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

Self-Consistent Equation-guided Neural Networks for Censored Time-to-Event Data Sehwan Kim 1, Rui Wang 1,2, and Wenbin Lu 3 1 Department of Population Medicine, Harvard Pilgrim Health Care Institute and Harvard Medical School, Boston, MA 2 Department of Biostatistics, Harvard School of Public Health, Boston, MA 3 Department of Statistics, North Carolina State University, Raleigh, NC March 13, 2025 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. 1 Introduction Censored time-to-event data are widely encountered in various fields where understanding the timing of events, such as failure rates or disease progression, is critical, but the exact event times Correspondence author: Wenbin Lu, email: wlu4@ncsu.edu 1 arXiv:2503.09097v1 For example, estimating survival probability based on covariate information is essential for risk prediction, which plays a key role in developing and evaluating personalized medicine. The Kaplan-Meier (KM) estimator (Kaplan and Meier, 1958), Cox proportional hazards model (Cox, 1972), and random survival forests (Ishwaran et al., 2008) are commonly-used methods for estimating survival functions. The KM estimator is a non-parametric method suitable for population-level analyses. However, its utility is limited when the objective is to estimate conditional survival probabilities at the individual level. The Cox proportional hazards model offers a semi-parametric approach for estimating conditional survival functions, accommodating the incorporation of covariates.