Semiparametric inference on the absolute risk reduction and the restricted mean survival difference

Abstract

For time-to-event data, when the hazards are non-proportional, in addition to the hazard ratio, the absolute risk reduction and the restricted mean survival difference can be used to describe the time-dependent treatment effect. The absolute risk reduction measures the direct impact of the treatment on event rate or survival, and the restricted mean survival difference provides a way to evaluate the cumulative treatment effect. However, in the literature, available methods are limited for flexibly estimating these measures and making inference on them. In this article, point estimates, pointwise confidence intervals and simultaneous confidence bands of the absolute risk reduction and the restricted mean survival difference are established under a semiparametric model that can be used in a sufficiently wide range of applications. These methods are motivated by and illustrated for data from the Women’s Health Initiative estrogen plus progestin clinical trial.

Appendices

Appendix A : Consistency

Throughout the Appendices, we assume the following regularity conditions, which is a little weaker than the conditions used in Yang and Prentice (2005).

• Condition 1. $\lim \frac{n_1}{n}=\rho \in (0,1)$.

• Condition 2. The survivor function $G_i$ of $C_i$ given $Z_i$ is continuous and satisfies

  $$\begin{aligned} \frac{1}{n}\sum _{i\le n_1} G_i(t) \rightarrow \varGamma _1,\ \frac{1}{n}\sum _{i> n_1} G_i(t) \rightarrow \varGamma _2, \end{aligned}$$

  uniformly for $t\le \tau$, for some $\varGamma _1, \varGamma _2$, and $\tau <\tau _0$ such that $\varGamma _j(\tau )>0,\ j=1,2$.

• Condition 3. The survivor functions $S_C$ and $S_T$ are absolutely continuous and $S_C(\tau )>0$.

Under these conditions, the strong law of large numbers implies that (3) is satisfied.

For $t\le \tau ,$ define

$$\begin{aligned} L(t)&= \varGamma _1 S_C+ \varGamma _2 S_T, \\ U_j(t;\mathbf{b})&= \int ^t_0\varGamma _1 dF_C+ \exp (-b_j) \int ^t_0\varGamma _2dF_T, \ j= 1,2, \\ \varLambda _j(t;\mathbf{b})&= \int ^t_0\frac{dU_j(s;\mathbf{b})}{L(s)},\ \ j= 1,2,\\ P(t;\mathbf{b})&= \exp \{-\varLambda _2(t;\mathbf{b})\}, \ R(t;\mathbf{b})=\frac{1}{P(t;\mathbf{b})}\int ^t_0P(s;\mathbf{b})d\varLambda _1(s;\mathbf{b}),\\ f_j^0(t;\mathbf{b})&= \frac{\exp (-b_j)R^{j-1}(t;\mathbf{b})}{\exp (-b_1)+ \exp (-b_2)R(t;\mathbf{b})},\ j=1,2, \\ m_j(\mathbf{b})&= \left\{ \int ^{\tau }_0f_j^0 \varGamma _2(t)dF_T(t) -\int ^{\tau }_0\frac{f_j^0\varGamma _2(t) S_T(t)dR(t;\mathbf{b})}{\exp (-b_1) +\exp (-b_2)R(t;\mathbf{b})}\right\} ,\ j=1,2, \end{aligned}$$

and $m(\mathbf{b})=(m_1(\mathbf{b}),m_2(\mathbf{b}))^{\prime }$. We will also assume

• Condition 4. The function $m(\mathbf{b})$ is non-zero for $b\in \mathcal {B}-\{\mathbf{\beta }\}$, where $\mathcal {B}$ is a compact neighborhood of $\mathbf{\beta }$.

Theorem 1

Suppose that Conditions $1\sim 4$ hold. Then, (i) the zero $\hat{\mathbf{\beta }}$ of $Q(\mathbf{b})$ in $\mathcal {B}$ is strongly consistent for $\mathbf{\beta }$; (ii) $\hat{\varPhi }(t)$ is strongly consistent for $\varPhi (t)$, uniformly for $t\in [0, \tau ]$, and $\hat{\varPsi }(t)$ is strongly consistent for $\varPsi (t)$, uniformly on $t\in [0, \tau ]$; (iii) $\hat{\varOmega }$ converges almost surely to a limiting matrix $\varOmega ^*$.

Proof

Under Conditions $1 \sim 3$, the limit of $\sum ^n_{i=1}I(X_i\ge t)/n$ is bounded away from zero on $t\in [0,\tau ]$. Thus, with probability 1,

$$\begin{aligned} \frac{\sum ^n_{i=1}\delta _i e^{-b_jZ_i}I(X_i=t)}{\sum ^n_{i=1}\delta _i I(X_i\ge t)}\rightarrow 0,\ j=1,2, \end{aligned}$$

(14)

uniformly for $t\in [0,\tau ]$ and $b\in \mathcal {B}.$ From this, one also has, with probability 1,

$$\begin{aligned} | \Delta \hat{P}(t;\mathbf{b})|\rightarrow 0,\ | \Delta \hat{R}(t;\mathbf{b})|\rightarrow 0, \end{aligned}$$

(15)

uniformly for $t\in [0,\tau ]$ and $b\in \mathcal {B},$ where $\Delta$ indicates the jump of the function in $t$.

