Assessing the Benefits and Burdens of Preventive Interventions

Abstract

Cancer prevention is recognized as a key strategy for reducing disease incidence, mortality, and the overall burden on individuals and society. However, determining when to begin preventive interventions presents a significant challenge: starting too early may lead to more interventions and increased lifetime burdens due to repeated administrations, while delaying may miss opportunities to prevent cancer. Evidence-based recommendations require a benefit-burden analysis that weighs life-years gained against the burden of interventions. With the growing availability of large-scale observational data, there is now an opportunity to empirically evaluate these trade-offs. In this paper, we propose a causal framework for assessing the benefit and burden of cancer prevention, using an illness-death model with semi-competing risks. Extensive simulations demonstrate that the proposed estimators are unbiased, with robust inference across realistic scenarios. We apply this approach a benefit-burden analysis of the preventive screening for colorectal cancer, utilizing data from the large-scale Women’s Health Initiative. Our findings suggest that initiating screening at age 50 years achieves the highest life-year gains with an acceptable incremental burden-to-benefit ratio compared to no screening, contributing valuable real-world evidence to the field of preventive cancer interventions.

1 |. INTRODUCTION

Prevention is widely recommended as a key strategy for reducing cancer incidence and is an integral component of cancer control efforts¹. Unlike early-detection cancer screening, which focuses on detecting cancer at an early stage, preventive interventions target cancer-free individuals by identifying and removing precancerous lesions. Since these interventions are applied broadly to cancer-free populations, they often involve large numbers of individuals, potentially resulting in substantial burden per cancer case prevented. To justify their implementation, it is essential to evaluate both their benefits and burdens². A common strategy to reduce these burdens is to target interventions to subgroups at higher risk, such as older individuals. This is well illustrated by colorectal cancer (CRC) prevention. CRC is the third most common cancer and the second leading cause of cancer-related mortality worldwide³. Yet, it is one of the most preventable cancers due to the effectiveness of endoscopic procedures, which can detect and remove precancerous lesions^4,5. Despite its preventive potential, determining the appropriate age to begin endoscopy is challenging. These procedures are typically recommended to be repeated every 10 years; starting too early increases cumulative costs and starting too late may miss opportunities to prevent cancer. Thus, a rigorous evaluation the trade-off between benefit and burden is essential to inform evidence-based recommendations on when to begin endoscopy. Microsimulation modeling has widely been used to evaluate such trade-offs. These models simulate individual life histories to capture the natural history of adenoma initiation, progression to preclinical cancer, clinical detection, and death, while explicitly modeling screening, surveillance, and treatment processes^5,6,7,8,9. Microsimulation models offer flexibility to incorporate biological and behavioral heterogeneity. However, they rely on unobserved assumptions (e.g., adenoma dwell time, transition rates), which can be difficult to validate, leading to uncertainties in evaluation of benefit and burden.

This paper aims to provide a framework for empirically evaluating health gains relative to the burdens of initiating a preventive procedure at various ages using observational data. Quantifying the benefits and burdens of cancer prevention involves metrics such as life-years gained for the benefit and cumulative lifetime interventions to represent the burden. However, diseases like cancer are relatively infrequent, occur later in life, and have a long incubation period. Reliable empirical estimates of these metrics requires large cohorts with extended follow-up. The Women’s Health Initiative (WHI)¹⁰, one of the largest disease prevention programs in the U.S., provides an ideal data set for evaluating prevention strategies. Established in 1991, the WHI has followed over 100,000 women aged 50 to 79 years until today, collecting extensive sociodemographic and epidemiologic factors, along with records of colorectal endoscopic status. This large sample size and long follow-up provide a unique opportunity to assess how the timing of preventive interventions influences lifetime CRC risk and mortality.

Despite its strengths, observational studies like the WHI present methodological challenges, including censored outcomes, competing events and confounding. Existing statistical approaches, such as the Cox proportional hazards regression model for restricted mean survival time¹¹, a propensity score approach for cost estimation¹², and doubly robust estimators for evaluating both cost and effectiveness¹³, address some of these issues. However, these studies typically focus on binary intervention, whereas our research examines the initiation of preventive intervention at various ages. Unlike traditional models for continuous interventions, such as the dose-response functions¹⁴, preventive interventions pose additional complexities. Timing is subject to censoring, as individuals may die or develop the disease before undergoing the prevention. Moreover, these interventions aim to delay or prevent disease development, extending the disease-free period and ultimately reducing mortality. Since diseases like CRC are infrequent, relying solely on overall survival may fail to fully capture the targeted impact of these interventions. To address this, it is crucial to evaluate their effects not only on overall survival, but also on disease development using a multi-state modeling framework.

Intervention decisions such as when to begin rely on presumed causal connections and their anticipated outcomes¹⁵. Traditional methodologies for causal inference in healthcare decision-making are primarily derived from randomized trials^16,17. While randomized trials provide robust evidence, observational data provide valuable insights under real-world conditions. For determining when to initiate an intervention, large-scale observational studies like the WHI are particularly invaluable. They ensure adequate representation of participants across various intervention timings and capture the long-term effects on lowincidence, late-onset diseases like CRC. By combining observational data and causal inference methods, empirical evidence can be generated to inform decisions about the timing for intervention initiation.

Recent works have formulated causal frameworks in the presence of competing events (e.g. Young et al.¹⁸) or semi-competing risks (e.g., Gao et al.¹⁹ and Nevo and Gorfine²⁰) under time-independent interventions. Assessing the causal effect of time-dependent intervention on multiple events introduces additional complexity, particularly in formulating identifiability conditions given the time-dependent nature of intervention. Several studies have examined the effect of intervention initiation timing, for example, Hu et al.²¹ proposed an estimation procedure based on marginal structural models weighted by the inverse probability of intervention timing, and Prosepe et al.²² modeled treatment initiation as an intermediate state before the primary event. However, the endpoint of these studies were limited to time to a single event. Our work extends this line of research by developing a causal illness-death framework with multiple time-to-event outcomes while treating intervention as a time-dependent exposure and formulating identifiability assumptions with this framework to evaluate estimands.

In this paper, we perform a causal benefit-and-burden analysis to determine the earliest age for starting a preventive intervention within a illness-death modeling framework. We propose a unified measure that evaluates both benefit and burden. This measure includes well-established quantities, such as restricted mean survival time and quality-adjusted life years^23,24,25, while also capturing the burden of interventions, including the cumulative number of procedures over a lifetime. Specifically, this paper makes several key contributions. First, we introduce a comprehensive measure that captures both the benefits and burdens of preventive interventions. Second, we extend this measure to a illness-death modeling framework, providing novel insights into how benefits and burdens vary with the initial intervention age. This framework explicitly models pathways to death either directly or through disease development. For infrequent but serious diseases, cause-specific pathways provide direct evidence of how the intervention impacts the targeted disease. Third, we define the causal estimand of interest and outline the necessary assumptions associated with the illness-death model for censored interventions and outcomes. Lastly, we apply the proposed estimands and estimation procedures to WHI data, assessing the benefits and burdens of initiating colorectal endoscopy at different ages. This analysis provides real-world evidence to support preventive CRC intervention.

The remainder of this paper is structured as follows. Section 2 presents the unified measure for benefits and burdens, along with the causal inference framework and estimation procedures. Section 3 describes the analysis of the WHI data, evaluating the impact of the initial age for endoscopy in terms of (quality-adjusted) life years and intervention burdens. Section 4 presents an extensive simulation study to assess the performance of our proposed framework. Finally, Section 5 concludes with a summary of key contributions and potential future directions.

2 |. METHODOLOGY

2.1 |. A Unified Measure for Benefits and Burdens

Consider an illness-death model (Figure 1), in which individuals can be in the healthy or diseased state during follow-up. Let $K$ represent the number of states; under the illness-death model, $K=2$. Further, let $T$ and $D$ be the age at disease diagnosis and death, respectively. We define the benefit or burden at time $t$ by

$$M(t)=\sum _{k=1}^K{\int }_0^t{\tilde{Y}}_k(u)d{W}_k(u),$$ (1)

where ${\tilde{Y}}_k(\cdot )$ is an indicator function equal to 1 if an individual is in the $k$th state and alive (i.e., $D\ge u$) and 0 otherwise, and $W_k(\cdot )$ is a cumulative weight function associated with the $k$th state, $k=1,\dots ,K.M(t)$ encompasses many measures for quantifying benefits or burdens. Below we provide several examples of $M(t)$ and provide additional details in Section 2.4.

FIGURE 1.

An illness-death model

Example 1: Restricted Mean Survival Time (RMST).

The RMST is a widely used measure for quantifying the average survival experience up to a specific time point, $t^{24}$. Mathematically, it is defined as the area under the survival curve up to time $t$. In this case, there is only one primary event, death. By setting $W_k(u)=u,k=1,\dots ,K$, we have $E\{M(t)\}=E\left\{{\int }_0^t\sum _{k=1}^K{\tilde{Y}}_k(u)du\right\}={\int }_0^t\text{Pr}(D\ge u)du$, which represents the RMST. While $K=1$ could suffice for calculating RMST, we opt for $K=2$ to more effectively capture the impact of intervention on the disease process, beyond just overall mortality.

Example 2: Cause-Specific Mean Life Years Lost.

Complementary to the RMST, life years lost provides an informative decomposition of the years lost due to specific cause pathways leading to mortality up to time $t$²⁶. Define the expected life years lost $E\{L(t)\}=t-E\{M(t)\}$. Under the illness-death model, an individual may die before the disease occurs (path 0 → 2) or first develop the disease and then die (path 0 → 1 → 3). Setting $W_k(t)=t$ for $k=\mathrm{1,2}$, and letting $f_D(\cdot \mid T,X)$ denote the conditional probability density function of $D$ given $T$ and $X$, the mean life years lost can be decomposed as:

$$\begin{gathered} E\{L(t)\}={\int }_0^t{\int }_0^u{f}_D(v\mid T\ge v,X)\text{Pr}(T\ge v\mid X)\mathit{\text{dvdu}} \\ +{\int }_0^t{\int }_0^u{f}_D\left(v∣T<v,X\right)\text{Pr}\left(T<v∣X\right)\mathit{\text{dvdu}}. \end{gathered}$$

The first term represents the mean life years lost through the path 0 → 2, while the second term represents the mean life years lost through the path 0 → 1 → 3. This decomposition is particularly useful for evaluating intervention performance, when a disease is rare. For example, while CRC is a common cancer, its incidence remains relatively infrequent at the population level. Consequently, population-level gains in all cause survival may not fully reflect the effectiveness of endoscopy. By examining cause-specific mean life years lost, the effectiveness of interventions targeting CRC becomes clearer.

Example 3: Quality-Adjusted Life Years (QALY).

In Example 1, $E\{M(t)\}$ represents the expected number of life years up to time $t$ with the time origin set at birth. By modifying $W_k(u)$ to account for the desirability of a health state such as a quality or utility score in the $k$th state, $E\{M(t)\}$ becomes the expected QALY up to $t$. Similarly, the cause-specific quality-adjusted mean life years lost can be defined by incorporating these weights.

