Counterfactual mediation analysis in the multistate model framework for surrogate and clinical time-to-event outcomes in randomized controlled trials

Abstract

In cancer randomized controlled trials, surrogate endpoints are frequently time-to-event endpoints, subject to the competing risk from the time-to-event clinical outcome. In this context, we introduce a counterfactual-based mediation analysis for a causal assessment of surrogacy. We use a multistate model for risk prediction to account for both direct transitions towards the clinical outcome and indirect transitions through the surrogate outcome. Within the counterfactual framework, we define natural direct and indirect effects with a causal interpretation. Based on these measures, we define the proportion of the treatment effect on the clinical outcome mediated by the surrogate outcome. We estimate the proportion for both the cumulative risk and restricted mean time lost. We illustrate our approach by using 18-year follow-up data from the SPCG-4 randomized trial of radical prostatectomy for prostate cancer. We assess time to metastasis as a surrogate outcome for prostate cancer-specific mortality.

1. Introduction

Surrogate outcomes are intermediate markers of disease development or progression that can be measured earlier than clinical outcomes. Using a surrogate outcome in a randomized controlled trial has the potential to shorten follow-up duration and decrease the required sample size, thereby reducing trial cost and minimizing participant burden.[1, 2] The evaluation of surrogate outcomes has received considerable attention in the literature.[3, 4] In this context, a measure of the proportion of treatment effect on the clinical outcome explained by the surrogate outcome is a useful way to convey how good a surrogate outcome is.[5, 6, 7]

In this paper, we consider a single-trial setting and we focus on the situation where both the surrogate and clinical outcomes are time-to-event outcomes subject to right censoring. In addition, the surrogate outcome is subject to semi-competing risk from the clinical outcome. This competing risk situation is particularly frequent in cancer trials, where time to progression is considered as a surrogate for time to death.[8, 9] Our work is motivated by data from the Scandinavian Prostate Cancer Group 4 (SPCG-4) randomized trial.[10, 11] This trial enrolled 695 men with clinically detected, localized prostate cancer at 14 centers in Sweden, Finland, and Iceland between 1989–1999. Participants were randomly allocated to radical prostatectomy (n=347) or watchful waiting (n=348) and followed for development of distant metastases and death from any cause. We aim to assess time to metastasis as a surrogate outcome for (i) time to death and (ii) time to prostate cancer-specific death. Here, the development of metastases is subject to the competing risk of death. In addition, prostate cancer death is subject to the competing risk of death from other causes.

Few methods have been proposed to assess surrogacy in this setting, where both outcomes are time-to-event outcomes subject to censoring and with potential competing risks. In the setting of a single trial, Ghosh has described a semi-parametric accelerated failure time model to derive the proportion of treatment effect explained.[12] In the multiple-study framework, Burzykowski et al. proposed bivariate Copula models focusing primarily on the proportional hazards models.[13] Moreover, Ghosh has extended his prior work based on the accelerated failure time models to the meta-analytic setting, allowing for analysis in the semi-competing risks setting.[14]

An alternative approach to the evaluation of surrogate outcomes is through a counterfactual-based causal mediation analysis. This framework is appropriate for evaluation of a special subclass of surrogate outcomes that are believed to be mediators on the causal pathway between treatment and the clinical outcome. The counterfactual outcome framework enables separating the direct effect of the treatment from the indirect effect through the intermediate surrogate outcome.[15, 16, 17] The ratio of the indirect effect to the total effect is interpreted as the proportion of the treatment effect on the clinical outcome explained by the surrogate outcome. Previous works have focused on causal mediation analysis when either the surrogate or clinical outcome is a time-to-event outcome but not both.[15, 18] We propose a method based on a multistate model to derive the indirect effect and the proportion of mediation when both the surrogate and clinical outcomes are time-to-event outcomes. We derive these quantities for two measures of the treatment effect: the difference in cumulative risks and the difference in restricted mean times lost. The latter quantifies the between-group gain or loss in the mean time lost due to a specific cause over a pre-specified time interval.[19] In a competing risk setting, the restricted mean time lost is the area below the cumulative incidence function.[20, 21] In the absence of competing risks, it comes down to the area above the survival function, or one minus the restricted mean survival time.[22, 23]

In Section 2, we define the notation for our motivating example in the SPCG-4 prostate cancer trial. In Sections 3 and 4, we describe the multistate model and define the indirect effect and the proportion of mediation. In Section, 5, we introduce estimators of these metrics based on the multistate model. In Section 6, we apply the proposed method to the prostate cancer trial example. We provide R code for reproducibility and implementation on GitHub (github.com/iweir/Mediation_Surrogacy).