Define the martingale residuals

$$\begin{aligned} \hat{M}_{i}(t;\mathbf{b})=\delta _iI(X_i\le t)-\int ^t_0I(X_i\ge s) \frac{\hat{R}(ds;\mathbf{b})}{e^{-b_1Z_i}+e^{-b_2Z_i} \hat{R}(s;\mathbf{b}) }, \ 1\le i \le n. \end{aligned}$$

From (14), (15), and the fundamental theorem of calculus, it follows that, with probability 1,

$$\begin{aligned} Q(\mathbf{b})=\sum ^n_{i=1}\int ^{\tau }_0\{f_{i}(t;\mathbf{b})+o(1)\}\hat{M}_i(dt;\mathbf{b}), \end{aligned}$$

(16)

uniformly in $t\le \tau , b\in \mathcal {B}$ and $i\le n$, where $f_i=(f_{1i},f_{2i})^T$, with

$$\begin{aligned} f_{1i}(t;\mathbf{b})=\frac{Z_ie^{-b_1Z_i}}{e^{-b_1Z_i}+ e^{-b_2Z_i}\hat{R}(t;\mathbf{b})},\ f_{2i}(t;\mathbf{b})=\frac{Z_ie^{-b_2Z_i}\hat{R}(t;\mathbf{b})}{e^{-b_1Z_i}+ e^{-b_2Z_i}\hat{R}(t;\mathbf{b})}. \end{aligned}$$

From the strong law of large numbers (Pollard 1990, p.41) and repeated use of Lemma A1 of Yang and Prentice (2005), one obtain, with probability 1,

$$\begin{aligned} \hat{P}(t;\mathbf{b})\rightarrow \hat{P}(t;\mathbf{b}),\ \hat{R}(t;\mathbf{b})\rightarrow R(t;\mathbf{b}),\ Q(\mathbf{b})/n \rightarrow m(\mathbf{b}), \end{aligned}$$

(17)

uniformly in $t\le \tau$ and $\mathbf{b}\in \mathcal {B}$. From these results and Condition 4, one obtains the strong consistency of $\hat{\mathbf{\beta }}, \hat{\varPhi }(t)$ and $\hat{\varPsi }(t)$, and almost sure convergence of $\hat{\varOmega }$. $\square$

Appendix B: Weak convergence

For $\ C(t),\ D(t),\ \mu _1(t), \mu _2(t), \nu _1(t),\ \nu _2(t)$, let $C^*(t), \ D^*(t),$ etc. be their almost sure limit. In addition, let $L_j$ be the almost sure limit of $K_j/n,\ j=1, 2.$ For $0\le s, t<\tau ,$ let

$$\begin{aligned}&\sigma _\varPhi (s,t)\nonumber \\&\quad = D^{*T}(s) \varOmega ^* \left( \int ^\tau _0 \frac{\mu _1^*\mu _1^{*T}}{1+R} L_1dR + \int ^\tau _0 \frac{\mu _2^*\mu _2^{*T}}{e^{-\beta _1}+e^{-\beta _2}R}L_2dR\right) \varOmega ^{*T} D^*(t)\nonumber \\&\quad + C^*(s)C^*(t) \left( \int ^s_0 \frac{\nu _1^{*2}}{1+R} L_1dR + \int ^s_0 \frac{\nu _2^{*2}}{e^{-\beta _1}+e^{-\beta _2}R}L_2dR\right)\nonumber \\&\quad + C^*(t) D^{*T}(s)\varOmega ^* \left( \int ^t_0 \frac{\mu _1^*\nu _1^*}{1+R} L_1dR + \int ^t_0 \frac{\mu _2^*\nu _2^*}{e^{-\beta _1}+e^{-\beta _2}R}L_2dR\right)\nonumber \\&\quad +C^*(s) D^{*T}(t) \varOmega ^* \left( \int ^s_0 \frac{\mu _1^*\nu _1^*}{1+R} L_1dR + \int ^s_0 \frac{\mu _2^*\nu _2^*}{e^{-\beta _1}+e^{-\beta _2}R}L_2dR\right), \end{aligned}$$

