While-alive regression analysis of composite survival endpoints

SUMMARY

Composite endpoints are frequently used in clinical trials to enhance the event rate and improve the statistical power. In the presence of a terminal event, the while-alive cumulative frequency measure offers a useful alternative to define composite survival outcomes, by relating the average event rate to the survival time. Although non-parametric methods have been proposed for two-sample comparisons, limited attention has been given to regression methods that directly address time-varying association effects in while-alive measures. We address this gap by developing a regression framework for exposure-weighted while-alive measures for composite survival outcomes that include a terminal component event. Our regression approach uses splines to model time-varying association between covariates and a generalized while-alive loss rate of all component events, and can be applied to both independent and clustered data. We derive the asymptotic properties of the regression estimator under both independent data and cluster-correlated data settings, and study the operating characteristics of our methods through simulations. Finally, we apply our regression method to analyze data two randomized clinical trials. The proposed methods are implemented in the WAreg R package.

1. INTRODUCTION

Composite endpoints are common in phase III clinical trials and consist of two or more component endpoints. For example, in cardiovascular disease research, a composite endpoint might combine time to death, myocardial infarction, stroke and hospitalization. The use of composite endpoints increases the efficiency of clinical trials by increasing the overall event rate, thereby improving the power to detect treatment effect (Freemantle et al. 2003). Regulatory guidelines have underscored the importance of carefully constructing, analyzing, and reporting composite endpoints. For instance, the ICH-E9 guideline, Statistical Principles for Clinical Trials (European Medicines Agency 2020), states: “If a single primary variable cannot be selected from multiple measurements associated with the primary objective, another useful strategy is to integrate or combine the multiple measurements into a single or composite variable.” The U.S. Food and Drug Administration guideline, Multiple Endpoints in Clinical Trials (Food and Drug Administration 2017), mentions that “composite endpoints are often assessed as the time to first occurrence of any one of the components” while also noting that “it may also be possible to analyze total endpoint events.”

Traditional analyses of composite endpoints typically rely on time-to-first-event models, which implicitly assign equal clinical weight to fatal and non-fatal outcomes. When both recurrent and terminal events are present, various regression approaches have been proposed that differ in the primary target parameters and how death is handled. When estimating the treatment effect in a regression model, the strategies for handling death can be mapped to methods for handling intercurrent events under the ICH E9 estimands framework (European Medicines Agency 2020). A summary of representative regression models and their strategies for handling death is provided in Table S1. In particular, the while-alive approach has recently been highlighted in the ICH-E9(R1) Addendum as a viable strategy to handle death as an intercurrent event. Under this framework, the occurrence of events is evaluated relative to the duration of survival. By normalizing the event burden with respect to survival, while-alive estimands provide interpretable measures of the average event rate among patients while they are alive. Two particular forms of the estimand have been defined (Schmidli et al. 2023): the patient-weighted while-alive estimand, which is the expectation of the event-to-survival ratio across individuals (Ragni et al. 2024), and the exposure-weighted while-alive estimand, which is the ratio of the expected number of events to the restricted mean survival time (Mao 2023; Wei et al. 2023).

There are several previous efforts in studying while-alive measures. Wei et al. (2023) established theoretical properties of exposure-weighted while-alive estimators under a gamma frailty model, Mao (2023) developed a nonparametric estimator for the exposure-weighted while-alive estimand, and Ragni et al. (2024) derived the efficient influence function and constructed efficient non-parametric estimators for the patient-weighted while-alive estimand. While these contributions are useful for estimating treatment effects, they do not address association analysis under a regression framework. For regression analysis of while-alive measures, it is often necessary to address time-varying association effects. To accommodate time-varying effects, several strategies have been previously employed in survival analysis with a single endpoint. For example, kernel-based methods have been developed for Cox models (Cai and Sun 2003) to estimate time-varying hazard ratios. Spline-based methods approximate time-varying coefficients with flexible basis functions, and have been applied in the context of mean residual life regression (Sun et al. 2012) and restricted mean survival time regression (Zhong and Schaubel 2022). Finally, landmark models (Van Houwelingen and Putter 2008) provide a complementary approach by fitting survival models at prespecified landmark times, allowing dynamic prediction and assessment of time-varying effects in a practical fashion.

In this paper, we propose a new regression framework that incorporates time-varying coefficients into the while-alive measure for composite survival endpoints. We focus on the exposure-weighted while-alive estimand, operationalized through the generalized while-alive loss rate. We model the association between covariates and while-alive loss rate with spline functions, allowing effects to vary smoothly over follow-up time. We first address independent data, and then discuss cluster-correlated data under the working independence assumption. We establish the asymptotic properties of the proposed regression estimators and investigate their finite-sample performance using simulations. Finally, we illustrate the use of our methods by analyzing two clinical trials. To facilitate implementation, an R package WAreg (https://github.com/fancy575/WAreg) has been developed, along with a short tutorial in Web Appendix S15.

2. MOTIVATING DATA EXAMPLES

The first motivating example is the Heart Failure: A Controlled Trial Investigating Outcomes of Exercise Training (HF-ACTION), a multicenter individually randomized trial. This study enrolled 2,331 patients with a median follow-up of 30 months to evaluate the effectiveness of exercise training in heart failure patients. Figure 1a shows the distribution of recurrent hospitalizations and mortality events for each subject, stratified by treatment group. The primary endpoint was a composite of all-cause mortality and all-cause hospitalization. Initial analysis using a Cox proportional hazards model for the time to the first event, adjusted for etiology, found statistically insignificant reductions in the exercise training group compared to usual care (hazard ratio = 0.92, with 95% confidence interval = (0.83, 1.03); p-value = 0.14) (O’Connor et al. 2009). However, this approach only considered the first occurrence of mortality or hospitalization, ignoring subsequent events and treating hospitalizations and deaths as equally important. To address these limitations, more advanced methods were applied to the HF-ACTION data. For example, Mao and Lin (2016) considered a semiparametric regression model under the proportional mean framework to analyze weighted endpoints, but did not directly address the issue of differential follow-up time across patients. In their analysis, there was no statistically significant reduction in the mean frequencies of the weighted composite events at the 0.05 level, regardless of weights assigned to hospitalization and death. However, the p-values are smaller compared to that under the time to first event analysis.

Fig. 1.

a) Distribution of hospitalization and fatal events per subject by intervention group in the HF-ACTION trial. b) Distribution of fall-related injuries and fatal events per subject by intervention group in the STRIDE trial. Each horizontal line shows one participant’s follow-up from randomization to the observed end of follow-up (death or censoring); circular markers indicate recurrent hospitalizations/injuries and triangular markers indicate deaths. The staircase-like right edge occurs because many participants share common administrative or scheduled end times, yielding a visual alignment. Figures were generated using the reReg R package (Chiou et al. 2023).

A second example is the Strategies to Reduce Injuries and Develop Confidence in Elders (STRIDE) trial, a pragmatic cluster randomized study evaluating the effectiveness of a multifactorial intervention for preventing fall-related injuries. A total of 86 primary care practices were randomly assigned to either the intervention or enhanced usual care (control) group, with 43 practices in each arm. There were 2,802 participants in the intervention group and 2,649 in the control group, with a maximum follow-up of 44 months. In the original analysis, a multistate survival model accounting for competing risks of death and clustering was used to evaluate time to first fall-related injury. The estimated hazard ratio was 0.90 (95% confidence Interval: (0.83, 0.99); p-value = 0.004). Figure 1b shows the distribution of recurrent fall injuries and mortality events for each subject, stratified by intervention group. This study requires consideration of both recurrent fall injuries and fatal events (Bhasin et al. 2020) in clustered settings. In both examples, there is interest in exploring the impact of treatment and other risk factors on the composite of recurrent and terminal events through a single regression model, to provide useful summary measures. This motivates the while-alive regression methods, which are specifically designed to provide a framework for regression-based analysis of composite survival endpoints in both independent and cluster-correlated data settings.

3. METHODS

3.1. While-alive measure

Let $D$ denote the time to the terminal event (death), and define $N_D\left(t\right)=\mathbb{I}(D\le t)$, where $\mathbb{I}(\cdot )$ is the indicator function. With recurrent events, let $N_k\left(t\right),k=1,\dots ,K$, denote the counting processes for $K(K\ge 1)$ distinct types of recurrent events. Since death is a terminal event, the counting processes for recurrent events are constrained by the survival time $D$, and are given by $N_k\left(t\right)=N_k(D\wedge t)={\sum }_{q=1}^{\infty }\mathbb{I}\left(T_{k,q}\le D\wedge t\right)$, where $T_{k,q}$ denotes the $q\text{th}$ recurrence time of the $k\text{th}$ event type, and $D\wedge t=\text{min}(D,t)$. This ensures that no recurrent events are observed after the terminal event. The complete event history up to time $t$ is $\mathscr{H}\left(t\right)$, which includes the times and types of all recurrent events up to $t$, as well as whether the terminal event has occurred.