2. Motivating example and notations

We use data from the SPCG-4 randomized trial to assess time to metastasis as a surrogate outcome for (i) time to death and (ii) time to prostate cancer death. Let M denote the time of metastasis, T the time of death, and C the time of right-censoring. Let A be the indicator for treatment arm with values A = 1 for radical prostatectomy and A = 0 for watchful waiting. We denote by n₁ and n₀ the numbers of participants randomly allocated to radical prostatectomy and to watchful waiting, respectively. We denote Z the vector of covariates at randomization. We assume that (M, T) and C are conditionally independent given Z and A. We observe n₀ + n₁ observations (X_l, $d_{X_l}$, Y_l, $d_{Y_l}$; Z_l, A_l) with l = 1,⋯, n₀, n₀ + 1⋯, n₀ + n₁ a random sample from (X, d_X, Y, d_Y; Z, A), where X = M ∧ T ∧ C and Y = T ∧ C, with corresponding censoring indicators, $d_X=1(M\le T\wedge C)$ and $d_Y=1(T\le C)$, denoting whether the time represents metastasis, death, or right-censoring. Metastasis is subject to the semi-competing risk of death. Thus, M is censored by the minimum of T and C, and not only by C.

In the case where we distinguish prostate cancer-specific death (ϵ = 1) from other causes of death (ϵ = 2), we further note T₁ the time to prostate cancer death and T₂ the time to death from other causes. We observe a random sample from (X, d_X, Y, d_Y; Z, A), where X = M ∧T₁ ∧T₂ Ĉ, $d_X=1\left(M\le T_1\wedge T_2\wedge C\right)$, Y = T₁ ∧ T₂ ∧ C, and $d_Y=ϵ\cdot 1\left(T_1\wedge T_2\le C\right)$. Prostate cancer death is the event of interest, and death from another cause is a competing event.

3. Multistate risk prediction model

We estimate the cumulative risk of all-cause death or prostate cancer death through multistate models without recovery. Let t denote the time elapsed since randomization. We let D(t) denote the occupied state at time t, where D(t) takes values in {0, 1, 2} corresponding to stable prostate cancer, metastatic prostate cancer, and death, respectively (Figure 1). All individuals start in state 0 at time t = 0 and can experience the following transitions at any time t > 0: 0 → 1, 0 → 2, and 1 → 2.

Figure 1:

Progressive illness-death and multistate models

Left: Illness-death framework in which states D(t) in {0, 1, 2} correspond to stable prostate cancer (PCa), metastatic PCa, and death.

Right: Multistate framework which states D′(t) in {0, 1, 2, 3} correspond to stable PCa, metastatic PCa, PCa-specific death, and death from other causes.

We can also distinguish between prostate cancer-specific death and death from other causes. Prostate cancer death is defined as death in the presence of metastatic disease. As a consequence, individuals who do not transition to metastasis cannot transition to prostate cancer death. We denote the different states by D′(t) taking values in {0, 1, 2, 3} corresponding to stable prostate cancer, metastatic prostate cancer, prostate cancer-specific death, and death from other causes (Figure 1). We model the following transitions: 0 → 1, 0 → 3, 1 → 2, and 1 → 3.

In the following section, we describe the non-parametric estimation of the risk function without covariates. We then describe a semi-parametric model to predict cumulative risks based on covariates at randomization and time-dependent covariates.

3.1. Non-parametric estimation

The cause-specific hazard of a transition from state j to state k at time t is defined by $h_{jk}(t)={\text{lim}}_{\Delta t\to 0}\frac{P(X(t+\Delta t)=j|X(t)=k)}{\Delta t}$. The cumulative transition hazard is then defined by $H_{jk}(t)={\int }_0^t{h}_{jk}(u)du$. We organize the cumulative hazards into the following square matrix with numbers of rows and columns equal to the number of states in the multistate model:

$$H(t)=\left(\begin{array}{ccc}-\sum _{k\ne j}{H}_{00}(t)& {\int }_0^t{h}_{01}(u)du& {\int }_0^t{h}_{02}(u)du\\ 0& -\sum _{k\ne j}{H}_{11}(t)& {\int }_0^t{h}_{12}(u)du\\ 0& 0& -\sum _{k\ne j}{H}_{22}(t)\end{array}\right).$$

The diagonal entries, $H_{jj}(t)=-\sum _{k\ne j}{H}_{jk}(t)$, indicate that no transition occurred and the individual remained in state j at time t. When using the model that distinguishes between prostate cancer-specific death and death from other causes, transition 0 → 2 is not possible, and the cumulative hazards matrix is given by:

$$H(t)=\left(\begin{array}{cccc}-\sum _{k\ne j}{H}_{00}(t)& {\int }_0^t{h}_{01}(u)du& 0& {\int }_0^t{h}_{03}(u)du\\ 0& -\sum _{k\ne j}{H}_{11}(t)& {\int }_0^t{h}_{12}(u)du& {\int }_0^t{h}_{13}(u)du\\ 0& 0& -\sum _{k\ne j}{H}_{22}(t)& 0\\ 0& 0& 0& -\sum _{k\ne j}{H}_{33}(t)\end{array}\right).$$

