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Deep Neural Networks for Survival Analysis Based on a Multi-Task Framework

Stephane Fotso

arXiv:1801.05512v1stat.MLcs.LG

TL;DR

Survival analysis helps organizations predict when events such as purchases, churn, or loan defaults will occur. The paper introduces N-MTLR, combining MTLR with a deep learning core, and reports consistent gains over MTLR plus an advantage over CoxPH for nonlinear dependencies.

  • Problem

    Survival analysis supports predicting when events such as purchases, churn, or loan defaults will occur, but MTLR and CoxPH rely on linear structures that cannot capture nonlinear data elements.

  • Method

    N-MTLR uses MTLR as its foundation and replaces its linear core with a deep learning framework based on a multi-layer perceptron.

  • Results

    N-MTLR consistently outperforms MTLR and yields similar or better results than CoxPH, with a clear advantage when data contain nonlinear dependencies.

  • Takeaways & Limitations

    N-MTLR provides a flexible survival-modeling approach without relying on CoxPH assumptions and benefits from contemporary deep learning techniques.

  • Takeaways & Limitations

    CoxPH relies on the proportional-hazard assumption, computational approximations, and an unspecified time component, limiting its suitability for direct survival-function prediction.

Abstract

from arXiv · show

Survival analysis/time-to-event models are extremely useful as they can help companies predict when a customer will buy a product, churn or default on a loan, and therefore help them improve their ROI. In this paper, we introduce a new method to calculate survival functions using the Multi-Task Logistic Regression (MTLR) model as its base and a deep learning architecture as its core. Based on the Concordance index (C-index) and Brier score, this method outperforms the MTLR in all the experiments disclosed in this paper as well as the Cox Proportional Hazard (CoxPH) model when nonlinear dependencies are found.

1 Introduction

Survival analysis predicts when events occur, supporting applications such as purchases, churn, and loan defaults. The paper proposes N-MTLR to address linearity in MTLR while retaining survival-function modeling, and reports gains over MTLR and CoxPH in nonlinear settings.

  • 1 Introduction: Survival analysis predicts when an event of interest will happen, including purchases, customer churn, and loan defaults.These applications extend survival analysis beyond its origins in medical research.
  • 1 Introduction: CoxPH predicts individual survival and hazard functions but assumes linear feature effects, proportional hazards, and an unspecified baseline function.Its handling of ties may also require computational approximations such as Efron’s or Breslow’s methods.
  • 1 Introduction: MTLR avoids CoxPH’s listed assumptions and approximations but remains limited by a linear core that cannot properly model nonlinear dataset dependencies.Prior neural survival approaches still rely heavily on CoxPH structure.
  • 1 Introduction: N-MTLR outperforms MTLR in all disclosed experiments and CoxPH when datasets contain nonlinear dependencies, using C-index and Brier score.The paper compares CoxPH, MTLR, and N-MTLR on artificially generated and real datasets.
  • 1 Introduction: The proposed N-MTLR combines MTLR with a deep learning architecture as its core.Its motivation is to address the remaining linearity problem in MTLR.

2 Survival Analysis, an Overview

Survival analysis models time until an event while accounting for right censoring, and estimates survival or hazard functions from observed data. The overview introduces core notation and contrasts cohort-level Kaplan-Meier estimation with feature-based CoxPH modeling.

  • 2.1 Notations: Survival analysis estimates time until an event, with right censoring indicating that the event time is only known to exceed the censoring time.Observed time combines the actual event time and censoring time through Ti = min(ti, ci).
  • 2.1 Notations: The survival function S(t) = P[T > t] gives the probability that the event has not occurred by time t, while h(t) is its instantaneous occurrence rate.These are the central quantities used throughout the paper’s survival-analysis formulation.
  • 2.1 Notations: A survival-analysis datapoint consists of a feature vector x_i, an event indicator δ_i, and an observed time T_i.The event indicator equals 1 when an event occurs and 0 under censoring.
  • 2.2.1 Kaplan-Meier Model: Kaplan-Meier is a non-parametric estimator of the survival function for a homogeneous cohort, using event counts and numbers at risk at distinct event times.It does not incorporate individual feature vectors.
  • 2.2.2 Cox Proportional Hazard Model: CoxPH models an individual’s hazard as ˆhCoxPH(t, x_i) = λ0(t)η(x_i), separating a baseline time component from a feature-based risk component.The baseline function is usually unspecified, while the risk function is usually represented linearly.

