SurvSurf: a partially monotonic neural network for first-hitting time prediction of intermittently observed discrete and continuous sequential events

TL;DR

SurvSurf tackles the challenge of predicting the first hitting times of sequential events from baseline predictors while preserving the natural monotone ordering between sequential CIFs. It introduces a partially monotonic neural network with time- and grade-dependent components and a LossDyDg objective to enforce event-order constraints, along with an implied-truth-imputed IBS for robust evaluation. The work provides a theoretical monotonicity guarantee, handles missing intermediate events, and unifies discrete and continuous time/event definitions, demonstrated by superior performance and zero monotonicity violations across simulated and real data. The approach offers reliable, interpretable time-to-event predictions from baseline data for applications in healthcare, economics, and other domains with intermittently observed sequential events.

Abstract

We propose a neural-network based survival model (SurvSurf) specifically designed for direct and simultaneous probabilistic prediction of the first hitting time of sequential events from baseline. Unlike existing models, SurvSurf is theoretically guaranteed to never violate the monotonic relationship between the cumulative incidence functions of sequential events, while allowing nonlinear influence from predictors. It also incorporates implicit truths for unobserved intermediate events in model fitting, and supports both discrete and continuous time and events. We also identified a variant of the Integrated Brier Score (IBS) that showed robust correlation with the mean squared error (MSE) between the true and predicted probabilities by accounting for implied truths about the missing intermediate events. We demonstrated the superiority of SurvSurf compared to modern and traditional predictive survival models in two simulated datasets and two real-world datasets, using MSE, the more robust IBS and by measuring the extent of monotonicity violation.

Figures (17)

• Figure 1: For simulated data, the transition matrix for a subject is generated by passing the feature values of this subject through a neural network. We used $\lambda_{prog} = 2$ and $\lambda_{stay}=1$ for the main simulated dataset (Sim-Main). A smaller dataset with little or no progression (Sim-LackProg) was simulated by setting $\lambda_{prog} = 0.02$ and $\lambda_{stay}=1$
• Figure 2: Clustering of subjects in the RW-TRAE dataset derived from the maximum severity grade, time taken to reach the maximum severity, and censoring time. Subject IDs are as provided in the raw data. The values in each column were mean-centered and normalized before plotting. The color gray, blue, orange, pink and green correspond to cluster ID 0, 1, 2, 3 and 4, respectively.
• Figure 3: Schematic summary of the SurvSurf network. Orange arrows represent non-decreasing monotonic transformations. Blue arrows represent non-increasing monotonic transformations. Black arrows represent non-monotonic transformations. Symbols are defined in Equation \ref{['eq:z_k_full']}.
• Figure 4: Model performance and the extent of monotonicity violation for the SurvSurf and the benchmarking models in the test set of different datasets. Points with the same color and marker are the same model trained by initializing with different random seeds. Points closer to the bottom-left corner represent models with better performance and less monotonicity violation.
• Figure 5: Test-set result: Spearman's rank correlation ($\rho$) between mean squared error and $IBS^{ipcw}_{naive}$ (left) and $IBS^{ipcw}_{iti}$ (right) on Sim-Main dataset.

Theorems & Definitions (8)

• Theorem 1
• proof
• Theorem 2: monotonicity in $t$
• proof
• Theorem 3: monotonicity in $g$
• proof
• Theorem 4
• proof