Assessing the value of a censored surrogate outcome

Abstract

Assessing the potential of surrogate markers and surrogate outcomes for replacing a long term outcome is an active area of research. The interest in this topic is partly motivated by increasing pressure from stakeholders to shorten the time required to evaluate the safety and/or efficacy of a treatment or intervention such that treatments deemed safe and effective can be made available to those in need more quickly. Most existing methods in surrogacy evaluation either require strict model assumptions or that primary outcome and surrogate outcome information is available for all study participants. In this paper, we focus on a setting where the primary outcome is subject to censoring and the aim is to quantify the surrogacy of an intermediate outcome, which is also subject to censoring. We define the surrogacy as the proportion of treatment effect on the primary outcome that is explained by the intermediate surrogate outcome information and propose two robust methods to estimate this quantity. We propose both a nonparametric approach that uses a kernel smoothed Nelson–Aalen estimator of conditional survival, and a semiparametric method that derives conditional survival estimates from a landmark Cox proportional hazards model. Simulation studies demonstrate that both approaches perform well in finite samples. Our methodological development is motivated by our interest in investigating the use of a composite cardiovascular endpoint as a surrogate outcome in a randomized study of the effectiveness of angiotensin-converting enzyme inhibitors on survival. We apply the proposed methods to quantify the surrogacy of this potential surrogate outcome for the primary outcome, time to death.

Appendix: Theoretical properties

The derivations below show that ${\widehat{\varDelta }}_Q(\tau ,t_0)$ is a consistent estimator of $\varDelta _Q(t,t_0)$ and that

$$\begin{aligned} \sqrt{n}\left( \begin{array}{c}{\widehat{\varDelta }}_Q(\tau ,t_0) -\varDelta _Q(\tau ,t_0) \\ {\widehat{\varDelta }}(\tau )-\varDelta (\tau )\end{array}\right) \rightarrow N(0, \varSigma _\varDelta ) \ \text{ in } \text{ distribution }, \quad \text{ as } \,\,\,n_A, n_B \rightarrow \infty . \end{aligned}$$

We first note that

$$\begin{aligned}&\sqrt{n_A}\left\{ {\widehat{\psi }}_A(\tau |t_0)-\psi _A(\tau |t_0)\right\} \\&\quad = \frac{1}{\sqrt{n_A}}\sum _{i=1}^{n_A} \left[ \frac{(T_{Ai}\wedge \tau ) I(S_{Ai}\wedge X_{Ai}>t_0)\delta _{Ai}^{\tau }}{\text{ pr }(S_{Ai}\wedge T_{Ai}>t_0)W_{A}^{C}(T_{Ai}\wedge \tau )}-\psi _A(\tau |t_0)\right] \\&\quad \quad -\frac{1}{\sqrt{n_A}}\sum _{i=1}^{n_A}E\left\{ \frac{(T_{Ai}\wedge \tau ) I(S_{Ai}\wedge X_{Ai}>t_0)\delta _{Ai}^{\tau }}{\text{ pr }(S_{Ai}\wedge T_{Ai}>t_0)W_{A}^{C}(T_{Ai}\wedge \tau )}\right\} \int _0^{t_0}\frac{dM_{Ai}^{C}(t)}{\text{ pr }(X_{Ai}\ge t)} \\&\quad \quad +\frac{1}{\sqrt{n_A}}\sum _{i=1}^{n_A} \frac{(T_{Ai}\wedge \tau ) I(S_{Ai}\wedge X_{Ai}>t_0)\delta _{Ai}^{\tau } }{\text{ pr }(S_{Ai}\wedge T_{Ai}>t_0)W_{A}^{C}(T_{Ai}\wedge \tau )} \int _0^\tau \frac{I( T_{Ai}>t)dM_{Ai}^{C}(t)}{\text{ pr }(X_{Ai}\ge t)}\\&\quad \quad -\frac{1}{\sqrt{n_A}} \sum _{i=1}^{n_A}E\left\{ \frac{W_A^{C}(t_0)(T_{Ai}\wedge \tau ) I(S_{Ai}\wedge X_{Ai}>t_0)\delta _{Ai}^{\tau } }{\text{ pr }(S_{Ai}\wedge X_{Ai}>t_0)^2W_{A}^{C}(T_{Ai}\wedge \tau )}\right\} \left\{ I(S_{Ai}\wedge X_{Ai}>t_0)\right. \\&\quad \quad \left. -\text{ pr }(S_{Ai}\wedge X_{Ai}>t_0)\right\} +o_p(1)\\&\quad =\frac{1}{\sqrt{n_A}}\sum _{i=1}^{n_A} \xi _{Ai}+o_p(1), \end{aligned}$$

