On estimation of covariate-specific residual time quantiles under the proportional hazards model

Abstract

Estimation and inference in time-to-event analysis typically focus on hazard functions and their ratios under the Cox proportional hazards model. These hazard functions, while popular in the statistical literature, are not always easily or intuitively communicated in clinical practice, such as in the settings of patient counseling or resource planning. Expressing and comparing quantiles of event times may allow for easier understanding. In this article we focus on residual time, i.e., the remaining time-to-event at an arbitrary time t given that the event has yet to occur by t. In particular, we develop estimation and inference procedures for covariate-specific quantiles of the residual time under the Cox model. Our methods and theory are assessed by simulations, and demonstrated in analysis of two real data sets.

Appendix

We restate conditions A–D of Andersen and Gill (1982) with minor modification as in Fleming and Harrington (2005):

• A. The time $\tau$ is such that $\int _0^{\tau }\lambda (t)dt<\infty$.

• B. For $\varvec{S}^{(j)}, j=0,1,2$, there exists a neighborhood $\mathscr {B}$ of $\varvec{\beta }$ and scalar, vector, and matrix functions $s^{(0)}$, $\varvec{s}^{(1)}$, and $\varvec{s}^{(2)}$ defined on $\mathscr {B} \times [0,\tau ]$ such that, for $j=0,1,2$,

  $$\begin{aligned} \sup _{t \in [0,\tau ],\varvec{\beta }\in \mathscr {B}}||\varvec{S}^{(j)} (\varvec{\beta },t)-\varvec{s}^{(j)}(\varvec{\beta },t)|| \rightarrow 0 \end{aligned}$$

  in probability as $n\rightarrow \infty$.

• C. There exists $\delta >0$ such that

  $$\begin{aligned} \sqrt{n} \sup _{i,t}|\varvec{Z}_i|Y_i(t)I (\varvec{\beta }^{\mathrm{T}}\varvec{Z}_i > -\delta |\varvec{Z}_i|) \rightarrow 0 \end{aligned}$$

  in probability as $n\rightarrow \infty$.

• D. Let $\mathscr {B}$ and $\varvec{s}^{(j)},j=0,1,2$ be defined as in Condition B and $\varvec{e}$ and $\mathbf {v}$ as in Sect. 2. Then for all $\varvec{\beta } \in \mathscr {B}$ and $t \in [0,\tau ]$:

  $$\begin{aligned} \varvec{s}^{(1)}(\varvec{\beta },t)=\frac{\partial }{\partial \varvec{\beta }}s^{(0)}(\varvec{\beta },t), \;\;\; \varvec{s}^{(2)}(\varvec{\beta },t)=\frac{\partial ^2}{\partial \varvec{\beta }^2}s^{(0)}(\varvec{\beta },t), \end{aligned}$$

  $s^{(0)}(\cdot ,t)$, $\varvec{s}^{(1)}(\cdot ,t)$, and $\varvec{s}^{(2)}(\cdot ,t)$ are continuous function of $\varvec{\beta } \in \mathscr {B}$, uniformly in $t \in [0,\tau ]$, $s^{(0)}$, $\varvec{s}^{(1)}$, and $\varvec{s}^{(2)}$ are bounded on $\mathscr {B} \times [0,\tau ]$; $s^{(0)}$ is bounded away from zero on $\mathscr {B} \times [0,\tau ]$, and the matrix $\varvec{\Sigma }$ (as defined in Sect. 2) is positive definite.

Proof

(Theorem 1) The proof for (1) follows directly from the established convergence of the individual estimators and the application of the continuous mapping theorem. Likewise for (3), where the consistency of the kernel-estimated baseline hazard has already been established (Ramlau-Hansen 1983; Yandell 1983). Therefore only (2) requires a detailed proofl.

