Estimation of average causal effect using the restricted mean residual lifetime as effect measure

Abstract

Although mean residual lifetime is often of interest in biomedical studies, restricted mean residual lifetime must be considered in order to accommodate censoring. Differences in the restricted mean residual lifetime can be used as an appropriate quantity for comparing different treatment groups with respect to their survival times. In observational studies where the factor of interest is not randomized, covariate adjustment is needed to take into account imbalances in confounding factors. In this article, we develop an estimator for the average causal treatment difference using the restricted mean residual lifetime as target parameter. We account for confounding factors using the Aalen additive hazards model. Large sample property of the proposed estimator is established and simulation studies are conducted in order to assess small sample performance of the resulting estimator. The method is also applied to an observational data set of patients after an acute myocardial infarction event.

Appendices

Appendix 1: Weak convergence of $n^{1/2}\{\hat{S}_a(t)-S_a(t)\}$

According to Martinussen and Scheike (2006, p. 118), $n^{1/2}\{\hat{B}(t)-B(t)\}$ can be written as

$$\begin{aligned} n^{1/2}\{\hat{B}(t)-B(t)\}=n^{-1/2} \sum _{i=1}^n \int _0^t\{n^{-1}Z^T(u)Z(u)\}^{-1}Z_i(u)dM_i(u), \end{aligned}$$

(5)

where

$$\begin{aligned} M_i(t)=N_i(t)-\int _0^tZ_i^T(u)dB(u), \end{aligned}$$

is the martingale process. When n is large, representation (5) is equivalent to a sum of independent and identically distributed zero-mean martingales

$$\begin{aligned} n^{1/2}\{\hat{B}(t)-B(t)\}=n^{-1/2} \sum _{i=1}^n \varepsilon _i^B(t)+o_p(1), \end{aligned}$$

(6)

where

$$\begin{aligned} \varepsilon _i^B(t)=\int _0^t w(u) Z_i(u)dM_i(u), \end{aligned}$$

and w(t) is the limit in probability of $\{n^{-1}Z^T(t)Z(t)\}^{-1}$.

We now turn to the proof for the weak convergence of $n^{1/2}\{\hat{S}_a(t)-S_a(t)\}$. Define

$$\begin{aligned} S_{a1}(t)=\exp \{-B_0(t)-B_A(t) \, a\}, \quad S_{a2}(t)=\int \exp \{-x^TB_X(t)\} \; dF_X(x), \end{aligned}$$

so that $S_a(t)=S(t \mid \hat{A}=a)=S_{a1}(t) \; S_{a2}(t)$, and likewise, define

$$\begin{aligned} \hat{S}_{a1}(t)=\exp \{-\hat{B}_0(t)-\hat{B}_A(t) \, a\}, \quad \hat{S}_{a2}(t)=n^{-1}\sum _{i=1}^n\exp \{-X_i^T\hat{B}_X(t)\}, \end{aligned}$$

where $\hat{S}_a(t)=\hat{S}_{a1}(t) \; \hat{S}_{a2}(t)$. Then it easily follows that

$$\begin{aligned} n^{1/2}\{\hat{S}_a(t)-S_a(t)\} =n^{1/2}\{\hat{S}_{a1}(t)-S_{a1}(t)\}\hat{S}_{a2}(t) +n^{1/2}\{\hat{S}_{a2}(t)-S_{a2}(t)\}S_{a1}(t). \end{aligned}$$

(7)

Taking the Taylor series expansion of $\hat{S}_{a1}(t)$, together with the consistency of $\hat{B}(t)$ yields that

$$\begin{aligned} n^{1/2}\{\hat{S}_{a1}(t)-S_{a1}(t)\} =&-n^{1/2}\left[ \{\hat{B}_0(t)-B_0(t)\}+\{\hat{B}_A(t)-B_A(t)\} \, a\right] S_{a1}(t)+o_p(1) \nonumber \\ =&\, n^{-1/2}\sum _{i=1}^n\varepsilon _i^{S_{a1}}(t)+o_p(1), \end{aligned}$$

(8)

where

$$\begin{aligned} \varepsilon _i^{S_{a1}}(t)=-S_{a1}(t)\{\varepsilon _i^{B_0}(t)+a \, \varepsilon _i^{B_A}(t)\}. \end{aligned}$$

In the latter display, $\varepsilon _i^{B_0}(t)$ and $\varepsilon _i^{B_A}(t)$ are defined as the influence function of $n^{1/2}\{\hat{B}_0(t)-B_0(t)\}$ and $n^{1/2}\{\hat{B}_A(t)-B_A(t)\}$, respectively, given in (6).

