Adaptive spline-based weighting functions for blended survival curve extrapolation

Abstract

Health technology assessment frequently requires survival predictions well beyond the observed trial follow-up, yet single parametric models fitted to short horizons can accumulate long-term bias. Current guidance also encourages the principled use of external evidence, such as registry summaries or expert anchors, while maintaining fidelity to the trial data. We introduce an adaptive spline-weighted blended extrapolation on the cumulative-hazard scale that unifies these aims. The observation side is fitted on the trial window using a piecewise-exponential Cox-type model with a smooth prior-driven continuation beyond the administrative cutoff, implemented via INLA. The external side is an anchored Gompertz tail identified by a prespecified survival level at a clinically relevant time. To blend the two components, we compare their cumulative hazards, learn a monotone P-spline score over time, and pass it through a logistic link to obtain a data-driven, time-varying weight. A simple “temperature” scaling controls the slope of this weight and provides a practical guarantee of non-negative blended hazards on a chosen grid, preserving adaptivity while ensuring feasibility. Across Monte Carlo scenarios spanning multiple tail shapes and censoring levels, the method delivers consistently lower absolute survival error, smaller restricted mean survival time error, and improved stability compared with fixed-schedule blending and single-family parametric models. In a SEER registry study with three-year observation and ten-year extrapolation, the blended curve tracks the Kaplan–Meier estimates within follow-up and transitions smoothly toward the anchored tail, yielding small long-horizon errors across cancer sites and age strata. The framework is modular, interpretable, and easily extended to alternative tails and multiple anchors. An open-source implementation is available in the R package survblendr at https://github.com/haohaostats/survblendr.