Moving From Individualized Risk‐Based Prevention to Benefit‐Based Prevention: Estimating Individualized Life‐Years Gained From Prevention Services as a Basis for Eligibility

ABSTRACT

The current bedrock of precision prevention is selecting high‐risk individuals for screening or other prevention services under the assumption that those at highest risk would have the highest benefit from prevention services. However, this may not hold when disease risk and competing mortality are highly correlated. In such cases, risk‐based prevention may preferentially select older individuals with multiple comorbidities who would have substantially reduced life‐years gainable from the service and increased risks of harm from any resulting surgical procedures. For such prevention services, we propose a benefit‐based selection strategy in which individuals are selected according to their expected gain in life‐years (i.e., difference in mean survival time with and without the prevention service). We estimate the expected gain in life‐years for individuals in a target screening population by combining data from a randomized trial, which may not be population‐representative, and data from a population‐representative survey that has larger sample size, more covariates, and longer follow‐up time to evaluate mortality than the trial. We derive the Taylor‐linearized variances for the estimated expected gain in life‐years that take into account the randomness due to both trial and survey sample. We show that benefit‐based selection of ever‐smokers for lung‐cancer screening can identify individuals with more favorable benefit‐harm trade‐offs compared to risk‐based selection. Using simulation studies, we examine the conditions in which one strategy may be preferable over the other.

Abbreviations

FP

finite population

MST

mean survival time

NHIS

US National Health Interview Survey

NLST

National Lung Screening Trial

RMST

restricted mean survival time

USPSTF

US Preventive Services Task Force

1. Introduction

The “precision prevention”, or “risk‐based strategy,” identifies individuals at high‐risk of a disease and intervenes only on them to prevent future morbidity and mortality from the disease [1]. With a discriminant risk model, such a risk‐based strategy can enhance the efficiency of disease screening without sacrificing sensitivity compared to population‐wide screening [2, 3]. Risk‐based strategies presume that individuals at highest disease risk also experience the greatest benefit, often measured as prevented deaths from the disease [4]. However, preventing disease‐specific mortality may not provide appreciable gains in life‐expectancy, another measure of benefit, when disease risk and competing mortality are highly correlated. In such cases, risk‐based selection can preferentially select older individuals with multiple disease‐correlated comorbid conditions. As such, many cancer screening guidelines declare a cessation age beyond which screening is considered more harmful than beneficial [5, 6, 7, 8].

To avoid selecting low‐benefit individuals (when using risk alone) or excluding high‐benefit older individuals (when using risk with a cessation age), we propose directly developing individualized calculators of life expectancy gained from a prevention service. Individualized life‐gained combines several important clinical considerations: disease risk, life expectancy, and the effect of the intervention on mortality [9]. In contrast to risk‐based selection, benefit‐based selection (i.e., selecting those with highest life‐gained from the prevention service) could circumvent the need for ages at initiation or cessation of screening by naturally excluding both younger individuals without high enough disease risk to benefit from disease screening and older individuals without enough remaining life expectancy to gain life‐years from prevention; guidelines may only need to specify a minimum life expectancy gain for screening to be worthwhile to the patient [10].

Benefit‐based selection has previously been suggested for lung cancer screening [10] to address concerns that risk‐based lung screening may not substantially increase life‐expectancy [11] or may preferentially select older individuals with smoking‐related comorbidities who are not eligible for curative lung cancer resection [12]. As such, an individualized life‐gained calculator is currently recommended by the American College of Chest Physicians (ACCP) to determine who to screen for lung cancer [13]. Despite growing interest in applying these approaches to other areas, there is scant literature on how to develop such models, their statistical properties, and under what conditions benefit‐based prevention strategies would be preferable over risk‐based prevention strategies.

The individualized life‐gained from a prevention service is the difference in covariate‐specific mean survival time (MST) for those who receive the prevention service and those who do not. The MST, restricted to a time $\tau$ (restricted mean survival time, RMST), has previously been suggested as an alternative to the hazard ratio for comparing intervention and control arms in randomized trials [14, 15]. For estimating individualized life‐gained from a prevention service, it may be necessary to project the RMST beyond the trial data because (1) the duration of the trial may be short and the trial‐estimated RMST may not be proportional to the MST, (2) trial participants tend to be healthier than similar individuals from the target population [16], and (3) some covariate values (e.g., age, comorbidities, race/ethnicity) present in the target population may be underrepresented in the trial due to trial inclusion/exclusion criteria or differential response to recruitment efforts.

In this article, we provide a formal statistical treatment for estimating individualized life‐gained from a prevention service by using component models for disease‐specific mortality, competing mortality, and treatment effects. To allow individualized life‐gained to be estimated for covariate combinations found in the target population, we develop mortality models using a population representative survey sample. We assume that the mean disease‐specific treatment effects estimated from the trial are portable to the target population. This assumption can be relaxed to estimate conditional and mean treatment effects for the target population using a non‐representative trial when certain identifiability conditions are met [17], but in our motivating example, some covariate values in the target population were not randomized into the trial, precluding use of these approaches. Instead, we focused on quantifying the treatment effect of a prevention service as a time‐dependent, disease‐specific mortality hazard ratio. For example, for early detection screening, the hazard ratio typically has a “bathtub” shape between the beginning of screening to a time after screening ends [18].

We revisit use of the individualized life‐gained approach in lung‐cancer screening. We estimate the individualized gain in life‐expectancy over 19 years after undergoing three rounds of annual lung screening with low‐dose computed tomography (CT), by combining the hazard ratio of lung screening on cause‐specific mortality as observed in the National Lung Screening Trial (NLST) [19], with the mortality experience of the target population using US‐representative surveys [20]. In comparison to the earlier effort [10], our RMST‐based approach avoids extrapolating the mortality experience beyond the 19 years observed in the survey data. We apply the prediction model to contrast risk‐based versus life‐gained‐based selection for lung screening in the contemporary US ever‐smoker population.

Using simulation studies, we compared the performance of using individualized life‐gained from a prevention service versus using absolute risk of death from the disease in identifying which individuals in a population most benefit from receiving the prevention service. We considered several settings including different length of follow‐up data for estimation of risk or life‐gained models, different duration of prevention service effect on disease‐specific mortality, and different directionality of prevention service effects on competing mortality.

2. Methods

2.1. Individual Life‐Expectancy Gained From a Prevention Service

To identify high‐benefit candidates for a prevention service, we need to estimate the individual life‐expectancy gained from the prevention service for all members, $j=1,\dots ,M$, of the target finite population (FP) under consideration for the service. Let $T$ be the time of death relative to the time the preventive service is to be offered, $\mathbf{x}$ be the covariate vector, and $z$ be the binary variable that indicates receiving the prevention service. We adopt a cause‐specific hazard framework as the preventive service may reduce death from some causes but not others. The life expectancy gained from $z$ is the change in the mean remaining life expectancy with and without the prevention service, and is the area between the two survival curves as follows:

$$\begin{array}{cc}\mathrm{\Delta }\mu (z;\mathbf{x})& =\mu (\mathbf{x},z=1)-\mu (\mathbf{x},z=0)\\ & ={\int }_0^{{\tau }_{\infty }}\{S(u;\mathbf{x},z=1)-S(u;\mathbf{x},z=0)\}\mathit{du},\end{array}$$

where $S(t;\mathbf{x},z)$ is the survival function of $T$ and ${\tau }_{\infty }=\inf \left\{t\in {\mathrm{ℝ}}_{(>0)}\mid S(t;\mathbf{x},z)=0\right\}$. The survival function is given as follows:

$$S(u;\mathbf{x},z)=\exp \left\{-{\int }_0^t\sum _{k=1}^K{\lambda }_k\left(v;{\mathbf{x}}_k,z\right)\mathit{dv}\right\},$$

where ${\lambda }_k\left(v;{\mathbf{x}}_{\mathbf{k}},z\right)$ is the hazard for the $k$th cause‐specific mortality, $k=1,\dots ,K$, under a Cox model as follows:

$${\lambda }_k\left(v;{\mathbf{x}}_k,z\right)={\lambda }_{k0}(t)\exp \left({\mathit{\beta }}_k^{\top }{\mathbf{x}}_k\right)\exp \left\{{\gamma }_k(t)z\right\},$$ (1)

${\lambda }_{k0}(t)$ is the baseline hazard function, ${\mathit{\beta }}_k$ are coefficients for covariates ${\mathbf{x}}_k$ (which is a subset of $\mathbf{x}$), and ${\gamma }_k(t)$ is the possibly time‐varying effect of prevention service $z$ (e.g., ${\gamma }_k(t)={\gamma }_{k0}$ when $t\le t_0$, and ${\gamma }_k(t)=0$ when $t>t_0$).