To characterize the cumulative events over time, we define $ℒ\left(\mathscr{H}\right)\left(dt\right)\ge 0$, as the instantaneous loss rate at time $t$. This definition satisfies: (C1) $ℒ\left(\mathscr{H}\right)\left(dt\right)=0$ for $t\ge D$, ensuring that no loss is incurred after death; (C2) $ℒ\left(\mathscr{H}\right)\left(dt\right)$ depends only on the event history up to time $t$. Condition (C1) aligns with the nature of recurrent event analysis. Condition (C2) is natural and exhibits the dependence of the loss rate on the observed history up to $t$ instead of the future. We follow Mao (2023) to define the generalized while-alive loss rate over the interval $[0,t]$ as:

$$l\left(\mathscr{H}\right)\left(t\right)=\frac{\mathbb{E}\left[ℒ\right(\mathscr{H}\left)\right(t\left)\right]}{\mathbb{E}[D\wedge t]},$$ (3.1)

where $ℒ\left(\mathscr{H}\right)\left(t\right)={\int }_0^tℒ\left(\mathscr{H}\right)du$, and $ℒ\left(\mathscr{H}\right)\left(du\right)=\sum _{k=1}^K{w}_{k}d{N}_k\left(u\right)+w_{D}d{N}_D\left(u\right)$. In this definition, $w_k$ is the weight assigned to the $k\text{th}$ type of recurrent event, $w_D$ is the weight assigned to the terminal event, and $\mathbb{E}[D\wedge t]$ is the restricted mean survival time (RMST). In definition (3.1), the numerator quantifies the expected cumulative weighted events up to time $t$. The denominator, $\mathbb{E}[D\wedge t]$, provides a time-at-risk adjustment by capturing the expected duration of survival over $[0,t]$. This definition accounts for both recurrent and terminal events as component endpoints, and adjusts the loss rate by the duration of survival. Particularly, (3.1) corresponds to the exposure-weighted while-alive loss rate (Wei et al. 2023), which quantifies the average burden per unit time alive across the cohort. In this formulation, individuals who survive longer contribute more exposure time and therefore receive greater weight in the overall average. This is to be differentiated from the patient-weighted while-alive loss rate (Ragni et al. 2024), defined as $l^{*}\left(\mathscr{H}\right)\left(t\right)=\mathbb{E}\left\{ℒ\right(\mathscr{H}\left)\right(t)/(D\wedge t\left)\right\}$, which averages each patient’s event-to-survival ratio and thus reflects the typical patient’s while-alive loss rate. The two quantities are linked by

$$l\left(\mathscr{H}\right)\left(t\right)=\mathbb{E}\left\{\frac{D\wedge t}{\mathbb{E}\left[D\wedge t\right]}\times \frac{ℒ\left(\mathscr{H}\right)\left(t\right)}{D\wedge t}\right\},$$

indicating that $l\left(\mathscr{H}\right)\left(t\right)$ equals the survival-time weighted average of the patient-level while-alive measure. Finally, in the special case where $w_k=0$ for all $k$ and $w_D=1$, (3.1) becomes the average hazard of the terminal event in Uno and Horiguchi (2023) and Uno et al. (2024).

3.2. While-alive generalized linear model

We denote ${\mathit{Z}}_i$ as a $p$-dimensional vector of bounded covariates for individual $i\in \{1,\dots ,n\}$. By convention, the first element may be set to 1 to include an intercept. The while-alive loss rate at truncation time $t$, conditional on ${\mathit{Z}}_i$, is denoted as $l\left({\mathscr{H}}_i\right)\left(t\mid {\mathit{Z}}_i\right)$, which we model as

$$\eta \left\{l\left({\mathscr{H}}_i\right)\left(t\mid {\mathit{Z}}_i\right)\right\}=\mathit{\beta }(t{)}^{\top }{\mathit{Z}}_i.$$ (3.2)

Throughout, we use the term truncation time to denote the evaluation horizon $t$ at which the while-alive loss rate is defined and modeled. Here ${\mathscr{H}}_i$ denotes the filtration generated by the event history up to time $t$ (recurrent counts and aliveness/terminal event). In (3.2), $\mathit{\beta }\left(t\right)={\left\{{\beta }_1\left(t\right),\dots ,{\beta }_p\left(t\right)\right\}}^{\top }$ is a $p$-dimensional vector of time-varying parameters, and $\eta (\cdot )$ is a specified, increasing, and differentiable link function. The choice of $\eta (\cdot )$ determines the interpretation of the regression coefficients. For example, with a binary treatment variable and a log-link function $\eta (\cdot )=\text{log}(\cdot )$, the exponentiated coefficient represents the ratio of while-alive loss rates between treatment groups. The vector $\mathit{\beta }\left(t\right)$ indexes covariate effects on the exposure-weighted while-alive loss rate at time $t$. Of note, a time-varying $\mathit{\beta }\left(t\right)$ does not by itself imply that the underlying treatment effect on the hazard of death or recurrent events is time-varying. Because the while-alive rate conditions on being alive, both its numerator and denominator evolve with the survivor cohort, so that even constant hazard-scale effects can translate into a time-varying contrast on the while-alive scale. We further illustrate this point in Web Appendix SA.2. Furthermore, we view ${\beta }_j\left(t\right)$ as a local or instantaneous covariate effect, in the sense that it captures the contribution of the $j\text{th}$ covariate to the while-alive loss rate at time $t$. Because clinical decision making is rarely based on an infinitesimal moment, one can use the window-averaged summaries of the local effect curve. For example, for an interval $\left[t_a,t_b\right]$, we define

$${\bar{\beta }}_j\left(t_a,t_b\right)=\frac{{\int }_{t_a}^{t_b}{\beta }_j(u)\omega (u)du}{{\int }_{t_a}^{t_b}\omega (u)du},$$

as the time-averaged effect of the $j\text{th}$ covariate, with weight $\omega (u)$ reflecting the effective information at time $u$ (for example, the probability of being alive and uncensored at $u$; in the absence of such adjustment we set $\omega \equiv 1$). Under the log link, $\text{exp}\left\{{\bar{\beta }}_j\left(t_a,t_b\right)\right\}$ is the weighted geometric mean of the instantaneous rate ratios $\text{exp}\left\{{\beta }_j(u)\right\}$ over $u\in \left[t_a,t_b\right]$. In practice, the time windows can be selected based on the scientific question or clinical judgment (eg 0–3 mo immediately after randomization, 3–6 mo during early recovery).

We let $C_i$ denote the censoring time due to either study termination or loss to follow-up. We assume covariate-dependent censoring such that $C_i\perp {\mathscr{H}}_i\mid {\mathit{Z}}_i$ and positivity such that ${\text{inf}}_{u\le \tau }G(u\mid \left.{\mathit{Z}}_i\right):={\text{inf}}_{u\le \tau }P\left(C_i\ge u\mid {\mathit{Z}}_i\right)\ge ϵ>0$ amost surely. Define $U_i=D_i\wedge C_i$ as the observed time and ${\Delta }_i=\mathbb{I}\left(D_i\le C_i\right)$ as the indicator for whether the terminal event occurred. We define the observed data for subject $i$ as ${𝒪}_i=\left\{{\tilde{\mathscr{H}}}_i,U_i,{\Delta }_i,{\mathit{Z}}_i\right\}$, where ${\tilde{\mathscr{H}}}_i$ denotes the observable portion of the event history up to time $U_i$. We further assume that $\mathit{\beta }(t)$ is a vector of continuous functions on $\left[0,t_V\right]$, parameterized via a finite-dimensional basis in $t$. Similar to Zhong and Schaubel (2022); Chen et al. (2023), we express the time-varying coefficients using

$${\beta }_j(t)=\sum _{r=1}^{R_j}{\gamma }_{jr}{J}_{jr}(t),{\mathit{J}}_j(t)={\left\{J_{j1}(t),\dots ,J_{j{R}_j}(t)\right\}}^{\top },$$

where ${\mathit{J}}_j(t)$ denotes the basis functions (e.g., piecewise-constant segments or spline functions with a chosen knot set) for covariate $j$. The stacked design vector can then be written as ${\tilde{\mathit{Z}}}_i(t)={\left(Z_{i1}{\mathit{J}}_1(t{)}^{\top },\dots ,Z_{ip}{\mathit{J}}_p(t{)}^{\top }\right)}^{\top }$, and the parameter vector as $\tilde{\mathit{\beta }}={\left({\mathit{\gamma }}_1^{\top },\dots ,{\mathit{\gamma }}_p^{\top }\right)}^{\top },{\mathit{\gamma }}_j={\left({\gamma }_{j1},\dots ,{\gamma }_{j{R}_j}\right)}^{\top }$. For convenience, we adopt a common basis across covariates, ie $R_j\equiv R$ and ${\mathit{J}}_j(t)\equiv \mathit{J}(t)={\left\{J_1(t),\dots ,J_R(t)\right\}}^{\top }$ with the same knot locations, in which case the representation simplifies to $\mathit{\beta }(t)=\sum _{r=1}^R{\mathit{\gamma }}_r{J}_r(t),{\tilde{\mathit{Z}}}_i(t)={\mathit{Z}}_i\otimes \mathit{J}(t),\tilde{\mathit{\beta }}={\left({\mathit{\gamma }}_1^{\top },\dots ,{\mathit{\gamma }}_R^{\top }\right)}^{\top }$.

