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Inference for competing risks based on area between curves statistics

Simon Mack, Marc Ditzhaus, Merle Munko, Markus Pauly

TL;DR

The paper addresses two-sample inference for cumulative incidence functions under competing risks when CIFs may cross. It shows that the previously proposed area-based test is not asymptotically normal and derives its true limit distribution, which depends on unknown quantities, then implements a wild bootstrap to obtain an asymptotically exact test. Through extensive simulations and a real-data EBMT example, the proposed wild bootstrap ABC test demonstrates favorable size control and power, especially for crossing CIFs, while offering a simple geometric interpretation and computational efficiency. These results provide a practical, robust method for assessing differences in CIFs between groups in the presence of competing risks.

Abstract

In competing risks models, cumulative incidence functions are commonly compared to infer differences between groups. Many existing inference methods, however, struggle when these functions cross during the time frame of interest. To address this problem, we investigate a test statistic based on the area between cumulative incidence functions. As the corresponding limiting distribution depends on quantities that are typically unknown, we propose a wild bootstrap approach to obtain a feasible and asymptotically valid two-sample test. The finite sample performance of the proposed method, in comparison with existing methods, is examined in an extensive simulation study.

Inference for competing risks based on area between curves statistics

TL;DR

The paper addresses two-sample inference for cumulative incidence functions under competing risks when CIFs may cross. It shows that the previously proposed area-based test is not asymptotically normal and derives its true limit distribution, which depends on unknown quantities, then implements a wild bootstrap to obtain an asymptotically exact test. Through extensive simulations and a real-data EBMT example, the proposed wild bootstrap ABC test demonstrates favorable size control and power, especially for crossing CIFs, while offering a simple geometric interpretation and computational efficiency. These results provide a practical, robust method for assessing differences in CIFs between groups in the presence of competing risks.

Abstract

In competing risks models, cumulative incidence functions are commonly compared to infer differences between groups. Many existing inference methods, however, struggle when these functions cross during the time frame of interest. To address this problem, we investigate a test statistic based on the area between cumulative incidence functions. As the corresponding limiting distribution depends on quantities that are typically unknown, we propose a wild bootstrap approach to obtain a feasible and asymptotically valid two-sample test. The finite sample performance of the proposed method, in comparison with existing methods, is examined in an extensive simulation study.
Paper Structure (14 sections, 9 theorems, 40 equations, 9 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 9 theorems, 40 equations, 9 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model and Notation
  3. Asymptotic Area based Test
  4. Wild Bootstrap Test
  5. Simulations
  6. Simulation Setup
  7. Results
  8. Real-Data Application
  9. Conclusion and Discussion
  10. Proofs
  11. Adjustments for Discontinuous CIFs
  12. Additional Simulation Results
  13. Further Plots and Tables
  14. Detailed Results of Model 1

Key Result

Theorem 2.1

Suppose yassumption and groupassumption hold. Then, as $n \to \infty$, we have convergence in distribution on the Skorokhod space $D([t_1,t_2]) \times D([t_1,t_2])$ equipped with the usual Skorokhod topology where $\mathbb{G}^{(j)}, j=1,2$ are independent, mean-zero Gaussian processes with almost surely continuous sample paths and covariance functions

Figures (9)

  • Figure 1: Estimated type I error under Model 2 stratified by sample sizes
  • Figure 2: Estimated power under Model 2 stratified by sample sizes
  • Figure 3: Estimated cumulative incidence for death after bone marrow transplantation for the different age and gender matching groups.
  • Figure 4: Estimated type I error under model 2 stratified by censoring parameters
  • Figure 5: Estimated power under model 2 stratified by censoring parameters
  • ...and 4 more figures

Theorems & Definitions (18)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Remark 1
  • Lemma A.1
  • proof
  • Lemma A.2
  • ...and 8 more