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.
For progressive diseases such as dementia or Alzheimer's disease (AD), the precise time of disease onset or progression is often unobserved due to the intermittent nature of assessments. For example, a patient may be disease-free at one clinical visit and then diagnosed with the disease at the subsequent visit. As a result, these non-fatal events of interest (e.g., onset of AD) are only known to occur within an interval between two assessment times, producing interval-censored data [8]. Because interval-censored data frequently arise in clinical and epidemiologic studies, accurately modeling and predicting disease progression trajectories is of practical importance to understand disease evolution and guide clinical decision-making.
Right censoring is a special case of interval censoring in which the subsequent visit time is infinite and has been thoroughly investigated in the literature. Traditional survival analysis techniques for right-censored data, such as the Kaplan-Meier estimator and regression-based models, have been adapted to accommodate interval-censored observations and are implemented in the IcenReg package [1]. This R package includes the non-parametric Turnbull estimator, the semi-parametric Cox-proportional hazards (PH) model, and a selection of parametric models. In the era of big data, the presence of high-dimensional predictors, such as genome-wide SNP data and high-resolution imaging data, presents significant challenges for these approaches. Recently, several methods have been developed for interval-censored data with high-dimensional predictors, such as the Cox PH model with adaptive lasso [10] and survival forest methods [23]. However, they cannot yet effectively handle thousands of SNPs in a dataset [18].
A neural network (NN) is a flexible and powerful method for modeling the complex relationships inherent in high-dimensional data. It can take high-dimensional data directly as input and approximate intricate associations from data without strong assumptions on the form of the model [2,16]. This flexibility allows NNs to learn complex, potentially nonlinear relationships among covariates.
Several NN-based survival analysis methods have been developed for right-censored data. Deep-Surv, among the first NN survival approaches, replaces the Cox proportional hazards model’s linear predictor with a neural network to capture complex covariate effects [7]. DeepHit and nnet-survival, two discretized-time survival NNs, do not rely on the PH assumption. Instead, they reformulate the continuous-time survival problem into a discrete-time problem, providing survival probabilities at predetermined times [9,6]. Recently, ordinary differential equation (ODE)-based NN frameworks have leveraged the intrinsic differential relationship between cumulative hazard and hazard functions, enabling continuous-time survival modeling through neural ODEs [20]. SuMo-net further employs a monotone NN to directly model the survival function [15]. Besides modeling the survival distribution, additional NN architectures have been applied for various medical applications, such as convolutional NNs and transformer-based architectures for clinical records and images with survival outcomes [11,24].
Recent studies have sought to extend these right-censoring-based methods to accommodate interval-censored observations. For example, BPNet adopts the PH assumption and models the baseline hazard function using Bernstein polynomials with a NN employed to learn the covariate effects [18]. There is also an accelerated failure time model-based NN method for interval-censored data [12]. These frameworks, however, typically depend on strong assumptions regarding the form of the underlying survival distribution.
In this work, we propose a flexible continuous-time framework for analyzing high-dimensional, interval-censored survival data without imposing any specific structure (such as the PH assumption) on the underlying hazard or survival functions. Furthermore, to accommodate left truncation, which is frequently encountered in clinical trial settings, we extend the model to jointly handle both interval censoring and left truncation within the same ODE-based neural architecture.
The article is organized as follows. In Section 2, we present the architecture of our proposed ODE-based NN for interval-censored survival data. Section 3 examines a series of simulation studies with comparisons to alternative methods. In Section 4, we apply our method to multiphase real-world research datasets of the Alzheimer’s Disease Neuroimaging Initiative (ADNI) and two Age-Related Eye Disease Studies (AREDS and AREDS2), and compare its performance against established approaches. The conclusion and discussion are presented in Section 5.
In this section, we introduce the proposed ODE-based NN for the interval-censored data method, namely, “ICODEN”. First, we introduce the notation for interval-censored data, which may be subject to left-truncation,
This content is AI-processed based on open access ArXiv data.