Example 4: Number of Interventions.

If $W_1(u)$ represents the cumulative number of interventions up to time $u$ and $W_2(u)=0$, then $E\{M(t)\}$ quantifies the expected number of interventions prior to disease or death, whichever occurs first. If cost information is available for interventions, it can be incorporated into $W_1(u)$ to reflect the associated total cost burden. However, such data are often challenging to obtain due to variations across healthcare organizations and regions. In this paper, for the real data analysis, we focus on the number of interventions rather than actual costs, as previously done in²⁷.

Having a unified measure for benefits or burdens is fundamental for evaluating the cost-effectiveness of an intervention. Decision-makers often face the challenge of determining the value of a particular intervention relative to its associated costs. This issue is particularly pressing when resources are limited, and choices must be made about how to allocate these resources to achieve the best outcomes. A unified measure provides a systematic approach to assess benefits, burdens, or their combination (e.g., the incremental cost effectiveness ratio, defined as incremental burden/incremental benefit) across various interventions.

2.2 |. Causal Inference of a Continuous Intervention under the illness-death Model

Our interest lies in assessing the effects of different initiation timings for an intervention on both benefits (e.g., increased survival years) and burdens (e.g., number of interventions) from a cost-effectiveness standpoint. Because the timing for intervention is continuous, the conventional counterfactual framework for binary interventions (e.g., treatment vs. control) cannot be directly applied. Instead, we approach this as a counterfactual problem with a continuous intervention. However, as previously mentioned, the timing of intervention initiation is subject to censoring, which introduces additional challenges. The central question that we aim to address is: what would the benefits and burdens be if the intervention were recommended to start at age $s$?

Let $T^{*}$ and $D^{*}$ denote the potential age at disease diagnosis and the potential age at death, respectively, had the individual never undergone the intervention. For $s<\text{min}\left(T^{*},D^{*}\right)$, we define $T^{(s)}$ and $D^{(s)}$ as the potential age at disease diagnosis and age at death, respectively, if the intervention were initiated at age $s$. If the individual had not undergone the intervention before the occurrence of disease or death, i.e., $s\ge \text{min}\left(T^{*},D^{*}\right)$, then $T^{(s)}=T^{*}$ and $D^{(s)}=D^{*}$. Define $X$ as a vector of baseline covariates. As shown in Figure 1, the illness-death model describes transitions between states: individuals start in state 0 (healthy) and then transition either to state 2 (death without disease), or first to state 1 (disease) followed by state 3 (death after disease). Given the continuous intervention age $s$, the illness-death model is characterized by three hazard functions based on the potential outcomes $T^{(s)}$ and $D^{(s)}$: the hazards for transitions from healthy to diseased and to death, as well as the hazard from disease to death. Specifically, the hazards functions from healthy to disease (0 → 1) and from healthy to death (0 → 2) under the competing risks framework are defined as:

$${\lambda }_{01}^{\left(s\right)}\left(t∣X\right)=\underset{\Delta t\to 0}{\text{lim}}\Delta t^{-1}\text{Pr}\left(t\le T^{\left(s\right)}<t+\Delta t\mid T^{\left(s\right)}\ge t,D^{\left(s\right)}\ge t,X\right),$$ (2)

$${\lambda }_{02}^{(s)}(t\mid X)=\underset{\Delta t\to 0}{\text{lim}}\Delta t^{-1}\text{Pr}\left(t\le D^{(s)}<t+\Delta t\mid T^{(s)}\ge t,D^{(s)}\ge t,X\right),$$ (3)

for $t>0$. For the hazard function describing the transition from disease to death (1 → 3), we define it in terms of the sojourn time, denoted as ${\tilde{D}}^{(s)}=D^{(s)}-T^{(s)}$, representing the time from disease diagnosis to death. The standard semi-Markov assumption commonly used in such models posits that ${\tilde{D}}^{(s)}$ is conditionally independent of $T^{(s)}$ given covariates $X$. However, this assumption may not always hold, as survival time following disease diagnosis often depends on the age at diagnosis²⁸. To account for this dependence, we model the hazard function for the transition from disease to death using a nonhomogeneous semi-Markov model, explicitly including age at disease diagnosis as a covariate,

$${\lambda }_{13}^{(s)}\left(\tilde{t}\mid X,T^{(s)}\right)=\underset{\Delta \tilde{t}\to 0}{\text{lim}}(\Delta \tilde{t}{)}^{-1}\text{Pr}\left(\tilde{t}\le {\tilde{D}}^{\left(s\right)}<\tilde{t}+\Delta \tilde{t}\mid {\tilde{D}}^{\left(s\right)}\ge \tilde{t},T^{\left(s\right)},X\right),$$ (4)

where $\tilde{t}$ represents the sojourn time starting from the age of disease diagnosis²⁹.

Both the potential time to disease diagnosis and the potential time to death are subject to right censoring at time $C$. We assume that $C$ is independent of these times, conditional on baseline covariates $X$, as stated below:

Assumption A1 (Conditional Independence of Censoring). The censoring time $C$ is independent of $\left\{T^{*},D^{*},T^{(s)},D^{(s)},0<\right.\left.s\le \text{min}\left(T^{*},D^{*}\right)\right\}$ given $X$.

This assumption also implies that, given $T^{(s)}$ and $X$, the sojourn time ${\tilde{D}}^{(s)}=D^{(s)}-T^{(s)}$ is independent of $C-T^{(s)}$. This allows for the estimation of ${\lambda }_{13}^{(s)}\left(\tilde{t}\mid X,T^{(s)}\right)$ in (4) by including $T^{(s)}$ as a covariate without requiring additional assumptions about the censoring mechanism. This approach also avoids issues related to induced informative censoring in the nonparametric estimation of the distribution function for the sojourn time³⁰. Now, define the potential observational time as $U^{(s)}=\text{min}\left(T^{(s)},D^{(s)},C\right)$. This represents the potential age at disease diagnosis if the disease indicator ${\delta }_1^{(s)}\equiv I\left(T^{(s)}\le \text{min}\left(D^{(s)},C\right)\right)$ equals 1, the potential age at death if the death indicator ${\delta }_2^{(s)}\equiv \left(1-{\delta }_1^{(s)}\right)I\left(D^{(s)}\le C\right)$ equals 1, and the time at last observation, i.e., the censoring time $C$, if both ${\delta }_1^{(s)}$ and ${\delta }_2^{(s)}$ are 0. For rare occasions where disease is diagnosed at the time of death, we assume $T^{(s)}<D^{(s)}$. Additionally, define $V^{(s)}={\delta }_1^{(s)}\left\{\text{min}\left(D^{(s)},C\right)-T^{(s)}\right\}$, which is the potential time to death since disease occurrence if ${\delta }_3^{(s)}\equiv {\delta }_1^{(s)}I\left(D^{(s)}\le C\right)$ equals 1, and the time to last observation since disease occurrence, otherwise. The following potential outcome counting processes are then defined,

$$N_{01}^{\left(s\right)}\left(t\right)=I\left(U^{\left(s\right)}\le t,{\delta }_1^{\left(s\right)}=1\right),N_{02}^{\left(s\right)}\left(t\right)=I\left(U^{\left(s\right)}\le t,{\delta }_2^{\left(s\right)}=1\right),N_{13}^{\left(s\right)}\left(\tilde{t}\right)=I\left(V^{\left(s\right)}\le \tilde{t},{\delta }_3^{\left(s\right)}=1\right),$$

$$Y_0^{(s)}(t)=I\left(U^{\left(s\right)}\ge t\right),Y_1^{\left(s\right)}\left(\tilde{t}\right)=I\left(V^{\left(s\right)}\ge \tilde{t}\right).$$

Here, the time-shifted counting process $N_{13}^{(s)}(\tilde{t})$ denotes the potential death count by sojourn time $\tilde{t}$ when the intervention occurs at time $s$, and $Y_1^{(s)}(\tilde{t})$ indicates whether an individual remains at risk in the disease state at sojourn time $\tilde{t}$. Under Assumption A1, the potential outcome hazard functions can be represented in terms of event counting and at-risk processes. Let $d{N}_{0k}^{(s)}(t)=N_{0k}^{(s)}\left(t^{-}+\Delta t\right)-N_{0k}^{(s)}\left(t^{-}\right),k=\mathrm{1,2}$, and $d{N}_{13}^{(s)}(\tilde{t})=N_{13}^{(s)}\left({\tilde{t}}^{-}+\Delta \tilde{t}\right)-N_{0k}^{(s)}(\tilde{t})$, Eq. (2) – (4) can be written as

$${\lambda }_{0k}^{(s)}(t\mid X)=\underset{\Delta t\to 0}{\text{lim}}\frac{1}{\Delta t}\text{Pr}\left\{d{N}_{0k}^{(s)}(t)=1\mid Y_0^{(s)}(t),X\right\}$$

$${\lambda }_{13}^{(s)}\left(\tilde{t}\mid T^{(s)},X\right)=\underset{\Delta \tilde{t}\to 0}{\text{lim}}\frac{1}{\Delta \tilde{t}}\text{Pr}\left\{d{N}_{13}^{(s)}(\tilde{t})=1\mid Y_1^{(s)}(\tilde{t}),T^{(s)},X\right\}.$$

Now, we define the observed data. Let $U=\text{min}(T,D,C)$ and $V={\delta }_1\{\text{min}(D,C)-T\}$ denote the observed time to the first event of disease or death and the observed time to death since disease, respectively, where ${\delta }_1=I(T\le \text{min}(D,C))$ indicates disease status. Let ${\delta }_2=\left(1-{\delta }_1\right)I(D\le C)$, and ${\delta }_3={\delta }_{1}I(D\le C)$ be the indicators of death without disease and death after disease, respectively. The observed counting processes are given by

$$N_{01}\left(t\right)=I\left(U\le t,{\delta }_1=1\right),N_{02}\left(t\right)=I\left(U\le t,{\delta }_2=1\right),N_{13}\left(\tilde{t}\right)=I\left(V\le \tilde{t},{\delta }_3=1\right),$$

$$Y_0(t)=I(U\ge t),Y_1(\tilde{t})=I(V\ge \tilde{t}),$$

where time $\tau$ is the end of a study or a pre-specified time point and $\text{Pr}\left(Y_0(\tau )Y_1(\tau )>0\right)>0$. Further let $S$ denote the time at which the individual initiates the intervention. The observational history for intervention up to age $U$ is denoted by $\bar{Z}(U;S)=\{Z(u;S)=I(S<u),0<u\le U\}$.