We use H to estimate the probability of transition from state j at time s into state k at time t, P_jk(s, t) = P (D(t) = k|D(s) = j). The transition probability matrix can be calculated with the product integral by $P(s,t)={\prod }_{(s,t]}(I+dH(u))$, where I is the identity matrix. The Aalen-Johansen estimator of the transition probability matrix is

$$\widehat{P}(s,t)=\prod _{u\in (s,t]}(I+\Delta \widehat{H}(u))$$ (1)

where u takes values among all event times in (s,t], $\widehat{H}(u)$ is the matrix of Nelson-Aalen estimators of the cumulative hazards, and ∏ denotes the finite product over event times.[24, 25]

When we focus on all-cause death as the clinical outcome, we estimate the cumulative risk of all-cause death between time 0 and t (occupation of state D = 2) by ${\widehat{P}}_{02}(0,t)$. When we focus on prostate cancer death as the clinical outcome, we estimate the cumulative incidence (or marginal probability) of prostate cancer death between time 0 and t (occupation of state D′ = 2) by ${\widehat{P}}_{02}(0,t)$.[26] In both cases, the functions take into account the probability of a transition through the intermediate state of metastasis.

3.2. Semi-parametric estimation with Cox regression

To assess the associations between covariates and transition hazards, we use a Cox model stratified by transition. Data are stacked such that for each participant, there is one row corresponding to each transition for which that participant is at risk. The row includes the start time, stop time, and indicator for the transition. Each stratum of the Cox model corresponds to a transition. Covariates are fixed at baseline but we let covariates have different effects on each transition hazard. The model for the hazard of transition from state j to k is given by

$$h_{jk}(t|Z,A)=h_{jk,0}(t)\text{\hspace{0.17em}exp}\left({\beta }_{jk}^{T}Z+{\theta }_{jk}^{T}A\right)$$ (2)

where h_jk,0(t) denotes the baseline hazard function for a transition from state j to state k at time t.

We predict the transition probabilities by plugging in the cumulative hazards estimated from the Cox regression models into equation (1). In particular, for participant l, we can estimate the cumulative risk of death (or prostate cancer death) by ${\widehat{P}}_{02}\left(0,t|Z_l,A_l\right)$, while accounting for a potential transition through the intermediate state of metastasis.

The β_jk are estimated by maximizing the partial log-likelihood described in [24]. We used the mvna and etm R packages for non-parametric estimation and the mstate R package for the semi-parametric Cox model.[27]

We also expand the semi-parametric Cox regression model to include a time-dependent covariate, m, to consider how the time of occurrence of the surrogate outcome or the time since the occurrence of the surrogate may influence the transition to death. We update the stratified Cox model (2) and equation (1) by using:

$$\begin{gathered} h_{12}(t|Z,A,m)=h_{12,0}(t)\text{\hspace{0.17em}exp}\left({\beta }_{12}^{T}Z+{\theta }_{12}^{T}A+{\gamma }^{T}m\right) \\ {h}_{13}(t|Z,A,m)=h_{13,0}(t)\text{\hspace{0.17em}exp}\left({\beta }_{13}^{T}Z+{\theta }_{13}^{T}A+{\gamma }^{T}m\right) \end{gathered}$$

or

$$\begin{gathered} h_{12}(t|Z,A,m)=h_{12,0}(t)\text{\hspace{0.17em}exp}\left\{{\beta }_{12}^{T}Z+{\theta }_{12}^{T}A+{\gamma }^T(t-m)\right\} \\ {h}_{13}(t|Z,A,m)=h_{13,0}(t)\text{\hspace{0.17em}exp}\left\{{\beta }_{13}^{T}Z+{\theta }_{13}^{T}A+{\gamma }^T(t-m)\right\}. \end{gathered}$$

We use the clock forward approach to estimate h_12,0(t) and h_13,0(t), in which case only individuals who have transitioned to the surrogate outcome contribute to the estimation.

4. Counterfactual-based mediation analysis

We now are interested in the causal effect of radical prostatectomy on (i) time to death and (ii) prostate cancer death. Different estimands can be used as target for inference, for example the cumulative risk of death (or cumulative incidence of prostate cancer death). We first give the counterfactual definition of the estimand. We denote by $F_{a{M}_{a^{\prime }}}(\tau )$ the cumulative risk of death between 0 and τ had, possibly contrary to fact, all participants been randomly allocated to treatment arm A = a but with the metastasis status they would have under allocation to A = a′.[28] Under this definition of the counterfactual estimand, the average causal effect of prostatectomy on death is $F_{1M_1}(\tau )-F_{0M_0}(\tau )$. In addition, $F_{1M_0}(\tau )$ denotes the counterfactual cumulative risk of death by time τ had all participants been randomly allocated to prostatectomy but with the metastasis status they would have under watchful waiting.

