June 2022 Deep learning for the partially linear Cox model
Qixian Zhong, Jonas Mueller, Jane-Ling Wang
Author Affiliations +
Ann. Statist. 50(3): 1348-1375 (June 2022). DOI: 10.1214/21-AOS2153


While deep learning approaches to survival data have demonstrated empirical success in applications, most of these methods are difficult to interpret and mathematical understanding of them is lacking. This paper studies the partially linear Cox model, where the nonlinear component of the model is implemented using a deep neural network. The proposed approach is flexible and able to circumvent the curse of dimensionality, yet it facilitates interpretability of the effects of treatment covariates on survival. We establish asymptotic theories of maximum partial likelihood estimators and show that our nonparametric deep neural network estimator achieves the minimax optimal rate of convergence (up to a polylogarithmic factor). Moreover, we prove that the corresponding finite-dimensional estimator for treatment covariate effects is n-consistent, asymptotically normal and attains semiparametric efficiency. Extensive simulation studies and analyses of two real survival data sets show the proposed estimator produces confidence intervals with superior coverage as well as survival time predictions with superior concordance to actual survival times.

Funding Statement

The first author was supported by National Science Foundation of China Grant NSFC-11931001 and Key Laboratory of Econometrics (Xiamen University), Ministry of Education.
The third author was supported by NSF Grant DMS-1914917 and NIH Grant UG3-0D023313 (ECHO Program).


The authors are grateful to the Editor, Associate Editor and referees for their helpful comments that led to numerous improvements of the paper. The first author also thanks Professor Ying Yang at Department of Mathematical Sciences, Tsinghua University for kind support.


Download Citation

Qixian Zhong. Jonas Mueller. Jane-Ling Wang. "Deep learning for the partially linear Cox model." Ann. Statist. 50 (3) 1348 - 1375, June 2022. https://doi.org/10.1214/21-AOS2153


Received: 1 May 2021; Revised: 1 August 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441123
zbMATH: 07547933
Digital Object Identifier: 10.1214/21-AOS2153

Primary: 62G20 , 62N02
Secondary: 62C20 , 62G08

Keywords: Censored data , minimax estimation , neural network , partial likelihood , Semiparametric efficiency , Survival analysis

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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