We begin with two equations: one based on the true values,

$$\begin{aligned} \varLambda _0 \{ \theta (t,q|\varvec{Z})+t \}=\varLambda _0(t)-\log {q}\times \exp {(-\varvec{\beta }^{\mathrm{T}} \varvec{Z})}, \end{aligned}$$

and one based on the estimated values,

$$\begin{aligned} \widehat{\varLambda }_0 \{ \widehat{\theta }(t,q|\varvec{Z})+t \}=\widehat{\varLambda }_0(t)-\log {q}\times \exp {(-\varvec{\widehat{\beta }^{\mathrm{T}}} \varvec{Z})}. \end{aligned}$$

Taking the differences between the left- and right-hand sides of both equations yields

$$\begin{aligned}&\widehat{\varLambda }_0 \{ \widehat{\theta }(t,q|\varvec{Z})+t \}-\varLambda _0 \{ \theta (t,q|\varvec{Z})+t \}\\&\quad \,=\,\widehat{\varLambda }_0(t)-\log {q}\times \exp {( -\varvec{\widehat{\beta }^{\mathrm{T}}} \varvec{Z})}-\varLambda _0(t)+\log {q}\times \exp {( -\varvec{\beta }^{\mathrm{T}} \varvec{Z})}. \end{aligned}$$

Examining the left-hand side first, we have

$$\begin{aligned}&\widehat{\varLambda }_0 \{ \widehat{\theta }(t,q|\varvec{Z})+t \} -\varLambda _0 \{ \theta (t,q|\varvec{Z})+t \}\\&\quad =\widehat{\varLambda }_0 \{ \widehat{\theta }(t,q|\varvec{Z})+t \} -\widehat{\varLambda }_0 \{ \theta (t,q|\varvec{Z})+t \}\\&\qquad + \widehat{\varLambda }_0 \{ \theta (t,q|\varvec{Z})+t \} -\varLambda _0 \{ \theta (t,q|\varvec{Z})+t \}\\&\quad \approx \widehat{\lambda }_0 \{ \theta (t,q|\varvec{Z})+t \} \{\widehat{\theta }(t,q|\varvec{Z})-\theta (t,q|\varvec{Z})\}\\&\qquad + \widehat{\varLambda }_0 \{ \theta (t,q|\varvec{Z})+t\} -\varLambda _0 \{ \theta (t,q|\varvec{Z})+t \}, \end{aligned}$$

where the approximation is due to Taylor’s expansion. Examining the right-hand side, we have

$$\begin{aligned}&\widehat{\varLambda }_0(t)-\varLambda _0(t)-\log {q}\times \{\exp {(-\varvec{\widehat{\beta }^{\mathrm{T}}} \varvec{Z})}-\exp {(-\varvec{\beta }^{\mathrm{T}} \varvec{Z})}\}\\&\quad \approx \widehat{\varLambda }_0(t)-\varLambda _0(t)\!+\!\log {q}\times \varvec{Z}\exp {(-\varvec{\widehat{\beta }^{\mathrm{T}}} \varvec{Z})}(\varvec{\widehat{\beta }}-\varvec{\beta }), \end{aligned}$$

again using Taylor’s approximation. Combining everything gives us the expression

$$\begin{aligned}&\widehat{\lambda }_0 \{ \theta (t,q|\varvec{Z})+t \} \{\widehat{\theta }(t,q|\varvec{Z})-\theta (t,q|\varvec{Z})\}\\&\qquad +\widehat{\varLambda }_0 \{ \theta (t,q|\varvec{Z})+t \} \!-\!\varLambda _0 \{ \theta (t,q|\varvec{Z})+t \}\\&\quad \approx \widehat{\varLambda }_0(t)-\varLambda _0(t)+\log {q}\times \varvec{Z}\exp {(-\varvec{\widehat{\beta }^{\mathrm{T}}} \varvec{Z})}(\varvec{\widehat{\beta }}-\varvec{\beta }). \end{aligned}$$

Rearranging yields the approximation

$$\begin{aligned} \widehat{\theta }(t,q|\varvec{Z})-\theta (t,q|\varvec{Z})\approx & {} \frac{ 1}{\widehat{\lambda }_0 \{ \theta (t,q|\varvec{Z})+t \}} \biggl ( \log {q}\times \varvec{Z}\exp { (-\varvec{\widehat{\beta }^{\mathrm{T}}} \varvec{Z})} (\varvec{\widehat{\beta }}-\varvec{\beta })\\&+\,\widehat{\varLambda }_0(t)\!-\!\varLambda _0(t) \!-\! \left[ \widehat{\varLambda }_0 \{ \theta (t,q|\varvec{Z})\!+\!t \}\!-\!\varLambda _0 \{ \theta (t,q|\varvec{Z})\!+\!t \}\right] \biggr ) . \end{aligned}$$

