Survival Models: Proper Scoring Rule and Stochastic Optimization with Competing Risks

TL;DR

SurvivalBoost not only outperforms 12 state-of-the-art models across several metrics on 4 real-life datasets, both in competing risks and survival settings, but also provides great calibration, the ability to predict across any time horizon, and computation times faster than existing methods.

Abstract

When dealing with right-censored data, where some outcomes are missing due to a limited observation period, survival analysis -- known as time-to-event analysis -- focuses on predicting the time until an event of interest occurs. Multiple classes of outcomes lead to a classification variant: predicting the most likely event, a less explored area known as competing risks. Classic competing risks models couple architecture and loss, limiting scalability.To address these issues, we design a strictly proper censoring-adjusted separable scoring rule, allowing optimization on a subset of the data as each observation is evaluated independently. The loss estimates outcome probabilities and enables stochastic optimization for competing risks, which we use for efficient gradient boosting trees. SurvivalBoost not only outperforms 12 state-of-the-art models across several metrics on 4 real-life datasets, both in competing risks and survival settings, but also provides great calibration, the ability to predict across any time horizon, and computation times faster than existing methods.

Figures (9)

• Figure 1: SurvivalBoost Algorithm with its Feedback Loop. After providing input to the algorithm, a random time is assigned, and the corresponding weights and target are computed. After each iteration, the feedback loop updates the censoring probability, $G^\star$ as defined in eq.\ref{['eqn:full_loss']}.
• Figure 2: Prediction performance / training time trade-off for competing risk on the synthetic dataset. Average IBS compared the fitting time for each model on 20k training data points, with a censoring rate of approximately 50% and a dependant censoring across 6 features.
• Figure 3: Prediction performance / training time trade-off for competing risks on SEER dataset. Average IBS versus fitting time for each model, with a maximum of 330k training points, except for Fine & Gray (50k) and RSF (100k). Table \ref{['tab:ibs_event_seer']} provides the IBS values for each event.
• Figure 4: Prediction accuracy at time $\zeta$ Accuracy of the Argmax of the Cumulative Incidence Functions across different time quantiles on the SEER dataset.
• Figure 5: Prediction performance / training time trade-off in survival analysis IBS (Integrated Brier score) function of fit time for each model on real-life datasets. For the big datasets, some algorithms exceeded computing resources.

Theorems & Definitions (27)

• definition 1: Quantities of interest
• definition 3: Proper Scoring Rule
• definition 4: PSR for competing risks settings
• definition 5: Competitive Weights Negative LogLoss
• lemma 5
• proof : Proof sketch
• theorem 6: Properness of the scoring rule
• proof : Proof sketch
• definition 7: Prediction accuracy at time $\zeta$
• lemma 7