A common scenario is where the prevention service reduces death from a targeted cause for some time ${\tau }_0$ and the effect of the prevention service on other causes of death is negligible (e.g., many vaccines or disease screening). In these scenarios, $\mathrm{\Delta }\mu (z;\mathbf{x})$ can be decomposed as followed (Derivation in Supporting Information: Appendix A),

$$\mathrm{\Delta }\mu (z;\mathbf{x})={\int }_0^{{\tau }_0}\mathrm{\Delta }{r}_u(z;\mathbf{x})\mathit{du}+\mathrm{\Delta }{r}_{{\tau }_0}(z;\mathbf{x}){\int }_{{\tau }_0}^{{\tau }_{\infty }}S\left(u|T>{\tau }_0;\mathbf{x}\right)\mathit{du},$$ (2)

where $\mathrm{\Delta }{r}_u(z;\mathbf{x})=r_u(z=1;\mathbf{x})-r_u(z=0;\mathbf{x})$ is the change in the absolute risk of death from the targeted cause at time $u$ after providing the prevention service.

Therefore in these common scenarios, the life‐gained from the prevention service is the sum of (1) the reduction in the absolute risk of death from the targeted cause integrated over the duration of the effect from the prevention service and (2) the total reduction in the absolute risk of death from the primary cause times the remaining life‐expectancy when the effect from the prevention services ends.

2.2. Estimation by Combining Trial Data and a Target Population Representative Survey

We consider estimation in the setting where the prevention service is new and its effect on mortality has only been assessed in $m$ individuals enrolled in a clinical trial, $s_A\subset \mathrm{FP}$, that may be a non‐random subset of the FP. We assume that we have a mortality‐linked representative survey, $s_B\subset \mathrm{FP}$, such as the US National Health Interview Survey (NHIS) [20], consisting of $n$ individuals who did not receive the preventive service and is randomly selected from the FP under a stratified multistage cluster sampling design. While this is just one possible data setting that might arise, we aim to describe estimation such that it can be adapted to other data settings.

Let $A=T\wedge C$, where $C$ is the censoring time and $D$ takes values of zero, indicating right‐censoring, or $1,\dots ,K$, indicating death from a specific cause. From $s_A$ we observe $\left(A_i,D_i,{\mathbf{x}}_i,z_i\right)$, $i,\dots ,m$. From $s_B$ we observe $\left(w_{\mathit{lji}},A_{\mathit{lji}},D_{\mathit{lji}},{\mathbf{x}}_{\mathit{lji}},z_{\mathit{lji}}=0\right)$, where $l=1,\dots ,L$ indexes the sampling strata in the population, $j=1,\dots ,n_l$ indexes the primary sampling units (PSU) randomly selected from stratum $l$, $i=1,\dots ,n_{\mathit{lj}}$ indexes individuals randomly selected from the $j$th PSU in stratum $l$, and $w_{\mathit{lji}}$ is the survey sample weight for individual‐$\mathit{lji}$ after accounting for undercoverage and nonresponse. We assume censoring is noninformative in the target population (represented by the weighted survey sample), that is, censoring times are independent of future event times conditional on observed covariates.

The parameters in Equation (1), ${\gamma }_k(t)$, $k=1,\dots ,K$, and $\left\{{\lambda }_{k0}(t),{\mathit{\beta }}_k\right\}$, $k=1,\dots ,K$, can be estimated by fitting separate Cox models to $s_A$ and $s_B$, respectively. Because mortality follow‐up in $s_B$ may be limited, we may only be able to estimate ${\lambda }_{k0}$ up to a time $\tau$ and $\mathrm{\Delta }{\mu }_{\tau }(z;\mathbf{x})$, the life expectancy gained from $z$ restricted to time $\tau$, rather than $\mathrm{\Delta }\mu (z;\mathbf{x})$, the full life‐expectancy gained. A semi‐parametric estimator of ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ obtained using the trapezoidal rule is as follows:

$${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})=\sum _{t_j}\left(t_{j+1}-t_j\right)\frac{\begin{array}{c}\left[\left\{\widehat{S}\left(t_j;\mathbf{x},z=1\right)-\widehat{S}\left(t_j;\mathbf{x},z=0\right)\right\}\right.\\ \left.+\left\{\widehat{S}\left(t_{j+1};\mathbf{x},z=1\right)-\widehat{S}\left(t_{j+1};\mathbf{x},z=0\right)\right\}\right]\end{array}}{2}$$ (3)

where $t_j:t_0=0,t_1,\dots ,t_J,t_{J+1}=\tau$ are the ordered, unique death times in $s_B$ and

$$\widehat{S}\left(t_j;\mathbf{x},z\right)=\exp \left[-\sum _{k=1}^K{\int }_0^{t_j}{\widehat{\lambda }}_{k0}(v)\exp \left({\widehat{\mathit{\beta }}}_k^{\top }{\mathbf{x}}_k\right)\exp \left\{{\widehat{\gamma }}_k(v)z\right\}\mathit{dv}\right]$$

is the estimated survival function, with $\left\{{\widehat{\gamma }}_k(t),{\widehat{\lambda }}_{k0}(t),{\widehat{\mathit{\beta }}}_k\right\}$, $k=1,\dots ,K$ as the plug‐in estimators of the intervention effects, cause‐specific baseline hazards, and log‐hazard ratios of ${\mathbf{x}}_k$.

The prevention service effect on cause‐specific mortality, ${\gamma }_k(t)$, $k=1,\dots ,K$, can be estimated from the intervention trial $s_B$ using Cox models for cause‐specific mortality, which can handle time‐varying coefficients. For example, we can use a step function ${\widehat{\gamma }}_k(t)={\sum }_{p=1}^P{\widehat{\gamma }}_{\mathit{kp}}{g}_{\mathit{kp}}(t)$, where $p=1,\dots ,P$ are disjoint time intervals covering $(0,\tau ]$ (i.e., $\left(0,{\tau }_{k,1}\right],\dots ,\left({\tau }_{(k,P-1},\tau \right]$), ${\widehat{\gamma }}_{\mathit{kp}}$ is a constant, and $g_{\mathit{kp}}(t)=1$ if $t\in \left({\tau }_{p-1},{\tau }_p\right]$ and 0 otherwise.

The hazard ratios and baseline hazard, $\left\{{\widehat{\lambda }}_{k0}(t),{\widehat{\mathit{\beta }}}_k\right\}$, $k=1,\dots ,K$, can be estimated by fitting Cox regression models for cause‐specific mortality to the survey sample $s_B$. We assume conditional independence of competing event times (i.e., $T_k\perp T_{k^{\prime }}\mid \mathbf{x},k\ne k^{\prime }$). Let $Y_{k,\mathit{lji}}(t)$ take a value of 1 if individual‐$\mathit{lji}$ is still at risk for death from cause $k$ at time $t$ and 0 otherwise. The ${\widehat{\mathit{\beta }}}_k$, are solutions to the sample‐weighted estimating equations,

$$\mathit{U}\left({\mathit{\beta }}_k\right)=\sum _{l=1}^L\sum _{j=1}^{n_j}\sum _{i=1}^{n_{\mathit{lj}}}{w}_{\mathit{lji}}\cdot {\delta }_{\mathit{lji}}\left(t_{\mathit{lji}}\right)\cdot \left\{{\mathbf{x}}_{k,\mathit{lji}}-{\bar{H}}_k\left({\mathit{\beta }}_k,t_{\mathit{lji}}\right)\right\}=\mathbf{0},$$

where ${\delta }_{\mathit{lji}}(t)=\left\{{\delta }_{1,\mathit{lji}}(t),\dots ,{\delta }_{K,\mathit{lji}}(t)\right\}$ is a vector of cause‐specific death indicators such that ${\delta }_{k,\mathit{lji}}(t)=1$ if death from $k$ occurs at time $t$ or 0 otherwise for individual‐$\mathit{lji}$ and

$${\bar{H}}_k\left({\mathit{\beta }}_k,t_{\mathit{lji}}\right)={\widehat{S}}_k^{(0)}{\left({\mathit{\beta }}_k,t\right)}^{-1}{\widehat{S}}_k^{(1)}\left({\mathit{\beta }}_k,t\right)$$

