Restricted Mean Survival Time Estimation: Nonparametric and Regression Methods

Abstract

In survival analyses, the log-rank test is the standard approach to comparison of survival distributions estimated from independent groups. The semiparametric proportional hazards model uses the hazard function as the conduit to assess the influence of covariates x on the survival distribution of an event time T. The accelerated failure time model aligned closely to standard linear regression can estimate summary features such as the mean and percentiles of the survival distribution as functions of x. However, a full specification of a parametric distribution is often needed to analyze a model for $E(\log T|{\mathbf{x}})$. A different approach is to model the restricted mean survival time $E(\hbox{min} (T,\tau )|{\mathbf{x}})$. The specified time horizon $\tau$ is informed by applications. All approaches must account for censoring in event times. We review analyses for restricted mean survival time based on the method of inverse-probability of censoring weighting, and on pseudo observations and a discussion on specified parametric models. As illustration, we apply the methods to a data set on relapse-free survival time in patients who underwent bone marrow transplantation.

Appendix

1.1 Adjustment for Estimated Weights in the IPCW Method

In the second term of Eq. (10) use $\hat{w}_{i} - w_{i} = - \delta_{i}^{ * } \left( {\frac{{\hat{G}(R_{i} - ) - G(R_{i} - )}}{{\hat{G}(R_{i} - )G(R_{i} - )}}} \right)$ and the representation for the KM estimator of the censoring distribution from Anderson et al., [2] or Lin [22],

$\left( {\frac{{\hat{G}(t) - G(t)}}{G(t)}} \right) = - \int_{0}^{t} {\frac{{\hat{G}(s - )}}{G(s)}} \frac{{dM^{c} (s)}}{Y(s)} \approx - \sum\nolimits_{i = 1}^{n} {\int_{0}^{t} {\frac{{dM_{i}^{c} (u)}}{Y(u)}} }$ where $Y_{i} (u) = [C_{i} \wedge T_{i} \ge u],\,\,\,Y(u) = \sum\nolimits_{i = 1}^{n} {Y_{i} (u)}$

$M_{i}^{c} (t) = [C_{i} \wedge T_{i} \le t,\delta_{i} = 0] - \int_{0}^{t} {[C_{i} \wedge T_{i} \ge u]} \,dA^{c} (u)$ and $A^{c}$ is the integrated hazard of C estimated by $\int_{0}^{t} {\frac{{dN^{c} (u)}}{Y(u)}}$ where $N^{c} (t) = \sum\nolimits_{i = 1}^{n} {[C_{i} \wedge T_{i} \le t,\delta_{i} = 0]}$. Next,

$$ \begin{aligned} \sum\limits_{i = 1}^{n} {(\hat{w}_{i} - w_{i} )} \left( {R_{i} - g^{ - 1} ({\mathbf{x^{\prime}}}_{i} \beta )} \right){\mathbf{x}}_{i} & = - \sum\limits_{i = 1}^{n} {\delta_{i}^{ * } \left( {\frac{{\hat{G}(R_{i} - ) - G(R_{i} - )}}{{\hat{G}(R_{i} - )G(R_{i} - )}}} \right)} \left( {R_{i} - g^{ - 1} ({\mathbf{x^{\prime}}}_{i} \beta )} \right){\mathbf{x}}_{i} \\ & = \sum\limits_{i = 1}^{n} {\hat{w}_{i} \left( {R_{i} - g^{ - 1} ({\mathbf{x^{\prime}}}_{i} \beta )} \right){\mathbf{x}}_{i} \left( {\sum\nolimits_{j = 1}^{n} {\int_{0}^{{R_{i} - }} {\frac{{dM_{j}^{c} (u)}}{Y(u)}} } } \right)} \\ \end{aligned} $$

$= \sum\nolimits_{i = 1}^{n} {w_{i} (R_{i} - g^{ - 1} ({\mathbf{x^{\prime}}}_{i} \beta )){\mathbf{x}}_{i} \left( {\sum\nolimits_{j = 1}^{n} {\int_{0}^{\infty } {\frac{{[u < R_{i} ]dM_{j}^{c} (u)}}{Y(u)}} } } \right)}$ replacing $\hat{w}_{i}$ by $w_{i}$.

Let ${\mathbf{Q}}(t) = \sum\nolimits_{i = 1}^{n} {w_{i} (R_{i} - g^{ - 1} ({\mathbf{x^{\prime}}}_{i} \beta ))[R_{i} > t]{\mathbf{x}}_{i} /Y(t)} \,$.

The previous term is (place all inside the integral)

$\sum\nolimits_{j = 1}^{n} {\int_{0}^{\infty } {\left\{ {\sum\nolimits_{i = 1}^{n} {w_{i} (R_{i} - g^{ - 1} ({\mathbf{x^{\prime}}}_{i} \beta )){\mathbf{x}}_{i} } [R_{i} > u]} \right\}\frac{{dM_{j}^{c} (u)}}{Y(u)}} } = \sum\nolimits_{j = 1}^{n} {\int_{0}^{\infty } {{\mathbf{Q}}(u)} dM_{j}^{c} (u)}$, and let $\psi_{j} = \int_{0}^{\infty } {{\mathbf{Q}}(u)\,} dM_{j}^{c} (u)$

Finally, $\begin{aligned} \sum\nolimits_{i = 1}^{n} {\hat{w}_{i} } (R_{i} - g^{ - 1} ({\mathbf{x^{\prime}}}_{i} \beta )){\mathbf{x}}_{i} & = \sum\nolimits_{i = 1}^{n} {w_{i} } (R_{i} - g^{ - 1} ({\mathbf{x^{\prime}}}_{i} \beta )){\mathbf{x}}_{i} + \sum\nolimits_{i = 1}^{n} {(\hat{w}_{i} - w_{i} } )(R_{i} - g^{ - 1} ({\mathbf{x^{\prime}}}_{i} \beta )){\mathbf{x}}_{i} \\ & = \sum\nolimits_{i = 1}^{n} {\left[ {\eta_{i} + \psi_{i} } \right]} \\ \end{aligned}$.