,n_1({\underset{\_}{t}}^{-})\right)=1|Z=z_a,{Ñ}_1({\underset{\_}{t}}^{-};z_b)\right.\\ \hfill & \left.=n_1({\underset{\_}{t}}^{-}),{Ñ}_2\left(t^{-};z_a,n_1({\underset{\_}{t}}^{-})\right)=0,W\right).\end{array}$$ (7)

  Equation (7) means that the hazard of the primary event, given the exposure and the history of the intermediate event, does not depend on how the exposure affects the potential process of the intermediate event, as long as the intermediate processes are identical, that is, $n_1({\underset{\_}{t}}^{-})$.

• B3.
  Sequential Ignorability II:

  $$\begin{array}{ll}\hspace{0.17em}& P\left({\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)=n_1({\underset{\_}{t}}^{-})|Z=z_b,{\tilde{N}}_2\left(t^{-};z_a,n_1({\underset{\_}{t}}^{-})\right)=0,W\right)\\ \hspace{0.17em}& =P\left({\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)=n_1({\underset{\_}{t}}^{-})|Z=z_b,{\tilde{N}}_2\left(t^{-};z_b,n_1({\underset{\_}{t}}^{-})\right)=0,W\right).\end{array}$$ (8)

  Equation (8) means that the history of the intermediate event among survivors of the primary event does not depend on how the exposure affects the potential process of the primary event.
• B4.Consistency: if $z=Z_i$ and $n_1({\underset{\_}{t}}^{-})={Ñ}_{1i}({\underset{\_}{t}}^{-})$, then $\mathrm{d}{Ñ}_{2i}(t)=\mathrm{d}{Ñ}_{2i}(t,z,n_1({\underset{\_}{t}}^{-}))$.

Assumption (B1) ensures that given $W$, there is no unmeasured confounding between $Z$ and ${Ñ}_1(t)$, as well as between $Z$ and ${Ñ}_2(t)$. Assumptions (B2) and (B3) ensure that given $W$, there is no unmeasured confounding between ${Ñ}_1({\underset{\_}{t}}^{-})$ and ${Ñ}_2(t)$ (Assumption (B2)) as well as between ${Ñ}_1(t)$ and ${Ñ}_2(t)$ (Assumption (B3)). The black component of Figure 1 depicts a single‐world intervention graph (SWIG), representing the causal structure implied by Assumptions (B1)‐(B3). The terms ${ϵ}_{N_1}$ and ${ϵ}_{N_2}$ denote the error terms associated with ${Ñ}_1(·)$ and ${Ñ}_2(·)$, respectively.

FIGURE 1.

No unmeasured mediator‐outcome confounder.

We show in Figure 1 that Assumptions (B2) and (B3) would be violated in the presence of a mediator‐outcome confounder $W^{\ast }$. In particular, Assumption (B2) is violated because ${Ñ}_1({\underset{\_}{t}}^{-};z_b)$ can reach ${Ñ}_2(t;z_a,n_1({\underset{\_}{t}}^{-}))$ through the path: ${Ñ}_1({\underset{\_}{t}}^{-};z_b)\leftarrow W^{\ast }\leftarrow {ϵ}_{W^{\ast }}\to W^{\ast }\to {Ñ}_2(t;z_a,n_1({\underset{\_}{t}}^{-}))$; Assumption (B3) is also violated because ${Ñ}_1(t^{-};z_b)$ can reach ${Ñ}_2(t^{-};z_a,n_1({\underset{\_}{t}}^{-}))$ through the path: ${Ñ}_1({\underset{\_}{t}}^{-};z_b)\leftarrow W^{\ast }\leftarrow {ϵ}_{W^{\ast }}\to W^{\ast }\to {Ñ}_2(t^{-};z_a,n_1({\underset{\_}{t}}^{-}))$. Assumptions (B2) and (B3) exclude the presence of a mediator‐outcome confounder induced by the exposure $Z$, $W^{\prime }$. As shown in Figure 2, the $Z$‐induced mediator‐outcome confounder $W^{\prime }$ contributes to both direct ($Z\to W^{\prime }\to {Ñ}_2(t)$) and indirect effects ($Z\to W^{\prime }\to {Ñ}_1({\underset{\_}{t}}^{-})\to {Ñ}_2(t)$). Even with the data on $W^{\prime }$, ${\mathrm{\Lambda }}_{DE}(t)$ and ${\mathrm{\Lambda }}_{IE}(t)$ controlling for $W^{\prime }$ would no longer have the interpretation of direct and indirect effects, respectively.

FIGURE 2.

No $Z$‐induced mediator‐outcome confounder.

Assumption (B4) states a standard consistency condition: for subjects whose observed exposure equals $z$ and whose history of the intermediate event equals $n_1({\underset{\_}{t}}^{-})$, the observed counting processes coincide with their corresponding individual‐level counterfactual processes under $z$ and $n_1({\underset{\_}{t}}^{-})$. It does not imply that all subjects with the same observed exposure $z=Z_i$ and intermediate‐event history $n_1({\underset{\_}{t}}^{-})={Ñ}_{1i}({\underset{\_}{t}}^{-})$ have identical realizations of the counting process; rather, the equality is defined at the individual level between the observed and counterfactual processes.

We present a structural causal model to describe the counterfactual data‐generating process in Web Appendix 15 of the Supporting Information. Based on this framework and under Assumptions (B1)‐(B4), we demonstrate that Equation (4) can be identified through the counting process in the following Theorem 1 (detailed in Subsection A). For simplicity, we omit $W$ in the subsequent development, but the related results concerning $W$ can be easily established.

Theorem 1

(Identification Formula) Under Assumptions (B1)–(B4),

$$\begin{array}{ll}\hfill \mathrm{d}\mathrm{\Lambda }(t;z_a,z_b)& ={\omega }_0(t|z_b)\mathrm{d}{\mathrm{\Lambda }}_0(t|z_a)+{\int }_0^{t^{-}}{\omega }_1(t_1|t,z_b)\mathrm{d}{\mathrm{\Lambda }}_1(t|t_1,z_a)\mathrm{d}{t}_1,\end{array}$$ (9)

where

$$\begin{array}{ll}\hfill {\omega }_0(t|z_b)& =P\left({Ñ}_1(t^{-})=0|{Ñ}_2(t^{-})=0,Z=z_b\right),\end{array}$$ (10)

$$\begin{array}{ll}\hfill \mathrm{d}{\mathrm{\Lambda }}_0(t|z_a)& =P\left(\mathrm{d}{Ñ}_2(t)=1|{Ñ}_1(t^{-})=0,{Ñ}_2(t^{-})=0,Z=z_a\right),\end{array}$$ (11)

$$\begin{array}{ll}\hfill {\omega }_1(t_1|t,z_b)& =\frac{\mathrm{d}}{\mathrm{d}{t}_1}P\left(T_1\le t_1|{Ñ}_2(t^{-})=0,Z=z_b\right),\end{array}$$ (12)

$$\begin{array}{ll}\hfill \mathrm{d}{\mathrm{\Lambda }}_1(t|t_1,z_a)& =P\left(\mathrm{d}{Ñ}_2(t)=1|\mathrm{d}{Ñ}_1(t_1)=1,{Ñ}_2(t^{-})=0,Z=z_a\right).\end{array}$$ (13)