(18)

and

$$\begin{aligned}&\sigma _\varPsi (s,t)\nonumber \\&\quad = \int _0^sD^{*T}(x)dx \varOmega ^* \left( \int ^\tau _0 \frac{\mu _1^*(w)\mu _1^{*T}(w)}{1+R(w)}L_1(w)dR (w)\right. \nonumber \\&\quad \left.+ \int ^\tau _0 \frac{\mu _2^*(w)\mu _2^{*T}(w)}{e^{-\beta _1}+e^{-\beta _2}R(w)}L_2(w)dR(w)\right) \varOmega ^{*T} \int _0^tD^{*T}(x)dx \nonumber \\&\quad + \int ^s_0 \frac{\nu _1^{*2}(w)}{1+R(w)} \left(\int _w^sC^*(x)dx\right)^2L_1(w)dR(w) \nonumber \\&\quad + \int ^s_0 \frac{\nu _2^{*2}(w)}{e^{-\beta _1}+e^{-\beta _2}R(w)} \left(\int _w^sC^*(x)dx\right)^2L_2(w)dR(w)\nonumber \\&\quad +\int _0^sD^{*T}(x)dx\varOmega ^* \int ^t_0 \frac{\mu _1^*(w)\nu _1^*(w)}{1+R(w)}\left(\int _w^tC^*(x)dx\right)L_1(w)dR(w) \nonumber \\&\quad + \int _0^sD^{*T}(x)dx\varOmega ^* \int ^t_0 \frac{\mu _2^*(w)\nu _2^*(w)}{e^{-\beta _1}+e^{-\beta _2}R(w)}\left(\int _w^tC^*(x)dx\right)L_2dR(w) \nonumber \\&\quad +\int _0^tD^{*T}(x)dx\varOmega ^* \int ^s_0 \frac{\mu _1^*(w)\nu _1^*(w)}{1+R(w)}\left(\int _w^sC^*(x)dx\right)L_1(w)dR(w) \nonumber \\&\quad + \int _0^tD^{*T}(x)dx\varOmega ^* \int ^s_0 \frac{\mu _2^*(w)\nu _2^*(w)}{e^{-\beta _1}+e^{-\beta _2}R(w)}\left(\int _w^sC^*(x)dx\right)L_2(w)dR(w). \end{aligned}$$

(19)

Theorem 2

Suppose that Conditions $1\sim 4$ hold and that the matrix $\varOmega ^*$ is non-singular. Then, (i) $U_n$ is asymptotically equivalent to the process $\tilde{U_n}$ in (8) which converges weakly to a zero-mean Gaussian process $U^*$ on $[0,\tau ]$, with covariance function $\sigma _\varPhi (s,t)$ in (18). In addition, $\hat{U}_n(s)$ given the data converges weakly to the same limiting process $U^*$. (ii) $V_n(t)$ is asymptotically equivalent to the process $\tilde{V}_n$ in (10) which converges weakly to the zero-mean Gaussian process $\int _0^tU^*(s)ds$ on $t\in [0, \tau ],$ with covariance function $\sigma _\varPsi (s,t)$ in (19). The process $\int _0^t\hat{V}_n(s)ds$ given the data also converges weakly to the same limiting process $\int _0^tU^*(s)ds$.

Proof

(i) As in the proof for Theorem A2 (ii) in Yang and Prentice (2005), from the strong embedding theorem and (16), $Q(\mathbf{\beta })/\sqrt{n}$ can be shown to be asymptotically normal. Now Taylor series expansion of $Q(\mathbf{b})$ around $\mathbf{\beta }$ and the non-singularity of $\varOmega ^*$ imply that $\sqrt{n}(\hat{\mathbf{\beta }}-\mathbf{\beta })$ is asymptotically normal. From the $\sqrt{n}$-boundedness of $\hat{\mathbf{\beta }}$,

$$\begin{aligned} \sqrt{n}( \hat{R}(t;\hat{\mathbf{\beta }}) -\hat{R}(t;\mathbf{\beta }) )&= \frac{\partial R(t;\beta )}{\partial \beta } \sqrt{n} (\hat{\mathbf{\beta }}-\mathbf{\beta })+o_p(1), \end{aligned}$$

uniformly in $t\le \tau .$ These results, some algebra and Taylor series expansion together show that $U_n$ is asymptotically equivalent to $\tilde{U}_n$. Similarly to the proof of the asymptotic normality of $Q(\mathbf{\beta })/\sqrt{n}$, one can show that $\tilde{U}_n$ converges weakly to a zero-mean Gaussian process. Denote the limiting process by $U^*$. From the martingale integral representation of $\tilde{U}_n$, it follows that the covariation process of $U^*$ is given by $\sigma (s,t)$ in (18), which can be consistently estimated by $\hat{\sigma }(s,t)$ in (8). By checking the tightness condition and the convergence of the finite-dimensional distributions, it can be shown that $\hat{U}_n(s)$ given the data also converges weakly to $U^*$. $\square$

(ii) From the results in (i), the assertions on $V_n$ and $\tilde{V}_n$ follow.