We make the standard assumptions for mediation analysis.[15, 16] Namely, that there are no unmeasured confounders for the associations between (i) the randomization arm and the true outcome, (ii) the randomization arm and surrogate outcome, and (iii) the surrogate outcome and true outcome. In addition, we assume that (iv) there is no confounder for the association between the surrogate and true outcome that is an effect of the randomization arm.

In the setting of a randomized trial, the exposure (treatment arm, A) is randomized, however, the mediator experience is not. Therefore, to satisfy assumption (iii), we assume that the baseline covariates in Z control for the confounding between the mediator and outcome. In some settings, assumption (iv) can be more markedly considered by ruling out potential confounders. In cases where the assumptions do not hold, sensitivity analysis techniques can be used for the direct and indirect effects. [29, 30]

We define the indirect effect and proportion of mediation for two measures of treatment effect, the difference in cumulative risks of the outcome and the difference in restricted mean times lost. In the case of competing risks, for example when focusing on prostate cancer-specific death in the SPCG-4 data, we focus on the cumulative incidence function instead of the cumulative risk.

4.1. Difference in cumulative risks

Using the counterfactual estimands, we define the direct effect of radical prostatectomy on cumulative risk at time τ by ${\delta }_d(\tau )=F_{1M_0}(\tau )-F_{0M_0}(\tau )$. Similarly, the indirect effect is defined by ${\delta }_i(\tau )=F_{1M_1}(\tau )-F_{1M_0}(\tau )$. The total effect is the sum of the direct and indirect effects, ${\delta }_t(\tau )=F_{1M_1}(\tau )-F_{0M_0}(\tau )$. We then define the proportion of mediation as the ratio of the indirect effect to the total effect:

$${\Pi }_1(\tau )=\frac{{\delta }_i(\tau )}{{\delta }_t(\tau )}=\frac{F_{1M_1}(\tau )-F_{1M_0}(\tau )}{F_{1M_1}(\tau )-F_{0M_0}(\tau )}.$$ (3)

The direct effect captures the effect of radical prostatectomy on the cumulative risk of death (or prostate cancer-specific death) if allocation to radical prostatectomy did not induce a change in time to metastasis. The natural indirect effect captures the effect if we were to assign all participants to radical prostatectomy and compare the cumulative risk of death (or prostate cancer-specific death) if the distribution of metastasis were set to what it would have been with radical prostatectomy versus watchful waiting. The total effect provides a measure of the total causal effect of changing the treatment arm from radical prostatectomy to watchful waiting for each participant. Finally, the proportion of mediation yields the proportion of the treatment effect on risk of death (or prostate cancer-specific death) that is explained by the treatment effect on time to metastasis.[31]

Note that the proportion of mediation is not bounded on the interval [0, 1]. When the direct and indirect effects are in opposite directions, the resulting proportion of mediation falls outside of this range. In such a case, the proportion of mediation does not allow its usual interpretation as a proportion of effect explained. However, the fact that the direct and indirect effects are in opposite directions suggests that there is no or poor surrogacy.

4.2. Difference in restricted mean times lost

We also can measure the treatment effect based on the restricted mean time lost $\mu (\tau )={\int }_0^{\tau }F(t)dt$. Using the counterfactual estimands, we then define the direct effect of radical prostatectomy on the restricted mean time lost at time τ by ${\delta }_d(\tau )={\mu }_{1M_0}(\tau )-{\mu }_{0M_0}(\tau )$. The indirect effect is ${\delta }_i(\tau )={\mu }_{1M_1}(\tau )-{\mu }_{1M_0}(\tau )$ and the total effect is ${\delta }_t(\tau )={\mu }_{1M_1}(\tau )-{\mu }_{0M_0}(\tau )$. The proportion of mediation is defined by:

$${\Pi }_2(\tau )=\frac{{\delta }_i(\tau )}{{\delta }_t(\tau )}=\frac{{\mu }_{1M_1}(\tau )-{\mu }_{1M_0}(\tau )}{{\mu }_{1M_1}(\tau )-{\mu }_{0M_0}(\tau )}.$$ (4)

5. Estimation of mediated proportion

5.1. Difference in cumulative risks

We use the multistate models described in Section 3 to estimate the counterfactual estimands and the proportion of mediation. We first fit the model using both the experimental and control groups and we estimate the predicted cumulative risk of outcome for each individual l, ${\widehat{P}}_{02}\left(0,\tau |Z_l,A_l\right)$. Note that these predictions account for the possibility of indirect transitions through metastasis.