Specifically, the derived quantities $\mathrm{d}{\mathrm{\Lambda }}_0(t|z_a)$ and $\mathrm{d}{\mathrm{\Lambda }}_1(t|t_1,z_a)$ are equivalent to the changes in the transition cumulative hazards from state 0 to state 2 and from state 1 to state 2, respectively. Therefore, the cumulative counterfactual hazard function can be obtained as

$$\begin{array}{ll}\hfill \mathrm{\Lambda }(t;z_a,z_b)& ={\int }_0^t\mathrm{d}\mathrm{\Lambda }(u;z_a,z_b)\\ \hfill & ={\int }_0^t{\omega }_0(u|z_b)\mathrm{d}{\mathrm{\Lambda }}_0(u|z_a)\\ \hfill & +{\int }_0^t{\int }_0^{u^{-}}{\omega }_1(t_1|u,z_b)\mathrm{d}{\mathrm{\Lambda }}_1(u|t_1,z_a)\mathrm{d}{t}_1\end{array}$$ (14)

$$\begin{array}{ll}\hfill & ={\int }_0^t{\omega }_0(u|z_b){\lambda }_0(u|z_a)\mathrm{d}u\\ \hfill & +{\int }_0^t{\int }_0^{u^{-}}{\omega }_1(t_1|u,z_b){\lambda }_1(u|t_1,z_a)\mathrm{d}{t}_1\mathrm{d}u.\end{array}$$ (15)

Equation (11) represents the transition hazard from the healthy state (state 0) to the absorbing state (state 2) in the illness‐death model. Equation (13) describes the transition hazard from state 1 to state 2, given that the intermediate event occurred at time $T_1=t_1$. The identification formula (14) consists of two components: the first represents the transition probability from state 0 to state 2, weighted by the probability of not experiencing the intermediate event, ${\omega }_0(·|z_b)$; the second accounts for the transition probability given the occurrence of the intermediate event, weighted by its occurrence probability density across all historical paths, ${\omega }_1(·|t,z_b)$. This formulation generalizes Huang [21] by relaxing the Markov assumption, making his identification formula a special case of our approach.

2.3. The Proposed Estimator

Consider an individual $i$ from a population of size $m$, where the observed data may involve potential independent right censoring at time $C_i$. Under the illness‐death model, the observation can be expressed as $U_i=\min (T_{2i},C_i)$, ${\delta }_{2i}=I(T_{2i}\le C_i)$, $T_{1i}^{\ast }=\min (T_{1i},U_i)$, and ${\delta }_{1i}=I(T_{1_i}\le U_i)$.

In Web Appendix 1 of the Supporting Information, we show that Equations (10), (11), (12), and (13) can be reformulated in terms of observed random variables accounting for right censoring. Based on these formulations, we propose the following non‐parametric estimators for ${\omega }_0(t|z_b)$, $\mathrm{d}{\mathrm{\Lambda }}_0(t|z_a)$, ${\omega }_1(t_1|t,z_b)$, $\mathrm{d}{\mathrm{\Lambda }}_1(t|t_1,z_a)$, respectively, using observable data.

$$\begin{array}{ll}\hfill {\widehat{\omega }}_0(t|z_b)& =\frac{\frac{1}{m}\sum _{i=1}^{m}I\left(Z_i=z_b,T_{1i}^{\ast }\ge t\right)}{\frac{1}{m}\sum _{i=1}^{m}I\left(Z_i=z_b,U_i\ge t\right)},\end{array}$$ (16)

$$\begin{array}{ll}\hfill \mathrm{d}{\widehat{\mathrm{\Lambda }}}_0(t|z_a)& =\frac{V_0(t)-V_0(t+\mathrm{\Delta }t)}{\frac{1}{m}\sum _{i=1}^{m}I\left(Z_i=z_a,T_{1i}^{\ast }\ge t\right)},\end{array}$$ (17)

$$\begin{array}{ll}\hfill {\widehat{\omega }}_1(t_1|t,z_b)& =\frac{\frac{1}{mh}\sum _{i=1}^{m}K\left(\frac{t_1-T_{1i}^{\ast }}{h}\right)I\left(Z_i=z_b,U_i\ge t,{\delta }_{1i}=1\right)}{\frac{1}{m}\sum _{i=1}^{m}I\left(Z_i=z_b,U_i\ge t\right)},\end{array}$$ (18)

$$\begin{array}{ll}\mathrm{d}{\widehat{\mathrm{\Lambda }}}_1(t|t_1,z_a)& =\frac{V_1(t,t_1)-V_1(t+\mathrm{\Delta }t,t_1)}{\frac{1}{mh}\sum _{i=1}^{m}K\left(\frac{t_1-T_{1i}^{\ast }}{h}\right)I\left(Z_i=z_a,U_i\ge t,{\delta }_{1i}=1\right)}\\ \hspace{0.17em}& ·I(t_1\le t),\end{array}$$ (19)

where

$$\begin{array}{ll}\hfill V_0(t)& =\frac{1}{m}\sum _{i=1}^{m}I(Z_i=z_a,U_i\ge t,{\delta }_{1i}=0,{\delta }_{2i}=1),\end{array}$$ (20)

$$\begin{array}{ll}\hfill V_1(t,t_1)& =\frac{1}{mh}\sum _{i=1}^{m}K\left(\frac{t_1-T_{1i}^{\ast }}{h}\right)I\left(Z_i=z_a,U_i\ge t,\right.\\ \hfill & \left.{\delta }_{1i}=1,{\delta }_{2i}=1\right).\end{array}$$ (21)