is the weighted average risk profile of those at risk for death from cause $k$ at time $t$ with ${\widehat{S}}_k^{(u)}\left({\mathit{\beta }}_k,t\right)={\sum }_{l=1}^L{\sum }_{j=1}^{n_l}{\sum }_{i=1}^{n_{\mathit{lj}}}{w}_{\mathit{lji}}{Y}_{k,\mathit{lji}}(t)\exp \left({\mathit{\beta }}_k^{\top }{\mathbf{x}}_{k,\mathit{lji}}\right)\cdot {\mathbf{x}}_{k,\mathit{lji}}^{(u)}$, where ${\mathbf{x}}_{k,\mathit{lji}}^{(0)}=1$ and ${\mathbf{x}}_{k,\mathit{lji}}^{(1)}={\mathbf{x}}_{k,\mathit{lji}}$ [21]. The weighted Nelson‐Aalen estimator [22] for the non‐parametric cause‐specific baseline hazard is

$${\widehat{\lambda }}_{k0}(t)={\widehat{S}}_k^{(0)}{\left({\mathit{\beta }}_k,t\right)}^{-1}{N}_k(t),$$ (4)

where $N_k(t)={\sum }_{l=1}^L{\sum }_{j=1}^{n_l}{\sum }_{i=1}^{n_{\mathit{lj}}}{w}_{\mathit{lji}}{\mathit{dN}}_{\mathit{lji}}(t)$ is estimated from the number of cause‐specific deaths at $t$ in the $\mathrm{FP}$.

2.3. Variance Estimation Using Influence Functions

We derive the variance using influence functions, which measure each individual's contribution to the estimator. We use $R_i^{(A)}$, where $R_i^{(A)}=1$ if $i\in s_A$ and $R_i^{(A)}=0$ if $i\in \mathit{FP}-s_A$, and $R_i^{(B)}$, where $R_i^{(B)}=1$ if $i\in s_B$ and $R_i^{(B)}=0$ if $i\in \mathit{FP}-s_B$, to denote the indicators of being included in the trial sample and in the survey sample from the $\mathit{FP}$, respectively. Let ${\mathrm{\Delta }}_i\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}$ denote the influence function ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$, which can be obtained by ${\mathrm{\Delta }}_i\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}=\partial {\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})/\partial w_i$ for $i\in s_A\cup s_B$, with $w_i$ being the survey weight for $i\in s_B$ and $w_i=1$ for $i\in s_A$. Following [21, 23, 24], the $\mathit{FP}$ variance of ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ can be approximated by

$$\mathit{Var}\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}\approx \mathit{Var}\left[\sum _{i\in \mathit{FP}}{R}_i^{(A)}{\mathrm{\Delta }}_i\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}+\sum _{i\in \mathit{FP}}{R}_i^{(B)}{\mathrm{\Delta }}_i\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}\right],$$

where the variance is expressed in terms of the randomness due to $R_i^{(A)}$ and $R_i^{(B)}$. We further assume that the trial participation and survey sampling are independent, that is, $\mathit{cov}\left(R_i^{(A)},R_i^{(B)}\right)=0$ for $i,j\in \mathit{FP}$ [24]. The Taylor linearized variance estimator for ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$, considering randomness of the trial sample $s_A$ and the general stratified clustering sampling design $s_B$ (Supporting Information: Appendix A), is given by,

$$\begin{array}{cc}{\upsilon }_{\mathit{TL}}\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}& =\sum _{l=1}^L\frac{n_l}{n_l-1}\sum _{j=1}^{n_l}{\left[{\mathrm{\Delta }}_{\mathit{lj}.}\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}-{\bar{\mathrm{\Delta }}}_{l..}\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}\right]}^2\\ & +\frac{m}{m-1}\sum _{i=1}^m{\left[{\mathrm{\Delta }}_i\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}-{\bar{\mathrm{\Delta }}}_A\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}\right]}^2.\end{array}$$ (5)

In the first summand of Equation (5), ${\mathrm{\Delta }}_{\mathit{lj}.}\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}={\sum }_{i=1}^{n_{\mathit{lj}}}{w}_{\mathit{lji}}{\mathrm{\Delta }}_{\mathit{lji}}\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}$ is the sample weighted cluster total of individual influence functions ${\mathrm{\Delta }}_{\mathit{lji}}\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}$ for individual‐$\mathit{lji}\in s_B$ and ${\bar{\mathrm{\Delta }}}_{l..}\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}=n^{-1}{\sum }_{j=1}^{n_l}{\mathrm{\Delta }}_{\mathit{lj}.}\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}$ is the mean of cluster totals in strata $l$. In the second summand of Equation (5), ${\bar{\mathrm{\Delta }}}_A\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}=m^{-1}{\sum }_{j=1}^{n_l}{\mathrm{\Delta }}_i\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}$ is the trial sample mean of the influence function ${\mathrm{\Delta }}_i\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})\right\}$ for $i\in s_A$. The individual influence function is given by the following,

(6)

where ${\mathrm{\Delta }}_i\left({\widehat{\mathit{\beta }}}_k\right)$, ${\mathrm{\Delta }}_i\left\{{\widehat{\lambda }}_{k0}\left(t_{\mathit{kl}}\right)\right\}$, and ${\mathrm{\Delta }}_i\left\{{\widehat{\gamma }}_k\left(t_{\mathit{kl}}\right)\right\}$ in Equation (6) are the influence functions of ${\widehat{\mathit{\beta }}}_k$, ${\widehat{\lambda }}_{k0}\left(t_{\mathit{kl}}\right)$, and ${\widehat{\gamma }}_k\left(t_{\mathit{kl}}\right)$ and are given in equations B10‐12, B17‐B18, and B20‐23 in the Supporting Information: Appendix B.

When the number of unique death times in $s_B$ is very large, estimation can be computationally expensive. A fast, reliable approximation can be achieved by taking a systematic subsample of size $M$ $(M<<K)$ from the $K$ death times (e.g., $M=100$). This subsample should include the first ($t_1$), last ($t_J$), and approximately evenly spaced intermediate ordered event times.

3. Application to Identifying High‐Benefit Individuals for Lung Cancer Screening

We consider the problem of identifying who to screen in the contemporary US population of ever‐smokers, aged 40–84 years (hereafter referred to as the target population). Current US Preventive Services Task Force (USPSTF) guidelines recommend screening for individuals, aged 50–80 years, who smoked within the past 15 years and have $\ge$ 20 pack‐years of exposure [8]. Targeted screening of individuals at highest lung cancer risk [25] or with highest life‐gained benefit from screening [10] have been proposed as alternative approaches to identifying screening candidates.

We developed models for life‐years gained from undergoing three annual lung CT screens by combining extended follow‐up data from the NLST [26] and the NHIS data from years 1997–2001, which included 44 111 ever‐smokers, aged 40–84 years, representing 55 328 229 US individuals. We also constructed a model for the absolute risk of lung cancer from the NHIS data. We then evaluated the effectiveness of using (a) a life‐gained model, (b) an absolute risk model and (c) USPSTF recommendations in determining who to screen. For inference in the target population, we used the NHIS, years 2015–2018, which included 33 566 ever‐smokers, aged 40–84 years, representing 60 768 292 individuals. The NHIS is an annual cross‐sectional, multistage probability sample representing the non‐institutionalized US population for that year [20]. Long‐term mortality outcomes for NHIS, years 1997–2001, were available through linkage with the National Death Index [27].

The NLST randomized 53 454 current and former heavy smokers ($\ge 30$ pack‐years), aged 55–74 years, into two screening arms: three annual rounds of screening with either low‐dose CT or standard chest X‐ray, which has previously been shown to have no effect on lung cancer or other cause mortality [28]. Individuals were followed for mortality endpoints for a median of 12.3 years. Three annual rounds of CT screening reduced lung‐cancer mortality in the NLST for the first 6 years (HR = 0.808; 95% CI = 0.708–0.923) but had little effect on lung cancer mortality beyond 6 years (HR = 0.998; 95% CI = 0.902–1.105) (Figure 1). Screening provided no significant reduction in mortality from other causes (HR = 0.989; 95% CI = 0.947–1.032). In estimating life‐years gained from screening, we assumed that the hazard‐ratios for CT screening estimated from the NLST were transportable to the target population.

FIGURE 1.

Impact of three rounds of annual low‐dose CT screening on relative cause‐specific mortality in the NLST. Relative mortalities for each year (circles for death from lung cancer; stars for death from all other causes) are estimated as the ratio of the yearly event rates. Hazard ratios are estimated by fitting cause‐specific Cox models with the time‐varying effect of screening modeled as a step function. For death from lung cancer, HR(t) = 0.808 (95% CI = 0.708–0.923) for (0,6] and HR = 0.998 (95% CI = 0.902–1.105) for (6,14]. For other cause mortality, HR = 0.989 (95% CI = 0.947–1.032).