From Fleming and Harrington (2005) we know that

$$\begin{aligned} \widehat{\varLambda }_0(t)-\varLambda _0(t) \approx - \left[ \int ^t_0\frac{\varvec{S}^{(1)}(\varvec{\widehat{\beta }},u)}{n\{S^{(0)}(\varvec{\widehat{\beta }},u)\}^2} d\overline{N}(u)\right] ^{\mathrm{T}}(\varvec{\widehat{\beta }} -\varvec{\beta } ) + \int ^t_0\frac{\sum ^n_{i=1}dM_i(u)}{S^{(0)} (\varvec{\widehat{\beta }},u)}, \end{aligned}$$

so we can rewrite the overall approximation as

$$\begin{aligned}&\widehat{\theta }(t,q|\varvec{Z})-\theta (t,q|\varvec{Z})\\&\quad \approx \frac{1}{\widehat{\lambda }_0 \{ \theta (t,q|\varvec{Z})+t\}} \Biggl \{-\int ^{\theta (t,q|\varvec{Z})+t}_t\frac{\sum ^n_{i=1}dM_i(u)}{S^{(0)}(\varvec{\widehat{\beta }},u)}+\Biggl [\log {q}\times \varvec{Z}\exp {(-\varvec{\widehat{\beta }^{\mathrm{T}}} \varvec{Z})} \\&\quad \qquad \qquad \qquad \qquad \qquad \quad \,+\,\int ^{\theta (t,q|\varvec{Z})+t}_t \frac{\varvec{S}^{(1)}(\varvec{\widehat{\beta }},u)}{n\{S^{(0)}(\varvec{\widehat{\beta }},u)\}^2}d \overline{N}(u)\Biggr ]^{\mathrm{T}}(\varvec{\widehat{\beta }} -\varvec{\beta }) \Biggr \}. \end{aligned}$$

We can also perform a substitution, setting $u=t+v$, and integrating with respect to v. This yields the approximation

$$\begin{aligned}&\widehat{\theta }(t,q|\varvec{Z})-\theta (t,q|\varvec{Z})\\&\quad \approx \frac{1}{\widehat{\lambda }_0 \{ \theta (t,q|\varvec{Z})+t \}}\Biggl \{-\int ^{\theta (t,q|\varvec{Z})}_0 \frac{\sum ^n_{i=1}dM_i(t+v)}{S^{(0)}(\varvec{\widehat{\beta }},t+v)}+ \,\left[ \log {q}\times \varvec{Z}\exp {(-\varvec{\widehat{\beta }^{\mathrm{T}}} \varvec{Z})} \right. \\&\left. \quad \qquad \qquad \quad \qquad \qquad \qquad \,+ \, \int ^{\theta (t,q|\varvec{Z}))}_0 \frac{\varvec{S}^{(1)}(\varvec{\widehat{\beta }},t+v)}{n\{S^{(0)}(\varvec{\widehat{\beta }},t+v)\}^2}d\overline{N} (t+v)\right] ^{\mathrm{T}}(\varvec{\widehat{\beta }}-\varvec{\beta }) \Biggr \}. \end{aligned}$$

Applying results from Fleming and Harrington (2005) about the asymptotic variance of $\varvec{\widehat{\beta }}$ and martingales, we have:

$$\begin{aligned} \mathrm {Var}\{\widehat{\theta }(t,q|\varvec{Z})- \theta (t,q|\varvec{Z})\} \approx \frac{\displaystyle \varvec{\widetilde{A}}^{\mathrm{T}}\mathrm {Var}(\varvec{\widehat{\beta }}) \varvec{\widetilde{A}} +\int ^{\theta (t,q|\varvec{Z})}_0\frac{d\widehat{\varLambda }_0(t+v)}{S^{(0)}(\varvec{\widehat{\beta }},t+v)}}{\displaystyle \widehat{\lambda }_0 \{ \theta (t,q|\varvec{Z})+t \}^2}, \end{aligned}$$