To estimate $\tilde{\mathit{\beta }}$, we create $V$ batches of data at time points $t_v,v=1,\dots ,V$, where the component $\left\{t_1,\dots ,t_V\right\}$ are sorted in ascending order within the interval $(0,\tau )$, and $\tau$ denotes the maximum follow-up time. The stacked estimating function is given by

$${\mathit{U}}_n\left(\tilde{\mathit{\beta }}\right)=n^{-1}\sum _{i=1}^n\sum _{v=1}^V\frac{\mathbb{I}\left(U_i\le t_v\right){\Delta }_i+\mathbb{I}\left(U_i>t_v\right)}{G\left(U_i\wedge t_v\mid {\mathit{Z}}_i\right)}{\tilde{\mathit{Z}}}_i\left(t_v\right)\left[\widetilde{ℒ}\left({\mathscr{H}}_i\right)\left(t_v\right)-{\eta }^{-1}\left\{{\tilde{\mathit{\beta }}}^{\top }{\tilde{\mathit{Z}}}_i\left(t_v\right)\right\}\left(U_i\wedge t_v\right)\right],$$ (3.3)

where $G(\cdot )$ is the censoring survival function given covariates, and $\widetilde{ℒ}\left({\mathscr{H}}_i\right)(t)={\int }_0^t\widetilde{ℒ}\left({\mathscr{H}}_i\right)(du)$, $\widetilde{ℒ}\left({\mathscr{H}}_i\right)(du)=\sum _{k=1}^K{w}_{k}d{N}_{i,k}\left(U_i\wedge u\right)+w_{D}d{\tilde{N}}_{i,D}(u)$, with $d{\tilde{N}}_{i,D}(u)=\mathbb{I}\left(U_i=u,{\Delta }_i=1\right)$. The weight in (3.3), $\frac{\mathbb{I}\left(U_i\le t_v\right){\Delta }_i+\mathbb{I}\left(U_i>t_v\right)}{G\left(U_i\wedge t_v\mid {\mathit{Z}}_i\right)}$ is the inverse probability of remaining uncensored up to the relevant exposure time $U_i\wedge t_v$, conditional on covariates. It restores the unbiasedness of the observed-data estimating function relative to its full-data counterpart under covariate-dependent censoring. Operationally, the numerator $\mathbb{I}\left(U_i\le t_v\right){\Delta }_i+\mathbb{I}\left(U_i>t_v\right)$ partitions the sample into those who die before $t_v$ (contribute at their death time $D_i$) and those who live beyond $t_v$ (contribute at $t_v$); those censored before $t_v$ contribute zero, and the denominator reweights the contribution by the inverse probability of being uncensored at the truncated follow-up time $U_i\wedge t_v$. The locations of the knots defining the basis functions $\mathit{J}(t)$ need not coincide with the stacking points $\left\{t_1,\dots ,t_V\right\}$ defining the estimating equations. However, for simplicity and convenience, the stacking points may often be chosen to coincide with the knot locations.

Assuming that $\left\{{𝒪}_i,\dots ,{𝒪}_n\right\}$ are independent and identically distributed, we can show that, as $n\to \infty ,{\mathit{U}}_n(\tilde{\mathit{\beta }})$ converges uniformly to a monotone limiting function $\mathit{u}(\tilde{\mathit{\beta }})=E\left[\sum _{v=1}^V{\tilde{\mathit{Z}}}_i\left(t_v\right)\left\{{ℒ}_i(\mathscr{H})\left(t_v\right)-{\eta }^{-1}\left({\tilde{\mathit{\beta }}}^{\top }{\tilde{\mathit{Z}}}_i\left(t_v\right)\right)\left(D_i\wedge t_v\right)\right\}\right]$. Let $\widehat{\mathit{\beta }}$ and ${\tilde{\mathit{\beta }}}^0$ denote the solutions to ${\mathit{U}}_n(\tilde{\mathit{\beta }})=0$ and $\mathit{u}(\tilde{\mathit{\beta }})=0$, respectively. In Web Appendix SA.3, we prove that $\widehat{\mathit{\beta }}\stackrel{p}{\to }{\tilde{\mathit{\beta }}}^0$ as $n\to \infty$, under correct specification of the while-alive regression model and $G(\cdot )$. Finally, the censoring survival function $G\left(\cdot \mid {\mathit{Z}}_i\right)$ can be estimated using familiar survival analysis tools, such as the Cox proportional hazards model:

$${\lambda }^C\left(t\mid {\mathit{Z}}_i\right)={\lambda }_0^C\left(t\right)\text{exp}\left({\tilde{\mathit{\theta }}}^{\top }{\mathit{Z}}_i\right),$$

and thus $G\left(t\mid {\mathit{Z}}_i\right)=\text{exp}\left\{-{\int }_0^t{\lambda }_0^C(u)\text{exp}\left({\tilde{\mathit{\theta }}}^{\top }{\mathit{Z}}_i\right)\right\}$. Under completely independent censoring, a Kaplan-Meier estimator can be used to estimate $G\left(t\mid {\mathit{Z}}_i\right)=G(t)$ without covariates.

Remark 3.1. If only the association effect at a specific time $t$ is of interest, the estimating equation (3.3) can be simplified by setting $V=1$ without stacking. That is, we can solve the time-specific coefficient $\mathit{\beta }$ from the unstacked estimating equation

$$n^{-1}\sum _{i=1}^n\frac{\mathbb{I}\left(U_i\le t\right)+\mathbb{I}\left(U_i>t\right)}{G\left(U_i\wedge t\mid {\mathit{Z}}_i\right)}{\mathit{Z}}_i\left[\widetilde{ℒ}\left({\mathscr{H}}_i\right)(t)-{\eta }^{-1}\left\{{\mathit{\beta }}^{\top }{\mathit{Z}}_i\right\}\left(U_i\wedge t\right)\right]=0.$$

This approach represents a localized, or “landmark” version of the while-alive regression model at the specified time horizon $t$, instead of borrowing information across multiple landmarks $\left\{t_1,\dots ,t_V\right\}$ via spline smoothing.

3.3. Asymptotic properties

We characterize the asymptotic properties of the estimator for the regression parameters $\widehat{\mathit{\beta }}$ with independent data under covariate-dependent censoring. The properties with completely independent censoring are provided in the Web Appendix SA.5 as a special case. We make the following regularity conditions:

1. $P\left(T_{i,k}\ge t\right)>0$ and $P\left(D_i\ge t\right)>0$ for $t\in (0,\tau )$.

2. The covariate $Z_{ij}$ is bounded for $i=1,\dots ,n,j=1,\dots ,p$.

3. $0<G\left(t\mid {\mathit{Z}}_i\right)\le 1$ is absolute continuous for $t\in (0,\tau ]$, and the cumulative hazard function for censoring ${\Lambda }^{\text{C}}\left(t\mid {\mathit{Z}}_i\right)={\int }_0^t{\lambda }^{\text{C}}\left(u\mid {\mathit{Z}}_i\right)du$ is absolute continuous for $t\in (0,\tau ]$

4. $\Omega (\tilde{\mathit{\beta }})=\mathbb{E}\left\{{\tilde{\mathit{Z}}}_i^{\otimes 2}\sum _{v=1}^V\left[{\dot{\eta }}^{-1}\left({\tilde{\mathit{\beta }}}^{\top }{\tilde{\mathit{Z}}}_i\right)\left(D\wedge t_v\right)\right]\right\}$ is positive definite for any $t\in (0,\tau ]$, where ${\dot{\eta }}^{-1}(.)$ is the derivative of the ${\eta }^{-1}(.)$, and $a^{\otimes 0}=1,a^{\otimes 1}=a,a^{\otimes 2}=a{a}^{\top }$ for vector $a$.