Due to trial exclusion criteria for various comorbidities [29], NLST participants were healthier than the target population represented in the NHIS, years 2015–2018 (mean number of comorbidities = 0.85 vs. 1.12; Table S1). Compared to the target population, NLST participants also fell into a narrower age range (55–74 vs. 40–84 years) and were not as racially and ethnically diverse (90% vs. 77% non‐Hispanic whites). In contrast, ever‐smokers, aged 40–84 years, in the NHIS, years 1997–2001, closely resembled the target population, but with slightly younger ages and slightly fewer non‐Hispanic whites. Therefore, we used the NHIS, years 1997–2001, to fit target population representative mortality models.

In our NHIS data, we had cause‐specific mortality through the end of 2006 (maximum follow‐up = 10 years) and overall mortality through the end of 2015 (maximum follow‐up = 19 years). Using the cause‐specific mortality data, we estimated ${\widehat{r}}_{\tau =10}(\mathbf{x})$, the 10‐year absolute risk of lung cancer death after accounting for competing mortality (derivation given in Supporting Information: Appendix C), and ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(y;\mathbf{x}),$ the gain in life expectancy after 10 years if screening began at the time of the NHIS survey (estimated using Equation 3). Selection using these models represents our primary comparison. Additionally, because our simulations in Section 4 show that using longer follow‐up $\tau$ to estimate ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(y;\mathbf{x})$ improves the resulting benefit‐based selection, we utilized the overall mortality data to estimate ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(y;\mathbf{x})$. Because 3 annual rounds of CT screening only reduce lung cancer mortality in the first 6 years (denoted as ${\tau }_1$), we can rewrite ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(y;\mathbf{x})$ as follows:

$$\begin{array}{l}{\widehat{\mathrm{\Delta }\mu }}_{\tau }(y;\mathbf{x})={\widehat{\mathrm{\Delta }\mu }}_{{\tau }_1}(y;\mathbf{x})+\left[\exp \left\{-{\widehat{\mathrm{\Lambda }}}_1\left({\tau }_1;\mathbf{x},z=1\right)\right\}-\exp \left\{-{\widehat{\mathrm{\Lambda }}}_1\left({\tau }_1;\mathbf{x},z=0\right)\right\}\right]\\ \hspace{0.17em}\times \exp \left\{-{\widehat{\mathrm{\Lambda }}}_2\left({\tau }_1;\mathbf{x}\right)+\widehat{\mathrm{\Lambda }}\left({\tau }_1;\mathbf{x},z=0\right)\right\}\times {\int }_{{\tau }_1}^{\tau }\exp \{-\widehat{\mathrm{\Lambda }}(u;\mathbf{x},z=0)\}\mathit{du},\end{array}$$ (7)

where ${\widehat{\mathrm{\Lambda }}}_1\left({\tau }_1;\mathbf{x},z\right)$, ${\widehat{\mathrm{\Lambda }}}_2\left({\tau }_1;\mathbf{x}\right)$, and $\widehat{\mathrm{\Lambda }}(u;\mathbf{x},z=0)$ are the estimated cumulative hazards of lung‐cancer mortality, other cause mortality, and overall mortality absent prevention services, respectively. Details of the derivation for Equation (7) are in the Supporting Information: Appendix D. Of note, while the survival functions constructed using the cumulative hazards from cause‐specific versus overall mortality are theoretically equivalent, the predicted survivals may differ, resulting in differences in the predicted life‐gained using Equations (3) versus (7). We compare survival and life‐gained predictions using the NHIS data in Supporting Information: Appendix E.

To estimate ${\mathrm{\Lambda }}_1(t;\mathbf{x},z=0)$ and ${\mathrm{\Lambda }}_2(t;\mathbf{x})$, we fit cause‐specific models to the sample weighted NHIS, years 1997–2001, with 10‐year follow‐up, subsetting to ever‐smokers, aged 40–84 years, who were lung cancer‐free at the time of the survey. For lung‐cancer mortality ($k=1$), we used the same predictors for ${\mathbf{x}}_{\mathbf{1}}$ as used in the previously published LCDRAT model: demographic variables (age, sex, race/ethnicity), socioeconomic variables (education levels, employment status), body mass index (BMI), smoking status (average smoking intensity measured in cigarettes per day, number of years in which the individual smoked, number of years since quitting, and total exposure measured in pack‐years), and emphysema [25]. For other‐cause mortality ($k=2$), predictors $x_2$ included $x_1$, whether participants had health problems requiring use of special equipment (e.g., cane, wheelchair, special bed), and 11 self‐reported smoking‐related comorbidities (hypertension, coronary heart disease, angina, heart attack, heart disease, stroke, diabetes, bronchitis, kidney disease, liver disease, and prior cancer). We also fitted the overall mortality model, $\mathrm{\Lambda }(t;\mathbf{x},z=0)$, with 19‐year follow‐up, using predictors $x_2$. The hazard ratios are given in Table S2.

We projected ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;\mathbf{x})$, ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;\mathbf{x})$, and ${\widehat{r}}_{\tau =10}(\mathbf{x})$ using the NHIS, years 2015–2018, treating the time of the NHIS survey as time zero (i.e., the time the preventive service would be offered). The median life‐gainable from undergoing 3 rounds of CT screening was 1.5 days after 10 years, 2.3 days after 19 years, and the median risk of dying from lung cancer after 10 years was 0.82%. The joint distribution of 10‐year risk and life‐gained from screening is shown in Figure S1.

Table 1 compares selection of individuals by the different approaches in the target population. In comparing selection strategies, we selected the highest p = 23.4% of the target population for screening by ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;\mathbf{x})$, ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;\mathbf{x})$, and ${\widehat{r}}_{\tau =10}(\mathbf{x})$, respectively, matching the percentage of the population that are USPSTF‐eligible. The threshold for being selected by the life‐gained‐based and the risk‐based approaches were ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;\mathbf{x})=3.7$ days, ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;\mathbf{x})=5.8$ days, and ${\widehat{r}}_{\tau =10}(\mathbf{x})=1.7\%$, respectively. Because individuals must be at high risk of dying from lung cancer to gain appreciable life‐expectancy from undergoing screening, individuals selected/not selected by life‐gained‐based approaches were often also selected/not selected by risk (19.4% selected by both ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;\mathbf{x})$ and ${\widehat{r}}_{\tau =10}(\mathbf{x})\%$, 72.6% selected by neither, and 8.0% selected by one approach but not the other). Compared to individuals selected by benefit‐only, individuals selected by risk‐only had greater absolute risk of lung cancer death after 10 years (mean = 2.8% vs. 1.5%) but gained fewer days of life from lung screening (mean = 2.2 days vs. 4.2 days) due to being older (mean age = 76 vs. 57 years) and having more comorbidities (median = 3.0 vs. 1.2). Concordance between risk and benefit‐based selections decreased when using ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;\mathbf{x})$ for benefit‐based selection (12.8% selected by one approach but not the other) but was still more closely aligned than when any of the risk‐ or benefit‐based selections were compared to the selection using the USPSTF guidelines.

TABLE 1.

Pairwise concordance between selection strategies for identifying individuals in the target population (US ever‐smokers, aged 40–84) for lung cancer screening.

Selection criteria	vs. Highest by ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;\mathbf{x})$		vs. Highest by ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;\mathbf{x})$		vs. Highest by ${\widehat{r}}_{\tau =10}(\mathbf{x})$		vs. USPSTF Eligible a
	Not selected (%)	Selected (%)	Not selected (%)	Selected (%)	Not selected (%)	Selected (%)	Not selected (%)	Selected (%)
Highest by ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;\mathbf{x})$
Selected	0.0	23.4	2.8	20.6	4.0	19.4	8.4	15.0
Not Selected	76.6	0.0	73.8	2.8	72.6	4.0	68.2	8.4
Highest by ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;\mathbf{x})$
Selected			0.0	23.4	6.4	17.0	8.6	14.8
Not Selected			76.6	0.0	70.2	6.4	68.1	8.6
Highest by ${\widehat{r}}_{\tau =10}(\mathbf{x})$
Selected					0.0	23.4	9.6	13.8
Not Selected					76.6	0.0	67.1	9.6
USPSTF Eligible a
Selected							0.0	23.4
Not Selected							76.6	0.0

Note: ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ is the life‐expectancy gain from screening after $\tau$ years; ${\widehat{r}}_{\tau }(\mathbf{x})$ is the risk of dying from lung cancer after $\tau$ years.