and

$$\begin{aligned}&\sqrt{n_B}\left\{ {\widehat{v}}_B(\tau |t_0)-v_B(\tau |t_0)\right\} \\&\quad = \frac{1}{\sqrt{n_B}}\sum _{i=1}^{n_B} \left[ \frac{(T_{Bi}\wedge \tau ) I(X_{Bi}>t_0)\delta _{Bi}^{\tau } }{\text{ pr }( T_{Bi}>t_0)W_{B}^{C}(T_{Bi}\wedge \tau )}-\psi _B(\tau |t_0)\right] \\&\quad \quad -\frac{1}{\sqrt{n_B}}\sum _{i=1}^{n_B}E\left\{ \frac{(T_{Bi}\wedge \tau ) I( X_{Bi}>t_0)\delta _{Bi}^{\tau }}{\text{ pr }( T_{Bi}>t_0)W_{B}^{C}(T_{Bi}\wedge \tau )}\right\} \int _0^{t_0}\frac{dM_{Bi}^{C}(t)}{\text{ pr }(X_{Bi}\ge t)} \\&\quad \quad +\frac{1}{\sqrt{n_B}}\sum _{i=1}^{n_B} \frac{(T_{Bi}\wedge \tau ) I( X_{Bi}>t_0)\delta _{Bi}^{\tau } }{\text{ pr }( T_{Bi}>t_0)W_{B}^{C}(T_{Bi}\wedge \tau )} \int _0^\tau \frac{I( T_{Bi}>t)dM_{Bi}^{C}(t)}{\text{ pr }(X_{Bi}\ge t)}\\&\quad \quad -\frac{1}{\sqrt{n_B}}\sum _{i=1}^{n_B} E\left\{ \frac{(T_{Bi}\wedge \tau ) I( X_{Bi}>t_0)\delta _{Bi}^{\tau } W_B^{C}(t_0)}{\text{ pr }( X_{Bi}>t_0)^2W_{B}^{C}(T_{Bi}\wedge \tau )}\right\} \\&\qquad \times \left\{ I( X_{Bi}>t_0) -\text{ pr }( X_{Bi}>t_0)\right\} +o_p(1)\\&=\frac{1}{\sqrt{n_B}}\sum _{i=1}^{n_B} \xi _{Bi}+o_p(1), \end{aligned}$$

where $E(\xi _{Ai})=E(\xi _{Bi})=0,$$\delta _{gi}^{\tau }=I(C_{gi}>T_{gi}\wedge \tau ),$$M_{gi}^{C}(t)=I(T_{gi}<t)(1-\delta _{gi})-\int _0^t I(X_{gi}>s)d\varLambda _{g}^{C}(s)$ and $\varLambda _{g}^{C}(\cdot )$ is the cumulative hazard function of the censoring distribution in group g. It then follows, using the same arguments as in Parast et al. (2017), that

$$\begin{aligned} \frac{1}{\sqrt{n_B}}\sum _{i=1}^{n_B}\left\{ \frac{I(X_{Bi}>t_0, S_{Bi}<t_0)}{{\widehat{W}}_B^{C}(t_0)}{\widehat{\phi }}_A(\tau |t_0, S_{Bi})-P_{t_0,2}^{(B)}\int _0^{t_0}\phi _A(\tau |t_0, s)dF_B(s)\right\} \end{aligned}$$

is asymptotically equivalent to the expansion

$$\begin{aligned} \frac{1}{\sqrt{n_B}}\sum _{i=1}^{n_B} \eta _{Bi}(\tau , t_0)+\frac{1}{\sqrt{n_A}}\sum _{j=1}^{n_A} \eta _{Aj}(\tau , t_0), \end{aligned}$$

where $\{\eta _{Bi}, i=1, \cdots , n_B\}$ and $\{\eta _{Aj}, j=1, \cdots , n_A\}$ are two sets of mean zero i.i.d random variables. Coupled with the fact that

$$\begin{aligned}&\sqrt{n_B}\left\{ \left( \begin{array}{c} \frac{1}{n_B} \sum _{i=1}^{n_B} I(S_{Bi}\wedge X_{Bi}>t_0)\\ \frac{1}{n_B} \sum _{i=1}^{n_B} I(X_{Bi}>t_0)\\ {\widehat{W}}_B^C(t_0) \end{array} \right) - \left( \begin{array}{c} \text{ pr }(S_{Bi}\wedge X_{Bi}>t_0)\\ \text{ pr }(X_{Bi}>t_0)\\ W_B^C(t_0) \end{array} \right) \right\} \\&\quad =\frac{1}{\sqrt{n_B}} \sum _{i=1}^{n_B} \left( \begin{array}{c} I(S_{Bi}\wedge X_{Bi}>t_0)-\text{ pr }(S_{Bi}\wedge X_{Bi}>t_0)\\ I(X_{Bi}>t_0)-\text{ pr }(X_{Bi}>t_0)\\ -W_B^C(t_0)\int _0^{t_0} \frac{dM_B^C(t)}{\text{ pr }(X_{Bi}>t)}\end{array}\right) +o_p(1), \end{aligned}$$

these results imply that

$$\begin{aligned} \sqrt{n}\left\{ {\widehat{\varDelta }}_Q(\tau , t_0)-\varDelta _Q(\tau , t_0)\right\} =\frac{1}{\sqrt{n_B}}\sum _{i=1}^{n_B} \zeta _{Bi}(\tau , t_0)+\frac{1}{\sqrt{n_A}}\sum _{j=1}^{n_A} \zeta _{Aj}(\tau , t_0)+o_p(1) \end{aligned}$$

using the delta method, where $\zeta _{gi}$ are i.i.d mean zero random variables, assuming that $n_g/n=\pi _g, g=A, B$ are constant as $n \rightarrow \infty .$ The asymptotical normality of $\sqrt{n}\left\{ {\widehat{\varDelta }}_Q(\tau , t_0)-\varDelta _Q(\tau , t_0)\right\}$ then follows from the central limit theorem.