The statistical properties of the kernel function $K(·)$ and bandwidth $h$ are further discussed in the Supporting Information. We obtain an estimator for the counterfactual cumulative hazard:

$$\begin{array}{l}\hfill \mathrm{d}\widehat{\mathrm{\Lambda }}(t;z_a,z_b)=\mathrm{d}\mathrm{\Lambda }\left({\widehat{\omega }}_0(t|z_b),\mathrm{d}{\widehat{\mathrm{\Lambda }}}_0(t|z_a),{\widehat{\omega }}_1(t_1|t,z_b),\mathrm{d}{\widehat{\mathrm{\Lambda }}}_1(t|t_1,z_a)\right)\end{array}$$ (22)

by plugging in. The proposed estimators of ${\mathrm{\Lambda }}_{DE}(t)$ and ${\mathrm{\Lambda }}_{IE}(t)$ are as follows:

$$\begin{array}{ll}\hfill {\widehat{\mathrm{\Lambda }}}_{DE}(t)& ={\int }_0^t\mathrm{d}\widehat{\mathrm{\Lambda }}(s;1,0)-{\int }_0^t\mathrm{d}\widehat{\mathrm{\Lambda }}(s;0,0),\end{array}$$ (23)

$$\begin{array}{ll}\hfill {\widehat{\mathrm{\Lambda }}}_{IE}(t)& ={\int }_0^t\mathrm{d}\widehat{\mathrm{\Lambda }}(s;1,1)-{\int }_0^t\mathrm{d}\widehat{\mathrm{\Lambda }}(s;1,0).\end{array}$$ (24)

The survival‐scale estimators are obtained by plugging the cumulative hazard estimators into the corresponding survival‐scale effect measures. Specifically, ${\widehat{\mathrm{\Delta }}}_{DE}(t)=e^{-\widehat{\mathrm{\Lambda }}(t;1,0)}-e^{-\widehat{\mathrm{\Lambda }}(t;0,0)}$, ${\widehat{\mathrm{\Delta }}}_{IE}(t)=e^{-\widehat{\mathrm{\Lambda }}(t;1,1)}-e^{-\widehat{\mathrm{\Lambda }}(t;1,0)}$, ${\widehat{\rho }}_{DE}(t)=e^{-{\widehat{\mathrm{\Lambda }}}_{DE}(t)}$, and ${\widehat{\rho }}_{IE}(t)=e^{-{\widehat{\mathrm{\Lambda }}}_{IE}(t)}$.

For implementation, we use a symmetric and bounded kernel function, such as the Gaussian kernel, satisfying the kernel regularity conditions in Assumption (A1). The bandwidth $h$ controls the smoothness of the estimated density of the intermediate event time in ${\widehat{\omega }}_1(t_1|t,z_b)$ and $\mathrm{d}{\widehat{\mathrm{\Lambda }}}_1(t|t_1,z_a)$. In our asymptotic development, the bandwidth is chosen to achieve undersmoothness because the weak convergence results in Theorems 4 and 5 are established for zero‐mean Gaussian processes. In particular, the bandwidth conditions in Assumption (A2) are mainly governed by $m{h}^5\to 0$ as $m\to \infty$ and ${\sum }_{m=1}^{\infty }{e}^{-\gamma m{h}^2}<\infty$ for every $\gamma >0$. The condition $m{h}^5\to 0$ ensures that the kernel smoothing bias is asymptotically negligible. The summability condition is used in the asymptotic proofs to establish almost sure uniform convergence of the relevant kernel estimators. These conditions suggest searching for the bandwidth over a range slightly smaller than the usual mean‐squared‐error optimal order $m^{-1/5}$, such as $m^{-1/2.5}$ to $m^{-1/4.5}$. In practice, we select $h$ by a maximum‐likelihood‐type criterion: we maximize a cross‐validated counterfactual log‐likelihood over a prespecified grid of bandwidths in this range, as illustrated in Section 4.2.

3. Asymptotic Results

In this subsection, we demonstrate the asymptotic properties of ${\widehat{\mathrm{\Lambda }}}_{DE}(t)$ and ${\widehat{\mathrm{\Lambda }}}_{IE}(t)$. Before establishing the uniform unbiasedness, we verify the consistency, theoretical boundedness, and variance of the estimators in Web Appendices 2, 3, and 4 of the Supporting Information, respectively. We then apply Proposition W.21, as presented by Helland [28], in Web Appendix 5. The proof of Theorem 2, which builds upon the triangle inequality and the results discussed in Web Appendix 5, is provided in Web Appendix 6.

Theorem 2

(Uniform Unbiasedness) Under Assumptions (A1–A7),

$$\begin{array}{l}\hfill \mathrm{E}\left[\underset{t\in [0,\zeta ]}{\sup }\left|{\widehat{\mathrm{\Lambda }}}_{DE}(t)-{\mathrm{\Lambda }}_{DE}(t)\right|\right]\to 0,\end{array}$$ (25)

and

$$\begin{array}{l}\hfill \mathrm{E}\left[\underset{t\in [0,\zeta ]}{\sup }\left|{\widehat{\mathrm{\Lambda }}}_{IE}(t)-{\mathrm{\Lambda }}_{IE}(t)\right|\right]\to 0\end{array}$$ (26)

as $m\to \infty$.

Based on the discussion in Web Appendix 5 of the Supporting Information and by applying Proposition W.25 from Gill [29], we prove the uniform consistency in Theorem 3 in Web Appendix 8, using the triangle inequality and the results established in Web Appendix 7.

Theorem 3

(Uniform Consistency) Under Assumptions (A1–A7),

$$\begin{array}{l}\hfill \underset{t\in [0,\zeta ]}{\sup }\left|{\widehat{\mathrm{\Lambda }}}_{DE}(t)-{\mathrm{\Lambda }}_{DE}(t)\right|\stackrel{p}{\to }0\end{array}$$ (27)

and

$$\begin{array}{l}\hfill \underset{t\in [0,\zeta ]}{\sup }\left|{\widehat{\mathrm{\Lambda }}}_{IE}(t)-{\mathrm{\Lambda }}_{IE}(t)\right|\stackrel{p}{\to }0\end{array}$$ (28)

as $m\to \infty$.

We then discuss the finite‐dimensional distribution (f.d.d.) and tightness in Web Appendices 9 and 10 of the Supporting Information. Based on these results, we derive ${𝒢𝒫}_{{\omega }_1(·|t,0)}$ and ${𝒢𝒫}_{\mathrm{d}{\mathrm{\Lambda }}_1(t|·,1)-\mathrm{d}{\mathrm{\Lambda }}_1(t|·,0)}$ in Lemmas W.40 and W.41, respectively. Applying the functional delta method discussed in Web Appendix 11, we then prove Theorem 4 in Web Appendix 12.

