Confidence bands for the difference between two median survival times as a function of covariates

Abstract

In this paper, the estimation of the difference between two median survival times is considered when two treatment groups of right-censored data and the associated covariates are available. To identify the possible range of covariates over which the two treatments would produce different median survival times, two confidence bands for the difference as a function of the covariates are proposed under the stratified and treatment-specific Cox models, respectively. The results of a simulation study indicate that the latter generally maintains its confidence level and the former holds its confidence level and preserves a narrower width when the two treatments satisfy the stratified Cox model. An application of the proposed confidence bands is finally illustrated with a data set in a two-arm lung cancer study.

Appendix: Proof of Theorem 1

Let $F_i (t)=1-\prod _{u\le t} {\{ {1-\Lambda _{0i} (du)} \}}$ and $F_i (t|\mathbf{x})=1-\prod _{u\le t} {\{ {1-\Lambda _{0i} (du)} \}} ^{\exp ({\varvec{{\upbeta }}}^\prime \mathbf{x})}$ be the baseline distribution function and conditional distribution function, respectively, for subjects with covariates x in group $i, i=0, 1$. The corresponding estimators under stratified Cox model (3) are then given by

$$\begin{aligned} \hat{{F}}_{si} (t)=1-\prod _{u\le t} {\left\{ {1-{\hat{\Lambda }}_{s0i} (du)} \right\} } \end{aligned}$$

(11)

and

$$\begin{aligned} \hat{{F}}_{si} (t|\mathbf{x})=1-\prod _{u\le t} {\left\{ {1-{\hat{\Lambda }}_{s0i} (du)} \right\} } ^{\exp \big ({\varvec{\hat{\upbeta }}}^\prime \mathbf{x}\big )} \end{aligned}$$

(12)

The results of Theorem 3.2 and 3.4 in Andersen et al. (1993) imply that, for $i=0, 1$, as $n\rightarrow \infty$,

$$\begin{aligned} \sqrt{n}\Big \{{\hat{\Lambda }}_{si} (t)-\Lambda _i (t),{\varvec{\hat{\upbeta }}}-{\varvec{\upbeta }}\Big \} \mathop {\rightarrow }^{d}\Big \{W_i (a_{si} (t))-\mathbf{b}_{si} (t{)}'\Sigma ^{-1/2}\mathbf{Z},\Sigma ^{-1/2}\mathbf{Z}\Big \}. \end{aligned}$$

(13)

Since

$$\begin{aligned} |\sqrt{n}\Big \{\hat{{S}}_{si} (t)-S_i (t)\Big \}-\sqrt{n}\Big \{\exp (-{\hat{\Lambda }}_{si} (t))-\exp (-\Lambda _i (t))\Big \}|\mathop {\rightarrow }^{p}0, \end{aligned}$$

by incorporating (A1)–(A3), we have

$$\begin{aligned} \sqrt{n}\Big \{\hat{{F}}_{si} (t)-F_i (t),{\varvec{{\hat{\upbeta }}}}-{\varvec{\upbeta }}\Big \} \mathop {\rightarrow }^{d}S_i (t)\Big \{W_i (a_{si} (t))-\mathbf{b}_{si} (t{)}'\Sigma ^{-1/2}\mathbf{Z},\Sigma ^{-1/2}\mathbf{Z}\Big \}. \end{aligned}$$

We then obtain, by using Theorem 1 in Doss and Gill (1992) along with the mean value theorem, that

$$\begin{aligned} \sqrt{n}\Big \{\hat{{\xi }}_{si} (\mathbf{x})-\xi _i (\mathbf{x})\Big \}=\frac{Q_i (\mathbf{x})}{F_i^d (\xi _i (\mathbf{x}))} -\left( {\frac{\exp (-{\varvec{\upbeta }}^\prime \mathbf{x})\ell n2}{\lambda _{0i} (\xi _i (\mathbf{x}))}} \right) ^{\prime }\sqrt{n}({\varvec{{\hat{\upbeta }}}}-{\varvec{\upbeta }})+R_n (\mathbf{x}), \end{aligned}$$

where $F^{d}$ is the derivative of $F$,

$$\begin{aligned} Q_i (\mathbf{x})=-\sqrt{n}\left\{ \hat{{F}}_{si} \Big (F_i^{-1} \Big (1-2^{-\exp (-\mathbf{{\beta }'x})}\Big )\Big )-\left[ 1-2^{-\exp (-{\varvec{\upbeta }}^\prime \mathbf{x})}\right] \right\} \end{aligned}$$

and

$$\begin{aligned} \mathop {\sup }\limits _{\mathbf{x}\in {\varvec{{\mathfrak {X}}}}} R_n (\mathbf{x})\mathop {\rightarrow }^{p}0. \end{aligned}$$

