Semiparametric Estimation of Proportional Mean Residual Life Model in Presence of Censoring

Abstract

A mean residual life function is the average remaining life of a surviving subject, as it varies with time. The proportional mean residual life model was proposed by Oakes and Dasu (1990, Biometrika77, 409-410) in regression analysis to study its association with related covariates in absence of censoring. In this article, we develop some semiparametric estimation procedures to take censoring into account. The proposed methodology is evaluated via simulation studies, and further applied to a clinical trial of chemotherapy in postoperative radiotherapy of lung cancer patients.

Estimating a Treatment Effect in Residual Time Quantiles Under the Additive Hazards Model

“…Assume that conditions A–C [1] hold. Denote $\hat{\theta }(t,qfalse|\mathrm{bold-italicZ})$ to be the solution to ${\stackrel{false^}{\Lambda }}_{0}false\{t+\hat{\theta }(t,qfalse|\mathrm{bold-italicZ})false\}+\hat{\theta }(t,qfalse|\mathrm{bold-italicZ}){\stackrel{false^}{\mathrm{bold-italic\beta }}}^{\mathrm{T}}\mathrm{bold-italicZ}={\stackrel{false^}{\Lambda }}_0(t)-\log q$.…”

“…Assume that conditions A-C [1] hold. Denote $\hat{\theta }(t,qfalse|\mathrm{bold-italicZ})$ to be the solution to ${\stackrel{false^}{\Lambda }}_{0}false\{t+\hat{\theta }(t,qfalse|\mathrm{bold-italicZ})false\}+\hat{\theta }(t,qfalse|\mathrm{bold-italicZ}){\stackrel{false^}{\mathrm{bold-italic\beta }}}^{\mathrm{T}}\mathrm{bold-italicZ}={\stackrel{false^}{\Lambda }}_0(t)-\log q$, then as n → ∞, for a given Z 1 and Z 2 and fixed t 1 , t 2 , q 1 , and q 2 , $\sqrt{n}\{(\hat{\theta }(t_1,q_{1}false|{\mathit{Z}}_1)-\hat{\theta }(t_2,q_{2}false|{\mathit{Z}}_2))-(\theta (t_1,q_{1}false|{\mathit{Z}}_1)-\theta (t_2,q_{2}false|{\mathit{Z}}_2))\}$ converges weakly to a zero-mean Gaussian process and whose variance function at t can be estimated consistently by $n\hat{W}(t)$ where …”

“…They are adapted from Chen et al [1]: There exists a time t 0 > 0 such that ${\lim }_{n\to \infty }{n}^{-1}true{\sum }_{i=1}^n{Y}_i(t_0)>0$.There exists an integrable function v ( t ) such that, for any t ∈ [0, t 0 ], $$n^{-1}{{\displaystyle \sum _{i=1}^n{Y}_{i}false(tfalse)false({\mathrm{bold-italicZ}}_i-\stackrel{bold-italic¯}{\mathrm{bold-italicZ}}false)}}^{\otimes 2}{\lambda }_i(tfalse|{\mathit{Z}}_i)-v(t)\to 0,$$in probability.For any ε > 0, $$n^{-1}true\sum _{i=1}^n{\displaystyle {\int }_0^{t_0}{Y}_i(s)\lambda (tfalse|{\mathit{Z}}_i){\Vert {\mathrm{bold-italicZ}}_i-\stackrel{bold-italic¯}{\mathrm{bold-italicZ}}\Vert }^{2}I\{n^{-1}\Vert {\mathit{Z}}_i-\overline{\mathit{Z}}\Vert >\epsilon \}ds\to 0,}$$in probability. …”

“…In particular, when the patients are still event-free at time t , say, then comparing their remaining event times after t can be more meaningful. In literature, similar comparisons have been done for the event time quantiles that were based on their empirical survival functions [1]. Dabrowska and Doksum [5] developed methods to estimate quantiles based on the Cox model.…”

Analysis on censored quantile residual life model via spline smoothing

“…In many clinical studies, especially when the associated diseases are chronic or/and incurable, knowing residual life is the major concern to patients. Modeling and estimating the mean of residual life has generated a large literature, for example, Oakes and Dasu (1990, 2003), Chen and Cheng (2005, 2006), Chen, Jewell and Cheng (2005), Müller and Zhang (2005), and Chen (2007). Compared with mean residual life models, quantile residual life models provide more complete and informative interpretation, especially when the distribution of the residual life is non-symmetric or skewed.…”