Theorem 4

(Weak Convergence) Under Assumptions (A1–A7) and selecting $h=o\left(m^{-1/5}\right)$,

$$\begin{array}{l}\hfill \sqrt{mh}\left({\widehat{\mathrm{\Lambda }}}_{DE}(·)-{\mathrm{\Lambda }}_{DE}(·)\right)\stackrel{w}{\to }{𝒢𝒫}_{DE}(·),\end{array}$$ (29)

where

$$\begin{array}{ll}\hfill {𝒢𝒫}_{DE}(·)& =-{\int }_0^{·}{\int }_0^{u^{-}}{𝒢𝒫}_{{\int }_0^{t_1}{\omega }_1(s|u,0)\mathrm{d}s}\left(\frac{\partial }{\partial t_1}{\lambda }_1(u|t_1,1)\right.\\ \hfill & \left.-\frac{\partial }{\partial t_1}{\lambda }_1(u|t_1,0)\right)\mathrm{d}{t}_1\mathrm{d}u\\ \hfill & +{\int }_0^{·}{\int }_0^{u^{-}}{𝒢𝒫}_{\mathrm{d}{\mathrm{\Lambda }}_1(u|t_1,1)-\mathrm{d}{\mathrm{\Lambda }}_1(u|t_1,0)}{\omega }_1(t_1|u,0)\mathrm{d}{t}_1\end{array}$$ (30)

is a zero‐mean Gaussian process with variance function ${\sigma }_{DE}^2(·)$, as $m\to \infty$.

Following the similar development as Theorem 4, we derive ${𝒢𝒫}_{{\omega }_1(·|t,1)-{\omega }_1(·|t,0)}$ and ${𝒢𝒫}_{\mathrm{d}{\mathrm{\Lambda }}_1(t|·,1)}$, which can be found in Lemmas W.42 and W.39, respectively, in the Supporting Information. Applying the functional delta method, we prove Theorem 5 in Web Appendix 13.

Theorem 5

(Weak Convergence) Under Assumptions (A1–A7) and selecting $h=o\left(m^{-1/5}\right)$,

$$\begin{array}{l}\hfill \sqrt{mh}\left({\widehat{\mathrm{\Lambda }}}_{IE}(·)-{\mathrm{\Lambda }}_{IE}(·)\right)\stackrel{w}{\to }{𝒢𝒫}_{IE}(·),\end{array}$$ (31)

where

$$\begin{array}{ll}{\mathcal{GP}}_{IE}(·)& =-{\int }_0^{·}{\int }_0^{u^{-}}{\mathcal{GP}}_{{\int }_0^{t_1}{\omega }_1(s|u,1)\mathrm{d}s-{\int }_0^{t_1}{\omega }_1(s|u,0)\mathrm{d}s}\\ \hspace{0.17em}& ·\frac{\partial }{\partial t_1}{\lambda }_1(u|t_1,1)\mathrm{d}{t}_1\mathrm{d}u\\ \hspace{0.17em}& +{\int }_0^{·}{\int }_0^{u^{-}}{\mathcal{GP}}_{\mathrm{d}{\mathrm{\Lambda }}_1(u|t_1,1)}\left({\omega }_1(t_1|u,1)-{\omega }_1(t_1|u,0)\right)\mathrm{d}{t}_1\end{array}$$ (32)

is a zero‐mean Gaussian process with variance function ${\sigma }_{IE}^2(·)$, as $m\to \infty$.

In practice, the variance functions, ${\sigma }_{DE}^2(·)$ and ${\sigma }_{IE}^2(·)$, are estimated using the bootstrap method [30]. Extensions of the asymptotic results to the survival‐scale causal effect measures are provided in Web Appendix 14. In particular, we establish the unbiasedness and consistency of ${\widehat{\mathrm{\Delta }}}_{DE}(t)$ and ${\widehat{\mathrm{\Delta }}}_{IE}(t)$, and briefly discuss the asymptotic properties of ${\widehat{\rho }}_{DE}(t)$ and ${\widehat{\rho }}_{IE}(t)$. Consistency follows from the continuous mapping theorem, whereas weak convergence follows from the functional delta method.

4. Numerical Studies

4.1. Simulation

To evaluate the performance of the proposed method, we conduct two simulation scenarios. The event times $T_1$ and $T_2$ in both scenarios are generated according to the following specifications:

$$\begin{array}{ll}\hfill {\lambda }_{01}(t|Z)& ={\lambda }_1{t}^2{e}^{{\beta }_{1}Z},\end{array}$$ (33)

$$\begin{array}{ll}\hfill {\lambda }_{02}(t|Z)& ={\lambda }_2{t}^2{e}^{{\beta }_{2}Z},\end{array}$$ (34)

where the exposure variable $Z$ is independently drawn from a Bernoulli distribution with $P(Z=1)=0.5$. We retain the original realization $t_2$ of $T_2$ and set $T_1=\infty$ if $T_2\le t_1$ (Assumption (A5)), where $t_1$ denotes the realization of $T_1$, indicating that the intermediate event does not occur before the primary event. Otherwise, we retain the original value of $t_1$ and regenerate $T_2$ from the cumulative hazard function:

$$\begin{array}{l}\hfill {\int }_0^{t_1}{\lambda }_{02}(s|Z)\mathrm{d}s+{\int }_{t_1}^t{\lambda }_{012}(s|t_1,Z)\mathrm{d}s,\end{array}$$ (35)

where ${\lambda }_{012}(t|t_1,Z)={\lambda }_3{t}^2{e}^{{\beta }_{3}Z+{\beta }_4{t}_1^{-4}}$. The censoring time $C$ is independently generated from an exponential distribution with rate 0.1. We consider two scenarios in our simulation study:

Scenario 1 (Markov): The parameter values are set as $({\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4,{\lambda }_1,{\lambda }_2,{\lambda }_3)=(-1,-2,-1,0,1,1,1.5).$ Under this setting, the 25th, 50th, and 75th percentiles of the primary event times are approximately 1.00, 1.44, and 1.93, respectively.

Scenario 2 (non‐Markov): The parameters are set as $({\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4,{\lambda }_1,{\lambda }_2,{\lambda }_3)=(-1,-1,-1,-2,1,1,1.5).$ In this scenario, the corresponding 25th, 50th, and 75th percentiles are approximately 0.99, 1.54, and 2.22, respectively.

Each scenario is replicated 100 times with a sample size of $m=1000$. For each estimator, we compute the average curve across replications and construct the 95% pointwise confidence intervals using the corresponding standard errors. The proposed non‐Markov‐based method is evaluated against the Markov‐based approach of Huang [21], using hazard difference metrics, and the semi‐Markov approach of Valeri et al. [23], using survival difference metrics.

We use the Gaussian kernel function for all kernel‐based estimations. In this section, we present results using two bandwidth choices: $h=m^{-1/4}$ and $h=m^{-1/5}$, and Web Appendix 16 provides additional results based on bandwidths $h=m^{-1/2}$, $h=m^{-1/3}$, $h=m^{-1/4}$, and $h=m^{-1/5}$.

Web Figures 3 and 4 in the Supporting Information present the survival effect estimators from the three approaches in the Markov setting (Scenario 1), where all methods are correctly specified. As expected, the three approaches provide approximately unbiased results under this setting. Our method is capable of handling the Markov condition. Although our approach exhibits relatively larger variance, this increased variability is anticipated due to its non‐parametric nature with the use of kernel smoothing.

FIGURE 3.