3 Multi-Task Logistic Regression (MTLR)

MTLR estimates survival by combining logistic regression models across time intervals, avoiding key CoxPH assumptions but retaining a linear core.

  • 3 Multi-Task Logistic Regression (MTLR): MTLR avoids the proportional-hazard assumptions and computational approximations required by CoxPH while enabling survival-function estimation.CoxPH assumes constant hazard ratios over time, uses approximations for ties, and leaves the baseline hazard unspecified.
  • 3 Multi-Task Logistic Regression (MTLR): MTLR estimates the probability that an event occurs within each of J time intervals using a series of logistic regression models.The time axis is divided into intervals a_j = [τ_j−1, τ_j), with τ_0 = 0 and τ_J = ∞.
  • 3 Multi-Task Logistic Regression (MTLR): The model encodes event status across intervals while keeping later statuses unchanged after an event, because recurrent events are not analyzed.This produces a structured response vector for the interval-based model.

4 Neural Multi-Task Logistic Regression (N-MTLR)

N-MTLR retains MTLR’s survival-analysis framework while replacing its linear core with a deep neural architecture. The design adds nonlinear modeling flexibility and a Python implementation built on modern deep-learning libraries.

  • 4 Neural Multi-Task Logistic Regression (N-MTLR): N-MTLR replaces MTLR’s linear core with a multilayer perceptron, adding nonlinear modeling flexibility without relying on CoxPH assumptions.The architecture uses a deep-learning framework while preserving the MTLR-based approach.
  • 4 Neural Multi-Task Logistic Regression (N-MTLR): N-MTLR is implemented in Python with TensorFlow and Keras, supporting modern initialization, optimization, activation, normalization, and dropout techniques.Examples include Xavier initialization, Adam or RMSprop optimization, Softplus or ReLU activations, batch normalization, and dropout.
  • 4 Neural Multi-Task Logistic Regression (N-MTLR): The paper presents the Python implementation of MTLR as a further contribution.The authors describe this as the first time MTLR was made available in Python, to their knowledge.
  • 4 Neural Multi-Task Logistic Regression (N-MTLR): The model maps a feature vector through a nonlinear transformation whose output corresponds to subdivisions of the time axis.The transformation ψ maps R^p to R^J, with output values associated with the J time subdivisions.
  • 4 Neural Multi-Task Logistic Regression (N-MTLR): A two-hidden-layer example uses tanh in the first layer and ReLU in the second layer.The layers contain M_1 and M_2 units, respectively.

5 Experiments

The experiments compare CoxPH, MTLR, and N-MTLR using C-index, Brier score, and IBS on simulated and real survival datasets. N-MTLR performs similarly to linear models when dependencies are not obviously nonlinear, but captures nonlinear risk patterns more effectively.

  • Evaluation metrics: The C-index evaluates ranking with censored data, while the Brier score evaluates survival-probability accuracy at a given time.For right-censored data, the Brier score uses inverse probability of censoring weights.
  • Experimental setup: The experiments evaluate predictive performance with C-index, Brier score, and Integrated Brier Score across simulated and real datasets.The benchmark includes linear, square, and gaussian simulated risk functions plus WHAS and veteran datasets.
  • Simulated Survival Data: In the linear-risk simulation, CoxPH performs well, while MTLR and N-MTLR achieve similar results and N-MTLR slightly outperforms MTLR.The linear and proportional-hazard assumptions are valid in this setting.
  • Simulated Survival Data: The N-MTLR outperforms linear peers when the simulated risk function is nonlinear, because its neural network captures nonlinear dependencies.For the square risk function, CoxPH and MTLR perform less satisfactorily, while N-MTLR outperforms them.
  • Simulated Survival Data: For the gaussian-risk simulation, N-MTLR outperforms the other two models.This experiment evaluates the models on a nonlinear gaussian risk function.
  • Real datasets: On the WHAS dataset, the three models perform similarly, with N-MTLR slightly outperforming MTLR and CoxPH.The dataset does not appear to contain obvious nonlinear dependencies.

6 Conclusion

The Neural Multi-Task Logistic Regression consistently outperforms MTLR and achieves similar or better results than CoxPH, with a clear advantage when data contain nonlinear dependencies.

  • N-MTLR consistently outperforms MTLR and yields similar or better results than CoxPH, with a clear advantage for nonlinear dependencies.The model also uses deep-learning techniques through its Python implementation with Keras and TensorFlow.
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