ICODEN: Ordinary Differential Equation Neural Networks for Interval-Censored Data

TL;DR

ICODEN introduces an ordinary differential equation neural network to model hazard rates for interval-censored survival data without relying on proportional hazards or predefined hazard forms. The hazard function is modeled by a neural network that receives the cumulative hazard as an input and is coupled to an ODE for the cumulative hazard, allowing continuous-time, flexible survival modeling in high-dimensional settings. Across simulations and real-world ADNI and AREDS/AREDS2 datasets, ICODEN achieves robust predictive accuracy, handles left truncation, and enables data-driven subgroup identification with meaningful biological signals. This approach provides a practical, assumption-lean tool for interval-censored survival analysis in biomedical research, with potential extensions to competing risks and improved interpretability.

Abstract

Predicting time-to-event outcomes when event times are interval censored is challenging because the exact event time is unobserved. Many existing survival analysis approaches for interval-censored data rely on strong model assumptions or cannot handle high-dimensional predictors. We develop ICODEN, an ordinary differential equation-based neural network for interval-censored data that models the hazard function through deep neural networks and obtains the cumulative hazard by solving an ordinary differential equation. ICODEN does not require the proportional hazards assumption or a prespecified parametric form for the hazard function, thereby permitting flexible survival modeling. Across simulation settings with proportional or non-proportional hazards and both linear and nonlinear covariate effects, ICODEN consistently achieves satisfactory predictive accuracy and remains stable as the number of predictors increases. Applications to data from multiple phases of the Alzheimer's Disease Neuroimaging Initiative (ADNI) and to two Age-Related Eye Disease Studies (AREDS and AREDS2) for age-related macular degeneration (AMD) demonstrate ICODEN's robust prediction performance. In both applications, predicting time-to-AD or time-to-late AMD, ICODEN effectively uses hundreds to more than 1,000 SNPs and supports data-driven subgroup identification with differential progression risk profiles. These results establish ICODEN as a practical assumption-lean tool for prediction with interval-censored survival data in high-dimensional biomedical settings.

Figures (4)

• Figure 1: The proposed NN structure for $\lambda(t|X_i)$. The inputs are time $t$, predictor variables $X_i$, and the cumulative hazard function $\Lambda(t|X_i)$. Hidden layers use ReLU activation, while the output layer applies Softplus to ensure positivity.
• Figure 2: Simulation with a binary predictor under interval censoring and proportional hazards violation. The left panel shows the fitted cumulative hazard functions. The right panel shows the corresponding fitted survival functions.
• Figure 3: Gaussian mixture model–based subgrouping of ADNI participants. The left panel shows the density distribution of predicted log cumulative hazard at age 80, with the fitted Gaussian components corresponding to low- and high-risk groups. The right panel shows the Turnbull survival estimates for age at AD onset in the two subgroups.
• Figure 4: Gaussian mixture model-based subgrouping in AREDS and AREDS2 Data. The left panel shows the density distribution of predicted log cumulative hazard at 10 years, with the fitted Gaussian components corresponding to three different risk groups. The right panel shows the Turnbull survival estimates for time-to-late AMD in different risk subgroups.