To connect observed processes with potential outcomes, we assume the following consistency assumption:

Assumption A2 (Consistency). For individuals who have initiate the intervention prior to $U$, i.e. $S<U$, the observed event and at-risk processes equal the potential processes under intervention age $S:N_{01}(t)=N_{01}^{(S)}(t),N_{02}(t)=N_{02}^{(S)}(t)$, and $N_{13}(\tilde{t})=N_{13}^{(S)}(\tilde{t})$ as well as $Y_0(t)=Y_0^{(S)}(t)$ and $Y_1(\tilde{t})=Y_1^{(S)}(\tilde{t})$. For individuals who have not undergone the intervention prior to $U$, the observed processes are consistent with any potential regime $s,s\ge U:N_{01}(t)=N_{01}^{(s)}(t)$, $N_{02}(t)=N_{02}^{(s)}(t)$, $N_{13}(\tilde{t})=N_{13}^{(s)}(\tilde{t})$, $Y_0(t)=Y_0^{(s)}(t)$, and $Y_1(\tilde{t})=Y_1^{(s)}(\tilde{t})$, for all $t>0$ and $\tilde{t}>0$.

This consistency assumption implies that for individuals who have not undergone the intervention before $U$, their observed processes align with the potential processes under any intervention regime $s,s\ge U$. In the absence of censoring, the potential processes would represent the times $T^{*}$ and $D^{*}$ had the individual never undergone the intervention. In addition, the following two identifiability assumptions are required.

Assumption A3 (No Unmeasured Confounding). Conditional on baseline risk factors $X$, the potential outcome hazard functions of disease and mortality at time $t$ with intervention time $s$ are independent of the observed intervention history up to and including time $t$. In other words, under Assumptions A1 and A3,

$${\lambda }_{0k}^{\left(s\right)}(t\mid X)=\underset{\Delta t\to 0}{\text{lim}}\frac{1}{\Delta t}\text{Pr}\left\{d{N}_{0k}^{\left(s\right)}\left(t\right)=1\mid Y_0^{\left(s\right)}\left(t\right),X,\bar{Z}\left(t;S\right)\right\},k=\mathrm{1,2},$$

$${\lambda }_{13}^{(s)}\left(\tilde{t}\mid T^{(s)},X\right)=\underset{\Delta \tilde{t}\to 0}{\text{lim}}\frac{1}{\Delta \tilde{t}}\text{Pr}\left(d{N}_{13}^{(s)}(\tilde{t})=1\mid Y_1^{(s)}(\tilde{t}),T^{(s)},X,\bar{Z}\left(T^{(s)};S\right)\right).$$

Assumption A4 (Positivity). There exists a positive constant $ϵ>0$ such that the hazard rate function of observed intervention at time $t$ given $X$ is bounded below by $ϵ$ and finite, i.e., $ϵ<{\text{lim}}_{\Delta t\to 0}P(t\le S<t+\Delta t\mid U\ge t,S\ge t,X)/\Delta t<\infty$.

Assumption A3 ensures that baseline risk factors $X$ are sufficient to account for self-selection in the timing of intervention in observational data. Together with Assumption A2, this establishes a connection between causal estimands and observed data, facilitating the inference of the causal effect of intervention timing. In this work, we consider $X$ to consist of only baseline covariates, because decision to recommend an individual when an individual should have an intervention is made at the time when the individual seeks advice, i.e., baseline. While time-dependent risk factors (e.g., body mass index) could also be incorporated, doing so requires projecting their trajectories, which would necessitate additional assumptions and modeling of this trajectory. We want to note that the framework proposed here can accommodate time-varying covariates. Specifically, Assumption A1 and A3 can be modified as follows. Let $X(t)$ denote the time-varying covariates and $\bar{X}(t)=\{X(u),0<u\le t\}$ represent the observational covariate history up to age $t$. The conditional independence of censoring (Assumption A1) with time-varying covariates is then extended such that the instantaneous risk of censoring at time $t$ given $\bar{X}(t)$, and the individual being at risk at time $t$ had the individual never undergone the intervention or had the intervention at time $s$ is independent of all the potential disease and death times. In other words, we have the following equality

$$\underset{{\Delta }_t\to 0}{\text{lim}}\frac{1}{{\Delta }_t}\text{Pr}\left(C\in \left[t,t+{\Delta }_t\right)\mid C\ge t,T^{*}\ge t,D^{*}\ge t,T^{(s)}\ge t,D^{(s)}\ge t,\bar{X}(t)\right)=\underset{{\Delta }_t\to 0}{\text{lim}}\frac{1}{{\Delta }_t}\text{Pr}\left(C\in \left[t,t+{\Delta }_t\right)\mid C\ge t,\bar{X}(t)\right).$$

The Assumption A3 with time-varying covariates states that

$${\lambda }_{0k}^{\left(s\right)}(t\mid \bar{X}(t))=\underset{\Delta t\to 0}{\text{lim}}\frac{1}{\Delta t}\text{Pr}\left\{d{N}_{0k}^{\left(s\right)}\left(t\right)=1\mid Y_0^{\left(s\right)}\left(t\right),\bar{X}\left(t\right),\bar{Z}\left(t;S\right)\right\},k=\mathrm{1,2},$$

$${\lambda }_{13}^{(s)}\left(\tilde{t}\mid T^{(s)},\bar{X}(t)\right)=\underset{\Delta \tilde{t}\to 0}{\text{lim}}\frac{1}{\Delta \tilde{t}}\text{Pr}\left(d{N}_{13}^{(s)}(\tilde{t})=1\mid Y_1^{(s)}(\tilde{t}),T^{(s)},\bar{X}(t),\bar{Z}\left(T^{(s)};S\right)\right).$$

where ${\lambda }_{0k}^{(s)}(t\mid \bar{X}(t))$ and ${\lambda }_{13}^{(s)}\left(\tilde{t}\mid T^{(s)},\bar{X}(t)\right)$ are defined as

$${\lambda }_{0k}^{(s)}(t\mid \bar{X}(t))=\underset{\Delta t\to 0}{\text{lim}}\frac{1}{\Delta t}\text{Pr}\left\{d{N}_{0k}^{(s)}(t)=1\mid Y_0^{(s)}(t),X\right\}$$

$${\lambda }_{13}^{(s)}\left(\tilde{t}\mid T^{(s)},\bar{X}(t)\right)=\underset{\Delta \tilde{t}\to 0}{\text{lim}}\frac{1}{\Delta \tilde{t}}\text{Pr}\left\{d{N}_{13}^{(s)}(\tilde{t})=1\mid Y_1^{(s)}(\tilde{t}),T^{(s)},X\right\}.$$

Assumption A4 ensures that every individual has a non-zero probability of undergoing the intervention at any age prior to disease diagnosis or death. Given our focus on the preventive effects of the intervention, we treat the time to intervention for individuals who have not undergone the intervention before disease diagnosis as censored at the time of diagnosis.

To evaluate the benefits and burdens of a recommended intervention age $s$, our interest lies in the following function,

$$\begin{gathered} E\left\{M^{(s)}(t)\right\}=E_X\left[{\int }_0^{t}E\left\{I\left(T^{(s)}\ge u,D^{(s)}\ge u\right)\mid X\right\}d{W}_1(u)\right] \\ +E_X\left[{\int }_0^{t}E\left\{I\left(T^{(s)}\le u,D^{(s)}\ge u\right)\mid X\right\}d{W}_2(u)\right]. \end{gathered}$$ (5)

Let ${\Lambda }_{0k}^{(s)}(t)={\int }_0^t{\lambda }_{0k}^{(s)}(u)du,{\Lambda }_{0k}(t)={\int }_0^t{\lambda }_{0k}(u)du,k=\mathrm{1,2},{\Lambda }_{13}^{(s)}(t)={\int }_0^t{\lambda }_{13}^{(s)}(u)du$, and ${\Lambda }_{13}(t)={\int }_0^t{\lambda }_{13}(u)du$. The first term can be expressed as:

$$\begin{gathered} E_X\left[{\int }_0^t\text{exp}\left\{-{\Lambda }_{01}^{(s)}(u\mid X)-{\Lambda }_{02}^{(s)}(u\mid X)\right\}d{W}_1(u)\right] \\ =E_X\left[{\int }_0^t\text{exp}\left\{-{\Lambda }_{01}^{(s)}(u\mid X,\bar{Z}(u;S))-{\Lambda }_{02}^{(s)}(u\mid X,\bar{Z}(u;S))\right\}d{W}_1(u)\right], \\ =E_X\left[{\int }_0^t\text{exp}\left\{-{\Lambda }_{01}(u\mid X,\bar{Z}(u;S=s))-{\Lambda }_{02}(u\mid X,\bar{Z}(u;S=s))\right\}d{W}_1(u)\right], \\ \equiv E_X\left\{{\int }_0^t\text{Pr}\left(\text{min}\left(T,D\right)\ge u∣X,\bar{Z}\left(u;S=s\right)\right)d{W}_1\left(u\right)\right\}, \end{gathered}$$ (6)

where the first equality is due to Assumption A3 and the second equality is due to Assumption A2. Following similar arguments, the second term on the right hand side of (5) equals

$$\begin{gathered} E_X\left[{\int }_0^t{\int }_0^u\text{Pr}\left(D^{(s)}\ge u\mid T^{(s)}=v,X\right)d{F}_{T^{(s)}}(v\mid X)d{W}_2(u)\right] \\ =E_X\left[{\int }_0^t{\int }_0^u\text{exp}\left\{-{\Lambda }_{13}^{(s)}\left(u-v\mid T^{(s)}=v,X,\bar{Z}(v;S)\right)\right\}d{F}_{T^{(s)}}(v\mid X,\bar{Z}(v;S))d{W}_2(u)\right] \\ =E_X\left[{\int }_0^t{\int }_0^u\text{exp}\left\{-{\Lambda }_{13}(u-v\mid T=v,X,\bar{Z}(v;S=s))\right\}d{F}_T(v\mid X,\bar{Z}(v;S=s))d{W}_2(u)\right], \\ \equiv E_X\left\{{\int }_0^t\text{Pr}\left(T<u,D\ge u∣X,\bar{Z}\left(u;S=s\right)\right)d{W}_2\left(u\right)\right\}, \end{gathered}$$ (7)

where $F_{T^{(s)}}(\cdot )$ is the cumulative incidence function (CIF) for the potential time at disease diagnosis, defined as $F_{T^{(s)}}(u\mid X)={\int }_0^u\text{exp}\left\{-{\Lambda }_{01}^{(s)}(v\mid X)-{\Lambda }_{02}^{(s)}(v\mid X)\right\}d{\Lambda }_{01}^{(s)}(v\mid X)={\int }_0^u\text{exp}\left\{-{\Lambda }_{01}(v\mid X,\bar{Z}(v;S=s))-{\Lambda }_{02}(v\mid X,\bar{Z}(v;S=s))\right\}d{\Lambda }_{01}(v\mid X,\bar{Z}(v;S=s))$, and the first and second equalities are due to Assumptions A3 and A2, respectively.

