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A Discontinuity Adjustment for Subdistribution Function Confidence Bands Applied to Right-Censored Competing Risks Data (with Erratum)

Dennis Dobler, Merle Munko

Abstract

The wild bootstrap is the resampling method of choice in survival analytic applications. Theoretic justifications rely on the assumption of existing intensity functions which is equivalent to an exclusion of ties among the event times. However, such ties are omnipresent in practical studies. It turns out that the wild bootstrap should only be applied in a modified manner that corrects for altered limit variances and emerging dependencies. This again ensures the asymptotic exactness of inferential procedures. An analogous necessity is the use of the Greenwood-type variance estimator for Nelson-Aalen estimators which is particularly preferred in tied data regimes. All theoretic arguments are transferred to bootstrapping Aalen-Johansen estimators for cumulative incidence functions in competing risks. An extensive simulation study as well as an application to real competing risks data of male intensive care unit patients suffering from pneumonia illustrate the practicability of the proposed technique.

A Discontinuity Adjustment for Subdistribution Function Confidence Bands Applied to Right-Censored Competing Risks Data (with Erratum)

Abstract

The wild bootstrap is the resampling method of choice in survival analytic applications. Theoretic justifications rely on the assumption of existing intensity functions which is equivalent to an exclusion of ties among the event times. However, such ties are omnipresent in practical studies. It turns out that the wild bootstrap should only be applied in a modified manner that corrects for altered limit variances and emerging dependencies. This again ensures the asymptotic exactness of inferential procedures. An analogous necessity is the use of the Greenwood-type variance estimator for Nelson-Aalen estimators which is particularly preferred in tied data regimes. All theoretic arguments are transferred to bootstrapping Aalen-Johansen estimators for cumulative incidence functions in competing risks. An extensive simulation study as well as an application to real competing risks data of male intensive care unit patients suffering from pneumonia illustrate the practicability of the proposed technique.

Paper Structure

This paper contains 12 sections, 6 theorems, 73 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Proposed Wild Bootstrap Adjustment for a Univariate Nelson-Aalen Estimator
  3. Extension to the Joint Convergence of Multiple Nelson-Aalen Estimators in Competing Risks Models
  4. Extension to Aalen-Johansen Estimators for Cumulative Incidence Functions in Competing Risks
  5. Small Sample Behaviour
  6. Real Data Example
  7. Discussion and Future Research
  8. Detailed Derivation of the Asymptotic Variance of the Nelson-Aalen Estimator
  9. Consistency of the Wild Bootstrap for the Univariate Nelson-Aalen Estimator
  10. Derivation of the Asymptotic Covariance of Multiple Nelson-Aalen Estimators for Cumulative Cause-Specific Hazards
  11. Consistency of the Discretization-Adjusted Wild Bootstrap for the Multivariate Nelson-Aalen Estimator
  12. Derivation of the Asymptotic Covariance Function of an Aalen-Johansen Estimator in the Competing Risks Set-Up

Key Result

Theorem 2.2

Assume Condition cond:main. Given $\mathcal{F}_0$ and as $n\rightarrow\infty$, we have the following conditional weak convergence on the càdlàg function space $D[0,K]$ equipped with the supremum distance topology, where $U_1$ is a Gaussian zero-mean martingale with variance function $t \mapsto \sigma^2_1(t)$. That is, the modified wild bootstrap succeeds in reproducing the same limit process of t

Figures (2)

  • Figure 1: Cumulative incidents functions $F_1^{p,k}$ underlying the present simulations.
  • Figure 2: Asymptotic $95\%$ equal precision (upper) and Hall-Wellner bands (lower panel) for the cumulative incidence function of the competing risk "alive discharge out of ICU" for male patients suffering from pneumonia.

Theorems & Definitions (10)

  • Theorem 2.2
  • Remark 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Remark 4.3: The weird bootstrap
  • Remark 4.4: Deduced confidence bands
  • proof
  • Theorem E.1: Corrected Theorem 4.1