Direct and indirect effects in the simulation study with $m=1000$ in the non‐Markov setting. The corresponding bandwidths are $h=m^{-1/4}\approx 0.178$ and $h=m^{-1/5}\approx 0.251$.

FIGURE 4.

Survival direct and indirect effects in the simulation study with $m=1000$ in the non‐Markov setting.

The results in Figures 3 (hazard difference) and 4 (survival difference) show that the average estimated direct and indirect effects in a non‐Markov setting (Scenario 2). All three methods closely align with their true values before the 50th percentile of the primary event times (i.e., for $t<1.54$). However, as time progresses, Huang's method tends to introduce substantial bias due to its dependence on the Markov assumption, which ignores the influence of the mediator's occurrence time on the outcome risk. The semi‐Markov estimator by Valeri et al. [23] may also lead to notable bias due to its misspecification of the transition hazard function ${\lambda }_{012}(t|t_1,Z)$. Therefore, the results demonstrate that our method performs reliably under both Markov and non‐Markov conditions, showing its robustness relative to the other two approaches.

Next, we examine the empirical coverage probabilities of our proposed method under the non‐Markov setting (Scenario 2). The variances of the direct and indirect effects are estimated using the bootstrap method with 1000 resamples. For each sample size $m$, 1000 replicated data sets are generated to compute the variances and assess the empirical coverage of the 95% pointwise confidence intervals.

Figure 5 demonstrates that the proposed method achieves appropriate coverage rates in regions where observations of both the mediator and outcome are relatively rich, particularly around the 25th and 50th percentiles of the primary event times (indicated by the first and second vertical dashed lines). In contrast, coverage tends to decline in both the early and late time regions, such as those before $t=0.5$ and after $t=1.5$, where data (either the mediator's or the outcome's occurrence) are sparse and estimation becomes more challenging. Specifically, coverage drops noticeably below 95% before time 0.5, likely due to limited information in this early period. For $t>1.5$, the performance of the estimators varies by bandwidth: the bandwidth $h=m^{-1/3}$ maintains relatively stable coverage (see Web Figures 9 and 10 in the Supporting Information), while $h=m^{-1/5}$ exhibits a substantial drop, with coverage falling below 0.6. This decline is likely attributable not only to data sparsity in the tail region but also to the fact that $h=m^{-1/5}$ fails to satisfy the condition $m{h}^5\to 0$, which is required for weak convergence to a zero‐mean Gaussian process, as established in Theorems 4 and 5.

FIGURE 5.

Empirical coverage probabilities of the 95% pointwise bootstrap confidence intervals under the non‐Markov setting.

4.2. Data Application

The REVEAL study is a community‐based, prospective cohort study conducted in Taiwan. A total of 23,820 participants enrolled between 1991 and 1992 [3] were followed through 2017. Chronic hepatitis C infection was defined as the presence of hepatitis C antibodies (anti‐HCV) at baseline. Information on septicemia incidence and mortality was obtained from the National Health Insurance Research Database (NHIRD) and the Death Certification Profile, respectively.

To reduce potential confounding, we restricted the analysis to male participants without a history of alcohol consumption ($m$ = 7,262), as both sex and alcohol consumption may be associated with septicemia and mortality [3, 31]. Although sequential ignorability is not empirically testable, this restriction makes the assumption more clinically plausible; however, unmeasured confounding cannot be ruled out.

In the data set, hepatitis C is treated as the exposure variable, septicemia as the mediator event, and death as the primary outcome. To assess the presence of non‐Markovian characteristics, we apply the global logrank‐based test proposed by Titman and Putter [9] (detailed in Web Appendix 19 of the Supporting Information.) The resulting $p$‐values is less than 0.01, providing strong evidence against the Markov assumption. In our analysis, we compare the results of three approaches. For all estimators, 95% confidence intervals are constructed using the bootstrap method. The primary objective is to assess whether the effect of hepatitis C on mortality is mediated through septicemia (indirect effects) or whether it operates independently (direct effects). By quantifying both effects, we aim to gain a deeper understanding of the causal pathways linking hepatitis C infection to mortality and to compare the aforementioned three methods under a non‐Markov setting.

To determine the optimal bandwidth, we employ a 5‐fold cross‐validation based on the counterfactual log‐likelihood function of the primary event. Specifically, the optimal bandwidth is defined as

$$\begin{array}{ll}\hfill h_{z_a,z_b}^{\text{opt}}& =\arg \underset{h\in \mathscr{H}}{\max }\sum _{k=1}^5\sum _{i\in {𝒟}_k}\log \left[e^{-{\widehat{\mathrm{\Lambda }}}_{h,-k}(T_{2i}^{\ast -},z_a,z_b)}\right.\left.-e^{-{\widehat{\mathrm{\Lambda }}}_{h,-k}(T_{2i}^{\ast },z_a,z_b)}+10^{-8}\right],\end{array}$$ (36)

where ${𝒟}_k$ denotes the $k$‐th validation set, and ${\widehat{\mathrm{\Lambda }}}_{h,-k}$ represents the estimated counterfactual cumulative hazard function computed excluding the $k$th validation set. The set of candidate bandwidths considered is $\mathscr{H}=\{h=C{m}^{-1/K}|C\in \{0.2,0.5,0.7\},K\in \{2.5,3,3.5,4,4.5\}\}$. The optimal bandwidth for this data set is 0.097.

As shown in the upper‐left panel of Figure 6, the 95% confidence interval for the natural direct effect does not include zero under both approaches after approximately the fourth year, suggesting evidence of a direct effect. The estimates from the proposed method initially closely align with those obtained using the method of Huang [21]. However, as time progresses, the difference between the two estimators becomes more apparent. The positive direct‐effect estimates suggest that the effect of hepatitis C on mortality may operate directly. The bandwidth sensitivity analysis in Web Figure 11 further supports the robustness of this conclusion, as the direct‐effect estimates remain qualitatively similar across the selected bandwidths.

FIGURE 6.

Direct and indirect effects of hepatitis C on mortality mediated through septicemia.

Regarding the natural indirect effect, shown in the lower‐left panel of Figure 6, the two methods show noticeable differences at the beginning of follow‐up. Under the proposed method, the confidence interval includes zero for most of the first 20 years, whereas the method of Huang [21] suggests statistical significance after approximately six years. However, the bandwidth sensitivity analysis in Web Figure 12 shows that the indirect effect estimator is more sensitive to the choice of bandwidth. This sensitivity is likely related to the sparsity of intermediate events. The data provide relatively robust evidence that the indirect effect of hepatitis C on mortality through septicemia is close to zero before approximately 15 years; however, the indirect effect during later follow‐up remains uncertain.

