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Deep Nonparametric Inference for Conditional Hazard Function

Wen Su, Kin-Yat Liu, Guosheng Yin, Jian Huang, Xingqiu Zhao

Abstract

We propose a novel deep learning approach to nonparametric statistical inference for the conditional hazard function of survival time with right-censored data. We use a deep neural network (DNN) to approximate the logarithm of a conditional hazard function given covariates and obtain a DNN likelihood-based estimator of the conditional hazard function. Such an estimation approach renders model flexibility and hence relaxes structural and functional assumptions on conditional hazard or survival functions. We establish the nonasymptotic error bound and functional asymptotic normality of the proposed estimator. Subsequently, we develop new one-sample tests for goodness-of-fit evaluation and two-sample tests for treatment comparison. Both simulation studies and real application analysis show superior performances of the proposed estimators and tests in comparison with existing methods.

Deep Nonparametric Inference for Conditional Hazard Function

Abstract

We propose a novel deep learning approach to nonparametric statistical inference for the conditional hazard function of survival time with right-censored data. We use a deep neural network (DNN) to approximate the logarithm of a conditional hazard function given covariates and obtain a DNN likelihood-based estimator of the conditional hazard function. Such an estimation approach renders model flexibility and hence relaxes structural and functional assumptions on conditional hazard or survival functions. We establish the nonasymptotic error bound and functional asymptotic normality of the proposed estimator. Subsequently, we develop new one-sample tests for goodness-of-fit evaluation and two-sample tests for treatment comparison. Both simulation studies and real application analysis show superior performances of the proposed estimators and tests in comparison with existing methods.

Paper Structure

This paper contains 18 sections, 8 theorems, 30 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Deep Neural Networks
  4. Estimation
  5. Nonasymptotic Error Bound
  6. Asymptotic Results
  7. Asymptotic Functional Normality
  8. Hypothesis Testing
  9. One-sample Tests
  10. Two-sample Tests
  11. Goodness-of-fit Tests
  12. Simulation Studies
  13. Evaluation of DNN Estimator
  14. Evaluation of One-sample Test
  15. Evaluation of Two-sample Test
  16. ...and 3 more sections

Key Result

Theorem 3.1

Suppose that conditions (C1)--(C3) hold. For any $N, M\in \mathbb{N}_{+}$, let ${\cal G}={\cal G}(L, {\bf p}, B)$ be the function class of ReLU multi-layer perceptrons with width $\|{\bf p}\|_\infty=38({\lfloor \alpha\rfloor}+1)^2p_0^{{\lfloor \alpha\rfloor}+1}N{\lfloor \log_2(8N)\rfloor}$ and dept where $c_0$ is a universal constant, and $c(B)$ is a constant depending on $B$.

Figures (4)

  • Figure 1: The pointwise averages of $\hat{H}(\cdot|x)$ estimated by the Cox model and the proposed DNN method.
  • Figure 2: The pointwise averages of $\hat{H}(\cdot|x)$ estimated by the AH model and the proposed DNN method.
  • Figure 3: The pointwise averages of $\hat{H}(\cdot|x)$ estimated by the AFT model and the proposed DNN method.
  • Figure 4: The conditional cumulative hazard functions $\hat{H}(t|x)$ of male and female patients in the four disease classes evaluated at the means of age, SPS, Scoma, which are 62.7, 25.5, and 12.1, respectively.

Theorems & Definitions (8)

  • Theorem 3.1: Nonasymptotic error bound
  • Theorem 4.1: Asymptotic normality
  • Theorem 4.2: Asymptotic distribution of the test statistic
  • Theorem 4.3: Asymptotic power function of the test
  • Theorem 4.4: Asymptotic distribution of the test statistic
  • Theorem 4.5: Asymptotic power function of the test
  • Theorem 4.6: Asymptotic distribution of the test statistic
  • Theorem 4.7: Asymptotic power function of the test