To estimate F_1M0, we specifically predict the cumulative risk of death among watchful waiting (A = 0) participants by using their observed covariate profile, their observed metastasis experience, but by modifying their allocation to radical prostatectomy (A = 1). This way, we estimate the risk of death had all participants been randomly allocated to prostatectomy but with the metastasis status they would have under watchful waiting, since the latter was indeed observed among participants randomized to watchful waiting. We define the estimator for $F_{1M_0}(\tau )$ by:

$${\widehat{F}}_{1M_0}(\tau )=\frac{1}{n_0}\sum _{l=1}^{n_0}{\widehat{P}}_{02}\left(0,\tau |Z_l,A_l=1\right)$$

where ${\widehat{P}}_{02}\left(0,\tau |Z_l,A_l\right)$ is given by

$${\widehat{P}}_{02}\left(0,\tau |Z_l,A_l\right)=1-e^{-{\int }_0^{\tau }{\widehat{h}}_{01}\left(u|Z_l,A_l\right)du-{\int }_0^{\tau }{\widehat{h}}_{02}\left(u|Z_l,A_l\right)du}-{\widehat{P}}_{01}\left(0,\tau |Z_l,A_l\right)$$

and ${\widehat{P}}_{01}\left(0,\tau |Z_l,A_l\right)={\int }_0^{\tau }{e}^{-{\int }_0^u{\widehat{h}}_{01}\left(u|Z_l,A_l\right)du-{\int }_0^u{\widehat{h}}_{02}\left(u|Z_l,A_l\right)du}\cdot {\widehat{h}}_{01}\left(u|Z_l,A_l\right)\cdot e^{-{\int }_u^{\tau }{\widehat{h}}_{12}\left(u|Z_l,A_l\right)du}du$.

We insist that, to estimate ${\widehat{F}}_{1M_0}(\tau )$, we average the predicted probabilities $\widehat{P}\left(1M_0<\tau |Z_l,A_l\right)$ only over the n₀ participants who were truly randomly allocated to the control arm, A = 0, but by counter-factually forcing their exposure to be (A = 1), indicating random allocation to the experimental arm. Because randomization ensures that the two treatment groups only differ with respect to treatment allocation, this method enables predicting the unobserved cumulative risk of death (or prostate cancer death) for participants randomly allocated to radical prostatectomy but with the time to metastasis they would have experienced under allocation to watchful waiting. In Appendix Table 1, we illustrate the counterfactual predicted risks of death at 18 years for an hypothetical participant profile and 8 combinations of time to metastasis and time to death.

We estimate ${\widehat{F}}_{1M_1}(\tau )$ and ${\widehat{F}}_{0M_0}(\tau )$ using the same strategy, by averaging predicted probabilities over the experimental and control arms, respectively, but without modifying the exposures of individuals (detailed formulas are provided in Appendix Section 2). We plug in the estimates of $F_{1M_0}(\tau )$, $F_{1M_1}(\tau )$, and $F_{0M_0}(\tau )$ in Equation (3) to estimate the proportion mediated according to the difference in cumulative risks.

5.2. Difference in restricted mean times lost

To obtain the estimate of the restricted mean time lost, we integrate the counterfactual cumulative risks or incidence functions over the time interval from [0, τ). Let us note t_m the event times within this interval. We have

$$\begin{gathered} {\widehat{\mu }}_{1M_0}(\tau )={\int }_0^{\tau }{\widehat{F}}_{1M_0}(t)dt \\ =\frac{1}{n_0}\sum _{l=1}^{n_0}\sum _{t_m\in 〚0,\tau 〛}\left(t_{m+1}-t_m\right){\widehat{P}}_{02}\left(0,t_m|Z_l,A_l=1\right). \end{gathered}$$ (5)

Similarly, we derive estimates of ${\widehat{\mu }}_1{M}_1(\tau )$ and ${\widehat{\mu }}_{0M_0}(\tau )$. We estimate the proportion mediated according to the difference in restricted mean times lost using Equation (4).

5.3. Confidence intervals

We derive confidence intervals on the mediation measures by using a perturbation-resampling approach.[32, 33] For $b\in 〚1,B〛$, let $V^{(b)}={\left(V_{01}^{(b)},\cdots ,V_{0n_0}^{(b)},V_{11}^{(b)},\cdots ,V_{1n_1}^{(b)}\right)}^T$ be a vector of independent random variables from a standard exponential distribution. Note that $V_{Al}^{(b)}=-\text{log}\left(U_{Al}^{(b)}\right)$, where $U_{Al}^{(b)}~\mathcal{U}(0,1)$.

Let ${\widehat{F}}_{1M_0}^{(b)}(\tau )=\frac{1}{n_0}\sum _{l=1}^{n_0}{V}_{Al}^{(b)}{\widehat{P}}_{02}\left(0,\tau |Z_l,A_l=1\right)$. This quantity is averaged over participants in the control group, with their observed covariate profile and metastasis experience, but by forcing the allocation to radical prostatectomy.