Additionally, we compared the survival causal effects estimated by the three approaches: the method by Valeri et al. [23], the method by Huang [21], and our proposed method. For the direct effects (upper‐right panel), the confidence interval of the semi‐Markov estimator proposed by Valeri et al. [23] is almost entirely contained within the confidence intervals of both our method and that of Huang [21]. Valeri's semi‐parametric estimator exhibits smaller variance and produces a smoother curve compared with the other two methods. For the indirect effects (lower‐right panel), the curve generated by Valeri's method lies below those of our method and Huang's estimator, indicating statistically significant indirect effects. Although the method of Valeri et al. [23] accounts for the non‐Markov condition, its results still diverge from ours for both the direct and indirect effects. This discrepancy may be attributable to model misspecification.

In real data applications, it is necessary to assess whether the Markov assumption holds at each time point within the study period. At certain times, the assumption may hold, whereas at others, it may not. Our proposed method is applicable to both Markov and non‐Markov data structures and, in particular, performs better when the Markov assumption is violated. This advantage eliminates the need to verify the assumption explicitly, allowing for direct application to survival data that may exhibit non‐Markov dynamics.

All computations were performed on a MacBook Pro equipped with an Apple M3 Pro chip, 11 CPU cores (5 performance cores and 6 efficiency cores), and 18 GB of memory. Using a time grid of 0.1, the proposed method required 14.36 min to complete the analysis, whereas the Markov‐based method of Huang [21] required 3.90 min. In contrast, the multistate approach of Valeri et al. [23] required approximately 582 min.

In summary, our analyses did not support the mechanism of hepatitis C‐related mortality that is mediated through septicemia during the first 15 years of follow‐up, particularly when accounting for the non‐Markov nature of the disease process.

5. Discussion

In this work, we propose a non‐parametric approach to estimate causal direct and indirect effects in a non‐Markov illness‐death model and implement it in our publicly available R package nmSurvMed. Our mediation analysis accounts for the full history of mediator occurrences, thereby better capturing the realistic nature of the data compared to the methods of Huang [21]. To relax the Markov assumption in illness‐death models, recent studies by Valeri et al. [23] and Breum et al. [20] adopt proportional hazards models [32] to specify transition hazards, incorporating the timing of the intermediate event as a covariate in the regression component. As illustrated in the numerical simulation and data application, such a model‐based approach to the non‐Markov condition is subject to model misspecification. In other words, the estimation would lead to a severe bias when the non‐Markov mechanism is incorrectly modeled. Moreover, both simulation and the REVEAL data analyses show that our estimated effects may initially resemble those obtained using the methods of Huang [21] (which assumes a Markov process) and Valeri et al. [23] (which assumes a semi‐Markov process); however, as time progresses, substantial differences emerge. The late divergence reflects the failure and inadequacy to account for the accumulating influence of long‐term mediator history by Huang [21] and Valeri et al. [23], respectively.

In the standard illness‐death model, all individuals are assumed to start in state 0 at time 0. However, Assumption (A7) allows individuals to begin in state 1 or state 2 at time 0. This relaxation introduces a trade‐off: it ensures that the conditioning event in $\mathrm{d}{\mathrm{\Lambda }}_1(t|t_1,z_a)$ is non‐empty. Without this assumption, the probability of the conditional event may be zero near time zero, rendering theoretical derivations and estimation procedures difficult or undefined at early time points.

In addition, the semi‐Markov estimates were implemented using the R package provided by Valeri et al. [23], which outputs effect estimates on the survival difference scale. Therefore, comparisons with this method are restricted to settings where the estimands are defined on the same scale‐specifically, the survival difference scale‐as shown in Figure 4 and Web Figures 3, 4, 7 and 8 of the Supporting Information.

Our work has several potential future directions. While the proposed non‐parametric method offers robustness against model misspecification, it may sacrifice estimation efficiency when confounders must be adjusted via stratification. This reflects an inherent trade‐off between robustness and statistical efficiency. A promising direction for future research is to extend our method to accommodate regression models that more flexibly incorporate covariates, thereby improving efficiency while retaining robustness. It is also possible to conduct a hypothesis test by comparing the proposed non‐Markov estimates with the Markov‐based estimates of Huang [21]. Such a testing procedure requires additional methodological development and theoretical justification, which warrants further research. Another natural extension of the current work is to consider two or more time‐to‐event mediators within a non‐Markov multi‐state modeling framework. One may aim to formalize causal pathways involving multiple intermediate events and define corresponding path‐specific effects [33], building upon recent developments in causal mediation analysis for multistate processes.

Funding

This work was supported by the National Science and Technology Council (Grant No. 113‐2118‐M‐001‐013‐MY3) and Academia Sinica (Grant No. AS‐IV‐114‐M05).

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Data S1: The supporting information (.pdf file) includes the mathematical proofs of the main results and additional simulation studies. Web Figure 1: Direct effects in the simulation study with $m=1000$ in the Markov setting. The bandwidth choices considered are $h=m^{-1/2}\approx 0.032$, $h=m^{-1/3}\approx 0.1$, $h=m^{-1/4}\approx 0.178$, and $h=m^{-1/5}\approx 0.251$. Web Figure 2: Indirect effects in the simulation study with $m=1000$ in the Markov setting. Web Figure 3: Survival direct effects in the simulation study with $m=1000$ in the Markov setting. Web Figure 4: Survival indirect effects in the simulation study with $m=1000$ in the Markov setting. Web Figure 5: Direct effects in the simulation study with $m=1000$ in the non‐Markov setting. Web Figure 6: Indirect effects in the simulation study with $m=1000$ in the non‐Markov setting. Web Figure 7: Survival direct effects in the simulation study with $m=1000$ in the non‐Markov setting. Web Figure 8: Survival indirect effects in the simulation study with $m=1000$ in the non‐Markov setting. Web Figure 9: Empirical coverage probabilities of the 95% pointwise bootstrap confidence intervals under the non‐Markov setting. Web Figure 10: Empirical coverage probabilities of the 95% pointwise bootstrap confidence intervals under the non‐Markov setting. Web Figure 11: Bandwidth sensitivity analysis for the natural direct effect in the REVEAL data analysis. Each panel corresponds to one candidate bandwidth. The solid curve represents the estimated natural direct effect, and the shaded region represents the 95% bootstrap confidence interval. The panel labeled CV corresponds to the bandwidth selected by the cross‐validation criterion. Web Figure 12: Bandwidth sensitivity analysis for the natural indirect effect in the REVEAL data analysis. Each panel corresponds to one candidate bandwidth. The solid curve represents the estimated natural indirect effect, and the shaded region represents the 95% bootstrap confidence interval. The panel labeled CV corresponds to the bandwidth selected by the cross‐validation criterion. Web Figure 13: Local and global tests for the Markov assumption in the simulation study. Web Figure 14: Local and global tests for the Markov assumption in the REVEAL data analysis.

Appendix A.