Therefore, to estimate $E\left\{M^{(s)}(t)\right\}$ for a given intervention age $s$, it has come down to estimating the hazard functions ${\lambda }_{0k}(t\mid X,\bar{Z}(v;S=s))$, $k=\mathrm{1,2}$, and ${\lambda }_{13}(t\mid X,\bar{Z}(v;S=s))$ with observed intervention history $\bar{Z}(v;S=s)$ and baseline risk factor $X$. These are modeled using Cox proportional hazards models

$${\lambda }_{0k}(t\mid \bar{Z}(t;S),X)={\lambda }_{0k,0}(t)\text{exp}\left({\beta }_{0k}Z\left(t;S\right)+{\gamma }_{0k}^{\prime }X\right),k=\mathrm{1,2},$$ (8)

$${\lambda }_{13}(\tilde{t}\mid T,\bar{Z}(T;S),X)={\lambda }_{\mathrm{13,0}}\left(\tilde{t}\right)\text{exp}\left(\alpha T+{\beta }_{13}Z\left(T;S\right)+{\gamma }_{13}^{\prime }X\right),$$ (9)

where ${\lambda }_{\mathrm{01,0}}(t)$, ${\lambda }_{\mathrm{02,0}}(t)$ and ${\lambda }_{\mathrm{13,0}}(\tilde{t})$ are unspecified baseline hazard functions, and ${\theta }_{01}={\left({\beta }_{01},{\gamma }_{01}^{\prime }\right)}^{\prime },{\theta }_{02}={\left({\beta }_{02},{\gamma }_{02}^{\prime }\right)}^{\prime }$ and ${\theta }_{13}={\left(\alpha ,{\beta }_{13},{\gamma }_{13}^{\prime }\right)}^{\prime }$ are the regression coefficients. Under these models, a key assumption is that covariates, including $Z(t;S)$, have time-invariant effect. In some situations, covariate effects may vary with time. To address such situations, a large body of work has extended the Cox proportional hazards model to accommodate timevarying effects, such as models with timebycovariate interactions, kernel-smoothing-based approaches, and spline-based varying coefficient models, see e.g., Therneau and Grambsch³¹; Kalbfleisch and Prentice³²; Tian et al.³³. These extensions allow flexible estimation of $\beta (t)$ and $\gamma (t)$ to capture departures from the proportional hazards assumption while preserving the interpretability of hazardbased inference.

2.3 |. Estimation

The observed data consist of $n$ independently and identically distributed random variables, denoted as ${𝒪}_i,i=1,\dots ,n$, where ${𝒪}_i=\left\{U_i,V_i,{\delta }_{1i},{\delta }_{2i},{\delta }_{3i},\bar{Z}\left(U_i;S_i\right)\right\}$. The estimation procedure under the Cox proportional hazards model with competing risks and time-dependent covariates is well established (see e.g.,³²). Hence, the maximum partial likelihood estimators ${\widehat{\theta }}_{01}$ and ${\widehat{\theta }}_{02}$ can be obtained by solving the following partial likelihood score equations:

$$\sum _{i=1}^n{\int }_0^{\tau }\left\{{\tilde{Z}}_i\left(t\right)-\frac{s_{01}^{\left(1\right)}\left(t;{\theta }_{01}\right)}{s_{01}^{\left(0\right)}\left(t;{\theta }_{01}\right)}\right\}{N}_{i01}(dt)=0,$$

and

$$\sum _{i=1}^n{\int }_0^{\tau }\left\{{\tilde{Z}}_i\left(t\right)-\frac{s_{02}^{\left(1\right)}\left(t;{\theta }_{02}\right)}{s_{02}^{\left(0\right)}\left(t;{\theta }_{02}\right)}\right\}{N}_{i02}(dt)=0,$$

where $N_{i01}(t)$, $N_{i02}(t)$, $Y_{i0}(t)$ are the realizations of $N_{01}(t)$, $N_{02}(t)$ and $Y_0(t)$ for the $i$th individual, ${\tilde{Z}}_i(t{)}^{\prime }=\left(Z\left(t;S_i\right),X_i^{\prime }\right),s_{01}^{(j)}\left(t;{\theta }_{01}\right)=\sum _{i=1}^n{Y}_{i0}(t){\tilde{Z}}_i^{\otimes j}\left(t;S_i\right)\text{exp}\left\{{\beta }_{01}Z\left(t;S_i\right)+{\gamma }_{01}^{\prime }{X}_i\right\}$, and $s_{02}^{(j)}\left(t;{\theta }_{02}\right)=\sum _{i=1}^n{Y}_{i0}(t){\tilde{Z}}_i^{\otimes j}(t)\text{exp}\left\{{\beta }_{02}Z\left(t;S_i\right)+{\gamma }_{02}^{\prime }{X}_i\right\}$ with $j=\mathrm{0,1}$ representing $Z^{\otimes 0}=1$ and $Z^{\otimes 0}=Z$. In the competing risks framework, the dependence between the time to disease and the time to death is unspecified and independence is not assumed³². Additionally, while the data are presented as right censoring, the estimation procedure can straightforwardly accommodate left truncation data by appropriately adjusting the risk set, where $Y_{i0}(t)=I\left(U_i\ge t>L_i\right)$ and $L_i$ is the left truncation time for the $i$th individual.

For the model describing the transition from disease to death, let $N_{i13}(\tilde{t})$ and $Y_{1i}(\tilde{t})$ represent the realizations of the time-shifted counting processes $N_{13}(\tilde{t})$ and $Y_1(\tilde{t})$, respectively, for the $i$th individual since disease diagnosis. Here, the time origin shifts from study entry to the time of disease diagnosis, and the partial likelihood is constructed using only individuals who have developed the disease. This framework allows us to estimate the parameters ${\theta }_{13}$ in model (9) based on the observed data $N_{i13}(\tilde{t})$ and $Y_{1i}(\tilde{t})$ for these individuals. A consistent estimator ${\widehat{\theta }}_{13}$ is obtained because model (9) explicitly includes $T$ and $X$ as covariates. Under Assumption A1, the independence between the sojourn time ($D-T$) and the (sojourn) censoring time ($C-T$), conditional on $T$ and $X$, is guaranteed. This ensures that the maximum partial likelihood estimator ${\widehat{\theta }}_{13}$ can be derived by solving the following score equation

$$\sum _{i=1}^n{\int }_0^{\tau }\left\{{\tilde{Z}}_{i1}-\frac{s_{13}^{\left(1\right)}\left(\tilde{t};{\theta }_{13}\right)}{s_{13}^{\left(0\right)}\left(\tilde{t};{\theta }_{13}\right)}\right\}{N}_{i13}(d\tilde{t})=0,$$

where ${\tilde{Z}}_{i1}^{\prime }=\left(T_i,Z\left(T_i;S_i\right),X_i^{\prime }\right),s_{13}^{(j)}\left(\tilde{t};{\theta }_{13}\right)=\sum _{i=1}^n{Y}_{i1}(\tilde{t}){\tilde{Z}}_{i1}^{\otimes j}\text{exp}\left\{\alpha T_i+{\beta }_{13}Z\left(T_i;S_i\right)+{\gamma }_{13}^{\prime }{X}_i\right\}$. The cumulative baseline hazard functions {${\Lambda }_{\mathrm{01,0}}(t)$, ${\Lambda }_{\mathrm{02,0}}(t)$, and ${\Lambda }_{\mathrm{13,0}}(t)$} are estimated using Breslow estimators, as detailed in Supplemental Materials (SM) Section 1.

An estimator of $E\left\{M^{(s)}(t)\right\}$ can then be obtained by substituting $\left\{{\widehat{\theta }}_{01},{\widehat{\theta }}_{02},{\widehat{\theta }}_{13}\right\}$ and the Breslow estimators $\left\{{\widehat{\Lambda }}_{\mathrm{01,0}}(t),{\widehat{\Lambda }}_{\mathrm{02,0}}(t),{\widehat{\Lambda }}_{\mathrm{13,0}}(t)\right\}$ into Eqs. (6) and (7). Specifically, denote $P_1(t\mid X,s)=\text{Pr}(\text{min}(T,D)\ge t\mid \bar{Z}(t;S=s),X)$, $P_{13}(t\mid r,X,s)=P(D\ge t\mid T=r,\bar{Z}(T;S=s),X)$ and $P_2(t\mid X,s)=\text{Pr}(D\ge t,T\le t\mid \bar{Z}(t;S=s),X)$. The corresponding estimators are

$${\widehat{P}}_1\left(t∣X,s\right)=\text{exp}\left\{-{\widehat{\Lambda }}_{01}\left(t∣\bar{Z}\left(u;S=s\right),X\right)-{\widehat{\Lambda }}_{02}\left(t∣\bar{Z}\left(u;S=s\right),X\right)\right\},$$

$${\widehat{P}}_{13}\left(t∣r,X,s\right)=\text{exp}\left\{-{\widehat{\Lambda }}_{13}\left(t-r∣T=r,\bar{Z}\left(r;S=s\right),X\right)\right\},$$

$${\widehat{P}}_2(t\mid X,s)={\int }_0^t{\widehat{P}}_{13}(t\mid r,X,s)d{\widehat{F}}_T(r\mid \bar{Z}(r;S=s),X),$$

where the cumulative hazard function estimators are ${\widehat{\Lambda }}_{0k}\left(t\mid \bar{Z}(t;S=s,X)={\int }_0^t\text{exp}\left\{{\widehat{\beta }}_{0k}Z(u;S=s)+{\widehat{\gamma }}_{0k}^{\prime }X\right\}{\widehat{\Lambda }}_{0k,0}(du)\right.$ for $k=\mathrm{1,2}$ and ${\widehat{\Lambda }}_{13}(t-r\mid T=r,\bar{Z}(r;S=s),X)={\int }_0^{t-r}\text{exp}\left\{\widehat{\alpha }r+{\widehat{\beta }}_{13}Z(r;S=s)+{\widehat{\gamma }}_{13}^{\prime }X\right\}{\widehat{\Lambda }}_{\mathrm{13,0}}(du)$, as well as cumulative disease incidence estimator ${\widehat{F}}_T(t\mid \bar{Z}(t;S=s),X)={\int }_0^t{\widehat{P}}_1(u\mid X,s)\text{exp}\left\{{\widehat{\beta }}_{01}Z(u;S=s)+{\widehat{\gamma }}_{01}^{\prime }X\right\}{\widehat{\Lambda }}_{\mathrm{01,0}}(du)$. The estimator of $E\left\{M^{(s)}(t)\right\}$ is thus expressed as:

$$\widehat{E}\left\{M^{(s)}(t)\right\}=\frac{1}{n}\sum _{i=1}^n\left({\int }_0^t{\widehat{P}}_1\left(u\mid X_i,s\right)d{W}_1(u)+{\int }_0^t{\widehat{P}}_2\left(u\mid X_i,s\right)d{W}_2(u)\right).$$ (10)

To obtain the variance estimator, we draw $B$ bootstrap samples from the observed data $\left\{{𝒪}_i,i=1,\dots ,n\right\}$ and obtain $\widehat{E}\left\{M^{(s)}(t)\right\}$ for each sample. The variance of $\widehat{E}\left\{M^{(s)}(t)\right\}$ is then estimated as the empirical variance across the $B$ bootstrap samples. The following theorem outlines the asymptotic properties of $\widehat{E}\left\{M^{(s)}(t)\right\}$:

Theorem 1. Under Assumptions (A1)–(A4) and regularity conditions (B1)–(B5) outlined in SM Section 2, for any $s\in [0,t]$, as $n\to \infty$,

1. $\widehat{E}\left\{M^{(s)}(t)\right\}$ converges uniformly to $E\left\{M^{(s)}(t)\right\}$ over $t\in [0,\tau ]$.