5. The matrix

   $${\mathit{I}}_{\theta }=\mathbb{E}\left\{{\int }_0^{t_v}\left[\left\{\frac{{\mathit{s}}^{(2)}(u,\tilde{\mathit{\theta }})}{s^{(0)}(u,\tilde{\mathit{\theta }})}-{\left(\frac{{\mathit{s}}^{(1)}(u,\tilde{\mathit{\theta }})}{s^{(0)}(u,\tilde{\mathit{\theta }})}\right)}^{\otimes 2}\right\}{s}^{(0)}(u,\tilde{\mathit{\theta }})d{\Lambda }_0^C(u)\right]\right\},$$

   is positive definite for $v=1,\dots ,V$, where ${\mathit{s}}^{(d)}(u;\tilde{\mathit{\theta }})=\mathbb{E}\left\{n^{-1}\sum _{i=1}^n\mathbb{I}\left(U_i\ge u\right){\mathit{Z}}_i^{\otimes d}\text{exp}\left({\tilde{\mathit{\theta }}}^{\top }{\mathit{Z}}_i\right)\right\}$ for $d=0,1,2$, and $s^{(0)}(u;\tilde{\mathit{\theta }})$ is bounded away from zero, and $d{\Lambda }_0^C(t)={\lambda }_0^C(t)$ is the baseline hazard for censoring time.

Theorem 3.2. Under the above regularity conditions, $\sqrt{n}\left(\widehat{\mathit{\beta }}-{\tilde{\mathit{\beta }}}^0\right)$ converge weakly to zero-mean Gaussian distribution with variance $\mathit{V}=\Omega {\left({\tilde{\mathit{\beta }}}^0\right)}^{-1}𝚺\left({\tilde{\mathit{\beta }}}^0\right){\left\{\Omega {\left({\tilde{\mathit{\beta }}}^0\right)}^{-1}\right\}}^{\top }$, where

$$\begin{gathered} \sum \left({\tilde{\mathit{\beta }}}^0\right)=E\left\{{\mathit{\varphi }}_i{\left({\tilde{\mathit{\beta }}}^0\right)}^{\otimes 2}\right\},\text{and} \\ {\varphi }_i\left({\tilde{\mathit{\beta }}}^0\right)=\sum _{v=1}^V\left[{\varpi }_i\left(t_v;{\tilde{\mathit{\beta }}}^0\right)+{\kappa }_{\theta }\left(t_v;{\tilde{\mathit{\beta }}}^0,\mathit{\theta },{\Lambda }_0^C\right){\xi }_i\left(\mathit{\theta }\right)\right.\left.+{\int }_0^{t_v}{\zeta }_i\left(u,\mathit{\theta }\right)d{\kappa }_{\Lambda }\left(u,t_v;{\tilde{\mathit{\beta }}}^0,\mathit{\theta },{\Lambda }_0^C\right)\right], \\ {\varpi }_i\left(t_v;{\tilde{\mathit{\beta }}}^0\right)=\frac{\mathbb{I}\left(U_i\le t_v\right){\Delta }_i+\mathbb{I}\left(U_i>t_v\right)}{G\left(U_i\wedge t_v\mid {\mathit{Z}}_i\right)}{\tilde{\mathit{Z}}}_i\left(t_v\right) \\ \left[{\tilde{L}}_i\left({\mathscr{H}}_i\right)\left(t_v\right)-{\eta }^{-1}\left\{{\left({\tilde{\mathit{\beta }}}^0\right)}^{\top }{\tilde{\mathit{Z}}}_i\left(t_v\right)\right\}\left(U_i\wedge t_v\right)\right], \\ {\xi }_i\left(\mathit{\theta }\right)={\int }_0^{\tau }\left\{{\mathit{Z}}_i-\overline{\mathit{z}}\left(r,\mathit{\theta }\right)\right\}d{M}_i^C\left(r\right), \\ {\zeta }_i\left(u,\mathit{\theta }\right)={\int }_0^u\frac{1}{s^{\left(0\right)}\left(s,\mathit{\theta }\right)}d{M}_i^C\left(s\right), \\ {\kappa }_{\mathit{\theta }}\left(t_v;{\tilde{\mathit{\beta }}}^0,\mathit{\theta },{\Lambda }_0^C\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{n}\sum _{i=1}^n\frac{\mathbb{I}\left(U_i\le t_v\right){\Delta }_i+\mathbb{I}\left(U_i>t_v\right)}{G\left(U_i\wedge t_v\mid {\mathit{Z}}_i\right)}{\tilde{\mathit{Z}}}_i\left(t_v\right) \\ \left[{\tilde{L}}_i\left({\mathscr{H}}_i\right)\left(t_v\right)-{\eta }^{-1}\left\{{\left({\tilde{\mathit{\beta }}}^0\right)}^{\top }{\tilde{\mathit{Z}}}_i\left(t_v\right)\right\}\left(U_i\wedge t_v\right)\right]\times \text{exp}\left({\mathit{\theta }}^{\top }{\mathit{Z}}_i\right)\left[-{\left\{{\int }_0^{t_v}\frac{{\mathit{s}}^{\left(1\right)}\left(u,\mathit{\theta }\right)}{s^{\left(0\right)}\left(u,\mathit{\theta }\right)}d{\Lambda }_0^C\left(u\right)\right\}}^{\top }+{\Lambda }_0^C\left(t_v\right){\mathit{Z}}_i^{\top }\right]{\mathit{I}}_{\mathit{\theta }}^{-1}, \\ {\kappa }_{\Lambda }\left(u,t_v;{\tilde{\mathit{\beta }}}^0,\mathit{\theta },{\Lambda }_0^C\right)=\underset{n\to \infty }{\text{lim}}\frac{1}{n}\sum _{i=1}^n\frac{\mathbb{I}\left(U_i\le t_v\right){\Delta }_i+\mathbb{I}\left(U_i>t_v\right)}{G\left(U_i\wedge t_v\mid {\mathit{Z}}_i\right)}\text{exp}\left({\mathit{\theta }}^{\top }{\mathit{Z}}_i\right){\tilde{\mathit{Z}}}_i\left(t_v\right)\left.\times \text{[}{\tilde{L}}_i\left({\mathscr{H}}_i\right)\left(t_v\right)-{\eta }^{-1}{\{{\tilde{\mathit{\beta }}}^0\text{)}}^{\top }{\tilde{\mathit{Z}}}_i\left(t_v\right)\}\left(U_i\wedge t_v\right)\right]\mathbb{I}\left(u\le t_v\right), \end{gathered}$$

where $M_i^C(t)=\mathbb{I}\left(U_i\le t,{\Delta }_i=0\right)-{\int }_0^t\mathbb{I}\left(U_i\ge u\right)d{\Lambda }_i^C(u)$ is the martingale for the censoring process.

The proof of Theorem 3.2 is provided in Web Appendix SA.4. In particular, the asymptotic variance matrix $\mathit{V}$ explicitly accounts for the variability in estimating the censoring survival function. Theorem 3.2 motivates a consistent sandwich variance estimator for $\widehat{\mathit{\beta }}$, which can be used to construct 95% pointwise confidence interval for the regression coefficient, as well as prediction intervals for the while-alive rate at a given time point (Web Appendix SA.4). In particular, the while-alive regression model generalizes the average hazard (AH) regression proposed by Uno et al. (2024) from a single terminal event to composite endpoints, and from a time-fixed association analysis to time-varying association analysis. Theorem 3.2 also motivates a global test for the $j\text{th}$ covariate effect based on the null hypothesis ${\tilde{\mathit{\beta }}}_j=\left\{{\gamma }_{j1},\dots ,{\gamma }_{j{R}_j}\right\}=0$. The corresponding Wald statistic is ${\chi }_{R_j-1}^2={\widehat{\mathit{\beta }}}_j^{\top }\widehat{𝚺}(\widehat{\mathit{\beta }}{)}_j^{-1}{\widehat{\mathit{\beta }}}_j$, where ${\widehat{\mathit{\beta }}}_j$ is the estimator of ${\tilde{\mathit{\beta }}}_j$, and $\widehat{𝚺}(\widehat{\mathit{\beta }}{)}_j$ denotes the estimated $R_j\times R_j$ submatrix of the full asymptotic covariance matrix $𝚺(\tilde{\mathit{\beta }})$. Under the null hypothesis, ${\chi }_{R_j-1}^2$ follows a Chi-squared distribution with $R_j-1$ degrees of freedom. The asymptotic results under the completely independent censoring scenario with Kaplan-Meier censoring estimator are provided in Web Appendix SA.5.