where

$$\begin{aligned} \varvec{\widetilde{A}}=\left[ \log {q}\times \varvec{Z}\exp {(-\varvec{\widehat{\beta }^{\mathrm{T}}} \varvec{Z})}+\int ^{\theta (t,q|\varvec{Z})}_0 \frac{\varvec{S}^{(1)}(\varvec{\widehat{\beta }},t+v)}{n\{S^{(0)}(\varvec{\widehat{\beta }},t+v)\}^2} d\overline{N}(t+v)\right] ^{\mathrm{T}}. \end{aligned}$$

Based on the consistency of the estimators used in this formulation, we have the asymptotic result $\sqrt{n}(\widehat{\theta }(t,q|\varvec{Z}) - \theta (t,q|\varvec{Z})) \rightarrow _d N(0,V(t))$, where

$$\begin{aligned} V(t) \approx \frac{1}{\lambda _0 \{ \theta (t,q|\varvec{Z})+t \}^2}\left[ \varvec{A}^{\mathrm{T}}\{\varvec{\Sigma } (\varvec{\beta },\tau )\}^{-1}\varvec{A} +\int ^{\theta (t,q|\varvec{Z})}_0\frac{\lambda _0(t+v)}{s^{(0)}(\varvec{\beta },v)}dv\right] \end{aligned}$$

and

$$\begin{aligned} \varvec{A}=\left[ \log {q}\times \varvec{Z}\exp {(-\varvec{\beta }^{\mathrm{T}} \varvec{Z})}+\int ^{\theta (t,q|\varvec{Z})}_0 \frac{\varvec{s}^{(1)}(\varvec{\beta },v)}{s^{(0)} (\varvec{\beta },v)}\lambda _0(t+v)d(v)\right] ^{\mathrm{T}}. \end{aligned}$$

$\square$

Proof

(Corollary 1) The proof for (1) follows directly from the established convergence of the individual estimators and the application of the continuous mapping theorem. Likewise for (3), where the consistency of the kernel-estimated baseline hazard has already been established (Ramlau-Hansen 1983; Yandell 1983). Therefore we need only prove (2) in detail.

From the proof of Theorem 1, we have

$$\begin{aligned}&\widehat{\theta }(t,q|\varvec{Z}_j)-\theta (t_j,q_j|\varvec{Z}_j)\\&\quad \approx \frac{1}{\widehat{\lambda }_0 ( \theta (t_j,q_j|\varvec{Z}_j)+t )} \left\{ -\int ^{\theta (t_j,q_j|\varvec{Z}_j)}_0 \frac{\sum ^n_{i=1}dM_j(t_j+v)}{S^{(0)}(\varvec{\widehat{\beta }}, t_j+v)} + \varvec{\widetilde{A}}_j^{\mathrm{T}}(\varvec{\widehat{\beta }}- \varvec{\beta })\right\} \end{aligned}$$

where

$$\begin{aligned} \varvec{\widetilde{A}}_j&=\Biggl [\log {q_j}\times \varvec{Z}_j\exp {(-\varvec{\widehat{\beta }^{\mathrm{T}}} \varvec{Z})}\\&\quad \qquad \,+ \,\int ^{\theta (t_j,q_j|\varvec{Z}_j)}_0 \frac{\varvec{S^{(1)}}(\varvec{\widehat{\beta }},t_j+v)}{n\{S^{(0)}(\varvec{\widehat{\beta }},t_j+v)\}^2} d\overline{N}(t_j+v)\Biggr ]^{\mathrm{T}}. \end{aligned}$$