2. $\sqrt{n}\left[\widehat{E}\left\{M^{(s)}(t)\right\}-E\left\{M^{(s)}(t)\right\}\right]$ converges weakly to a mean-zero Gaussian process over $t\in [0,\tau ]$.

3. The bootstrap variance estimator for $\widehat{E}\left\{M^{(s)}(t)\right\}$ is consistent for the asymptotic variance of $\widehat{E}\left\{M^{(s)}(t)\right\}$.

The main step of the proof is to show that Eq. (10) asymptotically equals a sum of independent and identically distributed martingales. The subsequent part of the proof utilizes martingale theory. Detailed proofs are provided in SM Section 2.

2.4 |. Some Common and Useful Estimands

This section provides several examples of $E\left\{M^{(s)}(t)\right\}$ under the illness-death model (Figure 1) illustrated in Section 2.1. These examples highlight how the framework can be used to evaluate important measures such as the RMST, cause-specific life years lost, and number of interventions. By considering these specific estimands, we demonstrate the flexibility and utility of the proposed framework in addressing diverse questions in benefit and burden analyses.

2.4.1 |. Example 1: RMST

The causal estimand for the RMST is derived by setting $W_1(u)=W_2(u)=u$ in Eq. (5) for $E\left\{M^{(s)}(t)\right\}$. This simplifies $E\left\{M^{(s)}(t)\right\}$ to $E_X\left\{{\int }_0^{t}E\left\{I\left(D^{(s)}\ge u\right)\mid X\right\}du\right\}$, which equals ${\int }_0^t\text{Pr}(D\ge u\mid \bar{Z}(u;S=s),X)du$. Traditionally, RMST focuses solely on mortality and models only the time to death²⁴. However, this framework extends the conventional approach by decomposing the overall survival probability into two components to account for the disease an individual could experience. This decomposition provides a better modeling of the intervention’s effect on both the disease process and subsequent death. The RMST can thus be expressed as $E\left\{M^{(s)}(t)\right\}=E_X\left\{{\int }_0^t{P}_1(u\mid X,s)du\right\}+E_X\left\{{\int }_0^t{P}_2(u\mid X,s)du\right\}$, which can be estimated by

$$\widehat{E}\left\{M^{(s)}(t)\right\}=\frac{1}{n}\sum _{i=1}^n\left({\int }_0^t{\widehat{P}}_1\left(u\mid X_i,s\right)du+{\int }_0^t{\widehat{P}}_2\left(u\mid X_i,s\right)du\right).$$

2.4.2 |. Example 2: Cause-Specific Mean Life Years Lost

The estimand for the mean life years lost for individuals recommended to initiate intervention at age $s$ is defined as $E\left\{L^{(s)}(t)\right\}=t-E\left\{M^{(s)}(t)\right\}$. This can be decomposed into two cause-specific estimands: (1) mean life years lost through the disease pathway 0 → 1 → 3), $E\left\{L_{0\to 1\to 3}^{(s)}(t)\right\}=E_X\left\{{\int }_0^t{\int }_0^u{f}_{D^{(s)}}\left(v\mid T^{(s)}<v,X\right)\text{Pr}\left(T^{(s)}<v\mid X\right)\mathit{\text{dvdu}}\right\}$; and (2) mean life years lost due to other causes (0 → 2), $E\left\{L_{0\to 2}^{(s)}(t)\right\}=E_X\left\{{\int }_0^t{\int }_0^u{f}_{D^{(s)}}\left(v\mid T^{(s)}\ge v,X\right)\text{Pr}\left(T^{(s)}\ge v\mid X\right)\mathit{\text{dvdu}}\right\}$. Here, $f_{D^{(s)}}\left(v\mid T^{(s)},X\right)$ is the conditional probability density function of $D^{(s)}$ given $T^{(s)}$ and $X$. Under Assumptions A2 and A3 (detailed in SM Section 3), these terms can be written as:

$$E\left\{L_{0\to 1\to 3}^{(s)}(t)\right\}=E_X\left\{{\int }_0^t{\int }_0^u\left\{1-P_{13}(u\mid v,X,s)\right\}{P}_1(v\mid X,s){\Lambda }_{01}(dv\mid X,Z(v;S=s))du\right\},$$

and

$$E\left\{L_{0\to 2}^{(s)}(t)\right\}=E_X\left\{{\int }_0^t{\int }_0^u{P}_1(v\mid X,s){\Lambda }_{02}(dv\mid X,Z(v;S=s))du\right\}.$$

The respective estimators for these quantities are given by $\widehat{E}\left\{L_{0\to 1\to 3}^{(s)}(t)\right\}=\frac{1}{n}\sum _{i=1}^n{\int }_0^t{\int }_0^u\{1-\left.{\widehat{P}}_{13}\left(u\mid v,X_i,s\right)\right\}{\widehat{P}}_1\left(v\mid X_i,s\right)d{\widehat{\Lambda }}_{01}\left(v\mid Z\left(v;S_i=s\right),X_i\right)du$, and $\widehat{E}\left\{L_{0\to 2}^{(s)}(t)\right\}=\frac{1}{n}\sum _{i=1}^n{\int }_0^t{\int }_0^u{\widehat{P}}_1\left(v\mid X_i,s\right)d{\widehat{\Lambda }}_{02}\left(v\mid Z\left(v;S_i=s\right),X_i\right)du$.

2.4.3 |. Example 3: QALY and Quality-Adjusted Years Lost

The estimand for the QALY extends the RMST by incorporating a quality score function $Q(u)$, which ranges from 0 (death) to 1 (health). Let $W(u)={\int }_0^{u}Q(v)dv$ denote the cumulative quality score up to time $u$. Using the unified cost-effectiveness measure $E\left\{M^{(s)}(t)\right\}$ from Eq. (5), the QALY estimand is defined as

$$E_X\left[{\int }_0^{t}E\left\{I\left(T^{(s)}\ge u,D^{(s)}\ge u\right)\mid X\right\}du\right]+E_X\left[{\int }_0^{t}E\left\{I\left(T^{(s)}\le u,D^{(s)}\ge u\right)\mid X\right\}Q(u)du\right].$$

The first term corresponds to the contribution from the healthy state, identical to the first term in the RMST. However, the second term incorporates the reduced quality of life in the diseased state.

The QALY concept can be extended to cause-specific QALY lost due to the disease as $E_X\left\{{\int }_0^t{\int }_0^u{f}_{D^{(s)}}\left(v\mid T^{(s)}\ge v,X\right)\text{Pr}\left(T^{(s)}\ge v\mid X\right)dvQ(u)du\right\}$, which can be estimated by

$$\frac{1}{n}\sum _{i=1}^n{\int }_0^t{\int }_0^u{\widehat{P}}_1\left(v\mid X_i,s\right)\left\{1-{\widehat{P}}_{13}\left(u\mid v,X_i,s\right)\right\}d{\widehat{\Lambda }}_{01}\left(v\mid Z\left(v;S_i=s\right),X_i\right)Q\left(u\right)du.$$

In practice, here, in our CRC example, the quality score $Q(\cdot )$ may vary depending on the phase of cancer care the individual is in. We further illustrate estimation for the QALY and quality adjusted life years lost using phase-specific quality scores in the real data analysis (Section 3).

2.4.4 |. Example 4: Number of Interventions

The estimand for the number of interventions is a special case of the unified measure in Eq. (5), with $W_1(u)$ defined as the cumulative count of interventions up to time $u$ and $W_2(\cdot )=0$. This reflects the assumption that preventive interventions occur before disease diagnosis or terminal events. For CRC endoscopy, the US Preventive Services Task Force recommends endoscopy should repeat every 10 years⁹. The cumulative intervention function can thus be written as $W_1(u;s)=\lfloor (u-s+10{)}_{+}/10⌋$, where $u$ is the age in years, $s$ is the starting age for the intervention (e.g., 50 years), $x_{+}$ equals $x$ if $x>0$ and 0 otherwise, and $\lfloor x\rfloor$ is the largest integer $\le x$. The estimand for the expected number of interventions by time $t$, given the intervention starting age $s$, is $E\left\{M^{(s)}(t)\right\}=E_X{\int }_0^{t}E\left\{I\left(T^{(s)}\ge u,D^{(s)}\ge u\right)\mid X\right\}d{W}_1(u;s)$. The corresponding estimator is given by $\widehat{E}\left\{M^{(s)}(t)\right\}=\frac{1}{n}\sum _{i=1}^n{\int }_0^t{\widehat{P}}_1\left(u\mid X_i,s\right)d{W}_1(u;s)$.

3 |. ANALYSIS OF BENEFITS AND BURDENS OF CRC ENDOSCOPY

In this section, we evaluate the benefits and burdens of CRC endoscopy using data from the WHI. The analysis focused on the 65,062 study participants who had not undergone endoscopy before study entry. These participants were followed over a mean duration of 16 years with a maximum follow-up of 25 years. Comprehensive baseline data, including socio-demographic and epidemiologic factors, were collected. During the follow-up period, 1,291 individuals (2.0%) were diagnosed with CRC, among whom 603 (1.0%) died following their diagnosis. An additional 15,485 participants (23.8%) died without developing CRC. Out of the total participants, 44,912 (69.0%) underwent endoscopy with an average age at the time of the procedure being 67.1 years. The primary outcomes of interest were the age at CRC diagnosis and the age at death, both subject to left truncation (age at enrollment) and right censoring (age at the last follow-up). We employed the proposed estimators to assess the following metrics across different ages of initiating endoscopy ($s=50,60,\text{and}70\text{years}$): RMST, QALY, cumulative number of endoscopies performed, and (quality-adjusted) mean life years lost, over time intervals spanning from ages 50 to $t=60$, 70, and 80 years.

The analysis incorporated six baseline risk factors, selected based on established CRC risk prediction models by³⁴ and³⁵. These factors were obesity (BMI < 30, ≥ 30kg/m²), family history of CRC among first-degree relatives (no, yes), exercise (0, 0–2, > 2 hours per week), use of aspirin and other nonsteroidal anti-inflammatory drugs (NSAIDs) (nonuser, regular user), vegetable consumption (# of servings per day), and estrogen status within the last 2 years (negative, positive). Two demographic factors were also included: education (high school or less, some college, college/graduate degree) and race and ethnicity (White, Asian/Pacific Islander, Black/African American, Hispanic/Latino, and others). The reference levels for categorical covariates were defined as the first value within each parenthesis. A summary of these risk factors is provided in SM Table S1.