Remark 3.3. Our regression framework extends the nonparametric estimator of Mao (2023), which focused only on estimating arm-specific exposure-weighted while-alive loss rate under arm-specific independent censoring. In the special case of a binary treatment $A\in \left\{\mathrm{0,1}\right\}$, the while-alive regression model specified by ${\mathit{Z}}_i={\left(1,A_i\right)}^{\top }$ in (2) is saturated and can be used to predict arm-specific while-alive loss rate; this is asymptotically equivalent to the nonparametric estimator of Mao (2023). Additional details are provided in Web Appendix SA.6.

4. EXTENSION TO CLUSTER-CORRELATED DATA

We next describe a marginal while-alive regression model as an extension to handle cluster-correlated data. Suppose there are $M$ independent clusters, where each cluster $i$ have $N_i$ individuals for $i=1,\dots ,M$. Denote $T_{ijk}$ as the $k\text{th}$ type of recurrent event, $D_{ij}$ as the fatal event, $C_{ij}$ as the censoring time for individual $j$ in cluster $i$. Then for individual $j$ in cluster $i$, we denote $U_{ij}=\text{min}\left(D_{ij},C_{ij}\right),{\Delta }_{ij}=\mathbb{I}\left(D_{ij}\le C_{ij}\right)$. The covariates ${\mathit{Z}}_{ij}$ can include both cluster-level and individual-level covariates. The complete history of event for subject $j$ in cluster $j$ is denoted as ${\mathscr{H}}_{ij}$, thus, we assume covariate-dependent censoring such that ${\mathit{C}}_i\perp {\mathcal{H}}_i\mid {\mathit{Z}}_i,{\mathit{N}}_i$, where ${\mathcal{H}}_i={\left({\mathscr{H}}_{i1},\dots ,{\mathscr{H}}_{i{N}_i}\right)}^{\top },{\mathit{C}}_i={\left(C_{i1},\dots ,C_{i{N}_i}\right)}^{\top }$, and ${\mathit{Z}}_i={\left({\mathit{Z}}_{i1},\dots ,{\mathit{Z}}_{i{N}_i}\right)}^{\top }$ be the collection of the observed information within cluster $i$. With clustered data, individuals within the cluster can be correlated. Let the observed data for cluster $i$ be denoted as ${𝒪}_i=\left\{{\tilde{\mathcal{H}}}_i,{\mathit{U}}_i,{𝚫}_i,{\mathit{Z}}_i\right\}$, where ${\tilde{\mathcal{H}}}_i=\left({\tilde{\mathscr{H}}}_{i1},\dots ,{\tilde{\mathscr{H}}}_{i{N}_i}\right)$, and ${\tilde{\mathscr{H}}}_{ij}$ denotes the observed event histories up to $U_{ij}$ for cluster $j$ in cluster $i=1,\dots ,m$. We assume the cluster-level observations ${𝒪}_i’\text{s}$ are identically independent distributed across clusters. The marginal while-alive regression for clustered data is defined as:

$$\eta \left\{l\left({\mathscr{H}}_{ij}\right)\left(t\mid {\mathit{Z}}_{ij}\right)\right\}=\mathit{\beta }(t{)}^{\top }{\mathit{Z}}_{ij},$$ (4.4)

where the while-alive loss rate $l\left({\mathscr{H}}_{ij}\right)(t)$ is defined analogously to (3.1) based on the individual event history up to $t$. Similar to (3.3), we propose the following unbiased estimating equation under the working independence assumption to estimate $\tilde{\mathit{\beta }}$ as

$$\begin{gathered} {\mathit{U}}_n^{*}(\tilde{\mathit{\beta }})=M^{-1}\sum _{i=1}^M\sum _{j=1}^{N_i}\sum _{v=1}^V\frac{\mathbb{I}\left(U_{ij}\le t_v\right){\Delta }_{ij}+\mathbb{I}\left(U_{ij}>t_v\right)}{G\left(U_{ij}\wedge t_v\mid {\mathit{Z}}_{ij}\right)}{\tilde{\mathit{Z}}}_{ij} \\ \left[\widetilde{ℒ}\left({\mathscr{H}}_{ij}\right)\left(t_v\right)-{\eta }^{-1}\left\{{\tilde{\mathit{\beta }}}^{\top }{\tilde{\mathit{Z}}}_{ij}\left(t_v\right)\right\}\left(U_{ij}\wedge t_v\right)\right]. \end{gathered}$$

In the above estimating equation, we can similarly estimate the censoring survival function $G\left(t\mid {\mathit{Z}}_{ij}\right)=P\left(C_{ij}\ge t\mid {\mathit{Z}}_{ij}\right)$ via the marginal proportional hazard model (Chen et al. 2023), given by ${\lambda }_{ij}^C(t)={\lambda }_0^C(t)\text{exp}\left({\mathit{\theta }}_C^{\top }{\mathit{Z}}_{ij}\right)$. We then denote $\widehat{G}\left(t\mid {\mathit{Z}}_{ij}\right)$ as the estimator of $G\left(t\mid {\mathit{Z}}_{ij}\right)$. If censoring is considered completely independent of covariates ${\mathit{Z}}_{ij}$, then the estimation of $G(t)$ can simply proceed with the Kaplan-Meier estimator. We then establish the asymptotic results in Web Appendix SA.7. With clustered-correlated data, besides accounting for the variability introduced by estimating the censoring survival function, the “meat” of the sandwich variance is additionally constructed in two steps. First, we compute the individual contributions as in the independent-data setting. Then, these are aggregated at the cluster level by summing over $j=1,\dots ,N_i$. Since clusters are assumed to be independent, the variance is constructed based on these independent cluster-level sums rather than the individual-level scores, which is a major difference from the asymptotic results in Theorem 3.2. A consistent variance estimator is obtained by replacing $\tilde{\mathit{\beta }}$ with its estimator $\overline{\mathit{\beta }}$, and $G\left(t\mid {\mathit{Z}}_{ij}\right)$ with its estimator $\widehat{G}\left(t\mid {\mathit{Z}}_{ij}\right)$ (Web Appendix SA.7). The asymptotic results under the completely independent censoring scenario with Kaplan-Meier censoring estimator are provided in Web Appendix SA.8. Similar to the case with independent data, by setting $w_k=0$ for all $k=1,\dots ,K$ and focusing on the parameter at a fixed time point $t$, the marginal while-alive regression reduces to an average hazard regression for cluster-correlated data (Web Appendix SA.9).

5. PRACTICAL CONSIDERATIONS

In our weighted estimation equation, smoothing parameters, including the degree of the spline basis function $d$, and the number and location of knots $R$, need to be chosen to balance flexibility and overfitting. It may often be impractical to optimize all components jointly. To simplify the process, we consider combinations of $d$ and $R$, where knots are placed at equally spaced quantiles of the observed time scale. The optimal pair $(d,R)$ is then selected using a data-driven criterion based on predictive performance tailored to while-alive loss rate estimation. To determine the optimal pair $(d,R)$, we adopt the B-fold cross-validation by minimizing the prediction error within the held-out validation data.

With independent data, we divide the data into $B$ approximately equal-sized subsets, denoted ${𝒟}_1,\dots ,{𝒟}_B$. For each fold $b=1,\dots ,B$, we fit the while-alive regression model with $R$ knots and degree of $d$, excluding the subset ${𝒟}_b$. Let ${\widehat{\mathit{\beta }}}^{(-b)}$ denote the estimator obtained from this training data (excluding fold $b$). We then evaluate the prediction error $P{E}_b(d,R)$ using the held-out data ${𝒟}_b$. Repeating this procedure over all folds gives the overall prediction error $PE(d,R)=\sum _{b=1}^{B}P{E}_b(d,R)$, and the optimal pair $(\widehat{d},\widehat{R})$ is selected as the pair $(d,R)$ that minimizes $PE(d,R)$. For while-alive regression, we define the fold-specific prediction error as

$$P{E}_b\left(d,R\right)=\sum _{i\in {𝒟}_b}{\int }_0^{\tau }{\left[\frac{\mathbb{I}\left(U_i\le s\right){\Delta }_i+\mathbb{I}\left(U_i>s\right)}{{\widehat{G}}^{\left(-b\right)}\left(U_i\wedge s\mid {\mathit{Z}}_i\right)}\left\{\widetilde{ℒ}\left({\mathscr{H}}_i\right)\left(s\right)-{\eta }^{-1}\left({\widehat{\mathit{\beta }}}^{\left(-b\right)}(s{)}^{\top }{\tilde{\mathit{Z}}}_i(s)\right)\left(U_i\wedge s\right)\right\}\right]}^{2}ds,$$

where ${\widehat{G}}^{-b}\left(t\mid {\mathit{Z}}_i\right)$ is the estimated censoring survival probability, obtained by excluding ${𝒟}_b$.

