Generalized Bayesian Ensemble Survival Tree (GBEST) model

TL;DR

Survival analysis with small samples and substantial censoring suffers from high variance and unstable tree-based predictions. The paper introduces GBEST, a bagging ensemble that replaces traditional bootstrap with Proper Bayesian bootstrap and Beta-Stacy bootstrap to create informative replicas for censored data, enabling generation of prior-informed observations and more stable survival estimates. Across simulated and real data, GBEST_BSB demonstrates lower Integrated Brier Score (IBS) and greater robustness than Cox and Survival Random Forest, especially at high censorship levels, with an accompanying R implementation for practical use. This approach broadens the toolkit for reliable survival prediction in challenging data regimes and offers avenues for extending to categorical covariates and complex covariance structures.

Abstract

This paper proposes a new class of predictive models for survival analysis called Generalized Bayesian Ensemble Survival Tree (GBEST). It is well known that survival analysis poses many different challenges, in particular when applied to small data or censorship mechanism. Our contribution is the proposal of an ensemble approach that uses Bayesian bootstrap and beta Stacy bootstrap methods to improve the outcome in survival application with a special focus on small datasets. More precisely, a novel approach to integrate Beta Stacy Bayesian bootstrap in bagging tree models for censored data is proposed in this paper. Empirical evidence achieved on simulated and real data underlines that our approach performs better in terms of predictive performances and stability of the results compared with classical survival models available in the literature. In terms of methodology our novel contribution considers the adaptation of recent Bayesian ensemble approaches to survival data, providing a new model called Generalized Bayesian Ensemble Survival Tree (GBEST). A further result in terms of computational novelty is the implementation in R of GBEST, available in a public GitHub repository.

Figures (5)

• Figure 1: Comparison of mean and non-parametric confidence intervals for IBS obtained for the 50 simulated datasets at a censorship level of 10%.. The models are the Cox model, the proposed model with different weights (GBEST_BSB w=0, GBEST_BSB w=0.1, GBEST_BSB w=0.2) and the Random forest, respectively.
• Figure 2: Comparison of mean and non-parametric confidence intervals for IBS obtained for the 50 simulated datasets at a censorship level of 70%. The models are the Cox model, the proposed model with different weights (GBEST_BSB w=0, GBEST_BSB w=0.1, GBEST_BSB w=0.2) and the Random forest, respectively.
• Figure 3: Comparison of mean and nonparametric confidence intervals for IBS obtained for the 100 simulated datasets. The models are the old model with different weights (GBEST w=0, GBEST w=0.25, GBEST w=0.50) and the proposed model with different weights (GBEST_BSB w=0, GBEST_BSB w=0.1, GBEST_BSB w=0.2), respectively.
• Figure 4: Comparison of mean and nonparametric confidence intervals for IBS obtained for the 100 simulated datasets. The models are the old model with different weights (GBEST w=0, GBEST w=0.25, GBEST w=0.50) and the proposed model with different weights (GBEST_BSB w=0, GBEST_BSB w=0.1, GBEST_BSB w=0.2), respectively.
• Figure 5: Comparison of mean and parametric confidence intervals for IBS obtained in a 5-fold cross validation exercise. The models are the Cox model, the proposed model with different weights (GBEST_BSB w=0, GBEST_BSB w=0.1, GBEST_BSB w=0.2) and the Random forest, respectively.