^a Eligible according to US Preventive Services Task Force (USPSTF) Guidelines Recommendations for Lung Cancer Screening: Ages 40–80 years, smoked within the past 15 years, $\ge 20$ pack‐years of exposure.

Table S3 gives examples of individuals in the NHIS, years 2015–2018, that would be selected by different approaches. A 56‐year old woman with 3 comorbidities who smoked an average of six cigarettes daily for 36 years would be predicted to gain an average of 4.2 days of life from lung cancer screening after 10 years (${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;\mathbf{x})$), 8.1 days after 19 years (${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;\mathbf{x})$), and has a 10‐year absolute risk of dying from lung cancer (${\widehat{r}}_{\tau =10}(\mathbf{x})\%$) of 1.45%. She is not eligible by USPSTF guidelines and would not be selected for screening by ${\widehat{r}}_{\tau =10}(\mathbf{x})\%$, but she would be selected by both ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;\mathbf{x})$ and ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;\mathbf{x})$. The seemingly small days of life‐gained after 10 years is a weighted average of two outcomes, a small chance (0.17%) of gaining substantial life‐expectancy (6.8 years) if lung cancer death is prevented by screening through early detection, and a large chance (99.83%) of gaining zero days of life if she would not die of lung cancer (or acquire lung cancer) or if screening would fail to prevent her death.

Table 2 projects the number of life‐years gainable by screening 23.4% of the target population of US ever‐smokers according to the different selection criteria. Notably both risk and benefit‐based approaches project to gain considerably more life‐years than the USPSTF recommendations. While these projections are optimistic as they are based on the same life‐gained estimates used for the benefit‐based selection, they highlight that while lung cancer screening may only improve individual life‐expectancy by a few days, the gains in life‐years can be considerable from a population perspective. Similarly, small improvements in identifying better candidates for lung cancer screening may lead to large population level gains in life‐years.

TABLE 2.

Projected gainable life‐years from screening all individuals selected by different strategies applied to the US population of ever‐smokers, aged 40–84 years.

Selection criteria	Individuals screened	LYG after 10 years	LYG after 19 years
	n in millions (%)	n in thousands (%)	n in millions (%)
All ever‐smokers	60.8 (100.0)	472.2 (100.0)	696.2 (100.0)
USPSTF Eligible a	14.2 (23.4)	243.1 (51.5)	353.1 (50.7)
Highest by ${\widehat{r}}_{\tau =10}(\mathbf{x})$	14.2 (23.4)	295.3 (62.5)	402.4 (57.8)
Highest by ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;\mathbf{x})$	14.2 (23.4)	308.8 (65.4)	438.0 (62.9)
Highest by ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;\mathbf{x})$	14.2 (23.4)	303.5 (64.3)	445.7 (64.0)

Note: ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ is the life‐expectancy gain from screening after $\tau$ years; ${\widehat{r}}_{\tau }(\mathbf{x})$ is the risk of dying from lung cancer after $\tau$ years; n, numbers.

Abbreviation: LYG, life‐years gainable.

^a US Preventive Services Task Force (USPSTF) Guidelines Recommendations for Lung Cancer Screening: Ages 40–80 years, smoked within the past 15 years, $\ge 20$ pack‐years of exposure.

Figure 2 compares the conditional probability of selection by age, when selecting $p=23.4\%$ of the population for screening. Importantly, under risk‐based selection, the probability of selection increased with age, with a 56.0% selection probability among ever‐smokers, aged 80–84 years. In practice, an age cutoff may be used to avoid selecting individuals too frail to undergo curative procedures following cancer diagnosis. But risk‐based selection with an age cutoff leads to individuals close to the cutoff age having either very large or zero probability of selection, depending on which side of the cutoff they fall. For example, using an eligibility risk threshold of ${\widehat{r}}_{\tau =10}(\mathbf{x})=1.7\%$ with an age cutoff of 80 would mean individuals 79 years of age would have a probability of selection of $\ge 50\%$ while 81‐year‐old individuals have zero probability of selection (Figure 2). In contrast, under benefit‐based selection, the probability of selection naturally decreased with age as the impact of prevention services on an individual's remaining life‐expectancy decreased, with ever‐smokers, aged 80 to 84 years, having an average selection probability of 25.1% when using ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;\mathbf{x})$ for selection and 15.4% when using ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;\mathbf{x})$.

FIGURE 2.

Percentage of US ever‐smokers in each age (in years) that would be selected for screening if using (a) benefit‐based screening using ${\widehat{\mathrm{\Delta }\mu }}_{\tau =10}(z;x)$, (b) benefit‐based screening using ${\widehat{\mathrm{\Delta }\mu }}_{\tau =19}(z;x)$, (c) risk‐based screening using ${\widehat{r}}_{\tau =10}(x)$, and (d) USPSTF guidelines recommendations.

4. Simulation Studies

We conducted simulation studies to (1) assess the bias and coverage of the proposed life‐gained estimator and Taylor‐linearized variance, (2) assess the performance of models for life‐expectancy gained from the prevention service and for disease‐specific mortality risk in identifying individuals who would highly benefit from receiving the prevention service, and (3) assess whether the performance of life‐gained‐based selections is robust to unaccounted for residual dependence in the competing risks. We considered several settings with different duration of effect on disease‐specific mortality provided by the prevention service and with different effects of the prevention service on competing mortality.

4.1. Generation of Finite Population, Trial, and Complex Survey

We generated a FP of size $M=\mathrm{500,000}$ with two covariates $x_1\sim \mathcal{N}\left(0,2^2\right)$ and $x_2\sim {\chi }^2(1.3)$. For everyone in FP, we generated $T_0$ and $T_1$, the counterfactual times to death without and with the prevention service $z\in \{0,1\}$, respectively. We considered $K=2$ competing causes of death, with $k=1$ being the primary cause of death that the effect of the prevention service is meant to reduce and $k=2$ all other causes of death.

To generate death times for the $K$ events, we used a Weibull distribution with shape $\alpha$ and scale $1/{\theta }_k$ for $k=1,2$, such that the survival function is as follows:

$$S(t;\mathbf{x},z)=\exp \left\{-\sum _{k=1}^K{\left({\theta }_{k}t\right)}^{\alpha }\right\},\text{with}{\theta }_k^{\alpha }=\exp \left\{{\beta }_{k0}+{\mathit{\beta }}_k^{\top }\mathbf{x}+{\gamma }_k(t)z\right\},$$

where ${\beta }_{k0}$ is the intercept, ${\mathit{\beta }}_k={\left({\beta }_{k1},{\beta }_{k2}\right)}^{\top }$ is the effect of covariates $\mathbf{x}={\left(x_1,x_2\right)}^{\top }$, and ${\gamma }_k(t)$ is the effect of the prevention service. The cause‐specific hazard of dying from cause $k$ was as follows:

$${\lambda }_k(t;\mathbf{x},z)={\lambda }_{k0}(t)\exp \left\{{\mathit{\beta }}_k^{\top }\mathbf{x}+{\gamma }_k(t)z\right\},{\lambda }_{k0}(t)=\alpha t^{\alpha -1}\exp \left({\beta }_{k0}\right).$$

The time of death from any cause, $T_z$, was generated as follows:

$$T_z=\left\{\begin{array}{ll}{\left[\frac{-\log (u)}{\sum _{k=1}^K\exp \left({\beta }_{k0}+{\mathit{\beta }}_k^{\top }\mathbf{x}+{\gamma }_{k}z\right)}\right]}^{1/\alpha },& \text{if}-\log (u)\le t^{*},\\ {\left[\frac{-\log (u)-t_1^{\alpha }\sum _{k=1}^K\exp \left({\beta }_{k0}+{\mathit{\beta }}_k^{\top }\mathbf{x}\right)\left\{\exp \left({\gamma }_{k}z\right)-1\right\}}{\sum _{k=1}^K\exp \left({\beta }_{k0}+{\beta }_k^{\top }\mathbf{x}\right)}\right]}^{1/\alpha },& \text{if}-\log (u)\ge t^{*},\end{array}\right.$$ (8)

where $t^{*}={\tau }_1^{\alpha }{\sum }_{k=1}^K\exp \left({\beta }_{k0}+{\mathit{\beta }}_k^{\top }\mathbf{x}+{\gamma }_k\right)$ and $u\sim \text{Unif}(0,1)$. (The derivation of Equation 8 is shown in the Supporting Information: Appendix F.)