With clustered data, we modify the cross-validation approach by partitioning the data at the cluster level. Specifically, we divide the $M$ clusters into $B$ non-overlapping subsets ${𝒞}_1,\dots ,{𝒞}_B$, each containing one or more clusters. For each fold $b=1,\dots ,B$, we fit the model using all clusters except those in ${𝒞}_b$, obtaining the estimator ${\overline{\mathit{\beta }}}^{(-b)}$ and compute the prediction error $P{E}_b(d,R)$ using only the data from the held-out clusters. The total prediction error is then $PE(d,R)=\sum _{b=1}^{B}P{E}_b(d,R)$, and the optimal pair is selected to minimize this sum. Generalizing the definition from the independent data case, the $P{E}_b(d,R)$ for clustered data is defined as

$$P{E}_b(d,R)=\sum _{i\in {𝒞}_b}\sum _{j=1}^{N_i}{\int }_0^{\tau }{\left[\frac{\mathbb{I}\left(U_{ij}\le s\right){\Delta }_{ij}+\mathbb{I}\left(U_{ij}>s\right)}{{\widehat{G}}^{-b}\left(U_{ij}\wedge s\mid {\mathit{Z}}_{ij}\right)}\left\{\widetilde{ℒ}\left({\mathscr{H}}_{ij}\right)(s)-{\eta }^{-1}\left({\overline{\mathit{\beta }}}^{(-b)}{(s)}^{\top }{\tilde{\mathit{Z}}}_{ij}(s)\right)\left(U_{ij}\wedge s\right)\right\}\right]}^{2}ds.$$

In practice, the integral over time in $P{E}_b(d,R)$ can be computed numerically. Taking the independent data as an example, we can define

$$f_i(s)={\left[\frac{\mathbb{I}\left(U_i\le s\right){\Delta }_i+\mathbb{I}\left(U_i>s\right)}{G^{(-b)}\left(U_i\wedge s\mid {\mathit{Z}}_i\right)}\left\{\widetilde{ℒ}\left({\mathscr{H}}_i\right)(s)-{\eta }^{-1}\left({\widehat{\mathit{\beta }}}^{(-b)}(s{)}^{\top }{\tilde{\mathit{Z}}}_i(s)\right)\left(U_i\wedge s\right)\right\}\right]}^2.$$

We discretize the time interval $[0,\tau ]$ into a fine grid of points $0=s_1<s_2,\dots ,s_{T-1}<s_T=\tau$, and approximate the integral using trapezoidal rule with ${\int }_0^{\tau }{f}_i(s)ds\approx \sum _{t=1}^{T-1}\frac{s_{t+1}-s_t}{2}\left[f_i\left(s_t\right)+f\left(s_{t+1}\right)\right]$.

6. SIMULATION STUDIES

We carry out simulations to examine the finite-sample performance of the proposed regression methods. Focusing on independent data, we follow the approach in Wei et al. (2023) and generate two types of recurrent events $(k=\mathrm{1,2})$ and a fatal event $(k=D)$ from a joint frailty model with $n=2000$ individuals (Toenges et al. 2021). Each individual has two baseline covariates $Z_{i1}~\text{Bernoulli}(0.5)$ and $Z_{i2}~𝒩(\mathrm{0,1})$, together with a multiplicative frailty $W_i~\text{Gamma}(\text{shape}=4.5,\text{rate}=4.5$) shared across all event types. Conditional on $({\mathit{Z}}_i,W_i)$ and being at risk, the $k\text{th}$ recurrent event for individual $i$ follows a nonhomogeneous Poisson distribution with intensity ${\mu }_k\left(t\mid {\mathit{Z}}_i,D_i\ge t\right)={\mu }_{0k}(t)W_i\text{exp}\left({\mathit{\alpha }}_k^{\top }{\mathit{Z}}_i\right),k=\mathrm{1,2}$, where ${\mu }_{0k}(t)$ is the baseline intensity for event type $k$. The fatal event $D_i$ has a hazard function ${\lambda }_D\left(d\mid {\mathit{Z}}_i\right)={\lambda }_{OD}(d)W_i\text{exp}\left({\mathit{\alpha }}_D^{\top }{\mathit{Z}}_i\right)$, and ${\lambda }_{0D}(d)$ is the baseline hazard function. We specify a Weibull baseline for type 1 recurrences, a three-interval piecewise exponential baseline for type 2 recurrences, and a Gompertz baseline for the terminal hazard. The while-alive estimand under this joint frailty model admits a closed-form expression (Wei et al. 2023), with full details provided in Web Appendix SA.2. Baseline and coefficient values for all scenarios are summarized in Table 1. Right censoring times are generated either from a covariate-dependent exponential distribution $C_i~\text{Exp}\left\{c_0\text{exp}\left({\mathit{\theta }}^{\top }{\mathit{Z}}_i\right)\right\}$, with $\left(c_0,\mathit{\theta }\right)$ chosen to achieve approximately 25% or 50% censoring rate (for the fatal event), or a completely independent exponential distribution $C_i~\text{Exp}\left({\lambda }_c\right)$ with ${\lambda }_c$ tuned to achieve approximately 50% censoring rate. We consider two scenario sets. Scenario Set I corresponds to a higher recurrent event intensity, whereas Scenario Set II corresponds to a lower recurrent event intensity. Within each set, we examine two weight specifications, $\left(w_1,w_2,w_D\right)=(\mathrm{1,1},1)$ representing equal weighting and (1, 2, 2) with greater emphasis on death and severe recurrences. A complete specification of all configurations is provided in Table 1.

Table 1.

Simulation scenarios for the independent data case under the joint frailty model.^a

Scenario	$(\kappa ,v)$	${\alpha }_1$	${\alpha }_2$	${\alpha }_D$	$(w_1,w_2,w_D)$	Censoring	Rates
I(a)	(0.45, 0.30)	(0.50, −0.80)	(0.30, 0.90)	(0.20, 1.00)	(1,1,1)	$\mathit{\theta }=\left(\mathrm{0.20,0.2,0.5}\right)(25\%)$	(1.780, 0.279, 0.514)
I(b)	(0.45, 0.30)	(0.50, −0.80)	(0.30, 0.90)	(0.20, 1.00)	(1,1,1)	$\mathit{\theta }=\left(\mathrm{0.45,0.5,0.5}\right)(50\%)$	(1.680, 0.304, 0.492)
I(c)	(0.45, 0.30)	(0.50, −0.80)	(0.30, 0.90)	(0.20, 1.00)	(1,2,2)	$\mathit{\theta }=\left(\mathrm{0.20,0.2,0.5}\right)(25\%)$	(1.780, 0.279, 0.514)
I(d)	(0.45, 0.30)	(0.50, −0.80)	(0.30, 0.90)	(0.20, 1.00)	(1,2,2)	$\mathit{\theta }=\left(\mathrm{0.45,0.5,0.5}\right)(50\%)$	(1.680, 0.304, 0.492)
II(a)	(0.10, 0.30)	(0.20, 0.50)	(0.80, 1.00)	(0.20, 1.00)	(1,1,1)	$\mathit{\theta }=\left(\mathrm{0.17,0.5,0.5}\right)(50\%)$	(0.599, 0.387, 0.188)
II(b)	(0.10, 0.30)	(0.20, 0.50)	(0.80, 1.00)	(0.20, 1.00)	(1,2,2)	$\mathit{\theta }=\left(\mathrm{0.17,0.5,0.5}\right)(50\%)$	(0.599, 0.387, 0.188)
IC(a)	(0.45, 0.30)	(0.50, −0.80)	(0.30, 0.90)	(0.20, 1.00)	(1,1,1)	$\lambda =0.55(50\%)$	(1.280, 0.385, 0.571)
IC(b)	(0.45, 0.30)	(0.50, −0.80)	(0.30, 0.90)	(0.20, 1.00)	(1,2,2)	$\lambda =0.55(50\%)$	(1.280, 0.385, 0.571)

^aType 1 recurrences: Weibull baseline (scale = 0.50, shape = 1.25); Type 2 recurrences: piecewise exponential distribution baseline with cuts (0, 1, 3) and rates (0.40, 0.22, 0.10); Death: Gompertz baseline with parameters $(\kappa ,v)$. Covariate effects are ${\mu }_1\left({\mathit{Z}}_i\right)={\mathit{\alpha }}_1^{\top }{\mathit{Z}}_i,{\mu }_2\left({\mathit{Z}}_i\right)={\mathit{\alpha }}_2^{\top }{\mathit{Z}}_i,{\mu }_D\left({\mathit{Z}}_i\right)={\mathit{\alpha }}_D^{\top }{\mathit{Z}}_i$. Censoring follows $C_i~\text{Exp}\left(c_0{e}^{{\mathit{\theta }}^{\top }{\mathit{Z}}_i}\right)$ or $C_i~\text{Exp}(\lambda )$ (independent). The Rates column reports realized event rates per person-time for type 1, type 2, and death.

