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Self-Consistent Equation-guided Neural Networks for Censored Time-to-Event Data

Sehwan Kim, Rui Wang, Wenbin Lu

TL;DR

This work proposes a novel deep learning approach to non-parametric estimation of the conditional survival functions using the generative adversarial networks leveraging self-consistent equations and establishes the convergence rate of the proposed estimator of the conditional survival function.

Abstract

In survival analysis, estimating the conditional survival function given predictors is often of interest. There is a growing trend in the development of deep learning methods for analyzing censored time-to-event data, especially when dealing with high-dimensional predictors that are complexly interrelated. Many existing deep learning approaches for estimating the conditional survival functions extend the Cox regression models by replacing the linear function of predictor effects by a shallow feed-forward neural network while maintaining the proportional hazards assumption. Their implementation can be computationally intensive due to the use of the full dataset at each iteration because the use of batch data may distort the at-risk set of the partial likelihood function. To overcome these limitations, we propose a novel deep learning approach to non-parametric estimation of the conditional survival functions using the generative adversarial networks leveraging self-consistent equations. The proposed method is model-free and does not require any parametric assumptions on the structure of the conditional survival function. We establish the convergence rate of our proposed estimator of the conditional survival function. In addition, we evaluate the performance of the proposed method through simulation studies and demonstrate its application on a real-world dataset.

Self-Consistent Equation-guided Neural Networks for Censored Time-to-Event Data

TL;DR

This work proposes a novel deep learning approach to non-parametric estimation of the conditional survival functions using the generative adversarial networks leveraging self-consistent equations and establishes the convergence rate of the proposed estimator of the conditional survival function.

Abstract

In survival analysis, estimating the conditional survival function given predictors is often of interest. There is a growing trend in the development of deep learning methods for analyzing censored time-to-event data, especially when dealing with high-dimensional predictors that are complexly interrelated. Many existing deep learning approaches for estimating the conditional survival functions extend the Cox regression models by replacing the linear function of predictor effects by a shallow feed-forward neural network while maintaining the proportional hazards assumption. Their implementation can be computationally intensive due to the use of the full dataset at each iteration because the use of batch data may distort the at-risk set of the partial likelihood function. To overcome these limitations, we propose a novel deep learning approach to non-parametric estimation of the conditional survival functions using the generative adversarial networks leveraging self-consistent equations. The proposed method is model-free and does not require any parametric assumptions on the structure of the conditional survival function. We establish the convergence rate of our proposed estimator of the conditional survival function. In addition, we evaluate the performance of the proposed method through simulation studies and demonstrate its application on a real-world dataset.

Paper Structure

This paper contains 36 sections, 7 theorems, 39 equations, 18 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Self-Consistent Equations for Conditional Survival Functions
  3. Notation and formulation
  4. Self-consistent equation for the KM estimator
  5. Self-consistent equations for conditional survival functions
  6. Reformulation of Self-Consistent Equations for Conditional Survival Functions
  7. Weighted self-consistent equations
  8. Min-max optimization for solving the class of weighted self-consistent equations
  9. Self-Consistent Equation-Guided Neural Net (SCENE)
  10. Proposed SCENE estimation
  11. Variable importance
  12. Simulations
  13. A Real Data Example
  14. Discussion
  15. Availability
  16. ...and 21 more sections

Key Result

Proposition 1

Given $X = x\in \mathcal{X}$, the support of $X$, if $S_T(t|x)$ satisfies the Equation (const:2) for all $t\in \mathcal{T}\subseteq \mathbb{R}^+$, then $S_T(t|x)=S_T^*(t|x)$ for $t\in \mathcal{T}$ almost surely.

Figures (18)

  • Figure 1: Illustration of the conditional generator used to approximate the survival function through generated samples.
  • Figure 2: Comparison of conditional survival function estimation for PH Model, $C=5$, $N=4000$, $d=5$: $(5\%, 95\%)$ empirical band for Test Subject 1 to Test Subject 4.
  • Figure 3: Comparison of conditional survival function estimation for PO Model, C=5, $N=4000$, $d=5$ : $(5\%,95\%)$ empirical bound for Test Subject 1 to Subject 4.
  • Figure 4: Comparison of conditional survival function estimation for PH Model, C=5, $N=4000$, $d=100$ : $(5\%,95\%)$ empirical bound for Test Subject 1 to Subject 4.
  • Figure 5: Comparison of conditional survival function estimation for PO Model, C=5, $N=4000$, $d=100$ : $(5\%,95\%)$ empirical bound for Test Subject 1 to Subject 4.
  • ...and 13 more figures

Theorems & Definitions (12)

  • Proposition 1: Uniqueness of solution
  • Proposition 2
  • Theorem 1
  • Remark 1
  • Theorem 2
  • proof
  • proof
  • proof
  • Lemma 1: Theorem 1 from yarotsky2017 and Lemma 7 from Farrell_2021
  • Lemma 2: Symmetrization, Theorem 2.1 in Bartlett_2005 or Lemma 5 in Farrell_2021
  • ...and 2 more