Statistical Assumptions

Here, we introduce the statistical assumptions required to establish the proposed theorems using the kernel smoothing approach under the survival outcome framework. These assumptions are listed as follows:

• A1.The kernel function $K(·)$ is symmetric, bounded, Lipschitz continuous, and of bounded variation. It satisfies $\int K(u)\mathrm{d}u=1$ and $\int |K(u){|}^p\mathrm{d}u<\infty$ for $p=1,2,3$, and has a finite second moment.
• A2.The bandwidth $h$ satisfies $h\to 0$, $mh\to \infty$, and $m{h}^5\to 0$ as $m\to \infty$. Additionally, for every $\gamma >0$, ${\sum }_{m=1}^{\infty }{e}^{-\gamma m{h}^2}<\infty$.
• A3.The densities $f_{T_1^{\ast },U}(·,·)$, $f_{T_1^{\ast }|U}(·|·)$ and $f_{U|T_1^{\ast }}(·|·)$ are bounded, Lipschitz continuous, square‐integrable, and twice continuously differentiable, with square‐integrable second derivatives.
• A4.Within the study period $[0,\zeta ]$, we assume that $P(C\ge t|\{{\tilde{N}}_1(s),{\tilde{N}}_2(s),0\le s\le \zeta \},W,Z)=P(C\ge t|W,Z),0\le t\le \zeta .$
• A5.We assume that $T_1=\infty$ if the subject dies before experiencing the intermediate event, that is, $T_2<T_1$.
• A6.The supports of the survival time $T_2$ and the potential censoring time $C$ include the study period $[0,\zeta ]$.
• A7.The conditional probability of the exposure factor $Z$, the intermediate event process ${Ñ}_1(t_1)$, and the death event process ${Ñ}_2(t)$ given the confounder $W$ satisfies the positivity assumption, that is, $P(Z=z,{Ñ}_1(t_1)=n_1,{Ñ}_2(t^{-})=n_2|W)>0$ for $0\le t_1\le t\le \zeta$.

Assumptions (A1) and (A2) are widely accepted for controlling the convergence rates of the bias and variance of the smoothing estimator. Assumption (A3) provides that the density function admits at least a second‐order Taylor expansion. Assumption (A4) states that, conditional on $Z$ and $W$, the censoring time is independent of both $T_1$ and $T_2$. Assumption (A5) is a regularity condition derived from the illness‐death model framework of Xu et al. [14], where the joint probability density function $f_{T_1,T_2}(t_1,t_2)$ of the fully observable data satisfies $0\le t_1\le t_2$. Assumption (A6) ensures that we can analyze the joint probability $(T_1,T_2)$ within the case study period $[0,\zeta ]$. Assumption (A7) ensures that the set $(Z,\mathrm{d}{Ñ}_1(t_1),{Ñ}_2(t))$ is possible in all stratifications.

Proof of Theorem 1

Define $\eta (n_1({\underset{\_}{t}}^{-}))=I\left(t_1<t\right),$ which indicates whether the intermediate event has occurred before time $t$. Based on this distinction, we decompose the jump of counterfactual cumulative hazard

$$\begin{array}{ll}\hspace{0.17em}& \mathrm{d}\mathrm{\Lambda }(t;z_a,z_b)\\ \hspace{0.17em}& =P\left(\mathrm{d}{\tilde{N}}_2\left(t;z_a,{\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)\right)=1|{\tilde{N}}_2\left(t^{-};z_a,{\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)\right)=0\right)\\ \hspace{0.17em}& =\sum _{\eta (n_1({\underset{\_}{t}}^{-}))}P\left(\mathrm{d}{\tilde{N}}_2\left(t;z_a,n_1({\underset{\_}{t}}^{-})\right)=1|{\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)=n_1({\underset{\_}{t}}^{-}),\right.\\ \hspace{0.17em}& \left.{\tilde{N}}_2\left(t^{-};z_a,n_1({\underset{\_}{t}}^{-})\right)=0\right)\\ \hspace{0.17em}& ·P\left({\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)=n_1({\underset{\_}{t}}^{-})|{\tilde{N}}_2\left(t^{-};z_a,n_1({\underset{\_}{t}}^{-})\right)=0\right).\end{array}$$

Applying ignorability (B1), sequential ignorability I (B2), and consistency (B4), we obtain

$$\begin{array}{ll}\hspace{0.17em}& \mathrm{d}\mathrm{\Lambda }(t;z_a,z_b)\\ \hspace{0.17em}& =\sum _{\eta (n_1({\underset{\_}{t}}^{-}))}P\left(\mathrm{d}{\tilde{N}}_2\left(t;z_a,n_1({\underset{\_}{t}}^{-})\right)=1|Z=z_a,{\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)\right.\\ \hspace{0.17em}& \left.=n_1({\underset{\_}{t}}^{-}),{\tilde{N}}_2\left(t^{-};z_a,n_1({\underset{\_}{t}}^{-})\right)=0\right)\\ \hspace{0.17em}& ·P\left({\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)=n_1({\underset{\_}{t}}^{-})|Z=z_b,{\tilde{N}}_2\left(t^{-};z_a,n_1({\underset{\_}{t}}^{-})\right)=0\right)\\ \hspace{0.17em}& =\sum _{\eta (n_1({\underset{\_}{t}}^{-}))}P\left(\mathrm{d}{\tilde{N}}_2\left(t;z_a,n_1({\underset{\_}{t}}^{-})\right)=1|Z=z_a,{\tilde{N}}_1({\underset{\_}{t}}^{-};z_a)\right.\\ \hspace{0.17em}& \left.=n_1({\underset{\_}{t}}^{-}),{\tilde{N}}_2\left(t^{-};z_a,n_1({\underset{\_}{t}}^{-})\right)=0\right)\\ \hspace{0.17em}& ·P\left({\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)=n_1({\underset{\_}{t}}^{-})|Z=z_b,{\tilde{N}}_2\left(t^{-};z_a,n_1({\underset{\_}{t}}^{-})\right)=0\right)\\ \hspace{0.17em}& =\sum _{\eta (n_1({\underset{\_}{t}}^{-}))}P\left(\mathrm{d}{\tilde{N}}_2(t)=1|Z=z_a,{\tilde{N}}_1({\underset{\_}{t}}^{-})=n_1({\underset{\_}{t}}^{-}),\right.\\ \hspace{0.17em}& \left.{\tilde{N}}_2\left(t^{-}\right)=0\right)\\ \hspace{0.17em}& ·P\left({\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)=n_1({\underset{\_}{t}}^{-})|Z=z_b,{\tilde{N}}_2\left(t^{-};z_a,n_1({\underset{\_}{t}}^{-})\right)=0\right).\end{array}$$

