Partially Public Data in Survival Analysis

• Partially public data in survival analysis is characterized by the public availability of certain event times or features while other components remain censored or privatized.
• Nonparametric inference and differential privacy techniques, including Dempster–Shafer analysis and local sensitivity measures, ensure robust evidence quantification amid data constraints.
• Advanced methods such as synthetic data generation, semi-supervised deep learning, and multi-modal fusion improve survival prediction by effectively handling incomplete and privatized datasets.

Partially public data in survival analysis refers to scenarios where certain elements of the data—such as event times, covariate details, or outcome labels—are accessible or shareable, while other components remain censored, privatized, or incomplete due to privacy, operational, or practical constraints. This paradigm encompasses right-censored survival data, imprecise or missing labels, privacy-preserving releases, semi-supervised augmentation from unlabeled sources, and multi-institutional settings with restricted data sharing. The methodological landscape spans nonparametric inference, advanced deep learning, differential privacy, and causal modeling, with real-world applications ranging from vaccine efficacy trials to collaborative medical research.

1. Theoretical Foundations and Types of Partial Publicity

Survival analysis fundamentally aims to characterize the time until an event occurs, often accommodating right-censored formats where some failure times are unknown due to paper design, loss to follow-up, or privacy restrictions. Partial publicity manifests in several forms:

• Right-censored data: Only the lower bound of time-to-event is known for certain observations.
• Partially observed labels: Instances are censored, unlabeled, or only baseline characteristics are available (Haredasht et al., 2022).
• Aggregated or transformed representations: Raw covariate data are not shareable, but summary features or reduced-dimension encodings are (Toyoda et al., 9 May 2025).
• Local privacy mechanisms: Sensitive failure indicators or event times are randomized or privatized while covariates and times may be public (Nguyên et al., 2017, Maxime et al., 2023).
• Multi-modal incompleteness: Observations consist of available modalities in some instances and missing modalities or incomplete annotation in others (Qu et al., 25 Jul 2024).

A recurring theme is that analyses must robustly handle not only fully observed and censored data but also those with missing, privatized, or surrogate features and outcomes.

2. Nonparametric and Dempster–Shafer Inference with Censored Data

Nonparametric approaches accommodate arbitrary distributions of survival times, avoiding strong parametric assumptions. Notably, Dempster–Shafer (DS) analysis provides a framework for quantifying evidence when observations are ambiguous due to right-censoring. The DS approach employs a triplet representation: (P, Q, R)—evidence for, against, and ambiguous regarding a hypothesis (e.g., vaccine efficacy surpassing a threshold).

Through the use of order statistics—where observed failure times x₁ ≤ x₂ ≤ … ≤ xₘ correspond to CDF heights Yᵢ (distributed as order statistics from uniforms)—the method supports interval quantification for population fractions failing within (tₗ, tᵤ), leveraging Beta distributions:

$$Y_j - Y_i \sim \mathrm{Beta}(j - i, m + 1 - (j - i))$$

Censoring-induced ambiguity is encoded as bounds on the number of events: lower (d₍ⱼ,ₖ₎) and upper (e₍ⱼ,ₖ₎), propagating to evidence calculations via Beta CDFs (Edlefsen et al., 2012).

This is essential in applications like the RV144 HIV-1 vaccine trial, where conclusions about efficacy are sensitive to assumptions about lost-to-follow-up failures—DS analysis precisely quantifies this ambiguity, shifting probability mass between evidence for, against, and ignorance.

3. Privacy-Preserving Mechanisms and Differential Privacy

Survival datasets often contain sensitive individual records. Several mechanisms have been devised to allow public release of survival statistics while ensuring privacy:

• Global and Local Sensitivity: The notion of local sensitivity is exploited to reduce noise when publishing survival model parameters, e.g., Weibull distribution shape and scale. The exponential mechanism with a ladder-shaped utility function enables accurate publication of parameter estimates under differential privacy constraints (Nguyên et al., 2017).
• Local Differential Privacy for Outcome Indicators: Noise is injected only onto failure indicators δ, using a channel $q_{\alpha}(z|b) = (\alpha / 2)\exp(-\alpha |z-b|)$, so that likelihoods of output differ by at most $e^{\alpha}$. A nonparametric kernel estimator generalizes the Nelson–Aalen estimator, maintaining consistency and minimax optimality under private failure labels (Maxime et al., 2023).
• Distributed Data Collaboration: Through anchor datasets and dimension-reduced intermediate representations, accurate propensity scores and survival curves are estimated without exchanging raw data, further reducing privacy risks and communication overhead (Toyoda et al., 9 May 2025).

