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.
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Acknowledgment of Priority:
After our paper (Crouch et al. 2015) was accepted and appeared online in May 2015, we became aware that some content of our paper overlaps with an article recently published by Lin et al. (doi: 10.1007/s10985-013-9289-x). The majority of the overlap occurs in Section 2 of both papers. This is perhaps unsurprising, as these sections present the earliest stages of our derivations, both of which are based on a fundamental relationship between residual time and the survival function and utilize the Cox proportional hazards framework. Beyond these sections, the papers are more divergent. Specifically, we present a “plug-in” variance estimator in addition to resampling methods, provide a calculation of confidence bands, consider different methods for comparing two covariate-specific quantiles of residual time, perform different simulations (notably varying sample size in addition to other parameters), use different examples, and include graphics to illustrate results of both simulations and examples.
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Appendix
Appendix
We restate conditions A–D of Andersen and Gill (1982) with minor modification as in Fleming and Harrington (2005):
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A.
The time \(\tau \) is such that \(\int _0^{\tau }\lambda (t)dt<\infty \).
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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 \(n\rightarrow \infty \).
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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 \(n\rightarrow \infty \).
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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 map** 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,
and one based on the estimated values,
Taking the differences between the left- and right-hand sides of both equations yields
Examining the left-hand side first, we have
where the approximation is due to Taylor’s expansion. Examining the right-hand side, we have
again using Taylor’s approximation. Combining everything gives us the expression
Rearranging yields the approximation
From Fleming and Harrington (2005) we know that
so we can rewrite the overall approximation as
We can also perform a substitution, setting \(u=t+v\), and integrating with respect to v. This yields the approximation
Applying results from Fleming and Harrington (2005) about the asymptotic variance of \(\varvec{\widehat{\beta }}\) and martingales, we have:
where
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
and
\(\square \)
Proof
(Corollary 1) The proof for (1) follows directly from the established convergence of the individual estimators and the application of the continuous map** 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
where
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
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
where
\(\square \)
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Crouch, L.A., May, S. & Chen, Y.Q. On estimation of covariate-specific residual time quantiles under the proportional hazards model. Lifetime Data Anal 22, 299–319 (2016). https://doi.org/10.1007/s10985-015-9332-1
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DOI: https://doi.org/10.1007/s10985-015-9332-1