We set $\alpha =1.2$ for a time‐varying increasing hazard;

$${\beta }_{10}=\log \left\{-\frac{\log (0.95)}{15}\right\};{\beta }_{20}=\log \left\{-\frac{\log (0.7)}{15}\right\};{\beta }_1=\left(\begin{array}{c}-0.2\\ 0.3\end{array}\right);{\beta }_2=\left(\begin{array}{c}-0.1\\ 0.2\end{array}\right).$$

The effect of the prevention service on the primary cause of death was as follows:

$${\gamma }_1(t)=\left\{\begin{array}{cc}a,& \text{if}t\le {\tau }_1,\\ 0,& \text{if}t>{\tau }_1,\end{array}\right.$$

and we vary $a\in \{-0.8,-0.22\}$ (corresponding to hazard ratios $\mathrm{HR}\approx 0.45$ and $\mathrm{HR}\approx 0.8$, respectively, the latter being comparable to the data example in Section 3). The patterns of results are similar; hence, we only include results for $a=-0.8$ in the following sections.

We considered five scenarios for how the prevention service affected cause‐specific mortality. In scenarios 1–3, we varied the duration of the disease‐specific mortality reduction provided by the prevention service, with ${\tau }_1\in \{\mathrm{5,12,46}\}$ corresponding to the 20th, 50th, and 95th percentiles of the death time $T_0$ in FP, respectively, and with ${\gamma }_2=0$. These different lengths of effect correspond to what might be expected from screening for cancers with short versus long sojourn times to cancer death or from a lifelong prevention agent such as human papillomavirus vaccination [30]. In scenarios 4 and 5, we allowed the prevention service to also affect competing mortality, but in opposite directions, with ${\gamma }_2=-0.06$ denoting a protective effect and ${\gamma }_2=0.06$ denoting a harmful effect.

At the simulated survival times $T_z$, we generated cause‐of‐death indicators $D_{z,k}$, for $z\in \{0,1\}$ and $k\in \{1,2\}$, according to a Bernoulli distribution with probabilities ${\lambda }_k\left(T_z;\mathbf{x},z\right)/\left\{{\lambda }_1\left(T_z;\mathbf{x},z\right)+{\lambda }_2\left(T_z;\mathbf{x},z\right)\right\}$.

We selected trial $s_A$ and survey samples $s_B$ from the FP with maximum follow‐up times ${\tau }_A=7$ and ${\tau }_B=15$, respectively. Since trials, in practice, are not representative of the FP, we randomly selected trial $s_A$ from the FP using probability‐proportional‐to‐size (PPS) sampling with the measure of size defined by $\mathit{sz}=\exp \left(-0.1x_1+0.07x_2\right)$. Under this PPS design, the implicit trial participation rate for $i\in \mathrm{FP}$ is proportional to ${\mathit{sz}}_i$, but because in practical analysis the participation mechanism is unknown, we estimated ${\widehat{\gamma }}_k(t)$, $k=1,2$, treating $s_A$ as a simple random sample of FP. The sample size of $s_A$ was $m=\mathrm{2,000}$ with assignment to prevention services $z\sim \text{Bernoulli}(0.5)$. The time of death was set as $T=T_z$. The observed survival time for the trial was defined as the minimum of survival and censoring times $t_A=\min \left\{T,C_A,{\tau }_A\right\}$, where the censoring times $C_A\sim \text{Weibull}\left\{-\log (0.8)/{\tau }_A,1\right\}$ are independent of the survival times and yield 20% dropout over the course of the trial. The observed case indicator for event $k$ in the trial is $D_{z,k,A}=I\left\{T\le \min \left(C_A,{\tau }_A\right),D_{z,k}=1\right\}$. By ${\tau }_A$, 30% of individuals had a recorded death in the trial.

The survey sample $s_B$ with size $n=\mathrm{8,000}$ was randomly selected using PPS sampling with the size defined by $\exp \left(0.1x_1-0.07x_2\right)$. In the analysis of the survey sample, the cause‐specific mortality models were fitted to the weighted survey sample accounting for the complex survey design. Since the survey sample did not have the intervention, the observed survival time and cause‐of‐death indicators were defined by $t_B=\min \left\{T_0,C_B,{\tau }_B\right\}$ and $D_{0,k,B}=I\left\{T_0\le \min \left(C_B,{\tau }_B\right),D_{0,k}=1\right\}$, respectively, where the censoring times $C_B\sim \text{Weibull}\left\{-\log (0.8)/{\tau }_B,1\right\}$ are independent of the survival times and are set for 20% lost to follow‐up. By the time of the mortality linkage ${\tau }_B$, 60% of individuals had a recorded death in the survey.

4.2. Simulation Results

4.2.1. Inference on $\mathit{\beta }$, $\mathit{\gamma }$, and $\mathrm{\Delta }{\mu }_{\tau }(z;\mathbf{x})$ From the Combined Trial and Survey Sample

For each of the five simulation scenarios described in Section 4.1, in which the effect of prevention services on cause‐specific mortality varied, we simulated $3000$ selections of $s_A$ and $s_B$ from the FP. For each selection, we applied the methods described in Section 2 to estimate ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ with $\tau =7$ years and $\tau =15$ years, for the quintile values of $x$ in the FP (denoted as $x_{01},\dots ,x_{05}$). We estimated ${\lambda }_{k0}(t)$ nonparametrically but assumed that the parameterization of ${\gamma }_k(t)$ (a step function with a jump point at ${\tau }_1$) was known in advance.

Table 3 shows that in Scenario 1, the sample estimates of $\mathit{\beta }$ and $\gamma$ were unbiased. The estimated effect of the prevention service, ${\widehat{\gamma }}_1$, had a larger variance compared to the $\widehat{\mathit{\beta }}$'s because ${\widehat{\gamma }}_1$ was estimated from the trial sample, which has a smaller sample size and a shorter maximum follow‐up time than the survey sample used to estimate the $\widehat{\mathit{\beta }}$'s. For the quintile values of $\mathbf{x}$ in the FP, both ${\widehat{\mathrm{\Delta }\mu }}_{\tau =7}(z;\mathbf{x})$ and ${\widehat{\mathrm{\Delta }\mu }}_{\tau =15}(z;\mathbf{x})$ were approximately unbiased and the coverage rate of the 95% confidence interval for $\widehat{\mathrm{\Delta }{\mu }_{\tau }}(z;\mathbf{x})$ maintained the nominal 95% level. The pattern of results was similar across the other scenarios (see Table S4).

TABLE 3.

Inference of log‐hazard ratios and life‐gain expectancy (in days) in Scenario 1: Prevention service reduces death from the primary cause for 5 years and has no effect on competing mortality.

Parameter	True value a	RelBias b (%)		EmpSE c	VarRatio d	95% CP e
Hazard ratios
${\beta }_{11}$	$-0.20$	$-0.71$		0.02	1.05	0.96
${\beta }_{12}$	$0.30$	$-0.36$		0.02	0.96	0.94
${\gamma }_1$	$-0.81$	1.89		0.33	0.95	0.95
${\beta }_{21}$	$-0.10$	$-0.31$		0.01	1.05	0.96
${\beta }_{22}$	$0.20$	0.21		0.01	0.99	0.95
		Standard f	Fast g			Standard	Fast
Life‐gain expectancy (in days) for five individuals ${\mathbf{x}}_{01},\dots ,{\mathbf{x}}_{05}$ h with $\tau =7$ years
$\mathrm{\Delta }{\mu }_{(\tau =7)}\left(z,{\mathbf{x}}_{01}\right)$	27.95	$-1.71$	$-0.42$	7.79	1.02	0.96	0.95
$\mathrm{\Delta }{\mu }_{(\tau =7)}\left(z,{\mathbf{x}}_{02}\right)$	23.59	$-1.80$	$-0.49$	6.48	1.01	0.95	0.95
$\mathrm{\Delta }{\mu }_{(\tau =7)}\left(z,{\mathbf{x}}_{03}\right)$	21.21	$-1.86$	$-0.55$	5.79	1.01	0.95	0.94
$\mathrm{\Delta }{\mu }_{(\tau =7)}\left(z,{\mathbf{x}}_{04}\right)$	21.92	$-1.86$	$-0.56$	6.00	1.01	0.95	0.94
$\mathrm{\Delta }{\mu }_{(\tau =7)}\left(z,{\mathbf{x}}_{05}\right)$	26.15	$-1.74$	$-0.48$	7.33	1.01	0.96	0.95
Life‐gain expectancy (in days) for five individuals ${\mathbf{x}}_{01},\dots ,{\mathbf{x}}_{05}$ f with $\tau =15$ years
$\mathrm{\Delta }{\mu }_{(\tau =15)}\left(z,{\mathbf{x}}_{01}\right)$	62.70	$-1.77$	$-2.39$	17.42	1.02	0.96	0.96
$\mathrm{\Delta }{\mu }_{(\tau =15)}\left(z,{\mathbf{x}}_{02}\right)$	54.09	$-1.85$	$-2.50$	14.84	1.01	0.95	0.95
$\mathrm{\Delta }{\mu }_{(\tau =15)}\left(z,{\mathbf{x}}_{03}\right)$	49.08	$-1.90$	$-2.56$	13.37	1.01	0.95	0.95
$\mathrm{\Delta }{\mu }_{(\tau =15)}\left(z,{\mathbf{x}}_{04}\right)$	49.98	$-1.89$	$-2.53$	13.67	1.01	0.95	0.95
$\mathrm{\Delta }{\mu }_{(\tau =15)}\left(z,{\mathbf{x}}_{05}\right)$	57.12	$-1.77$	$-2.36$	15.98	1.01	0.96	0.96

^a True values estimated from set parameters in the finite population.