Similarly, we define ${\widehat{F}}_{1M_1}^{(b)}(\tau )=\frac{1}{n_1}\sum _{l=1}^{n_1}{V}_{Al}^{(b)}{\widehat{P}}_{02}\left(0,\tau |Z_l,A_l=1\right)$, averaged over participants in the experimental group, and ${\widehat{F}}_{0M_0}^{(b)}(\tau )=\frac{1}{n_0}\sum _{l=1}^{n_0}{V}_{Al}^{(b)}{\widehat{P}}_{02}\left(0,\tau |Z_l,A_l=0\right)$, averaged over participants in the control group, in both cases with their observed covariate profile, metastasis experience, and allocation.

We further define the direct, indirect, and total effects by

$${\delta }_d^{(b)}(\tau )={\widehat{F}}_{1M_0}^{(b)}(\tau )-{\widehat{F}}_{0M_0}^{(b)}(\tau ),$$

$${\delta }_i^{(b)}(\tau )={\widehat{F}}_{1M_1}^{(b)}(\tau )-{\widehat{F}}_{1M_0}^{(b)}(\tau ),$$

$${\delta }_t^{(b)}(\tau )=F_{1M_1}^{(b)}(\tau )-F_{0M_0}^{(b)}(\tau ),$$

and the proportion of mediation by

$${\Pi }_1^{(b)}(\tau )=\frac{{\delta }_i^{(b)}(\tau )}{{\delta }_t^{(b)}(\tau )}=\frac{F_{1M_1}^{(b)}(\tau )-F_{1M_0}^{(b)}(\tau )}{F_{1M_1}^{(b)}(\tau )-F_{0M_0}^{(b)}(\tau )}.$$

We can estimate the distribution of

$$\left(\begin{array}{c}{\widehat{\delta }}_d(\tau )-{\delta }_d(\tau )\\ {\widehat{\delta }}_i(\tau )-{\delta }_i(\tau )\\ {\widehat{\delta }}_t(\tau )-{\delta }_t(\tau )\\ {\widehat{\Pi }}_1(\tau )-{\Pi }_1(\tau )\end{array}\right)$$

by the empirical distribution of

$$\left(\begin{array}{c}{\widehat{\delta }}_d^{(b)}(\tau )-{\widehat{\delta }}_d(\tau )\\ {\widehat{\delta }}_i^{(b)}(\tau )-{\widehat{\delta }}_i(\tau )\\ {\widehat{\delta }}_t^{(b)}(\tau )-{\widehat{\delta }}_t(\tau )\\ {{\widehat{\Pi }}_1}^{(b)}(\tau )-{\widehat{\Pi }}_1(\tau )\end{array}\right);b=1,\cdots ,B.$$

To obtain the 95% confidence interval for each mediation statistic, we calculate the 2.5^th and 97.5^th empirical percentiles of ${\delta }_d^{(b)}(\tau )$, ${\delta }_i^{(b)}(\tau )$, ${\delta }_t^{(b)}(\tau )$, and ${\Pi }_1^{(b)}(\tau )$, respectively. We apply the same method to derive the confidence interval for the measures based on the restricted mean time lost.

As previously noted, Π₁(τ) and Π₂(τ) are not restricted to the closed interval [0, 1]. As a consequence, we may also find confidence interval bounds outside of [0, 1] indicating that there is no evidence that the direct and indirect effects share the same direction.

6. Results in the SPCG-4 randomized trial

We analyzed data from the SPCG-4 trial collected through 18 years of follow-up. We performed 6 analyses in total. We evaluated time to metastasis as a surrogate outcome for all-cause mortality (D = 2) and for prostate cancer-specific mortality (D′ = 2). For both cases, we fit three models: (i) a multistate model with covariates at randomization (age, prostate-specific antigen level, Gleason score, tumor stage, WHO grade, and year of diagnosis); (ii) model (i) plus time of metastasis occurrence as a time-dependent covariate; (iii) model (i) plus time spent in metastasis as a time-dependent covariate. For all analyses, we used complete-case data, with 297 radical prostatectomy participants and 285 watchful waiting participants. Appendix Table 2 shows characteristics of participants by treatment arm. Below, we present the analysis for prostate cancer-specific death based on multistate model (ii), incorporating time of metastasis occurrence as a time-dependent covariate. We report the 5 remaining analyses in Appendix Tables 3–7 and Appendix Figures 1–10.

Table 1 shows the transition-specific hazard ratios for each covariate in the multistate model. There is evidence that radical prostatectomy decreases the hazard of metastasis (HR=0.56, 95% CI from 0.40 to 0.77, p < 0.001) but no evidence of an association for the remaining three transitions. Older age is associated with increased hazard of transition from stable prostate cancer and from metastatic prostate cancer to death from other causes (HR=1.57, 95% CI from 1.34 to 1.85, p < 0.001 and HR=2.21, 95% CI from 1.28 to 3.82, p = 0.005, respectively).

