SPOT-IC: Improving prediction for interval-censored data via survival probability transfer
Source: arXiv:2607.09640 · Published 2026-07-10 · By Jingyi Hu, Yu Gu
TL;DR
This paper addresses the challenge of prediction with interval-censored survival data, which arises when failure times are only known to lie within examination intervals rather than exact times. This is common in chronic disease studies with infrequent follow-ups. Existing transfer learning methods for survival data either rely on restrictive assumptions such as shared model parameters and require access to individual-level source data, or scale poorly in multi-source settings. The authors propose SPOT-IC, a novel transfer learning framework that transfers survival probability information—rather than model parameters—from arbitrary source models without requiring source data sharing. SPOT-IC uses a cross-entropy based penalty to align the target survival function with source survival probability estimates. An efficient EM algorithm enables scalable computation. When multiple sources are available with unknown informativeness, the method aggregates candidate estimators via a Q-aggregation procedure, providing robustness against negative transfer. Theoretical results show the estimator achieves faster convergence rates than a target-only estimator if at least one source is informative. Extensive simulations under diverse scenarios demonstrate consistent gains over competing methods. A real data application on Alzheimer's Disease Neuroimaging Initiative data validates the approach’s practical effectiveness.
Key findings
- SPOT-IC outperforms competing methods such as target-only, combined pooling, Trans-Cox-Lasso, and Trans-Cox-MIC across multiple simulation scenarios with interval-censored data (Table 1).
- SPOT-IC achieves median L2-distance in survival function estimation reduced by approximately 50% relative to target-only in single-source settings.
- The multi-source SPOT-IC procedure achieves convergence rates comparable to the best informative source with an additive O_p(n^{-1}) term (Theorem 3).
- The proposed cross-entropy penalty allows flexible survival probability transfer without requiring aligned covariates or models between target and source studies.
- The EM algorithm for SPOT-IC avoids inversion of large matrices and depends only on averages of latent Bernoulli variables, improving computational stability.
- In simulations, SPOT-IC demonstrates robustness to negative transfer from non-informative sources by adaptively weighting multiple source estimators.
- Theoretical analysis proves faster convergence than target-only estimators when source survival functions are sufficiently similar and estimators converge at adequate rates (Theorem 2).
- SPOT-IC accommodates multiple source studies via a two-stage screening and Q-aggregation procedure, mitigating the adverse effects of unknown source informativeness.
Threat model
n/a — this paper is a statistical methodology contribution focused on survival prediction with interval-censored data, not a security threat model.
Methodology — deep read
The authors formulate the problem assuming that the target failure times are interval-censored, observed only to lie within intervals defined by examination times. The target survival function is modeled via a semiparametric transformation model with a frailty-induced log-likelihood, covering models such as Cox proportional hazards and proportional odds. The target data consist of interval-censored outcomes and potentially time-dependent covariates.
They propose SPOT-IC to incorporate auxiliary information from one or multiple source studies that provide estimated survival probabilities conditional on covariates. Crucially, source models are arbitrary and source individual-level data are not shared—only the estimated survival probabilities are available.
The key methodological idea is to add a penalty term based on the negative cross-entropy between the target survival function and the source estimator of survival probability at randomly sampled covariates and time points drawn from the union of examination times. This penalty encourages the target survival estimator to align with the transferred survival probabilities. This penalty is integrated into a joint objective combining the target log-likelihood and the penalty weighted by a tuning parameter.
To maximize this penalized objective, the authors derive an EM algorithm exploiting a Poisson data augmentation scheme. Latent Poisson variables are introduced to represent the interval-censored data likelihood and the cross-entropy penalty in a unified weighted log-likelihood framework. The E-step computes posterior expectations of latent variables, and the M-step updates model parameters and the cumulative baseline hazard function in a tractable manner without large matrix inversions.
In the multi-source setting, the method uses a two-stage approach: (1) screening by applying single-source SPOT-IC separately with each source and the target data to obtain candidate estimators including a target-only baseline, and (2) aggregation via convex combination of candidates using Q-aggregation minimizing a penalized criterion with an entropy penalty to promote stability and avoid overconcentration.
Theoretical results establish convergence rates for both single- and multi-source settings, showing that the transfer estimator converges at least as fast as the better of the target-only estimator and informative source estimators. The penalty tuning parameter is data-adaptively selected, with asymptotically optimal choices provided.
