Online Learning and Inference for Cox Proportional Hazards Model Using Renewable Sieve Estimation

Online learning for the Cox model is challenging because its partial likelihood is non-decomposable, with each risk set requiring a summation over all samples. We propose Collaborative Operation of Linked Survival Analysis (COLSA), an online learning framework that replaces the partial likelihood with the full likelihood using sieve approximation of the baseline hazard. The proposed likelihood function is decomposable and eliminates the need to store historical data in memory, enabling efficient online updates. Moreover, COLSA maintains sufficient statistics for a higher-order basis and employs data-driven basis projection to adaptively scale model complexity to the effective sample size. Unlike existing online Cox methods, COLSA achieves asymptotic normality and attains the same statistical efficiency as the pooled-data partial likelihood estimator, without accessing full data and only requiring constant memory. Simulation studies and application to kidney transplant data demonstrate that COLSA outperforms existing online methods and matches the performance of full-data estimation.

Modern healthcare and scientific studies often collect survival data that contains information about how long it takes for an event such as death, disease recurrence, or equipment failure to occur. A major challenge in analyzing this type of data is that the standard statistical approach relies on a non-decomposable loss function, that is, evaluating one patient’s risk requires comparing that patient against all previously observed individuals. As a result, traditional methods typically require storing the entire historical dataset, making them difficult to scale to streaming or privacy-sensitive settings. An additional challenge is that the baseline hazard function, which captures how risk changes over time, may become increasingly complex as more data are collected. To accurately model this, the statistical model itself must gradually increase in flexibility and parameter size. Existing online learning methods typically assume a fixed model complexity and therefore struggle to adapt as the parameter space grows over time. We developed COLSA, a new online learning framework for survival analysis that addresses both challenges simultaneously. COLSA avoids storing historical patient data by maintaining only compact summary statistics, while dynamically expanding the model complexity as additional data arrive. Despite operating in a streaming setting, COLSA achieves nearly the same statistical accuracy and reliability as methods that require full access to all past data. We validated COLSA using large-scale simulations and two real-world data examples, demonstrating accurate and privacy-conscious survival analysis for streaming data.