Consistency of survival tree and forest models: splitting bias and correction

Abstract

Random survival forests and survival trees are popular models in statistics and machine learning. However, there is a lack of general understanding regarding consistency, splitting rules and influence of the censoring mechanism. In this paper, we investigate the statistical properties of existing methods from several interesting perspectives. First, we show that traditional splitting rules with censored outcomes rely on a biased estimation of the within-node failure distribution. To exactly quantify this bias, we develop a concentration bound of the within-node estimation based on non i.i.d. samples and apply it to the entire forest. Second, we analyze the entanglement between the failure and censoring distributions caused by univariate splits, and show that without correcting the bias at an internal node, survival tree and forest models can still enjoy consistency under suitable conditions. In particular, we demonstrate this property under two cases: a finite-dimensional case where the splitting variables and cutting points

Survival trees based on heterogeneity in time‐to‐event and censoring distributions using parameter instability test

“…Very recently, some survival tree methods [27] have been extended for left truncated right-censored data [20,35]. Random survival forests (RSF) in the context of survival tree are also proposed [15,16,25,28,31]. In this manuscript, we focus on the survival tree construction, and the discussion on the associated RSF is deferred to future research work.…”

“…A key feature of the proposed SurvCART algorithm is its flexibility to extend the concept of heterogeneity to the censoring distribution. Accounting for censoring heterogeneity in a survival tree can be useful when censoring mechanism is dependent on baseline covariates inducing “conditionally independent censoring” [16] (also referred to as “dependent censoring” or “marker dependent censoring” [38, 45, 51]). Under this type of censoring, censoring () depends on the baseline attributes () but assumes that time‐to‐event ( T *) distribution remains independent of conditional on [54].…”

ROC‐guided survival trees and ensembles

“…Theoretical properties of random survival forests have been studied in Ishwaran and Kogalur (2010) and Cui et al . (2019).…”

The Impact of Churn on Client Value in Health Insurance, Evaluation Using a Random Forest Under Various Censoring Mechanisms

“…While the theoretical properties of tree and forest-based learning algorithms are not fully understood yet, the consistency of RSF was first investigated in under the assumption that X takes a finite number of values. These properties were further studied in and Cui, Zhu, Zhou, & Kosorok (2017) which provided a theoretical framework to consider the consistency of survival forests, and established consistency under specific conditions that include random splitting rules and splitting rules with marginal signal checking. Cui, Zhu, Zhou, & Kosorok (2017) also underlined the problem of non optimal split selection for usual survival tree methods.…”

“…These properties were further studied in and Cui, Zhu, Zhou, & Kosorok (2017) which provided a theoretical framework to consider the consistency of survival forests, and established consistency under specific conditions that include random splitting rules and splitting rules with marginal signal checking. Cui, Zhu, Zhou, & Kosorok (2017) also underlined the problem of non optimal split selection for usual survival tree methods. The method we study in this article does not suffer from such problem since it requires less assumptions.…”