We fit the illness-death models (8) and (9) in Section 2.1 with time-dependent endoscopic status and adjusting for the aforementioned demographic and risk factors. For mortality following CRC diagnosis (model (9)), the age at diagnosis was also included as a covariate. The risk sets were adjusted for left truncation by age at enrollment. The results are summarized in Table 1. Undergoing endoscopy was associated with a 57% reduction in the risk of developing CRC. Obesity and a positive family history were associated with an increased risk of CRC, while regular exercise and a positive estrogen status were linked to a reduced risk. For mortality following CRC diagnosis, undergoing endoscopy was associated with a 17% reduction, possibly due to cancer detected at an earlier stage. Obesity remained a significant risk factor for mortality, and older age at CRC diagnosis was associated with a higher risk of earlier mortality.

TABLE 1.

Summary results of the illness-death models

	Healthy to CRC		Healthy to death		CRC to death
	HR (95% CI)*	p-value	HR (95% CI)	p-value	HR (95% CI)	p-value
Endoscopy	0.43 (0.38–0.48)	<1.0e-3	0.93 (0.90–0.96)	<1.0e-3	0.83 (0.69–0.99)	0.04
Obesity	1.25 (1.11–1.40)	<1.0e-3	1.31 (1.26–1.35)	<1.0e-3	1.22 (1.03–1.44)	0.02
Family hx	1.37 (1.18–1.59)	<1.0e-3	1.04 (0.99–1.08)	0.13	0.97 (0.78–1.21)	0.78
Exercise	0.90 (0.84–0.97)	<1.0e-3	0.93 (0.91–0.95)	<1.0e-3	0.99 (0.89–1.10)	0.88
NSAIDS use	0.93 (0.82–1.05)	0.23	1.14 (1.11–1.18)	<1.0e-3	1.07 (0.90–1.27)	0.45
Hormone use	0.82 (0.73–0.92)	<1.0e-3	0.93 (0.90–0.96)	<1.0e-3	1.03 (0.87–1.23)	0.72
Vegetable intake	1.00 (0.95–1.04)	0.84	0.97 (0.96–0.99)	<1.0e-3	0.95 (0.89–1.02)	0.16
Racial or ethnic group
Asian/Pacific Islander	1.26 (0.91–1.74)	0.17	0.78 (0.69–0.89)	<1.0e-3	0.73 (0.41–1.30)	0.28
Black/African American	1.15 (0.95–1.39)	0.16	1.12 (1.05–1.19)	<1.0e-3	1.03 (0.74–1.42)	0.87
Hispanic/Latino	0.87 (0.63–1.19)	0.38	0.85 (0.77–0.95)	<1.0e-3	0.95 (0.58–1.56)	0.84
Others	1.02 (0.66–1.59)	0.93	1.08 (0.95–1.23)	0.24	0.83 (0.41–1.67)	0.59
Education
Some college	1.04 (0.91–1.20)	0.55	0.99 (0.95–1.03)	0.62	0.99 (0.81–1.22)	0.95
College/graduate degree	1.00 (0.87–1.16)	0.98	0.89 (0.85–0.93)	<1.0e-3	0.83 (0.67–1.03)	0.10
Age at diagnosis (year)					1.07 (1.06–1.08)	<1.0e-3

^*HR (95%CI): hazard ratio (95% confidence interval)

3.1 |. Restricted Mean Survival Time (RMST)

Based on these models, we estimated the RMST for scenarios under no endoscopy and under endoscopy initiation at age 50, 60 and 70 years old (Table 2(a)). Standard errors (SE) were computed using 100 bootstrap samples. We also present the total RMST estimates in Figure 2 and the RMST estimates for disease-free and disease-to-death stages in Supplemental Figure S1.

TABLE 2.

Summary of RMST for $t=60,70,\text{and}80$ years for no endoscopy (No Endo) and endoscopy at ages 50 (Endo 50), 60 (Endo 60) and 70 (Endo 70) years.

(a) Proposed Approach
	Disease-free				Disease to death				Total RMST
RMST	No Endo	Endo 50	Endo 60	Endo 70	No Endo	Endo 50	Endo 60	Endo 70	No Endo	Endo 50	Endo 60	Endo 70
$t=\mathit{60}$
Est.	9.916	9.934	-*	-	0.021	0.009	-	-	9.937	9.942	-	-
SE	0.011	0.008	-	-	0.005	0.003	-	-	0.008	0.008	-	-
$t=\mathit{70}$
Est.	19.449	19.551	19.495	-**	0.118	0.053	0.088	-	19.567	19.604	19.584	-
SE	0.025	0.022	0.026	-	0.012	0.006	0.012	-	0.021	0.021	0.021	-
$t=\mathit{80}$
Est.	28.098	28.408	28.317	28.175	0.309	0.142	0.194	0.271	28.407	28.550	28.511	28.446
SE	0.042	0.040	0.043	0.042	0.018	0.011	0.017	0.018	0.037	0.038	0.038	0.036
(b) Overall Mortality Approach
RMST	No Endo	Endo 50	Endo 60	Endo 70
$t=\mathit{60}$
Est.	9.936	9.941	–	–
SE	0.009	0.008	–	–
$t=\mathit{70}$
Est.	19.560	19.598	19.578	–
SE	0.021	0.021	0.022	–
$t=\mathit{80}$
Est	28.409	28.544	28.510	28.452
SE	0.036	0.038	0.038	0.036

^*endoscopy at 60 or 70 for $t=60$ is same as no endoscopy.

^**endoscopy at 70 for $t=70$ is same as no endoscopy.

FIGURE 2.

Total RMST estimates by screening status and age evaluated over age [50, 60) (top panel), [50, 70) (middle panel), and [50, 80) (bottom panel), respectively.

Throughout this section, we report life years gained per 1000 individuals, following conventions in previous studies of cost-effectiveness analyses²⁷. Compared to no endoscopy, initiating endoscopy at age 50 would improve the RMST for the disease-free stage over a 30-year follow-up period $(t=80)$ by (28.408 – 28.098)*1000 = 310 years per 1000 individuals (95% CI, 196 to 424, p-value < 0.001). For individuals who developed CRC, RMST was reduced by (0.142 – 0.309)*1000= −167 years per 1000 individuals (95% CI, −208 to −126, p-value < 0.001). This reduction is because those who underwent endoscopy and still developed CRC tended to have cancer at an older age. Despite the reduction in post-diagnosis RMST, the total RMST would still improved by (28.550–28.407) * 1000 = 143 years per 1000 individuals (95% CI, 39 to 247, p-value = 0.005). Initiating endoscopy at older ages, such as 60 or 70 years, would also improve the total RMST and disease-free RMST, though to a lesser degree. Similar benefits were observed with shorter follow-up periods at $t=60$ and 70 years.

For comparison, we fit Cox proportional hazards model treating death as a single primary event, using the same covariates as in the illness-death models (SM Table S2) and refer to it as “Overall Mortality Approach”. The RMST estimates from this model were consistent with the total RMST obtained from the illness-death model (Table 2(b)). For example, over a 30 year period, the RMST for initiating endoscopy at 50 years was 28.544, compared to the total RMST 28.550 in the illness-death model. This similarity is likely attributable to CRC being relatively rare, with most participants dying without the disease. However, when the disease is more common, the illness-death model’s estimators are more efficient and less biased, as shown in one of the simulation scenarios in Section 4.

3.2 |. Quality-adjusted Life Years (QALY)

In cancer studies, QALY are commonly used to account for the reduced quality of life (QoL) during the cancer stage¹. Our proposed measure incorporates a time-varying QoL score to calculate QALY. For the cancer-free stage, we set $Q(u)=1$, as there was no loss in QoL during this stage. For the cancer stage, we derived the QoL score based on Table 1 in³⁶, dividing the CRC care into three clinically relevant phases: initial, continuing, and terminal care. Initial care was defined as the first year after diagnosis; terminal care was the last year of life; and continuing care included all years in between. For participants surviving ≤ 2 years, the final year was categorized as terminal care, with the remaining time allocated to initial care. For participants surviving ≤ 1 year, the entire period was considered terminal care. QoL scores for these phases were measured on a continuous scale, depending on the duration between the age at diagnosis and age at death. Let $a,b$ and $c$ represent the quality scores for initial, continuing and terminal care, respectively (SM Table S3). The QoL score is defined as

$$Q\left(u;T,D\right)=\left\{\begin{array}{c}aI\left(T\le u\le T+1\right)+bI(T+1<u\le D-1)\hfill \\ \begin{array}{cc}+cI(D-1<u\le D),\hfill & D-T>2,\hfill \\ aI\left(T\le u\le D-1\right)+cI(D-1<u\le D),\hfill & 1<D-T\ge 2,\hfill \\ cI(T<u\le D),\hfill & D-T\le 1.\hfill \end{array}\hfill \end{array}\right.$$

We incorporated $Q(u;T,D)$ into our estimators to calculate $\widehat{E}\{M(t)\}$ (details in SM Section 4.1). The results are presented in Table 3. The difference in QALY during the cancer-free stage was the same as that of RMST. However, for the cancer stage, the QALY gain for individuals undergoing endoscopy at age 50 compared to no endoscopy would be 121 years per 1000 individuals (95% CI, 92 to 150). The overall QALY gain from initiating endoscopy at age 50 compared to no endoscopy would be 189 years per 1000 individuals (95% CI, 83 to 295), surpassing the increase simply in total life years (~ 143 years per 1000 individuals), as indicated by the RMST.

TABLE 3.

Summary of QALY at $t=60,70,\mathrm{and}80$ for no endoscopy (No Endo) and for endoscopy at age 50 (Endo 50), 60 (Endo 60) and 70 (Endo 70).

QALY	Disease-free				Disease to death				Total QALY
	No Endo	Endo 50	Endo 60	Endo 70	No Endo	Endo 50	Endo 60	Endo 70	No Endo	Endo 50	Endo 60	Endo 70
$t=60$
Est.	9.916	9.934	-*	-	0.015	0.007	-	-	9.931	9.941	-	-
SE	0.011	0.008	-	-	0.004	0.002	-	-	0.008	0.007	-	-
$t=70$
Est.	19.449	19.551	19.495	-**	0.085	0.038	0.064	-	19.535	19.589	19.559	-
SE	0.025	0.022	0.026	-	0.009	0.004	0.008	-	0.021	0.021	0.022	-
$t=80$
Est.	28.098	28.408	28.317	28.175	0.224	0.103	0.140	0.196	28.322	28.511	28.457	28.371
SE.	0.042	0.040	0.043	0.042	0.013	0.008	0.012	0.013	0.037	0.039	0.039	0.037

^*endoscopy at 60 or 70 for $t=60$ is same as no endoscopy.

^**endoscopy at 70 for $t=70$ is same as no endoscopy.