Table 1:

Multistate model for the SPCG-4 trial accounting for time of metastasis occurrence

	Transition 0 → 1			Transition 0 → 3			Transition 1 → 2			Transition 1 → 3
	HR	95% CI	P-value	HR	95% CI	P-value	HR	95% CI	P-value	HR	95% CI	P-value
RP vs WW	0.56	(0.40, 0.77)	<0.001	1.00	(0.76, 1.33)	0.98	0.86	(0.58, 1.27)	0.44	0.47	(0.17, 1.28)	0.14
Age, SD	0.93	(0.81, 1.06)	0.28	1.57	(1.34, 1.85)	<0.001	1.07	(0.88, 1.29)	0.51	2.21	(1.28, 3.82)	0.005
Prostate-specific antigen, ng/mL	1.03	(1.02, 1.04)	<0.001	1.01	(0.99, 1.02)	0.29	0.99	(0.97, 1.00)	0.09	0.99	(0.95, 1.03)	0.71
Gleason score, points (range 3–9)	1.45	(1.24, 1.69)	<0.001	1.09	(0.95, 1.25)	0.22	1.03	(0.86, 1.24)	0.76	1.02	(0.64, 1.62)	0.95
Tumor Stage
T2 vs. T1b	1.01	(0.60, 1.68)	0.98	0.94	(0.61, 1.44)	0.76	0.96	(0.52, 1.76)	0.89	1.14	(0.30, 4.40)	0.85
T1c vs. T1b	0.44	(0.19, 1.02)	0.06	0.94	(0.51, 1.73)	0.84	1.81	(0.62, 5.33)	0.28	7.35	(0.71, 76.50)	0.10
Year of Diagnosis	0.90	(0.84, 0.97)	0.006	0.93	(0.87, 0.99)	0.03	0.98	(0.90, 1.07)	0.62	0.96	(0.77, 1.18)	0.68
WHO grade 2 vs. 1	1.38	(0.96, 1.99)	0.08	1.00	(0.73, 1.35)	0.98	1.38	(0.89, 2.14)	0.15	1.31	(0.47, 3.67)	0.60
Time of transition to metastatic PCa	-	-	-	-	-	-	1.07	(1.00, 1.16)	0.06	0.92	(0.80, 1.06)	0.26

RP: radical prostatectomy; WW: watchful waiting; PCa: prostate cancer.

Transitions 0 → 1: from stable prostate cancer to metastatic prostate cancer; 0 → 3: from stable prostate cancer to other cause death; 1 → 2: from metastatic prostate cancer to prostate cancer death; 1 → 3: from metastatic prostate cancer to other cause death.

Figure 2 shows the direct, indirect, and total effects over time with perturbation-resampling 95% confidence intervals for the difference in cumulative risks and in restricted mean times lost. With the difference in cumulative risks, the indirect effect indicates a 2.5% reduction in cumulative risk of prostate cancer death (95% CI from −4.7% to −0.45%) at 18 years. Furthermore, at 18 years, 40% (95% CI from 14% to 53%) of the treatment effect on prostate cancer death is mediated by the treatment effect on development of metastases (Figure 3). With the difference in restricted mean times lost, the indirect effect suggests an improvement of 2.4 months in life-years lost to prostate cancer over 18 years of follow-up. We find that the proportion of mediation is 35% (95% CI from 5.3% to 47%) at 18 years. Both estimates of the proportion of mediation suggest that time to the development of metastases is a satisfactory surrogate for the clinical outcome of time to prostate cancer death. For both measures, we see that the proportion mediated increases over time. The confidence intervals on the indirect effect are wide. As a consequence, the confidence intervals for the proportion mediated are also wide.

Figure 2: Direct, indirect, and total effects in the SPCG-4 trial.

Top: difference in cumulative risks of prostate cancer death. Bottom: difference in restricted mean times lost to prostate cancer death. Solid lines: point estimate; dotted lines: perturbation-resampling 95% confidence intervals.

Figure 3: Proportion mediated by metastasis for the difference in cumulative risks of prostate cancer death (left) and for the difference in restricted mean times lost to prostate cancer death in the SPCG-4 trial.

7. Discussion

We have introduced a new model in the causal mediation framework for surrogate outcome evaluation when both the surrogate and clinical outcomes are time-to-event outcomes and the surrogate outcome is on the causal pathway between the exposure and the clinical outcome. The method handles censoring and competing risks. We used the framework of counterfactual-based mediation analysis, which offers a causal interpretation of surrogacy in the bivariate survival context using a single randomized trial. We defined the direct, indirect, total effect, and proportion of mediation with respect to differences in cumulative risks and differences in restricted mean times lost. In the SPCG-4 randomized trial, we found that a considerable proportion of the treatment effect on all-cause death was explained by the treatment effect on metastasis. However, we found substantial uncertainty on our estimates, as evidenced by wide confidence intervals. Our findings are consistent with results from the Intermediate Clinical Endpoints in Cancer of the Prostate (ICECaP) individual participant data meta-analysis which found metastasis-free survival a strong surrogate for overall survival for localized prostate cancer.[34]