By a Taylor series expansion of $\hat{S}_{a2}(t)$ along with the consistency of $\hat{B}(t)$ and the uniform strong law of large numbers, it further follows that

$$\begin{aligned} n^{1/2}\{\hat{S}_{a2}(t)-S_{a2}(t)\}= n^{-1/2}\sum _{i=1}^n\varepsilon _i^{S_{a2}}(t)+o_p(1), \end{aligned}$$

(9)

where

$$\begin{aligned} \varepsilon _i^{S_{a2}}(t)=\exp \{-X_i^TB_X(t)\}-\mu (t) \; \varepsilon _i^{B_{X}}(t)-S_{a2}(t), \end{aligned}$$

with $\mu (t)$ being the limit in probability of $n^{-1}\sum _{i=1}^n\exp \{-X_i^TB_X(t)\}X_i^T$. In the latter display, $\varepsilon _i^{B_{X}}(t)$ is defined as the influence function of $n^{1/2}\{\hat{B}_X(t)-B_X(t)\}$, given in (6).

Therefore, combining the results from Eqs. (7), (8) and (9) follows that $n^{1/2}\{\hat{S}_a(t)-S_a(t)\}$ can be decomposed as a sum of independent and identically distributed terms as

$$\begin{aligned} n^{1/2}\{\hat{S}_a(t)-S_a(t)\}=n^{-1/2}\sum _{i=1}^n\varepsilon _i^{S_a}(t)+o_p(1), \end{aligned}$$

(10)

where

$$\begin{aligned} \varepsilon _i^{S_a}(t)=S_{a2}(t)\varepsilon _i^{S_{a1}}(t) +S_{a1}(t)\varepsilon _i^{S_{a2}}(t). \end{aligned}$$

Appendix 2: Weak convergence of $n^{1/2}\{\hat{\delta }(t)-\delta (t)\}$

To find the influence function for $n^{1/2}\{\hat{\delta }(t)-\delta (t)\}$, it follows that

$$\begin{aligned} n^{1/2}\{\hat{\delta }(t)-\delta (t)\}=&\, n^{1/2}\left\{ \frac{1}{\hat{S}_1(t)}\int _t^L\hat{S}_1(u)du-\frac{1}{S_1(t)}\int _t^L S_1(u)du \right\} \\ -&\, n^{1/2}\left\{ \frac{1}{\hat{S}_0(t)}\int _t^L\hat{S}_0(u)du-\frac{1}{S_0(t)}\int _t^L S_0(u)du \right\} . \end{aligned}$$

It can also be written that, for $a=0,1$,

$$\begin{aligned} \frac{1}{\hat{S}_a(t)}\int _t^L\hat{S}_a(u)du-\frac{1}{S_a(t)}\int _t^L S_a(u)du, \end{aligned}$$

is equal to

$$\begin{aligned} \frac{1}{\hat{S}_a(t)}\int _t^L\left\{ \hat{S}_a(u) -S_a(u)\right\} du-\frac{\left\{ \hat{S}_a(t)-S_a(t)\right\} }{S_a(t)\hat{S}_a(t)}\int _t^LS_a(u)du. \end{aligned}$$

Using expression (10), asymptotic approximation for $n^{1/2}\{\hat{\delta }(t)-\delta (t)\}$ can be displayed by

$$\begin{aligned} n^{1/2}\{\hat{\delta }(t)-\delta (t)\} =n^{-1/2}\sum _{i=1}^n\varepsilon _i^{\delta }(t)+o_p(1), \end{aligned}$$

where $\varepsilon _i^{\delta }(t)=\varepsilon _i^{\delta _1}(t) -\varepsilon _i^{\delta _0}(t)$ with

$$\begin{aligned} \varepsilon _i^{\delta _a}(t) =\frac{1}{S_a(t)}\int _t^L\varepsilon _i^{S_a}(u)du -\frac{\varepsilon _i^{S_a}(t)}{S_a(t)^2}\int _t^LS_a(u)du, \quad \text {for} \; a=0,1. \end{aligned}$$