tdCoxSNN: Time-Dependent Cox Survival Neural Network for Continuous-time Dynamic Prediction

TL;DR

This paper introduces tdCoxSNN, a time-dependent Cox survival neural network that merges a neural risk-score function with a Cox-type hazard to enable dynamic, continuous-time prediction using time-varying covariates, including high-dimensional inputs like longitudinal fundus images. The model minimizes a negative log partial likelihood and can incorporate pre-trained CNNs to process images, providing end-to-end dynamic risk estimates that update as new data arrive. Through simulations and two real datasets (AMD AREDS and PBC2), tdCoxSNN demonstrates robust calibration and discrimination, outperforming joint modeling, landmarking, and several deep-learning baselines, particularly in nonlinear or high-dimensional settings. The approach offers a practical, scalable framework for personalized prognosis in settings with irregular observation times and rich longitudinal data, with potential extensions to model continuous-time dynamics and time-varying effects.

Abstract

The aim of dynamic prediction is to provide individualized risk predictions over time, which are updated as new data become available. In pursuit of constructing a dynamic prediction model for a progressive eye disorder, age-related macular degeneration (AMD), we propose a time-dependent Cox survival neural network (tdCoxSNN) to predict its progression using longitudinal fundus images. tdCoxSNN builds upon the time-dependent Cox model by utilizing a neural network to capture the non-linear effect of time-dependent covariates on the survival outcome. Moreover, by concurrently integrating a convolutional neural network (CNN) with the survival network, tdCoxSNN can directly take longitudinal images as input. We evaluate and compare our proposed method with joint modeling and landmarking approaches through extensive simulations. We applied the proposed approach to two real datasets. One is a large AMD study, the Age-Related Eye Disease Study (AREDS), in which more than 50,000 fundus images were captured over a period of 12 years for more than 4,000 participants. Another is a public dataset of the primary biliary cirrhosis (PBC) disease, where multiple lab tests were longitudinally collected to predict the time-to-liver transplant. Our approach demonstrates commendable predictive performance in both simulation studies and the analysis of the two real datasets.

Figures (5)

• Figure 1: (a) Workflow of tdCoxSNN illustrated using the AMD example. (b) Dynamic prediction using tdCoxSNN for the AMD example. The y-axis on the left denotes the risk score estimated from the tdCoxSNN at each visit. The y-axis on the right is the predicted probability of developing late AMD since the latest visit time.
• Figure 2: Low dimensional simulations: Boxplots of BS and cdAUC at $\Delta t=1,2,3,4$ from landmark time $s=1$ for four simulation settings. Models included in the comparison are the time-dependent Cox model (tdCoxPH), the joint modeling (JM), the LM with RSF (LM-RSF), the Dynamic-DeepHit (Dyn-DeepHit), and the proposed model (tdCoxSNN).
• Figure 3: High dimensional simulations: Boxplots of BS and cdAUC at $\Delta t=1,2,3,4$ from landmark time $s=1$ in simulation with longitudinal $28\times28$ handwriting digit images. Models included in the comparison are time-independent Cox SNN (CoxSNN), time-dependent Cox SNN (tdCoxSNN), and the true model with the true risk score (Truth).
• Figure 4: (a) Mean and standard deviation of BS and cdAUC for the 5-fold cross-validation analysis of AREDS data analysis. (b) Saliency map for the left eye at year 2.3 of the subject with baseline age 71.4, at least high-school education, and was a smoker at baseline.
• Figure 5: The KM estimators of the disease-free probabilities (predicted by tdCoxSNN in the test dataset) for high-risk and low-risk groups identified by the baseline fundus image. The histograms show the estimated baseline risk score with subgroups identified by the Gaussian mixture model (log-rank test $p<2\times 10^{-16}$).