Next, applying sequential ignorability II (B3) and consistency (B4), we further identify the counterfactual hazard in a counting‐process form:

$$\begin{array}{ll}\hspace{0.17em}& \mathrm{d}\mathrm{\Lambda }(t;z_a,z_b)\\ \hspace{0.17em}& =\sum _{\eta (n_1({\underset{\_}{t}}^{-}))}P\left(\mathrm{d}{\tilde{N}}_2(t)=1|Z=z_a,{\tilde{N}}_1({\underset{\_}{t}}^{-})=n_1({\underset{\_}{t}}^{-}),\right.\\ \hspace{0.17em}& \left.{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& ·P\left({\tilde{N}}_1({\underset{\_}{t}}^{-};z_b)=n_1({\underset{\_}{t}}^{-})|Z=z_b,{\tilde{N}}_2\left(t^{-};z_b,n_1({\underset{\_}{t}}^{-})\right)=0\right)\\ \hspace{0.17em}& =\sum _{\eta (n_1({\underset{\_}{t}}^{-}))}P\left(\mathrm{d}{\tilde{N}}_2(t)=1|Z=z_a,{\tilde{N}}_1({\underset{\_}{t}}^{-})=n_1({\underset{\_}{t}}^{-}),\right.\\ \hspace{0.17em}& \left.{\tilde{N}}_2(t^{-})=0\right)P\left({\tilde{N}}_1({\underset{\_}{t}}^{-})=n_1({\underset{\_}{t}}^{-})|Z=z_b,{\tilde{N}}_2\left(t^{-}\right)=0\right).\end{array}$$

We discretize the interval $[0,t)$ into small subintervals, each representing a potential jump time of the intermediate event process. By letting the mesh size go to zero, this construction captures all possible jump times of the intermediate event in $[0,t)$. Formally,

$$\begin{array}{ll}\hspace{0.17em}& \mathrm{d}\mathrm{\Lambda }(t;z_a,z_b)\\ \hspace{0.17em}& =\underset{n\to \infty }{\lim }\sum _{i=1}^{n}P\left({\tilde{N}}_2(t)=1|Z=z_a,\mathrm{d}{\tilde{N}}_1(t_i)=1,\right.\\ \hspace{0.17em}& \left.{\tilde{N}}_2(t^{-})=0\right)P\left(\mathrm{d}{\tilde{N}}_1(t_i)=1|Z=z_b,{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& +P\left({\tilde{N}}_2(t)=1|Z=z_a,{\tilde{N}}_1(t^{-})=0,{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& ·P\left({\tilde{N}}_1(t^{-})=0|Z=z_b,{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& =\underset{n\to \infty }{\lim }\sum _{i=1}^{n}P\left({\tilde{N}}_2(t)=1|Z=z_a,T_1\in [t_i,t_{i+1}),\right.\\ \hspace{0.17em}& \left.{\tilde{N}}_2(t^{-})=0\right)P\left(T_1\in [t_i,t_{i+1})|Z=z_b,{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& +P\left({\tilde{N}}_2(t)=1|Z=z_a,{\tilde{N}}_1(t^{-})=0,{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& ·P\left({\tilde{N}}_1(t^{-})=0|Z=z_b,{\tilde{N}}_2(t^{-})=0\right),\end{array}$$

where $0=t_1<\cdots <t_{n+1}=t.$ By the mean value theorem, we obtain

$$\begin{array}{ll}\hspace{0.17em}& \mathrm{d}\mathrm{\Lambda }(t;z_a,z_b)\\ \hspace{0.17em}& =\underset{n\to \infty }{\lim }\sum _{i=1}^n\frac{P\left({\tilde{N}}_2(t)=1,Z=z_a,T_1=t_i^{\ast },{\tilde{N}}_2(t^{-})=0\right)}{P\left(Z=z_a,T_1=t_i^{\ast },{\tilde{N}}_2(t^{-})=0\right)}\\ \hspace{0.17em}& ·P\left(T_1=t_i^{\ast }|Z=z_a,{\tilde{N}}_2(t^{-})=0\right)\left(t_{i+1}-t_i\right)\\ \hspace{0.17em}& +P\left({\tilde{N}}_2(t)=1|Z=z_a,{\tilde{N}}_1(t^{-})=0,{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& ·P\left({\tilde{N}}_1(t^{-})=0|Z=z_b,{\tilde{N}}_2(t^{-})=0\right),\end{array}$$

where $t_i\le t_i^{\ast }<t_{i+1}.$ We reformulate the expression and apply a Riemann‐sum argument to obtain the identification formula:

$$\begin{array}{ll}\hspace{0.17em}& \mathrm{d}\mathrm{\Lambda }(t;z_a,z_b)\\ \hspace{0.17em}& =\underset{n\to \infty }{\lim }\sum _{i=1}^n\frac{P\left({\tilde{N}}_2(t)=1,Z=z_a,T_1=t_i^{\ast },{\tilde{N}}_2(t^{-})=0\right)}{P\left(Z=z_a,T_1=t_i^{\ast },{\tilde{N}}_2(t^{-})=0\right)}\\ \hspace{0.17em}& ·\frac{\mathrm{d}}{\mathrm{d}{t}_1}P{\left.\left(T_1\le t_1|Z=z_b,{\tilde{N}}_2(t^{-})=0\right)\right|}_{t_1=t_i^{\ast }}\left(t_{i+1}-t_i\right)\\ \hspace{0.17em}& +P\left({\tilde{N}}_2(t)=1|Z=z_a,{\tilde{N}}_1(t^{-})=0,{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& ·P\left({\tilde{N}}_1(t^{-})=0|Z=z_b,{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& ={\int }_0^{t^{-}}\frac{P\left({\tilde{N}}_2(t)=1,Z=z_a,\mathrm{d}{\tilde{N}}_1(t_1)=1,{\tilde{N}}_2(t^{-})=0\right)}{P\left(Z=z_a,\mathrm{d}{\tilde{N}}_1(t_1)=1,{\tilde{N}}_2(t^{-})=0\right)}\\ \hspace{0.17em}& ·\frac{\mathrm{d}}{\mathrm{d}{t}_1}P\left(T_1\le t_1|Z=z_b,{\tilde{N}}_2(t^{-})=0\right)\mathrm{d}{t}_1\\ \hspace{0.17em}& +P\left({\tilde{N}}_2(t)=1|Z=z_a,{\tilde{N}}_1(t^{-})=0,{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& ·P\left({\tilde{N}}_1(t^{-})=0|Z=z_b,{\tilde{N}}_2(t^{-})=0\right)\\ \hspace{0.17em}& ={\omega }_0(t|z_b)\mathrm{d}{\mathrm{\Lambda }}_0(t|z_a)+{\int }_0^{t^{-}}{\omega }_1(t_1|t,z_b)\mathrm{d}{\mathrm{\Lambda }}_1(t|t_1,z_a)\mathrm{d}{t}_1.\end{array}$$

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.