3.3 |. Cause-Specific (Quality-Adjusted) Mean Life Years Lost

An advantage of illness-death model is its ability to quantify mean life years lost due to different causes, such as CRC and other causes. This helps us understand the impact of endoscopy on preventing CRC and, consequently, reducing CRC-related years lost. This is particularly valuable in low-incidence but severe diseases where the benefits might be overlooked in total RMST or QALY analyses. As shown in Figure 3 and Table 4, initiating endoscopy at age 50 would significantly reduce years lost due to CRC (0.099–0.037)/0.099 = 62.6% (p-value< 0.001) over a 30-years follow-up compared to no endoscopy. Starting endoscopy at older ages would also reduce years lost to CRC but to a lesser extent. There were no statistically significant differences in years lost to other causes between those who had endoscopy and those who did not (p-value > 0.05). When accounting for QoL scores, endoscopy would similarly reduce QALY lost due to CRC compared to no endoscopy (see Table 4 and SM Figure S2).

FIGURE 3.

Mean life years lost due to CRC (left panel) and other causes (right panel) estimates and 95% CI by screening status and age evaluated over [50, 60) (top panel), [50, 70) (middle panel), and [50, 80) (bottom panel), respectively.

TABLE 4.

Summary of (quality-adjusted) cause-specific mean life years lost at $t=60,70,\text{and}80$ for no endoscopy (No Endo) and endoscopy starting at ages 50 (Endo 50), 60 (Endo 60), and 70 (Endo 70).

Years lost	Other Causes				CRC				Quality-Adjusted CRC
	No Endo	Endo 50	Endo 60	Endo 70	No Endo	Endo 50	Endo 60	Endo 70	No Endo	Endo 50	Endo 60	Endo 70
$t=60$
Est	0.061	0.057	–*	–	0.002	0.001	–	–	0.001	4e-4	–	–
SE	0.008	0.008	–	–	0.001	2e-4	–	–	4e-4	2e-4	–	–
$t=70$
Est	0.418	0.392	0.406	–**	0.018	0.006	0.013	–	0.014	0.005	0.010	–
SE .	0.020	0.021	0.021	–	0.003	0.001	0.003	–	0.002	0.001	0.002	–
$t=80$
Est	1.509	1.426	1.446	1.480	0.099	0.037	0.056	0.086	0.076	0.028	0.043	0.066
SE	0.034	0.038	0.037	0.034	0.011	0.004	0.008	0.010	0.008	0.003	0.006	0.008

^*endoscopy at 60 or 70 for $t=60$ is same as no endoscopy.

^**endoscopy at 70 for $t=70$ is same as no endoscopy.

3.4 |. Burden-to-Benefit Analysis

Although earlier endoscopy extends lifetime and reduces mean life years lost to CRC, it also increases the burden on individuals and societal resources due to more frequent lifetime interventions, adhering to the guideline of endoscopy every 10 years⁹. To understand the balance of benefits relative to burden, we estimated the number of endoscopies assuming adherence to CRC guidelines, as described in Section 2.4.4. As expected, earlier intervention would result in a higher total number of endoscopies during the follow-up (SM Table S4) and greater QALY gains (left panel in Figure 4). To quantify the burden-to-benefit tradeoff, we calculated the ratio of incremental endoscopies to incremental QALY (right panel in Figure 4), analogous to the incremental cost-effectiveness ratio (ICER). When the WHI was initiated, endoscopy was not widely adopted in the U.S., with recommendations for starting endoscopy at age 50 introduced over a decade later by the US Preventive Services Task Force. Considering this, we used no endoscopy as the benchmark strategy and evaluated starting ages of 50, 55, 60, 65 and 70 years, adding intermediate ages for a clearer trend. The ICER ranged from 13.9 to 18.9 on average, with all strategies falling within the typically acceptable threshold of <50²⁷. Among the evaluated strategies, starting endoscopy at age 50 would yield the highest QALY. These findings provide robust real-world evidence supporting CRC screening with endoscopy at an early age. We also evaluated the ICER under alternative screening schedules: every 5 years and every 3 years. The results are presented in Supplemental Table S4 and Figure S3 in SM Section 4.4). Despite the increasing number of endoscopies with the 5-year repeated schedule, the ICER estimates remain below the commonly used willingness-to-pay threshold of 50 across all initiation ages and the greatest QALY gain is observed when screening is initiated at age 50. However, when the screening schedule is reduced to every 3 years, initiating screening at age 55 becomes the preferred strategy, as it yields the highest QALY gain while maintaining the ICER below the threshold.

FIGURE 4.

Scatterplot of QALY gained vs. total endoscopies per 1000 individuals (left) and ICER for initial intervention ages 50–70 years compared to no screening (right).

We assessed the proportional hazards assumptions for all models, and model (9) showed a modest violation (p-value = 0.04). Further investigation suggested this was due to the non-proportional effect of age at diagnosis (p-value < 0.01, SM Figure S4). This may reflect the fact that cancer is a major life event and has a strong impact on mortality. In addition, age at diagnosis is a continuous variable spanning a wide range, and the assumption of a constant hazard ratio may be more susceptible to violation. To account for this, we fit a natural cubic spline to model the time-varying effect of age at diagnosis (see Supplemental Figure S6 in SM Section 4.5). This modeling had minimal impact on the regression coefficient estimates for other risk factors (SM Table S5), and the estimates for RMST, QALY and cause-specific mean life years lost remained consistent, although with slightly larger SEs than those from the illness-death models (SM Table S6).

4 |. SIMULATION

We conducted simulation studies to examine the finite-sample performance of the proposed illness-death modeling approach. Two simulation settings were considered. The first setting (Setting I) aims to assess the finite-sample properties of proposed approach by modeling the disease process under a wide range of scenarios. The second setting (Setting II) is designed to mimic the real-data example with a relatively low disease incidence. In both settings, we assessed the RMST and cause-specific mean years lost due to the disease or other causes comparing no intervention and initiating intervention at age 50 years.

4.1 |. Simulation Setting I

In this setting, we generated data under a wide range of scenarios. Specifically, the transition hazard functions were modeled using Weibull distributions: ${\lambda }_{01}(u\mid Z(u;S),X)=5.2/{56.3}^{5.2}{t}^{5.2-1}\text{exp}\{-1.4Z(u;S)+0.5X\}$ and ${\lambda }_{02}(u\mid Z(u;S),X)=5.9/{83.0}^{5.9}{t}^{5.9-1}\text{exp}\{-0.05Z(u;S)+0.4X\}$. Here, $X$ was a risk score representing aggregated lifestyle, environmental, and genetic risk factors. While individual factors may be discrete, the aggregation of these factors was typically continuous. Thus, we generated $X~\text{Normal}(\mathrm{0,1})$. The age at starting intervention, $S$, was modeled as a function of the risk score $X$ from an exponential distribution with mean $50\text{exp}(0.53X)$. For individuals who developed the disease, the time from diagnosis to death was generated from an exponential distribution with mean $45\text{exp}(-0.3X+0.05T)$. For the censoring distribution, we considered two scenarios: (i) censoring time $C$ was independent of $T,D,S$ and $X$, sampled from Uniform(40, 100); (ii) $C$ was conditionally independent of $T$ and $D$ given $X$, generated using a Cox model with a baseline hazard function reflecting the US population age distribution and hazard ratio $\text{exp}(X)$. Both scenarios resulted in similar incidence and mortality rates: ~25% of individuals developed the disease, ~5% died before disease onset, and the overall death rate was about 21%. The sample size was set to be $n=\mathrm{2,500}$ and a total of 300 datasets were generated under each scenario. For evaluating bootstrap-based inference, 50 bootstrap samples were generated for each simulated dataset.

We compared the proposed illness-death approach with an “overall-mortality” approach that models only death as the terminal event and ignores the intermediate disease process. Specifically, for the overall-mortality approach, we considered time to death as the outcome and fitted a Cox proportional hazards model, ${\lambda }^{*}(t\mid \bar{Z}(t;S),X)={\lambda }_0^{*}(t)\text{exp}\left(\beta Z(t;S)+{\gamma }^{\prime }X\right)$, and subsequently estimated the RMST based on this model. For both approaches, RMST was evaluated over the age range [40, 70] with intervention at 50 years and no intervention.

Table 5 summarizes the results from both approaches. Under both censoring scenarios, the proposed estimator exhibited no bias, with bootstrap-based SEs closely matching with empirical standard deviations (ESD), and coverage probabilities close to 95% across all states (disease-free and disease) and overall. In comparison, the overall mortality approach yielded biased RMST estimates for the intervention group, with coverage probabilities dropping to ~30.0% under both censoring scenarios, respectively. For the no-intervention group, while both the proposed and overall-mortality approaches were unbiased, the proposed estimators were more efficient. We also examined the performance of the proposed approach for estimating the cause-specific mean life years lost due to disease and other causes (Table 6). These estimators were unbiased, with bootstrap-based SEs closely matching ESD and coverage probabilities near 95%.

TABLE 5.

Summary of RMST estimates under intervention (Int) and no intervention (No Int) under Simulation Setting I.

Scenario (i): Independent censoring
(a) Proposed approach
	Disease-free				Disease to death				Total
	True		Proposed		True		Proposed		True		Proposed
	No Int	Int	No Int	Int	No Int	Int	No Int	Int	No Int	Int	No Int	Int
Mean*	11.833	14.646	11.812	14.617	3.014	2.333	3.082	2.416	14.846	16.979	14.894	17.033
ESD*			0.470	0.454			0.202	0.179			0.433	0.407
SE*			0.458	0.448			0.203	0.181			0.433	0.407
CP*			0.933	0.950			0.953	0.943			0.937	0.945
(b) Overall mortality approach
	True		Overall death
			No Int	Int
Mean	14.846	16.979	14.940	17.995
ESD			0.472	0.408
SE			0.491	0.403
CP			0.947	0.300
Scenario (ii): Conditional independent censoring
(a) Proposed approach
	Disease-free				Disease to Death				Total
	True		Proposed		True		Proposed		True		Proposed
	No Int	Int	No Int	Int	No Int	Int	No Int	Int	No Int	Int	No Int	Int
Mean	11.833	14.646	11.813	14.611	3.014	2.333	3.087	2.422	14.846	16.979	14.900	17.033
ESD			0.479	0.465			0.210	0.186			0.436	0.405
SE			0.463	0.449			0.208	0.184			0.441	0.402
CP			0.947	0.937			0.950	0.950			0.950	0.943
(b) Overall mortality approach
	True		overall death
			No Int	Int
Mean	14.846	16.979	14.830	17.946
ESD			0.458	0.400
SE			0.489	0.401
CP			0.947	0.333

^*Mean, ESD, SE and CP are the mean, empirical standard deviation of estimates, mean of the bootstrap-based SE, and coverage probability of 95% CI over 300 datasets, respectively.

TABLE 6.

Summary of cause-specific mean life years lost under intervention (Int) and no intervention (No Int) in Simulation Setting I.