Previous works focused on surrogate endpoint evaluation in the counterfactual mediation framework for different surrogate and clinical endpoint types. For example, Oba et al. used marginal structural models to estimate direct and indirect effects in the setting of a continuous mediator and binary outcome.[35] In their illustrative example, they assessed systolic blood pressure as a surrogate endpoint for cardiovascular, cerebrovascular, and cardiac events. They found wide 95% confidence intervals for the direct, indirect, and total effects. Hsu et al. outlined a conceptual framework for a counterfactual mediation model for time-varying exposures and time-varying surrogates.[36] Vandenberghe et al. described a counterfactual-based model for surrogate evaluation with a binary mediator and a continuous or binary true endpoint.[37] They proposed several estimators using propensity score-based models and generalized linear models. Subsequently, Vandenberghe et al. described a counterfactual-based model for evaluation of a binary surrogate endpoint for a time-to-event clinical endpoint.[38] They defined restricted maximum likelihood and locally efficient estimators for each mediation measure on the risk ratio scale. In another paper the authors introduced methods for surrogate evaluation for a continuous mediator measured repeatedly over time in the setting of a time-to-event true endpoint. They used a generalised dynamic path analysis framework to estimate the mediation measures using the hazard ratio scale.[39] Weber and Titman applied copula-based models to explore the association between progression-free and overall survival in oncology trials using the illness-death model for estimation of Kendall’s τ. [40] They found the Clayton copula to be appropriate for many oncology scenarios but acknowledged that the use of copula models in practice may be a limitation.

In our work, we used a multistate model approach which allows us to handle competing risks data and to explain the treatment effect on the clinical outcome with intermediate outcomes reflecting different parts of the disease process. Our method is unique because it is grounded in the causal mediation framework and addresses the specific setting where both the surrogate and clinical outcomes are time-to-event endpoints. To our knowledge, there are no existing methods for surrogate outcome evaluation that estimate the causal mediation measures in this bivariate time-to-event setting and therefore we did not pursue comparisons with other methods. We include the details and results of a small simulation study in Appendix Section 3 showing a comparison of the estimated mediation measures between our method and a Cox model which treats the surrogate outcome as binary, not time-to-event, for increasing proportions of censoring. Future work will connect the true values for the direct, indirect, total effects, and the proportion of mediation to the simulation parameters and will assess the statistical performance of the proposed method in a comprehensive simulation study.

The vast majority of applied work on surrogacy evaluation has focused on the hazard ratio as a measure of treatment effect.[41] Stensrud et al. have discussed how hazard ratios have a difficult causal interpretation, whereas the difference in restricted mean times lost offers a more accessible interpretation as the causal difference in life-years lost between two groups over a restricted period of time.[42] Parast et al. have also defined a model to assess surrogacy using the difference in restricted mean survival times, however it is not in the counterfactual context.[43] Our proposed method focuses on the evaluation of surrogate outcomes that are causal mediators. The mediation measures with the difference in restricted mean times lost provide a causal interpretation of treatment effect on the time scale. However, our estimation model relies on the hazard functions to be proportional over time. This condition appeared consistent with the data in our motivating example, but may be violated in other randomized trials. An alternative approach would be to use parametric multistate survival models with time-dependent effects.[44]

Conventional surrogacy evaluation methods focus on the relationship between the treatment effect on the surrogate endpoint and the treatment effect on the true endpoint. In our motivating example, our causal measures provide insight into the tumor-dependent and tumor-independent mechanisms through which treatment affects survival. Our method provides insight into the biological mechanisms through which treatment influences progression to lethal prostate cancer. This idea can be applied more broadly to other clinical applications.

In the SPCG-4 trial, radical prostatectomy showed a benefit for all-cause mortality, prostate cancer-specific death, as well as metastasis.[10] This gave ground for the application of our method to explore metastasis as a potential surrogate for all-cause mortality and prostate cancer mortality. We found that the treatment effect on metastasis explained a considerable proportion of the treatment effect on all-cause death, however the corresponding confidence interval was wide. Despite controlling for covariates at randomization, there could be residual confounding between the surrogate and clinical outcome due to the fact that our all-cause death can potentially occur at a much later time point than metastasis. We could apply our method at lagged time points, i.e. assuming that data on metastasis are truncated at an earlier time point than data on mortality. This way, one would assess the temporal relationship between the treatment effect on the mediator at an earlier time point and the treatment effect on the clinical endpoint at a later time point.

Parast et al. addressed a similar idea in which they suggest to assess whether the treatment effect for the clinical endpoint at an earlier time point is a surrogate for the treatment effect on the same clinical endpoint at a later time point.[45] However, in their method, if an individual experiences both the surrogate and the primary outcome, only the primary outcome is incorporated into the analysis. Thus the method ignores indirect paths to the primary outcome and in such cases, the surrogate event does not contribute to the proportion of mediation.

Data Availability

Research data are not shared.