In order to calculate $\mathrm {Var}(\widehat{\theta }(t_1,q_1|\varvec{Z}_1)- \widehat{\theta }(t_2,q_2|\varvec{Z}_2))$, we note that

$$\begin{aligned}&\mathrm {Var}(\widehat{\theta }(t_1,q_1|\varvec{Z}_1) -\widehat{\theta }(t_2,q_2|\varvec{Z}_2))\\&\quad =\mathrm {Var}\left[ \{\widehat{\theta }(t_1,q_1|\varvec{Z}_1) -\theta (t_1,q_1|\varvec{Z}_1)\}-\{\widehat{\theta }(t_2,q_2| \varvec{Z}_2)-\theta (t_2,q_2|\varvec{Z}_2)\}\right] \\&\quad = \mathrm {Var}\{\widehat{\theta }(t_1,q_1|\varvec{Z}_1) -\theta (t_1,q_1|\varvec{Z}_1)\} + \mathrm {Var}\{\widehat{\theta } (t_2,q_2|\varvec{Z}_2)-\theta (t_2,q_2|\varvec{Z}_2)\} \\&\qquad -\, 2\times \mathrm {Cov} \{\widehat{\theta }(t_1,q_1|\varvec{Z}_1) -\theta (t_1,q_1|\varvec{Z}_1),\widehat{\theta } (t_2,q_2|\varvec{Z}_2)-\theta (t_2,q_2|\varvec{Z}_2)\}. \end{aligned}$$

Now

where $\eta _{\mathrm {min}}=\min \{ \theta (t_1,q_1|\varvec{Z}_1) + t_1 ,\theta (t_2,q_2|\varvec{Z}_2) + t_2\}$ and $t'=\max \{ t_1 , t_2\}$.

So, combining the above with results from Theorem 1 and taking into account the consistency of the estimators used in this formulation, we have the asymptotic result

$$\begin{aligned} \sqrt{n}[\{\widehat{\theta }(t_1,q_1|\varvec{Z}_1) - \widehat{\theta }(t_2,q_2|\varvec{Z}_2)\} - \{\theta (t_1,q_1|\varvec{Z}_1) - \theta (t_2,q_2|\varvec{Z}_2)\}] \rightarrow _d N(0,W(t)), \end{aligned}$$

where

$$\begin{aligned} W(t)= & {} \frac{\displaystyle \varvec{A_1}^{\mathrm{T}}\{\varvec{\Sigma } (\varvec{\beta },\tau )\}^{-1} \varvec{A_1} + \int ^{\theta (t_1,q_1|\varvec{Z_1})}_0 \frac{\lambda _0(t_1+v)}{s^{(0)}(\varvec{\beta },t_1+v)}dv}{\displaystyle \lambda _0 ( \theta (t_1,q_1|\varvec{Z_1})+t_1 )^2} \\&+\, \frac{\displaystyle \varvec{A_2}^{\mathrm{T}}\{\varvec{\Sigma } (\varvec{\beta },\tau )\}^{-1} \varvec{A_2} +\int ^{\theta (t_2,q_2|\varvec{Z_2})}_0 \frac{\lambda _0(t_2+v)}{s^{(0)}(\varvec{\beta },t_2+v)}dv}{\displaystyle \lambda _0 ( \theta (t_2,q_2|\varvec{Z_2})+t_2 )^2}\\&- \, \frac{\displaystyle 2 \left\{ \int ^{\eta _{\mathrm {min}}-t'}_0\frac{\lambda _0(t'+v)}{s^{(0)} (\varvec{\beta },t'+v)}dv + \varvec{A_1}^{\mathrm{T}} \{\varvec{\Sigma }(\varvec{\beta },\tau )\}^{-1} \varvec{A_2} \right\} }{\displaystyle \lambda _0 ( \theta (t_1,q_1|\varvec{Z_1})+t_1 )\lambda _0 ( \theta (t_2,q_2|\varvec{Z_2})+t_2 )},\\ \varvec{A_j}= & {} \Biggl [\log {q_j}\times \varvec{Z_j} \exp {(-\varvec{\beta }^{\mathrm{T}} \varvec{Z_j})}\\&+\,\int ^{\theta (t_j,q_j|\varvec{Z_j})}_0 \frac{\varvec{s}^{(1)}(\varvec{\beta },t_j+v)}{s^{(0)} (\varvec{\beta },t_j+v)}\lambda _0(t_j+v)d(t_j+v)\Biggr ]^{\mathrm{T}}. \end{aligned}$$

$\square$