Examining marginal properness in the external validation of survival models with squared and logarithmic losses
This paper addresses the theoretical validity of two widely used external validation metrics for survival analysis models—Integrated Survival Brier Score (ISBS) and Right-Censored Log-Likelihood (RCLL)—by introducing “marginal propriety” as a novel formal criterion for scoring rule appropriateness. We prove theoretically that neither metric satisfies marginal propriety. However, Monte Carlo simulations and extensive experiments across diverse right-censored survival modeling scenarios demonstrate that RCLL consistently satisfies this property empirically, while ISBS exhibits only negligible violations under extremely small sample sizes, remaining robust in practice. This reveals a critical dissociation between theoretical impropriety and empirical robustness—a key insight with important implications for metric selection and design. Building on this finding, we propose a new class of loss-function frameworks for survival prediction, grounded in marginal propriety, thereby providing both theoretical guidance and practical foundations for developing future survival scoring rules.