We fit the while-alive regression model described in Section 3.2 using a step-function basis with $R=7$ equally spaced knots at {1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0}. That is, ${\beta }_j(t)={\sum }_{r=1}^R{\gamma }_{j,r}{J}_r(t)$, and $J_r(t)=1\left\{t\ge t_{r-\mathbf{1}}\right\},j=\mathrm{1,2}$. We further let the stacking points coincide with the knot locations, and consider a log link $\eta (\cdot )=\text{log}(\cdot )$ in the regression model. The true association effects $\mathit{\beta }(t)={\left\{{\beta }_1(t),{\beta }_2(t)\right\}}^{\top }$ are evaluated directly from the joint frailty data-generating model using Monte Carlo approximation with a super population of size $n={10}^6$ with censoring removed; see Web Appendix SA.10 for further details. We consider 1000 simulation replicates for each setting, and evaluate the following performance metrics at the knot locations: absolute bias (ABias), Monte Carlo standard deviation (MCSD), average estimated standard error (AESE), and 95% coverage probability (CP) of the confidence interval estimator.

Tables 2 and 3 summarize the simulation results under two censoring scenarios: covariate-dependent censoring (Scenarios I(b) and I(d), censoring rate = 50%) and completely independent censoring (Scenarios II(a) and II(b), censoring rate = 50%). For each selected time point, we report the Abias, MCSD, AESE and CP of 95% confidence intervals of the regression coefficient estimators. The results show that the proposed regression estimators are approximately unbiased across all time points and under both censoring settings. The mean standard error estimates obtained from the proposed sandwich variance estimator closely match the empirical Monte Carlo standard deviation throughout, leading to close to nominal coverage.

Table 2.

Simulation results under Scenario I(b) and I(d) with $R=7$ equally spaced knots and stacking time at (1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0) and covariate-dependent censoring (50% censoring rate).^a

		$\left(w_1,w_2,w_D\right)=(\mathrm{1,1},1)$						$\left(w_1,w_2,w_D\right)=(\mathrm{1,2},2)$
Time	$\beta (t)$	True	ABias	MCSD	AESE	CP	True	ABias	MCSD	AESE	CP
1.0	${\beta }_1$	0.904	0.001	0.043	0.042	0.951	1.341	0.001	0.043	0.043	0.955
	${\beta }_2$	−0.008	0.004	0.056	0.056	0.941	0.355	0.004	0.064	0.063	0.940
1.5	${\beta }_1$	0.871	0.001	0.047	0.047	0.949	1.333	0.001	0.046	0.047	0.953
	${\beta }_2$	−0.146	0.004	0.055	0.055	0.938	0.195	0.004	0.066	0.066	0.937
2.0	${\beta }_1$	0.839	0.002	0.052	0.053	0.952	1.312	0.001	0.053	0.054	0.963
	${\beta }_2$	−0.238	0.004	0.056	0.054	0.942	0.077	0.004	0.070	0.069	0.942
2.5	${\beta }_1$	0.812	0.003	0.057	0.058	0.952	1.287	0.003	0.061	0.062	0.944
	${\beta }_2$	−0.304	0.004	0.056	0.054	0.940	−0.013	0.005	0.072	0.070	0.945
3.0	${\beta }_1$	0.789	0.003	0.062	0.064	0.953	1.261	0.004	0.067	0.069	0.951
	${\beta }_2$	−0.354	0.005	0.056	0.055	0.934	−0.084	0.006	0.072	0.072	0.940
3.5	${\beta }_1$	0.768	0.005	0.069	0.069	0.946	1.235	0.007	0.076	0.076	0.941
	${\beta }_2$	−0.392	0.006	0.057	0.055	0.929	−0.140	0.007	0.074	0.073	0.933
4.0	${\beta }_1$	0.751	0.008	0.074	0.074	0.940	1.211	0.010	0.082	0.081	0.940
	${\beta }_2$	−0.421	0.007	0.058	0.056	0.929	−0.184	0.010	0.076	0.075	0.929

^aSample size $n=2000$. Metrics: ABias (absolute bias), MCSD (Monte Carlo SD), AESE (asymptotic SE), CP (coverage of the 95% CI). Column blocks compare weights $\left(w_1,w_2,w_D\right)=(1,1,1)$ vs. $(1,2,2)$.

Table 3.

Simulation results under Scenario IC(a) and IC(b) with $R=7$ equally spaced knots and stacking time at (1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0) and completely independent censoring.^a

		$\left(w_1,w_2,w_D\right)=(\mathrm{1,1},1)$					$\left(w_1,w_2,w_D\right)=(\mathrm{1,2},2)$
Time	$\beta (t)$	True	ABias	MCSD	AESE	CP	True	ABias	MCSD	AESE	CP
1.0	${\beta }_1$	0.904	0.002	0.037	0.038	0.958	1.341	0.002	0.039	0.039	0.961
	${\beta }_2$	−0.008	0.004	0.059	0.060	0.950	0.355	0.004	0.063	0.064	0.954
1.5	${\beta }_1$	0.871	0.000	0.042	0.042	0.953	1.333	0.002	0.039	0.040	0.961
	${\beta }_2$	−0.146	0.008	0.063	0.062	0.939	0.195	0.007	0.071	0.072	0.938
2.0	${\beta }_1$	0.839	0.003	0.048	0.049	0.947	1.312	0.000	0.044	0.045	0.963
	${\beta }_2$	−0.238	0.010	0.070	0.071	0.930	0.077	0.009	0.083	0.084	0.928
2.5	${\beta }_1$	0.812	0.005	0.055	0.056	0.953	1.287	0.002	0.050	0.052	0.956
	${\beta }_2$	−0.304	0.012	0.077	0.078	0.929	−0.013	0.011	0.094	0.095	0.919
3.0	${\beta }_1$	0.789	0.005	0.064	0.065	0.940	1.261	0.002	0.060	0.062	0.951
	${\beta }_2$	−0.354	0.015	0.086	0.087	0.929	−0.084	0.015	0.106	0.107	0.928
3.5	${\beta }_1$	0.768	0.007	0.075	0.076	0.944	1.235	0.004	0.071	0.073	0.953
	${\beta }_2$	−0.392	0.020	0.095	0.096	0.932	−0.140	0.021	0.118	0.119	0.930
4.0	${\beta }_1$	0.751	0.010	0.086	0.087	0.935	1.211	0.007	0.084	0.085	0.935
	${\beta }_2$	−0.421	0.025	0.105	0.106	0.932	−0.185	0.027	0.131	0.132	0.930

^aSample size $n=2000$; censoring rate 50%. Metrics: ABias (absolute bias), MCSD (Monte Carlo SD), AESE (asymptotic SE), CP (coverage of the 95% CI). Column blocks compare weights (1, 1, 1) vs. (1, 2, 2).

Comparing the two weight specifications, $\left(w_1,w_2,w_D\right)=(\mathrm{1,1},1)$ versus (1, 2, 2), the bias patterns are essentially unchanged. Under covariate-dependent censoring (Table 2), MCSD and AESE tend to be larger with (1, 2, 2) than with (1, 1, 1), and the association effects are positively shifted under the (1, 2, 2) weight specification. Under completely independent censoring (Table 3), MCSD and AESE are lower compared to the covariate-dependent censoring case, and these uncertainty metrics appear to be more similar when different weight specifications are considered. The similarity between the uncertainty metrics arises likely because the censoring survival functions are modeled by Kaplan-Meier estimator without covariates.

