AI Insight
This paper introduces a mathematical framework for evaluating and training probabilistic survival predictions when event times are incompletely observed due to right censoring (when follow-up ends before the event occurs). The authors propose mapping predicted distributions through the censoring mechanism before applying proper scoring rules, which recovers existing methods like IPCW while enabling censored versions of standard metrics including CRPS, Brier score, and energy score. They demonstrate that their approach correctly ranks forecasts across different censoring scenarios, while alternative methods can produce incorrect rankings, and show that their "censored engression" method substantially improves multivariate survival modeling compared to naive approaches.
Why it matters
This framework provides a theoretically sound foundation for developing and evaluating machine learning models in medical prognosis, clinical trials, and other applications where researchers need to predict time-to-event outcomes but observations are incomplete due to patients dropping out, studies ending, or events not yet occurring.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: Proper scoring rules provide a rigorous theoretical basis for the training and evaluation of probabilistic forecasts. In survival analysis, such forecasts describe the distribution of the time until an event occurs. However, this event time is often only partially observed because follow-up may end before the event occurs, resulting in right censoring. We propose a framework for proper scoring of right-censored survival outcomes based on a simple idea: first, map the predictive distribution through the censoring mechanism, then apply the underlying proper score on the induced observed-data law. This yields localized scores for fixed censoring times and marginalized scores when the censoring time is random or only partially observed. The resulting construction recovers familiar right-censored likelihood and IPCW-type criteria within a coherent framework, while also yielding right-censored versions of the CRPS, pinball loss, Brier score, and energy score. We show that the marginalized construction is proper under conditional independent censoring and, for strictly proper base scores, identifies the latent joint CDF on the identifiable region. The same principle also leads to censored engression, a sample-based learning objective for multivariate right-censored survival modeling. In experiments, our scores correctly rank the oracle forecast across several censoring regimes, whereas forecast-dependent plug-in weighted scores can exhibit ranking reversals. Censored engression likewise substantially improves over naive training on censored outcomes.
Source: Proper Scoring Rules for Right-Censored Survival Data