ICTSurF: Implicit Continuous-Time Survival Functions with Neural Networks

TL;DR

ICTSurF tackles censoring and limitations of traditional survival models by proposing a continuous-time survival framework that uses implicit representation to model the hazard $h(t)$ and survival $S(t)$ directly in continuous time. The method parameterizes $\hat{h}(t,x)$ with a neural network fed by covariates and time embeddings (Time2Vec), optimized via a discretized trapezoidal integral of the hazard and a likelihood-based loss, and extended to handle competing risks. Empirical results across METABRIC, SUPPORT, and synthetic datasets show ICTSurF achieving competitive $C^{td}$-index and strong calibration (Brier scores), outperforming several baselines, with time-embedding and discretization flexibility contributing to gains. The approach enables precise, continuous-time survival predictions and offers practical advantages in discretization and interpretability, with public code available for replication and extension to time-varying or multimodal data.

Abstract

Survival analysis is a widely known method for predicting the likelihood of an event over time. The challenge of dealing with censored samples still remains. Traditional methods, such as the Cox Proportional Hazards (CPH) model, hinge on the limitations due to the strong assumptions of proportional hazards and the predetermined relationships between covariates. The rise of models based on deep neural networks (DNNs) has demonstrated enhanced effectiveness in survival analysis. This research introduces the Implicit Continuous-Time Survival Function (ICTSurF), built on a continuous-time survival model, and constructs survival distribution through implicit representation. As a result, our method is capable of accepting inputs in continuous-time space and producing survival probabilities in continuous-time space, independent of neural network architecture. Comparative assessments with existing methods underscore the high competitiveness of our proposed approach. Our implementation of ICTSurF is available at https://github.com/44REAM/ICTSurF.

Figures (4)

• Figure 1: Discretization schemes: Illustration of discretization schemes with black dots denoting time points corresponding to events ($T^*_i$). (a) Individual discretization is applied to each time point for every sample. (b) Attempted uniform discretization for all samples, with variations occurring only at the times of events.
• Figure 2: Model architecture. The model takes covariates and a specific time of interest as input, producing hazard rates, denoted as $\hat{h}(x,t)$, for each event at the specified time as its output. Note that in this figure, we assume two competing risks, hence two hazard functions in the output. Conversely, in scenarios involving a single risk, there would be a single hazard function in the output.
• Figure 3: 5-fold CV. Hyperparameters were first selected using a holdout validation set. Performance metrics were then evaluated using 5-fold CV.
• Figure 4: Outcomes (mean and standard error of $C^{td}$-index) from Discretization Experiments at 25%, 50%, and 75% event percentiles in METABRIC and SUPPORT datasets. The green solid line represents the discretization scheme as demonstrated in Figure \ref{['fig:cutequal']}, while the blue dashed line corresponds to the scenario illustrated in Figure \ref{['fig:cutsame']}.