Simulations involved generating interval-censored failure times from Cox and alternative survival models under varying degrees of source-target similarity and censoring structure. Validation was performed on large uncensored datasets to precisely measure survival function estimation error, supremum absolute errors, integrated Brier scores, and concordance indices.
Overall, the methodology enables flexible transfer of survival probabilities from heterogeneous sources without data sharing, leveraging a novel penalty and efficient EM estimation, along with robust aggregation in the presence of multiple sources.
Technical innovations
- Introduction of a cross-entropy type penalty on survival probability functions to transfer survival knowledge from arbitrary source estimators without requiring parameter or covariate alignment.
- Development of an EM algorithm leveraging Poisson data augmentation to efficiently maximize the penalized likelihood incorporating interval censoring and survival probability transfer.
- Extension to multi-source transfer learning via a two-stage procedure combining screening of candidate estimators and robust aggregation through Q-aggregation with entropy penalties.
- Theoretical proof that the transfer estimator achieves a faster convergence rate than the target-only estimator when at least one source study provides sufficiently informative survival estimates.
Datasets
- Simulated interval-censored data — 100 to 150 target samples, 500 to 1,000 source samples — synthetic
- Alzheimer’s Disease Neuroimaging Initiative (ADNI) — real-world interval-censored survival data on Alzheimer’s disease progression
Baselines vs proposed
- Target-only: L2-distance = 0.106 vs SPOT-IC: 0.048 (Scenario 1)
- Combined pooling: L2-distance = 0.031 vs SPOT-IC: 0.048 (Scenario 1)
- Trans-Cox-Lasso: L2-distance = 0.065 vs SPOT-IC: 0.048 (Scenario 1)
- Trans-Cox-MIC: L2-distance = 0.066 vs SPOT-IC: 0.048 (Scenario 1)
- SPOT-IC outperforms all other methods in Scenarios 2–5 with greater L2-distance improvements over target-only
- In multi-source setting, aggregated SPOT-IC attains convergence rate matching the best informative source plus O_p(n^{-1}) (Theorem 3)
Limitations
- Method relies on the assumption that survival probabilities in source and target studies share sufficient similarity within the target domain; performance degrades otherwise.
- The tuning parameter selection involves cross-validation which may be computationally expensive in practice.
- The current work focuses on mixed-case interval censoring and semiparametric transformation models; extensions to other censoring types or model classes remain unexplored.
- The asymptotic theory assumes certain smoothness and regularity conditions on covariates and censoring times that may not hold in all real datasets.
- Negative transfer protection depends on the Q-aggregation step whose performance may deteriorate when sources are highly heterogeneous or when target sample size is very small.
Open questions / follow-ons
- How does SPOT-IC perform with high-dimensional covariates or complex time-dependent covariate structures beyond the current semiparametric transformation model?
- Can the survival probability transfer penalty be adapted to accommodate competing risks or multi-state survival models?
- How to effectively combine SPOT-IC with modern machine learning survival models such as deep neural networks while preserving interpretability and efficiency?
- What are the finite sample guarantees and robustness properties of the Q-aggregation step when the number of sources is large relative to the target sample size?
Why it matters for bot defense
While this work focuses on survival analysis with interval-censored data, the core principle—transferring predictive knowledge across heterogeneous datasets without sharing raw data—is relevant to bot-defense models that aggregate signals from multiple noisy sources. The cross-entropy penalty to align predicted distributions is analogous to calibration or domain adaptation approaches commonly used in bot detection. Also, the multi-source aggregation strategy addresses practical challenges of negative transfer when integrating different models. Bot-defense systems dealing with interval-censored or sparse temporal event data (e.g., clickstreams with missing timestamps) may consider adopting similar survival probability transfer penalties to improve prediction accuracy without exposing sensitive source data. The EM algorithm and Q-aggregation technique may inspire scalable techniques for aggregating threat signals from multiple domains under privacy constraints. However, direct application would require adapting the methodology to classification tasks and the event types encountered in bot detection scenarios.
Cite
@article{arxiv2607_09640,
title={ SPOT-IC: Improving prediction for interval-censored data via survival probability transfer },
author={ Jingyi Hu and Yu Gu },
journal={arXiv preprint arXiv:2607.09640},
year={ 2026 },
url={https://arxiv.org/abs/2607.09640}
}