These mechanisms are essential for medical studies, epidemiological modeling, and reliability analysis, where raw data sharing is either prohibited or operationally infeasible.

4. Handling Incomplete and Unlabeled Data: Semi-supervised, Multi-modal, and Deep Learning Approaches

Real-world survival data frequently suffers from incomplete follow-up, missing outcome labels, and multimodal incompleteness:

• Semi-supervised integration of unlabeled data: Random Survival Forests expanded by self-training—where confident predictions for unlabeled instances are iteratively promoted to the labeled set—demonstrably improve predictive performance even when most data are censored or unlabeled. Correction rules based on partial supervision from censored data mitigate overfitting and instability (Haredasht et al., 2022).
• Multi-modal fusion in the presence of missing data: Deep frameworks encode each available modality (e.g., images, reports) via dedicated foundation models and aggregate them at intra- and inter-modality levels, using attention mechanisms. Progressive survival disambiguation with Gaussian warm-up weighting and pseudo-label generation ensures robust learning with censored labels, yielding improved concordance and stratification (Qu et al., 25 Jul 2024).
• Deep Partially Linear Transformation Models: These models blend interpretable parametric covariates with nonlinear predictors estimated via deep neural networks, enabling flexibility and optimal convergence even when only subsets of features are publicly sharable, and supporting efficient, asymptotically normal inference (Yin et al., 10 Dec 2024).

Handling incompleteness robustly is crucial in large-scale clinical trials, multi-center studies, and electronic health record (EHR) applications, ensuring estimation and prediction power is sustained in the face of partial data availability.

5. Advancements in Synthetic Data Generation, Feature Augmentation, and Survival Prediction

The emergence of synthetic data generation and advanced feature engineering enhances both the privacy and utility of survival datasets with partial publicity:

• SurvivalGAN: A conditional GAN architecture, augmented with “imbalanced samplers” and a dedicated time regressor, generates synthetic survival datasets that preserve censoring and event horizon characteristics. New metrics—optimism, short-sightedness, and KM divergence—formally assess generative fidelity against real distributions, improving downstream survival prediction while reducing privacy risks (Norcliffe et al., 2023).
• Contrastive Learning for EHR Data: Ontology-aware contrastive survival (OTCSurv) utilizes temporal distinctiveness, forming contrastive pairs from both censored and observed durations, and hardness-aware negative sampling to create discriminative, interpretable embeddings for survival prediction (Kerdabadi et al., 2023).
• Geographic Feature Augmentation: Incorporating location-based public health statistics, e.g., State-based Expected Survival Rate, into models such as CoxPH and Deep Survival Machines yields statistically significant improvements in discrimination, enabling more accurate personalized predictions and regional stratification (Seidi et al., 2023).

These approaches facilitate data democratization, obviate the need for sensitive raw data releases, and empower analyses over restricted, incomplete, or simulated datasets.

6. Practical Applications, Performance Evaluation, and Limitations

Methodologies for partially public data have been validated on real-world datasets spanning vaccine efficacy, cancer survival, and multi-center treatment evaluation. Key performance metrics include:

• Concordance Index (C-index): Assessing discrimination between predicted and actual event times—improved by feature augmentation and deep learning frameworks.
• Brier Scores and Gap Metrics: Evaluating calibration and approximation fidelity of survival curves, especially in privacy-preserving or distributed contexts.
• Statistical Significance: Paired t-tests and area-under-curve measures demonstrate consistent improvements over baselines.

Limitations exist: local sensitivity and differential privacy mechanisms trade-off accuracy for privacy; dimensionality reduction for collaboration may induce information loss; pseudo-label generation and attention mechanisms depend on quality of foundation models and adequate tuning; minimax optimality depends on bandwidth selection in kernel estimators; and multi-modal frameworks may require sufficient data richness to generalize.

7. Future Directions

Future lines of inquiry include extending privacy-preserving inference to semiparametric and complex deep models, developing richer representations for time-dependent and multi-level treatments, integrating local differential privacy for additional outcome types, and adapting frameworks to broader, more heterogeneous datasets and multi-center collaborations. Advancements in attention aggregation, pseudo-labeling, and robust estimation strategies promise further improvements in predictive power, interpretability, and applicability across distributed, partially public survival datasets in medical, engineering, and actuarial domains.