Upgrading survival models with CARE

Clinical risk prediction models are regularly updated as new data, often with additional covariates, become available. We propose CARE (Convex Aggregation of relative Risk Estimators) as a general approach for combining existing “external” estimators with a new data set in a time-to-event survival analysis setting. Our method initially employs the new data to fit a flexible family of reproducing kernel estimators via penalised partial likelihood maximisation. The final relative risk estimator is then constructed as a convex combination of the kernel and external estimators, with the convex combination coefficients and regularisation parameters selected using cross-validation. We establish high-probability bounds for the $L_2$-error of our proposed aggregated estimator, showing that it achieves a rate of convergence that is at least as good as both the optimal kernel estimator and the best external model. Empirical results from simulation studies align with the theoretical results, and we illustrate the improvements our methods provide for cardiovascular disease risk modelling. Our methodology is implemented in the Python package care-survival.

💡 Research Summary

The paper introduces CARE (Convex Aggregation of relative Risk Estimators), a unified framework for upgrading clinical time‑to‑event prediction models when new data, possibly containing additional covariates, become available. The authors start by noting that many established risk scores (e.g., SCORE2, QRISK, PREDICT) are often released as fixed sets of hazard ratios derived from a proprietary training cohort. When a new cohort is collected—perhaps with richer measurements—the original training data may be inaccessible, yet the original model can still be queried for predictions. CARE addresses this situation by (1) fitting a flexible relative‑risk function on the new data using reproducing kernel Hilbert space (RKHS) methods, and (2) convexly aggregating this new estimator with one or more pre‑existing “external” estimators.

The RKHS component generalises the Cox proportional‑hazards model: a linear kernel reproduces the classic Cox model, while nonlinear kernels (Gaussian, polynomial, spline, etc.) allow the relative risk to vary in a highly flexible manner. The authors penalise the partial likelihood with a ridge‑type term γ‖f‖²_H, where γ is a regularisation parameter controlling smoothness. The penalised objective is minimised via first‑ and second‑order optimisation (e.g., L‑BFGS), yielding an estimator f̂ that adapts to the size of the new sample and the complexity of the kernel.

Next, CARE forms a convex combination
  f̂_CARE = w₀ f̂ + Σ_{m=1}^M w_m g_m,
where g_m are the external risk estimators and the weights satisfy w_m ≥ 0, Σ w_m = 1. Both the kernel regularisation γ and the convex weights w are selected jointly by maximising a “joint partial‑likelihood cross‑validation” criterion. This procedure splits the data into K folds, computes the penalised partial likelihood on the training folds, and evaluates predictive performance (partial likelihood and C‑index) on the held‑out fold, iterating over a grid of (γ, w) values. The authors prove that this data‑driven selection attains an oracle inequality: the chosen parameters perform almost as well as if the optimal (γ, w) were known in advance.

Theoretical contributions are threefold. Theorem 7 provides a high‑probability bound on the L₂ error of the RKHS estimator, explicitly involving the sample size n, the regularisation γ, and the eigenvalue decay of the kernel integral operator. Theorem 10 establishes that the cross‑validation scheme yields an estimator whose risk is within a small multiplicative factor of the oracle risk. Theorem 11, the central result, shows that the CARE estimator’s risk is bounded by the minimum of (i) the risk of the best external model and (ii) the risk of the optimally tuned RKHS estimator, up to a negligible excess term. Consequently, CARE automatically adapts: when the new cohort is small or noisy, the weights shift toward the external model; when the new data are abundant, the RKHS component dominates.

Empirical validation proceeds in two stages. First, extensive simulations vary sample size, signal‑to‑noise ratio, and kernel choice. The simulations confirm the predicted convergence rates and demonstrate that CARE consistently tracks the better of the two information sources. Second, a real‑world application upgrades the SCORE2 cardiovascular risk model using UK Biobank data (≈500 k participants) that include additional predictors such as genetic risk scores, detailed lifestyle variables, and novel biomarkers. After applying CARE, the concordance index improves by 1.21 % for women and 2.74 % for men relative to the original SCORE2 predictions, a statistically and clinically meaningful gain.

Implementation is provided in the open‑source Python package care‑survival, which offers a scikit‑learn‑compatible API for kernel selection, regularisation path exploration, joint cross‑validation, and model persistence. The package is designed for scalability (multi‑core parallelism) and ease of integration into existing clinical analytics pipelines.

In summary, CARE delivers a principled, theoretically justified, and practically usable solution for model upgrading in survival analysis. By unifying RKHS‑based flexible risk estimation with convex aggregation of legacy models and a rigorous cross‑validation scheme, it enables clinicians and data scientists to improve risk predictions without needing to re‑access original training data. Future work may extend CARE to competing‑risk settings, Bayesian weight estimation, and online updating for continuously arriving data streams.