^b Percent relative bias, computed as $100(\theta -\widehat{\theta })/\theta$, where $\widehat{\theta }$ and $\theta$ denote estimated and true parameter values.

^c Empirical standard error across 3000 simulations.

^d Ratio of empirical standard deviation to average asymptotic variance.

^e Frequency with which $\theta$ is contained within the estimated 95% confidence interval.

^f Estimation and inference using all time points in the sample.

^g Estimation and inference using a systematic subsample of 100 time points to obtain a fast approximation.

^h ${\mathbf{x}}_{01}={(-\mathrm{2.57,0.05})}^{\top }$, ${\mathbf{x}}_{02}={(-\mathrm{1.35,0.22})}^{\top }$, ${\mathbf{x}}_{03}={(\mathrm{0.00,0.72})}^{\top }$, ${\mathbf{x}}_{04}={(\mathrm{1.35,1.78})}^{\top }$, ${\mathbf{x}}_{05}={(\mathrm{2.56,3.32})}^{\top }$.

We also evaluated estimation and inference of ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ when using a systematic subsample of 100 death times (i.e., “fast” estimation). Estimates of ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ were approximately unbiased and the 95% confidence intervals provided nominal coverage (Table 3).

4.2.2. Using Estimated Life‐Gained Versus Risk of Disease‐Specific Mortality in Selecting Individuals Who Benefit Most From a Prevention Service

Next, we evaluated the effectiveness of using ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ and of using ${\widehat{r}}_{\tau }(\mathbf{x})$, the individual $\tau$‐year absolute risk for death from the primary cause, to select individuals in the FP who would benefit most from the prevention service. We defined benefit as the individual's true life‐expectancy gained from prevention services, $\mathrm{\Delta }\mu (z;\mathbf{x})$, estimated using the set values for ${\lambda }_{k0}(t)$, ${\mathit{\beta }}_k$, and ${\gamma }_k(t)$.

To reduce computational burden, we conducted the comparison using a large simple random sample of $\mathrm{13,000}$ individuals (denoted $s_C$) from the FP. For each individual in $s_C$, we calculated ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ and ${\widehat{r}}_{\tau }(\mathbf{x})$, with $\tau$ corresponding to the 5th ($\tau =1.4$), 30th ($\tau =7$), 60th ($\tau =15$), and 90th ($\tau =35$) percentiles of $T_0$ in the FP.

We then calculated the ratio of gain, defined as the fraction of maximum benefit achieved by selecting the top $P\%$ of individuals in $s_C$, ranked by the highest value of $U$ (where $U$ is either ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ or ${\widehat{r}}_{\tau }(\mathbf{x})$), to receive the prevention service, compared to selecting the top $P\%$ of individuals ranked by the true $\mathrm{\Delta }\mu (z;\mathbf{x})$. The ratio of gain, denoted by $R_U(P)$, was calculated as follows:

$$R_U(P)=\frac{\sum _{i\in s_C}\mathbb{I}\left\{U_i>Q_U(P)\right\}\mathrm{\Delta }\mu \left(z;{\mathbf{x}}_i\right)}{\sum _{i\in s_C}\mathbb{I}\left\{\mathrm{\Delta }\mu \left(z;{\mathbf{x}}_i\right)>Q_{\mathrm{\Delta }\mu }(P)\right\}\mathrm{\Delta }\mu \left(z;{\mathbf{x}}_i\right)},$$ (9)

where $Q_U(P)$ is the quantile function for the distribution of $U$.

Figure 3 and Table S5 present $R_{{\widehat{r}}_{\tau }}(P)$ and $R_{{\widehat{\mathrm{\Delta }\mu }}_{\tau }}(P)$ across the five simulation scenarios. Both $R_{{\widehat{r}}_{\tau }}(P)$ and $R_{{\widehat{\mathrm{\Delta }\mu }}_{\tau }}(P)$ consistently increased as $\tau$ became larger, regardless of the duration of the prevention effect (${\tau }_1$) or the effect on competing mortality (${\gamma }_2$). This indicates that a longer follow‐up period (allowing estimation of risk or life‐years gained up to a greater time $\tau$) improves the ability of ${\widehat{r}}_{\tau }(\mathbf{x})$ and ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ to prioritize individuals with the greatest lifetime benefit from the prevention service.

FIGURE 3.

Ratio of life‐years gain from selection versus optimal selection ($\mathit{R}$) when selecting $\mathit{p}\%$ of the population to receive a prevention service according to (i) highest estimated absolute risk of death from the primary cause or (ii) highest estimated life‐years gain from the prevention service. Five scenarios were examined: prevention service reduces death from primary cause for (a) 5 years with no effect on competing mortality; (b) 12 years with no effect on competing mortality; (c) 46 years with no effect on competing mortality; (d) 5 years and reduces competing mortality; (e) 5 years but increases competing mortality.

Across all five scenarios, when only short‐term follow‐up data were available (allowing estimation of ${\widehat{r}}_{\tau }(\mathbf{x})$ and ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ with a small $\tau$), both approaches performed similarly. However, when estimates based on longer $\tau$ were available, the improvement in selection effectiveness was more pronounced for ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$ than for ${\widehat{r}}_{\tau }(\mathbf{x})$.

When the prevention service had only a short‐term effect on mortality, estimating risk or life‐years gained up to a short time $\tau$ was sufficient to correctly identify individuals who would benefit most. This is reflected by the simulation results showing $R_{{\widehat{r}}_{\tau }}(P)\ge 0.96$ and $R_{{\widehat{\mathrm{\Delta }\mu }}_{\tau }}(P)\ge 0.96$ when $P\ge 5$. However, when the prevention service had a nearly lifelong effect on mortality (${\tau }_1=46$ in Scenario 3), a longer $\tau$ was necessary. For example, using ${\widehat{\mathrm{\Delta }\mu }}_{\tau =35}(z;\mathbf{x})$ to select 5% of the population for treatment achieved $R_{{\widehat{\mathrm{\Delta }\mu }}_{\tau }}(P)=97\%$, compared with $R_{{\widehat{\mathrm{\Delta }\mu }}_{\tau }}(P)=57\%$ for ${\widehat{\mathrm{\Delta }\mu }}_{\tau =7}(z;\mathbf{x})$ and $R_{{\widehat{r}}_{\tau }}(P)=63\%$ for ${\widehat{r}}_{\tau =35}(\mathbf{x})$.

When the prevention service had no effect on competing mortality (Scenarios 1–3) or reduced competing mortality (${\gamma }_2<0$, Scenario 4), both $R_{{\widehat{r}}_{\tau }}(P)$ and $R_{{\widehat{\mathrm{\Delta }\mu }}_{\tau }}(P)$ increased monotonically with $P$, indicating that maximum population‐level benefit is achieved when the prevention service is offered to everyone, even if some individuals gain only small benefits. However, when the prevention service increased the risk of competing mortality (${\gamma }_2>0$, Scenario 5), such that some individuals experienced net harm (negative life‐expectancy gain), $R_{{\widehat{r}}_{\tau }}(P)$ continued to increase monotonically with $P$, but $R_{{\widehat{\mathrm{\Delta }\mu }}_{\tau }}(P)$ did not, indicating that life‐gained‐based selection recognizes that maximum population‐level benefit is achieved by not selecting certain individuals to receive the prevention service.

