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R-Estimation with Right-Censored Data

Glen A. Satten, Mo Li, Ni Zhao, Robert L. Strawderman

TL;DR

This work develops a censored-data generalization of R-estimation for linear/AFT models by imputing ranks via mid-CDFs and leveraging a self-consistent residual distribution. It establishes exact connections to Ritov's and Tsiatis' estimating equations and extends to a broad class of score functions, providing accessible asymptotic theory and practical variance estimation. Through simulations, the proposed methods demonstrate robust performance and improved efficiency relative to Gehan-based approaches, with the Wilcoxon score often delivering strong power under right censoring. The framework broadens the R-estimation toolbox for censored data and paves the way for incorporating nonlinear and bounded score transformations in more complex survival and regression settings.

Abstract

This paper considers the problem of directly generalizing the R-estimator under a linear model formulation with right-censored outcomes. We propose a natural generalization of the rank and corresponding estimating equation for the R-estimator in the case of the Wilcoxon (i.e., linear-in-ranks) score function, and show how it can respectively be exactly represented as members of the classes of estimating equations proposed in Ritov (1990) and Tsiatis (1990). We then establish analogous results for a large class of bounded nonlinear-in-ranks score functions. Asymptotics and variance estimation are obtained as straightforward consequences of these representation results. The self-consistent estimator of the residual distribution function, and the mid-cumulative distribution function (and, where needed, a generalization of it), play critical roles in these developments.

R-Estimation with Right-Censored Data

TL;DR

This work develops a censored-data generalization of R-estimation for linear/AFT models by imputing ranks via mid-CDFs and leveraging a self-consistent residual distribution. It establishes exact connections to Ritov's and Tsiatis' estimating equations and extends to a broad class of score functions, providing accessible asymptotic theory and practical variance estimation. Through simulations, the proposed methods demonstrate robust performance and improved efficiency relative to Gehan-based approaches, with the Wilcoxon score often delivering strong power under right censoring. The framework broadens the R-estimation toolbox for censored data and paves the way for incorporating nonlinear and bounded score transformations in more complex survival and regression settings.

Abstract

This paper considers the problem of directly generalizing the R-estimator under a linear model formulation with right-censored outcomes. We propose a natural generalization of the rank and corresponding estimating equation for the R-estimator in the case of the Wilcoxon (i.e., linear-in-ranks) score function, and show how it can respectively be exactly represented as members of the classes of estimating equations proposed in Ritov (1990) and Tsiatis (1990). We then establish analogous results for a large class of bounded nonlinear-in-ranks score functions. Asymptotics and variance estimation are obtained as straightforward consequences of these representation results. The self-consistent estimator of the residual distribution function, and the mid-cumulative distribution function (and, where needed, a generalization of it), play critical roles in these developments.
Paper Structure (26 sections, 5 theorems, 106 equations, 3 figures, 1 table)

This paper contains 26 sections, 5 theorems, 106 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. R-Estimation and Inference
  4. Brief review: complete data setting
  5. Extending R-Estimation to Right-Censored Data
  6. Estimating $\beta_0$: the case of the Wilcoxon score
  7. Connections between \ref{['R.est.cens']} and the Ritov and Tsiatis classes
  8. Asymptotics: a brief summary
  9. A decomposition of $\Sigma$
  10. R-Estimation and Inference in Practice
  11. Estimating $\Sigma$ and the Quasi-Score Test
  12. Estimating $\beta_0$
  13. Estimating $\Omega$, Wald Tests and Confidence Intervals
  14. Extending the R-estimator to general scores
  15. Bounded score functions
  16. ...and 11 more sections

Key Result

Theorem 1

Define $F$ to be the CDF of $\epsilon,$ and let where $\tilde{E}_{\beta_0} = Y - X^T \beta_0$ and $\Delta$ are respectively the true observed residual and failure status. Then, and $E(\mathcal{R}_{\beta_0} \mid X) = E(\mathcal{R}_{\beta_0}) = 1/2.$ In the case where $F$ is continuous, it is additionally true that where $E_{\beta_0} = \log C^* - X^T \beta_0,$ in which case $\hbox{Var}(\mathcal{R

Figures (3)

  • Figure 1: Power of the Wald and Quasi-Score tests to reject the hypothesis $\beta_0=(0,0)^\top$ for varying weight functions $a(u)$ and for the Fygenson and Ritov estimator (fraft) for alternatives of the form $\beta_0 = (b,-b)^\top,$$b \in [-1,1].$
  • Figure 2: Empirical versus nominal coverage of (marginal) Wald-type confidence intervals for $\beta_1$ (1st column) and $\beta_2$ (2nd column) when $\beta_0 = (b,-b)^\top,$$b \in [-1,1],$ when using the optimal weight function $a(u)=-1-\log(1-\frac{200}{201}u)$. Red dashed line corresponds to equality of nominal and emprical coverage rates.
  • Figure 3: Empirical versus nominal coverage of (marginal) Wald-type confidence intervals for $\beta_1$ (1st column) and $\beta_2$ (2nd column) when $\beta_0 = (b,-b)^\top,$$b \in [-1,1],$ using generalized $F$ weight function with $(m_1,m_2)=(10,1)$. Red dashed line corresponds to equality of nominal and emprical coverage rates.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Remark 2
  • Remark 3