	Death due to CRC				Death due to other causes
	True		Proposed		True		Proposed
	No Int	Int	No Int	Int	No Int	Int	No Int	Int
Scenario (i): Independent censoring
Mean	13.559	11.125	13.500	11.081	1.754	2.033	1.742	2.021
ESD			0.442	0.426			0.267	0.193
SE			0.454	0.429			0.274	0.196
CP			0.950	0.943			0.950	0.940
Scenario (ii): Conditional independent censoring
Mean	13.559	11.125	13.493	11.082	1.754	2.033	1.744	2.020
ESD			0.445	0.423			0.284	0.199
SE			0.456	0.424			0.291	0.209
CP			0.957	0.940			0.953	0.950

4.2 |. Simulation Setting II

In this setting, we generated the data to mimic the CRC study described in Section 3. Time to CRC was generated based on a Weibull model fitted to CRC age-specific SEER incidence rates, and age at death was generated using a Weibull model fitted to US mortality data. All other settings were same as in Setting I. Each simulation included 100 datasets with $n=\mathrm{20,000}$ individuals under the conditional independence censoring scenario. On average, 2.0% of individuals developed the disease, 15.0% died before developing the disease, and 1.4% died after developing the disease.

Table 7 summarizes the RMST estimators from both the proposed illness-death and overall mortality approaches. Both methods performed well in estimating the total RMST. For the proposed approach, RMST estimates for each state (disease-free and disease) also showed good performance. Both estimators showed that intervention extended total lifetime, although modestly.

TABLE 7.

Summary of RMST under intervention (Int) and no intervention (No Int) under Simulation Setting II.

(a) Proposed approach
	Disease-free				Disease to death				Total
	True		Proposed		True		Proposed		True		Proposed
	No Int	Int	No Int	Int	No Int	Int	No Int	Int	No Int	Int	No Int	Int
Mean	21.483	22.006	21.487	22.016	0.786	0.514	0.782	0.516	22.269	22.520	22.269	22.532
ESD			0.188	0.123			0.063	0.041			0.177	0.115
SE			0.198	0.123			0.066	0.040			0.189	0.120
CP			0.950	0.960			0.960	0.930			0.950	0.980
(b) Overall mortality approach
	True		Overall death
			No Int	Int
Mean	22.269	22.520	22.446	22.731
ESD			0.179	0.118
SE			0.191	0.122
CP			0.950	0.960

However, by modeling the entire process, the illness-death approach provided additional insights, revealing that the intervention primarily prolonged the disease-free state, consistent with its intention to prevent or delay disease onset. This observation is further supported by the analysis of cause-specific mean life years lost (Table 8), where the intervention significantly reduced mean life years lost due to CRC, with minimal changes in years lost to other causes. These findings were consistent with the results from the WHI CRC data.

TABLE 8.

Summary of cause-specific mean life years lost under intervention (Int) and no intervention (No Int) in Simulation Setting II.

	Death due to disease				Death due to other causes
	True		Proposed		True		Proposed
	No Int	Int	No Int	Int	No Int	Int	No Int	Int
Mean	0.721	0.518	0.714	0.510	7.007	6.958	7.019	6.953
ESD			0.059	0.041			0.183	0.113
SE			0.062	0.041			0.190	0.117
CP			0.970	0.950			0.950	0.960

In summary, when the disease is common, as shown in Simulation I, the overall-mortality approach is likely to suffer from model misspecification of the intervention effect on the healthy-to-death process if the intervention primarily affects the healthy-to-disease transition, thereby misrepresenting its impact on post-disease mortality and yielding biased estimators. In this situation, the proposed illness-death approach is also more efficient, as it models the healthy to disease and disease to death processes more accurately for both the intervention effect and the effect of age at disease diagnosis on mortality. For rare-diseases, as shown in Simulation II, the overall-mortality approach shows minimal bias because relatively few death are mediated through the disease process. Nonetheless, the illness-death model offers additional insight by decomposing mean life years lost into cause-specific components, providing a clearer picture of the intervention’s impact that cannot be obtained from the overall-mortality approach.

We further conducted additional simulation studies to examine the robustness of the proposed estimators to the violation of no unmeasured confounding assumption (A3) and the results are presented in SM Section 5. Specifically, we focused on our WHI application to assess the potential impact of unmeasured factors that may influence both screening behavior and outcomes. Our primary WHI analysis already adjusts for a wide range of lifestyle and health-related covariates. Recent findings from the 2023 National Health Interview Survey (NHIS) indicate that, in addition to education and race/ethnicity, access to care, including insurance coverage and having a usual care provider, is among the strongest predictors of screening uptake³⁷. To approximate the effects of these factors, we generated two binary covariates representing insurance coverage and usual care provider, each with a prevalence of approximately 3% and odds ratios of 4.4 and 3.5 for intervention uptake, respectively. We further allowed these covariates to be associated with a 1.2-to-2.0-fold increased risk of colorectal cancer and mortality. Under these scenarios, our proposed estimators performed well with bias less than 5% (Supplemental Table S7). We also examined a setting where the unmeasured confounders were relatively common with 20% but had moderate association with intervention uptake. The proposed estimators remained robust with minimal bias in this scenario as well (Supplemental Table S8).

We also examined the impact of violating the proportional hazards assumption (SM Section 6) by generating data from models with time-varying coefficients. As shown in Supplemental Table S9, the RMST estimates obtained using the proportional hazards models exhibited minimal biases, indicating the robustness of our approach even when the proportional hazards assumption is not satisfied.

5 |. CONCLUSIONS

This paper presents a causal framework for evaluating the burdens and benefits of the timing of an intervention. Our approach introduces a unified measure that encompasses common metrics (e.g., RMST and number of interventions). Using a illness-death model, both benefits and burdens can be estimated across different health and disease states. Importantly, it provides a clearer picture of the benefits of intervention through cause-specific mean life years lost, particularly when the disease is relatively less frequent and the impact on disease prevention is obscured in overall mortality analyses. It is also able to capture the nuanced effects of interventions on targeted diseases. In comparison to the traditional microsimulation models commonly used in benefit-burden studies, which rely on unobservable assumptions about disease progression, our approach provides a data-driven alternative that leverages observational data with a causal modeling framework to achieve interpretable, empirically grounded estimates. We establish the asymptotic properties of the proposed estimators including consistency and asymptotic normality. Extensive simulation results confirm that the proposed estimator accounting for illness-death are more accurate than conventional overall mortality analyses. Application to CRC data demonstrated that preventive intervention improves overall life expectancy, particularly disease-free years. Initiating intervention at age 50 would offer the greatest improvement compared to no intervention, while remaining within acceptable ICERs.

The Cox proportional hazards model remains a cornerstone of biomedical research due to its practical advantages and established utility. However, its validity hinges on the proportional hazards assumption, which, if violated, can compromise accuracy. In our application, we detected a modest violation related to age at diagnosis in the sojourn model (9) and addressed it using splines to allow time-varying intervention effects. There are minimal changes to the findings in the application. For more complex relationships to account for heterogeneous intervention effects, advanced techniques such as survival trees³⁸ or inverse probability weighting³⁹ could be considered, though these approaches introduce additional challenges, particularly with semi-competing risks data. Future research in this area may yield valuable advancements.

This work focuses on the benefits and burdens of preventive intervention, distinguishing them from early detection screening. Early detection involves identifying cancer during its pre-clinical phase, often leading to better prognoses but also introducing potential biases such as length bias and over-diagnosis. In contrast, preventive intervention aims to prevent cancer from occurring in the first place, for instance, by removing precursor lesions before they can develop into cancer. For example, in CRC, endoscopic procedures can eliminate lesions, effectively preventing cancer development and obviating the need for further treatment. Preventive interventions target healthy individuals without cancer, rendering biases like length bias irrelevant in this context. Furthermore, we focused on the age to initial screening. Initial interventions are advised to individuals before the study entry at which time-varying confounders are not necessarily entailed. Once screening is initiated, individuals will follow a pre-determined schedule to repeat the procedure. Therefore, the time-invariant confounders collected at baseline are adequate to account for the self-selection in the intervention. While our method can be readily extended to incorporate time-varying confounders as stated in Section 2.2, we develop the methodology with time-invariant confounders to reflect the practical need for informing the initial age to screening decisions that are typically made at a single point in time based on baseline risk factors.

We applied our method to WHI data to empirically estimate the expected life years gained from preventive screenings, focusing on age at disease diagnosis and mortality using an illnessdeath model. A limitation of this model is that it does not explicitly incorporate precursor states such as adenomas or advanced adenomas. Although, in principle, the framework could be extended to include such precursor states (e.g., healthy → adenoma → CRC), modeling the corresponding transition hazard functions based on observational data poses substantial challenges. Specifically, precursor states are only detectable through endoscopy, and once detected, they are typically removed, thereby truncating the natural disease history and complicating model identification⁸. Our primary goal is to estimate gains in life years from CRC prevention rather than to reconstruct the unobserved natural history. From this perspective, the simplified illnessdeath model remains informative and can provide unbiased estimates, provided that models (8) and (9) adequately capture the intervention effect. In the WHI data analysis, we assumed perfect adherence to the USPSTF-recommended repeat screening schedule and a constant protective effect following screening initiation. While these assumptions simplify modeling, they may lead to optimistic estimates of benefit for individuals who undergo only a single screening and may overestimate the total number of screenings performed. To more accurately capture screening effects, data on repeated screenings and findings such as polyps or adenomas at screening examinations are needed. Future work in this direction would be especially valuable.

There are limitations to the WHI cohort. First, it includes only participants aged 50 years and older. Recent guidelines from the US Preventive Services Task Force recommend lowering the starting age for CRC preventive intervention to 45 years⁵, in response to the rising incidence among younger individuals⁴⁰. Thus, our study cannot directly assess the benefits and burdens of starting screening before age 50. Even if WHI had enrolled women younger than 50, preventive screening prior to this age was rarely performed before the guideline changed, as the insurance did not cover endoscopies without symptoms prompting diagnostic evaluation, which introduces potential selection bias. Unfortunately, observational data on the impact of screening in younger population will take years to accumulate, as these recommendations have only recently been adopted. A second limitation is that WHI includes only women who participated the study in mid-1990s. The recommendation based on the WHI analysis here should therefore be interpreted in context, that is, as guidance on what might have been optimal for that cohort. Generalizing these findings to contemporary populations requires careful consideration of shifts in risk factor distributions over time, as well as potential changes in the effect of risk factors on disease processes, although the latter is likely more stable, the former (e.g., sedentary life style) may be substantial. Nevertheless, the methodological framework we propose is broadly applicable and could be extended to cohorts that include men and/or individuals younger than 50.

Cost-effectiveness analysis for cancer preventive intervention is complex and requires considerations different from those for early detection or treatment. While microsimulation modeling remains a primary tool in this domain, real-world data, such from the WHI, offer invaluable insights despite their complexities and limitations. Empirical data-based modeling should be viewed as complementary to microsimulation. In this context, our proposed causal inference framework for analyzing the cost-effectiveness of time-varying intervention, such as the age of first endoscopy for CRC, provides a starting point for addressing these important questions using real-world data.