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Competing-risk Weibull survival model with multiple causes

Kai Wang, Yuqin Mu, Shenyi Zhang, Zhengjun Zhang, Chengxiu Ling

TL;DR

The paper addresses survival data with multiple contributing causes by introducing a competing-Weibull, min-structured model that yields time-varying, data-driven winning probabilities for each cause. It develops an EM-based maximum likelihood estimation framework with group-wise updates and sparsity penalties, and proves consistency and asymptotic normality under standard regularity conditions. Key contributions include the latent time representation, the explicit winning-probability formula, and the ability to identify informative biomarker combinations while handling high-dimensional covariates. The Alzheimer’s disease application demonstrates improved short-term prediction and interpretable biomarker ordering, highlighting the method’s potential for cause-mixture analysis and biomarker discovery in aging and neurodegenerative disease contexts.

Abstract

The failure of a system can result from the simultaneous effects of multiple causes, where assigning a specific cause may be inappropriate or unavailable. Examples include contributing causes of death in epidemiology and the aetiology of neurodegenerative diseases like Alzheimer's. We propose a parametric Weibull accelerated failure time model for multiple causes, incorporating a data-driven, individualized, and time-varying winning probability (relative importance) matrix. Using maximum likelihood estimation and the expectation-maximization (EM) algorithm, our approach enables simultaneous estimation of regression coefficients and relative cause importance, ensuring consistency and asymptotic normality. A simulation study and an application to Alzheimer's disease demonstrate its effectiveness in addressing cause-mixture problems and identifying informative biomarker combinations, with comparisons to Weibull and Cox proportional hazards models.

Competing-risk Weibull survival model with multiple causes

TL;DR

The paper addresses survival data with multiple contributing causes by introducing a competing-Weibull, min-structured model that yields time-varying, data-driven winning probabilities for each cause. It develops an EM-based maximum likelihood estimation framework with group-wise updates and sparsity penalties, and proves consistency and asymptotic normality under standard regularity conditions. Key contributions include the latent time representation, the explicit winning-probability formula, and the ability to identify informative biomarker combinations while handling high-dimensional covariates. The Alzheimer’s disease application demonstrates improved short-term prediction and interpretable biomarker ordering, highlighting the method’s potential for cause-mixture analysis and biomarker discovery in aging and neurodegenerative disease contexts.

Abstract

The failure of a system can result from the simultaneous effects of multiple causes, where assigning a specific cause may be inappropriate or unavailable. Examples include contributing causes of death in epidemiology and the aetiology of neurodegenerative diseases like Alzheimer's. We propose a parametric Weibull accelerated failure time model for multiple causes, incorporating a data-driven, individualized, and time-varying winning probability (relative importance) matrix. Using maximum likelihood estimation and the expectation-maximization (EM) algorithm, our approach enables simultaneous estimation of regression coefficients and relative cause importance, ensuring consistency and asymptotic normality. A simulation study and an application to Alzheimer's disease demonstrate its effectiveness in addressing cause-mixture problems and identifying informative biomarker combinations, with comparisons to Weibull and Cox proportional hazards models.

Paper Structure

This paper contains 10 sections, 9 theorems, 80 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Model Specification
  3. Estimation and Prediction
  4. Estimation via EM algorithm
  5. Penalization
  6. Theoretical Results
  7. Expected survival time
  8. Simulation
  9. Real Data Applications
  10. Discussions and Conclusion

Key Result

Theorem 3.1

Denote $\boldsymbol\theta_0$ as the true parameter and $\widehat{\boldsymbol\theta}_n$ as the MLE. Under assumptions A1$\sim$A3, as $n\to \infty$, If assumptions A1$\sim$A5 are satisfied, then as $n\to \infty$, where $I(\boldsymbol\theta_0)$ is the Fisher information matrix.

Figures (2)

  • Figure 1: Time-dependent ROCs for competing-Weibull, Cox, and Weibull models at their median survival times. All cases refer to the 10% random censorship for Example 1 (left), Example 2 (middle), and Example 3 (right).
  • Figure 2: Time-dependent ROCs (left, middle) for competing-Weibull, Cox, and Weibull models at 10 percentile survival time ($t=357$) and at 2 years ($t=730$). Estimated time-varying winning probability (right) of four competing groups for uncensored samples.

Theorems & Definitions (15)

  • Theorem 3.1
  • Theorem 3.2
  • Lemma 6.1
  • Lemma 6.2: Jennrich1969
  • proof : Proof of Lemma \ref{['Lemma: 2.1']}
  • Lemma 6.3
  • proof
  • Lemma 6.4: Extremum Consistency Theorem, newey1994large
  • Lemma 6.5
  • proof
  • ...and 5 more