After some algebraic manipulation, we have that, for $i=0$, 1, as $n\rightarrow \infty$,

$$\begin{aligned} \sqrt{n}\Big \{\hat{{\xi }}_{si} (\mathbf{x})-\xi _i (\mathbf{x})\Big \}\mathop {\rightarrow }^{d}U_{si} (\mathbf{x})\, \hbox {in}\, C_q ({\varvec{{\mathfrak {X}}}}), \end{aligned}$$

where, for $i=0, 1$,

$$\begin{aligned} U_{si} (\mathbf{x})=\frac{W_i \{a_{si} (\xi _i (\mathbf{x}))\}}{\lambda _{0i} (\xi _i (\mathbf{x}))}+{\varvec{\upomega }}_{si} (\mathbf{x})\Sigma ^{-1/2}\mathbf{Z} \end{aligned}$$

with

$$\begin{aligned} {\varvec{\upomega }}_{si} (\mathbf{x})=\frac{\mathbf{b}_{si} (\xi _i (\mathbf{x}))-\mathbf{x}\exp (-{\varvec{\upbeta }}^\prime \mathbf{x})\ell n2}{\lambda _{0i} (\xi _i (\mathbf{x}))}, \quad i= 0,1. \end{aligned}$$

Therefore, the continuous mapping theorem (Billingsley 1999) implies that

$$\begin{aligned}&\sqrt{n}\Big \{\hat{{\Delta }}_s (\mathbf{x})-\Delta (\mathbf{x})\Big \}=\sqrt{n}\Big \{\hat{{\xi }}_{s1} (\mathbf{x})-\xi _1 (\mathbf{x})\Big \}-\sqrt{n}\Big \{\hat{{\xi }}_{s0} (\mathbf{x})-\xi _0 (\mathbf{x})\Big \}\\&\quad \quad \quad \mathop {\rightarrow }^{d}U_s (\mathbf{x})=U_{s1} (\mathbf{x})-U_{s0} (\mathbf{x}) \,\hbox {in} \,C_q ({\varvec{{\mathfrak {X}}}}), \end{aligned}$$

where

$$\begin{aligned} U_s (\mathbf{x})&= \frac{W_1 \{a_{s1} (\xi _1 (\mathbf{x}))\}}{\lambda _{01} (\xi _1 (\mathbf{x}))}+{\varvec{\upomega }}_{s1} (\mathbf{x})\Sigma ^{-1/2}\mathbf{Z}-\frac{W_0 \{a_{s0} (\xi _0 (\mathbf{x}))\}}{\lambda _{00} (\xi _0 (\mathbf{x}))}-{\varvec{\upomega }}_{s0} (\mathbf{x})\Sigma ^{-1/2}\mathbf{Z}\\&= \frac{W_1 \{a_{s1} (\xi _1 (\mathbf{x}))\}}{\lambda _{01} (\xi _1 (\mathbf{x}))}-\frac{W_0 \{a_{s0} (\xi _0 (\mathbf{x}))\}}{\lambda _{00} (\xi _0 (\mathbf{x}))}+\{{\varvec{\upomega }}_{s1} (\mathbf{x})-{\varvec{\upomega }}_{s0} (\mathbf{x})\}^{\prime }\Sigma ^{-1/2}\mathbf{Z} \end{aligned}$$

is a zero mean Gaussian process in $C_q ({\varvec{{\mathfrak {X}}}})$. Since the variance of $W_i \{a_{si} (\xi _i (\mathbf{x}))\}$ is $a_{si} (\xi _i (\mathbf{x})), i= 0,1$, and the variance of $\mathbf{Z}$ is an identity matrix, the variance function of $U_s (\mathbf{x})$ is obtained as

$$\begin{aligned} \sigma _s^2 (\mathbf{x})&= \quad \frac{a_{s1} (\xi _1 (\mathbf{x}))}{\{\lambda _{01} (\xi _1 (\mathbf{x}))\}^{2}}+\frac{a_{s0} (\xi _0 (\mathbf{x}))}{\{\lambda _{00} (\xi _0 (\mathbf{x}))\}^{2}}\nonumber \\&+\{{\varvec{\upomega }}_{s1} (\mathbf{x})-{\varvec{\upomega }}_{s0} (\mathbf{x})\}^{\prime }\Sigma ^{-1}\{{\varvec{\upomega }}_{s1} (\mathbf{x})-{\varvec{\upomega }}_{s0} (\mathbf{x})\}. \end{aligned}$$