Dependence-robust evaluation of survival models under dependent censoring: misspecification robustness and stable estimation of the latent event-censoring copula

Right-censored survival data record, per subject, only the smaller of the event time $E$ and the censoring time $C$ and which of the two occurred, so the dependence between $E$ and $C$ - even conditional on covariates - cannot be identified from the data alone without additional assumptions (Tsiatis, 1975). Copula-informed evaluation metrics such as the dependent Brier score of Lillelund, Qi and Greiner therefore rest on an Archimedean working copula that cannot be validated against the truth. The open problem, made explicit in the limitations of that work, is to make survival-model evaluation under dependent censoring reliable when the true event-censoring copula is unknown and some misspecification is therefore unavoidable: (i) quantify how copula-family, dependence-strength, and non-Archimedean misspecification biases the dependent Brier score and the model rankings it induces, beyond the synthetic-only robustness evidence currently available; and (ii) supply stable, uncertainty-aware estimation of the dependence parameter under weak identification, with detection and updating under temporal distribution shift. Both components are currently open: identification results exist only inside parametric model classes and address estimation rather than evaluation, and no follow-up work has taken up the evaluation-robustness question.

Background

Survival analysis studies the time until an event (death, relapse, machine failure), but in most datasets the event time is right-censored: for each subject one observes only $t_i=\min(e_i,c_i)$ and the indicator $\delta_i=\mathbb{1}(e_i\le c_i)$ recording whether the event or the censoring (loss to follow-up, study end, dropout) happened first. Most methods and evaluation metrics assume independent censoring: the event time $E$ and the censoring time $C$ are independent, or at least conditionally independent given the measured covariates $X$. Dependent (informative) censoring - for example, patients dropping out of a study precisely because their health deteriorates - violates this assumption and biases both survival estimates and performance evaluations.

The bias reaches model evaluation through the Brier score, the mean squared error of a predicted survival probability $\hat S(t\mid x_i)$ against the observed event status at time $t$. With censored data it is computed by inverse probability of censoring weighting (IPCW), which reweights each contribution by the reciprocal of the estimated probability of remaining uncensored (Graf et al., 1999); the integrated Brier score (IBS) averages this over time. IPCW weights are valid only when the censoring distribution is (conditionally) independent of the event time, so under dependent censoring they are misspecified and model comparisons can be distorted.

The standard device for modeling such dependence is a copula: the function that couples the marginal distributions of $E$ and $C$ into their joint distribution (unique when the marginals are continuous). Archimedean copulas - those built from a one-dimensional generator applied to the sum of transformed marginal ranks, with families such as Clayton (Clayton, 1978) and Frank (Frank, 1979) - are the workhorses here, with dependence strength summarized by Kendall's tau, a rank correlation. Zheng and Klein (1995) showed for competing-risks data - the setting in which two or more latent event times compete and only the smallest is observed together with its identity, censoring being one such competing event - that if the copula linking the latent times is known, the marginal survival functions become identifiable; this yields the Copula-Graphic (CG) estimator, whose asymptotic theory - including its behavior under a misspecified copula - was developed by Rivest and Wells (2001) and extended to conditional settings by Braekers and Veraverbeke (2005). The copula-based treatment of dependent censoring is systematized in the book of Emura and Chen (2018).

Building on this, Lillelund, Qi and Greiner (arXiv:2502.19460) proposed a dependent Brier score (and its integrated version, IBS-Dep): an Archimedean working copula is fit jointly with marginal models, the CG estimator imputes a margin time $E[E\mid E>c]$ for each censored subject, and a dependence-informed weighting replaces IPCW. They prove consistency and asymptotic normality of the margin-time estimator and report a 12-16% average reduction in estimation error relative to IPCW on a semi-synthetic framework over 12 datasets, in which realistic dependent censoring is injected while true event times remain known for scoring.

The obstacle that survives all of this is classical: data containing only $(x_i, t_i, \delta_i)$ per subject cannot identify the joint law of $(E,C)$ - and hence its copula - without additional assumptions, because for any candidate dependence structure there is an alternative joint law generating the same observable distribution (Tsiatis, 1975). Identification of the dependence is restored only under restrictions. Czado and Van Keilegom (2023) proved that all parameters, including the copula association parameter, are identifiable when the copula is parametric and the marginals follow regression models, a route pursued further by Deresa and Van Keilegom (2019), Delhelle and Van Keilegom (2024), and D'Haen, Van Keilegom and Verhasselt (2024). In machine learning, Zhang, Ling and Zhang (2024) learn the Archimedean generator itself as a neural mixture of negative exponentials and prove identifiability, still under parametric survival marginals; Foomani Gharari et al. (2023) and Liu, Zhang and Zhang (2025) build deep copula survival models but either require a specified copula or give guarantees internal to their model class. When dependence recovery is abandoned, Willems, Beyhum and Van Keilegom (2025) derive bounds on regression parameters with the censoring dependence left fully unspecified, and Shih, Lo and Wilke (2026) confirm that nonparametric time-to-event models remain non-identifiable under unknown dependent censoring absent strong, untestable restrictions. No existing result recovers or sharply bounds the latent dependence for weakly-structured survival data, and none transfers such guarantees to evaluation metrics: the source paper's own misspecification study is synthetic-only and is described by its authors as a robustness analysis, not a parameter-recovery diagnostic. Since the true copula therefore cannot be validated against the data, the open problem is to make the evaluation itself reliable despite this unavoidable misspecification.

Problem Statement

Make copula-informed evaluation of survival models under dependent censoring reliable when the true event-censoring copula is unknown and untestable. The data are right-censored observations $(x_i, t_i, \delta_i)$ with $t_i=\min(e_i,c_i)$ and $\delta_i=\mathbb{1}(e_i\le c_i)$, where the event time $E$ and the censoring time $C$ may remain dependent even conditionally on the covariates $X$. The open problem comprises the following two linked components.

1. Quantified misspecification robustness of the evaluation metric. For a dependence-aware metric in the sense of the dependent (integrated) Brier score of Lillelund, Qi and Greiner - or a successor metric designed for this setting - quantify how misspecification of the working copula (at minimum a wrong Archimedean family and a wrong dependence strength, plus either a non-Archimedean or a time-varying truth) propagates to the metric value and to the model rankings it induces. This quantification must be validated beyond purely synthetic settings on realistic censoring regimes of the kind arising in biomedical survival datasets.

2. Stable and adaptive dependence estimation. Provide stable, uncertainty-aware estimation of the dependence parameter of the joint likelihood (copula together with marginal models) under weak identification - small samples and extreme censoring - and a method to detect and update the assumed dependence under temporal or distributional shift in $P(E\mid X)$, $P(C\mid X)$, or their joint law.

The common objective uniting the two components is that comparisons of survival models under dependent censoring be trustworthy - correct, or equipped with explicit, quantified error bounds - without requiring the true copula to be known in advance.

References

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