4.2.3. Assessing the Impact of Unaccounted‐for Residual Dependence in Competing Risks

In estimating ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$, our methods assume conditional independence of the competing event times in Equation (1) given covariates $x$. In practice, this assumption may not hold if the survey did not collect the requisite covariates or if the use of certain predictors (e.g., socioeconomic status) are not allowed for clinical decision‐making. In this section, we evaluate the impact of unaccounted for residual dependence between competing risk models on the resulting benefit‐based selection.

We generate the simulated data as in Section 4.1 but with the cause‐specific hazards of dying from cause $k$ including an unobserved frailty term $U$ such that

$${\lambda }_k(t;\mathbf{x},z,U)={\lambda }_{k0}(t)\exp \left\{{\mathit{\beta }}_k^{\top }\mathbf{x}+{\gamma }_k(t)z+U\right\},U\sim N\left(0,{\sigma }^2\right).$$

We consider values for $\sigma$ of 0.05, 0.1, 0.15, and 0.2. For context, these frailties represent 1%–9% of the variance in the linear predictor for ${\lambda }_1(t;\mathbf{x},z=0,U)$ and 2%–22% of the variance in the linear predictor for ${\lambda }_2(t;\mathbf{x},z=0,U)$. We assume the effect of the prevention service follows that of Scenario 1 (i.e., ${\gamma }_1(t)=-0.8,t\le 5;{\gamma }_1(t)=0,t>5$; and ${\gamma }_2=0$).

In Figure S2, we present comparisons of the ratio of gain $R_{{\widehat{\mathrm{\Delta }\mu }}_{\tau }}(P)$ under correctly specified cause‐specific models that include $(\mathbf{x},z,U)$ versus incorrectly specified models that include only $(\mathbf{x},z)$. The differences between the two were small, even under $\sigma =0.2$ for $U$, illustrating that benefit‐based selection can still be effective under moderate amounts of unaccounted residual dependence between the competing risks.

5. Discussion

We proposed a novel benefit‐based approach to precision prevention, in which individuals are selected for prevention services according to their estimated gain in life‐expectancy. We show how to construct models for individualized gain in life‐expectancy using the time‐dependent treatment effect of the prevention service on cause‐specific mortality, estimated from an intervention trial which may not be representative of the target population, and the risk of cause‐specific mortality, estimated from a target population‐representative survey sample. In our lung cancer screening example, reassuringly most individuals at high risk were at high benefit from screening, as might be expected. However, many individuals were also either high‐risk or high benefit, which happens when disease‐specific mortality is highly correlated with competing mortality (the major risk factors, age and smoking, are strongly associated with both lung cancer death and with other‐cause mortality). A risk‐based approach can select individuals for whom receiving the prevention service may save their life but would not appreciably extend their life. In contrast, a life‐gained‐based approach would identify high‐benefit individuals by naturally balancing the trade‐offs between higher risk of disease‐specific death and lower life‐expectancy.

We illustrated our approach in addressing who to screen for lung cancer, developing a life‐gained model for three annual rounds of CT screening using the NLST and NHIS 1997–2001, and applying the approach to the NHIS 2015–2018 population of ever‐smokers, aged 40–84 years. We demonstrated that (1) a life‐gained approach would replace low‐benefit individuals selected by a risk‐based approach with high‐risk, high‐benefit individuals, and (2) the life‐gained approach will naturally curtail screening at older ages as the benefit of screening reduces.

Our simulation studies examined when the proposed life‐gained estimator and selection strategy may be preferable over a selection strategy prioritizing those at high absolute risk of death from the disease of interest. When the duration of the prevention service effect on disease‐specific mortality is short, then risk‐based and life‐gained‐based selection strategies performed similarly in identifying high‐benefit individuals. When the duration of the prevention service effect is longer or when the prevention service increased the risk of competing mortality, then life‐gained‐based selection strategies outperform risk‐based selection strategies.

The performance of life‐gained‐based strategies improved with greater follow‐up time available in the survey sample, although these considerations would need to be balanced against whether past observations on cause‐specific mortality can accurately be used to predict mortality far in the future. Neither risk‐based strategies nor life‐gained‐based strategies are applicable in the absence of a discriminant risk model. When the prevention service does not prevent death or when the target population is not at high‐risk for competing mortality, life‐gained‐based strategies may not be appropriate.

A crucial assumption of the proposed method is that the effect of the prevention service estimated from the trial is portable to the target population. The portability assumption is commonly used in medical studies [31, 32] but can be violated when the effect of the prevention service is different under the controlled trial environment when administered to the target population. Additionally, unmeasured effect modifiers (e.g., socioeconomic factors, adherence differences) or trial exclusion criteria could further bias estimates. The NLST is the only large‐scale randomized trial for low‐dose CT lung screening in the US, but subsets of the target population were not represented. As additional data become available through observational studies, the burgeoning literature on estimating population‐representative causal effects of treatments [17, 33, 34] could be utilized to relax the portability assumption, so long as identifiability conditions are met. While our methods can accommodate flexible forms of the time‐varying effect of prevention services on mortality, we utilized a step function due to its simplicity and fit to NLST data. Although the analytic variance may be more involved, future work could consider the use of penalized smoothing splines to characterize the effect of prevention services on mortality.

Our methods assume that the covariates in the cause‐specific hazard models fully account for the correlation between the cause of death targeted by the prevention service and other competing mortality. In practice, there is often residual dependence even after adjusting for covariates. For example, the NHIS did not collect health biomarkers, and while other determinants were available (e.g., socioeconomic status), including it in models used for clinical decision‐making may not be ethical. While our simulation results show that benefit‐based selection can still be effective under a moderate amount of unaccounted for residual dependence, there is potential to explore robustness extensions in future work using frailty models.

Our calculators for the individualized life‐gained from prevention services may have applicability beyond determining screening eligibility, such as in prediction‐augmented shared decision‐making incorporating patient preferences [35] or for estimating the benefit in cost‐effectiveness studies that also consider the harms and costs of the prevention service [36, 37]. In some cost effectiveness studies, quality‐adjusted life years (QALY) are of interest. Our methods and R package can estimate the individual life‐expectancy gained from a prevention service up to any specified time $\tau$ supported by the data, which can be utilized to estimate QALY (e.g., $\text{QALY}={\sum }_{j=1}^{J}w(\mathbf{x},\tau )\left\{{\widehat{\mathrm{\Delta }\mu }}_{\tau =j}-{\widehat{\mathrm{\Delta }\mu }}_{\tau =j-1}\right\}$, where $w(\mathbf{x},\tau )$ is a weight function that depends on current age, comorbidities, and years from the current age).

Funding

This work was supported by NIH Intramural Research Program.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Appendix A. Decomposition of $\widehat{\mathrm{\Delta }\mu }(z;\mathbf{x})$.

Appendix B. Variance estimator of ${\widehat{\mathrm{\Delta }\mu }}_{\tau }(z;\mathbf{x})$.

Appendix C. Absolute risk of the primary event.

Appendix D. Estimating life‐gained from cause‐specific and overall mortality.

Appendix E. Evaluating predicted survival and life‐gained from screening using cause‐specific mortality vs. overall mortality approaches.

Appendix F. Generating Weibull competing event times with dichotomous time‐varying effect of the intervention.

Table S1: Comparison of Covariate Distribution between Ever‐smoker Populations in the NLST and the Weighted NHIS.

Table S2: Hazard ratio estimates and 95% confidence interval for lung cancer death and other‐cause mortality.

Figure S1: Joint distribution of absolute risk of lung cancer death and life‐expectancy gained from lung cancer screening in the US population of ever‐smokers, aged 40–84 years.

Table S3: Sample individuals in NHIS 2015‐2018.

Table S4: Inference of log‐hazard ratios and life‐expectancy gained in scenarios 2‐4.

Table S5: Ratio of gain (R) under risk‐based vs. life‐gained based selection.

Figure S2: Assessing the impact of unaccounted‐for residual dependence in competing risk on benefit‐based selection.

Data Availability Statement

All simulation and analytic code are available on Github (https://github.com/Lingxiao‐Wang/LYG). To protect the privacy of study participants, authors cannot directly NLST and NHIS data. NLST data can be requested from the Cancer Data Access System (https://cdas.cancer.gov/nlst). National Health Interview Survey data is publicly available from the National Center for Health Statistics website (https://cdc.gov/nchs/nhis); the NHIS cause‐specific mortality linkage must be requested from the Research Data Center (https://cdc.gov/rdc).