In Web Appendix SA.10, we report additional simulations under the independent data setting, including Scenarios I(a) and I(c) under covariate-dependent censoring with 25% censoring rate, Scenarios I(b)/I(d) and II(a)/II(b) with 50% censoring rate, coupled with low event rates. The findings are consistent with the main results. Web Appendix SA.11 extends the data-generating mechanism to cluster-correlated data settings. With $M=60$ clusters of varying sizes, our proposed estimators continue to perform well, with negligible bias and close to nominal coverage. Finally, in Web Appendix SA.12, we provide an additional sensitivity analysis when the locations of spline knots and stacking times differ, under Scenarios I(b) and I(d) with 50% censoring rate. Specifically, we fit the while-alive regression with spline knots at {1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0} but consider a finer grid of stacking times at {1.0, 1.3, 1.6, 1.9, 2.2, 2.5, 2.8, 3.1, 3.4, 3.7, 4.0}. This alternative implementation of the regression estimator maintains small bias and nominal coverage, but the MCSD and AESE appear slightly smaller compared to our default implementation.

7. WHILE-ALIVE REGRESSION ANALYSIS OF THE HF-ACTION TRIAL

HF-ACTION is a randomized controlled clinical trial to evaluate the efficacy and safety of exercise training among patients with heart failure with median follow-up 30 months (O’Connor et al. 2009). Patients were randomly assigned to usual care alone or usual care plus aerobic exercise training that consists of 36 supervised sessions followed by home-based training. The primary endpoint was a composite of all-cause mortality or all-cause hospitalization. Secondary endpoints included all-cause mortality, the composite of cardiovascular mortality or cardiovascular hospitalization, and the composite of cardiovascular mortality or heart failure hospitalization. We analyzed the data using our proposed method, adjusting for exercise duration from the CPX test (in minutes), best available baseline left ventricular ejection fraction (LVEF), and history of depression (binary), using complete cases. Our analysis focused on a high-risk subgroup consisting of 719 nonischemic patients with a baseline cardiopulmonary test duration of less than 12 minutes (the analysis of the full study cohort was reported Web Appendix SA.13). We specified a log link function and assigned weights $w_H=1,w_D=2$ for all-cause of hospitalization and all-cause of death, respectively, emphasizing the greater importance of fatal events. To assess potentially covariate-dependent censoring, we fit a Cox proportional hazards model at a significance level of 0.05 and found that the CPX test was significantly associated with censoring time; accordingly, CPX was included in the censoring model. For this analysis, we modeled time-varying covariate effects using a B-spline basis (Gordon and Riesenfeld 1974). We formed polynomial with order-$d$ B-spline basis ${\mathit{B}}_j(t)$ for $j=1,\dots ,p$ with $R$ interior knots. Thus the coefficient ${\mathit{J}}_i(t)=t{\mathit{B}}_j(t)$ is smooth on $[0,\tau ]$. The number of interior knots $R$ and the degree of spline $d$ were selected by 5-fold cross-validataion over $R\in \{\mathrm{2,3},\mathrm{4,5},6\}$ and $d\in \{\mathrm{1,2},\mathrm{3,4}\}$, and the optimal choice minimizing out-of-sample prediction error was $R=4$ and $d=2$. We report both the treatment effect and the covariate effects $\mathit{\beta }(t)$ under this optimal choice, along with their pointwise corresponding 95% confidence intervals.

Figure 2 presents the time-varying effects of covariates on the while-alive loss rate among this high-risk subgroup. The results reveal substantial variation in the treatment effect over time. Notably, the treatment leads to a significant reduction in the log while-alive loss rate throughout the study period, with the 95% confidence interval consistently excluding zero. This finding aligns with results from the nonparametric approach proposed by Mao (2023). The treatment effect becomes more pronounced after 1 year, suggesting that the benefits accumulate over time. These results underscore the limitations of traditional models that assume constant effects, as such approaches may fail to detect meaningful time-varying patterns in treatment efficacy, especially during the mid-study period. For instance, at 1 year, the estimated log-rate difference is −0.257, corresponding to an approximate 22.7% $\left(\approx 1-e^{-0.257}\right)$ reduction in the while-alive loss rate for the intervention group relative to control, with a $p$-value of 0.009. By year three, the effect increased to −0.293, corresponding to 25.4% reduction in while-alive loss rate, with a $p$-value of 0.006. The global test for the treatment effect gives a $p$-value of < 0.001. Additionally, exercise duration and baseline left ventricular ejection fraction (LVEF) show statistically significant and beneficial effects throughout the follow-up period, but with different trajectories over time. In contrast, the effect due to history of depression oscillates around zero, with confidence intervals consistently including zero.

Fig. 2.

Analysis of the HF-ACTION trial with high-risk subgroup using the while-alive regression model. Each panel presents the estimated covariate effect as a function of time.

In Web Appendix SA.14, we further apply the proposed while-alive regression framework to the pragmatic cluster-randomized trial STRIDE, introduced in Section 2.

8. DISCUSSION

Recent regulatory guidance, including the ICH E9(R1) addendum (European Medicines Agency 2020), has emphasized transparency in defining estimands and handling intercurrent events such as death. As summarized in Table S1, existing regression methods for recurrent and terminal events can be mapped to different strategies to handle death as an intercurrent event. Under that classification, the while-alive regression model adopt a combination of composite and while-alive strategies by directly incorporating death into the outcome and evaluating the burden of events relative to the time patients remain alive (Schmidli et al. 2023). This yields interpretable measures of the average event rate among individuals while they remain alive, providing a complementary view to estimands that condition on survival or model death separately. Our proposed regression framework expands this while-alive perspective by modeling the time-varying association with the while-alive loss rate, capturing the instantaneous event burden over time prior to death.

One implementation consideration is the selection of the degree of spline and the number of knots. To balance model fit and prevent overfitting, we select the number of knots via cross-validation. While this data-driven approach is flexible, our sandwich variance treats the selected configuration as fixed, as is standard in the spline regression literature, and it does not address the additional uncertainty arising from cross-validation. As noted by Chatfield (1995), treating the selected model as fixed may lead to an underestimation of variability. Nevertheless, simulations from the spline regression literature and ours indicate that its practical impact on coverage is typically modest. Recently, Yang et al. (2023) proposed a semi-smoothing estimating equation for selecting knots in linear splines, which avoids the need for tuning parameters common in traditional smoothing methods. However, extending this approach to composite endpoints presents additional challenges, particularly in constructing influence functions to estimate the knots, an intriguing avenue for future research. Although cross-validation is used to choose the number of knots, one can still pre-specify knot locations based on the study context. That is, one can choose interior knots at clinical landmarks where effects may plausibly change (e.g., scheduled visits) or calendar-time anchors when secular shifts are expected.

Choosing the weights for the composite endpoint can be controversial and critically depends on the study context and objective. Following Mao and Lin (2016), our method allows one to set weight reflecting the clinical severity or priority of each event type. This flexibility is useful but introduces subjectivity, and different weight choices may lead to different conclusions. Prior studies have shown that when less frequent but more critical outcomes are combined with more frequent but less severe events, the frequent event can overshadow the composite endpoint unless appropriately weighted (e.g., Baracaldo-Santamaría et al. 2023). Regulatory guidance emphasizes that components of a composite should be of comparable clinical importance, and if mortality is included, it should not be down-weighted relative to less severe events (Food and Drug Administration 2017; Baracaldo-Santamaría et al. 2023). Specification based on $w_D<{\text{max}}_k\left\{w_k\right\}$ may thus be less clinically coherent, whereas dramatic difference between $w_D$ and $w_k$ may also allow one component to dominate the analysis results (Ozga and Rauch 2022). In our applications, we used $\left(w_H,w_D\right)=(\mathrm{1,2})$ to address severity ordering but also to avoid dominance by either component. In practice, assigning weights to different component outcomes depends on the clinical context and requires a collective effort from an interdisciplinary study team.

Finally, our method requires specifying the stacking time points $t_v\in \left\{t_1,\dots ,t_V\right\}$. We allow the differentiation between the stacking times points in the estimating equation from the spline knots and recommend using the knots or finer points for stacking time points. A principled consideration is the number of events per time interval generated by the stacking time points. One may choose stacking time points to align with clinical landmarks, but it is often necessary to ensure that each interval contains a sufficient number of events, for numerical stability in the estimation procedure. Although there is no consensus, a reasonable rule of thumb is aiming for at least one or a few events of each type per interval, and sensitivity analysis can be considered based on alternative specifications of the stacking time points.

Supplementary Material

Supplementary material is available at Biostatistics Journal online.