DOI: 10.6084/m9.figshare.30836600.v1
title: Supplementary material: Simplifying fractional polynomials in Bayesian network meta-analysis via variable powers

description: <b>These are peer-reviewed supplementary materials for the article</b><b> </b><b>'</b><b>Simplifying fractional polynomials in Bayesian network meta-analysis via variable powers</b><b>'</b><b> </b><b>published in the</b><b> </b><b><i>Journal of Comparative Effectiveness Research</i></b><b>.</b><b>Supplementary Figure 1: </b>Example of two first-order FP survival curves, where β is -3 and p is either 0.1 or -0.1.<b>Formulation STAN model with fixed powers</b><b>Formulation STAN model with variable powers</b><b>Formulation STAN model with variable powers and general population mortality</b><b>Validation</b><b>Supplementary Figure 2: </b>Visualization of the LOOIC results for aNSCLC.<b>Supplementary Figure 3: </b>Visualization of the LOOIC results for mPC.<b>Supplementary Figure 4: </b>Visualization of the LOOIC results for eBC.<b>Supplementary Figure 5:</b> Posterior distribution of the powers, showing the aNSCLC dataset results<b>Supplementary Figure 6: </b>Posterior distribution of the powers, showing the mPC dataset results<b>Supplementary Figure 7: </b>Posterior distribution of the powers, showing the eBC dataset results<b>Aim: </b>Fractional Polynomial (FP) models are widely used in survival analysis for health technology assessment and network meta-analysis (NMA). However, current implementations rely on a fixed set of pre-specified powers, which may constrain model flexibility, limit predictive performance and increase computational cost in Bayesian settings. This study introduces and evaluates a Bayesian FP modeling approach in which the powers are estimated as continuous parameters rather than fixed, aiming to simplify model selection and improve fit. <b>Materials & methods:</b> Second-order Bayesian FP models were implemented in STAN, allowing the time transformation powers (p1, p2) to be estimated from the data. Model performance was evaluated across three oncology NMA datasets; in advanced non-small cell lung cancer, metastatic prostate cancer and early breast cancer. The performance was assessed using visual fit, leave-one-out-information-criteria, root mean square error, incremental survival estimates and computational efficiency. Validation steps included posterior predictive checks, sensitivity analyses and long-term extrapolation. <b>Results:</b> Across all datasets, variable power models consistently achieved better statistical fit (lower leave-one-out-information-criteria and root mean square error) than fixed power models. Incremental survival estimates were also more stable and clinically plausible, particularly in datasets with complex hazard dynamics. While variable models required slightly more time per run, the approach greatly reduced the number of required model configurations, leading to lower overall computational burden. <b>Conclusion:</b> Bayesian FP models with variable powers not only improve model fit and simplify model selection but also reduce structural uncertainty by replacing exhaustive grid searches with a unified, data-driven estimation of transformation powers, while retaining interpretability and computational efficiency. By producing robust, well-calibrated survival projections and streamlining model selection, this approach strengthens survival analysis for health technology assessment and supports more reliable decision-making in comparative effectiveness research.