Bayesian Simultaneous Credible Bands for Polynomial Regression

2026-06-26

AI Summary

This study addresses the challenge of quantifying uncertainty across the entire dose–response curve and integrating prior knowledge in dose–response studies. The authors propose a unified Bayesian framework that constructs Bayesian simultaneous credible bands (BSCBs) within a univariate polynomial regression model, achieving the first computationally efficient and practical implementation of such bands. By combining analytical derivations with posterior sampling, the method accurately determines critical constants. Under mild regularity conditions, the proposed BSCBs are asymptotically equivalent to frequentist simultaneous confidence bands, thereby offering both Bayesian flexibility and guaranteed frequentist coverage. Simulation studies demonstrate that the BSCBs attain accurate simultaneous posterior coverage across diverse scenarios and successfully identify the minimum effective dose in Phase II clinical trials.

Abstract

Quantifying efficacy uncertainty across the entire dose range is crucial in dose-response studies. Although the frequentist simultaneous confidence band (FSCB) is widely used for this purpose, it does not readily incorporate prior knowledge. The Bayesian simultaneous credible band (BSCB) offers a natural alternative, yet practical methods for constructing BSCBs remain scarce in the literature. In this paper, we propose a unified framework for constructing a BSCB for the regression curve in a univariate polynomial model over a finite covariate interval. An efficient simulation-based procedure is developed to determine the critical constant of a BSCB. The framework accommodates inference under different levels of prior information and can be implemented either analytically or via posterior sampling methods. Notably, we prove that under mild regularity conditions, the BSCB is asymptotically equivalent to the FSCB, thereby attaining the nominal frequentist coverage for a broad class of priors. Simulation studies confirm that the BSCB attains the exact posterior simultaneous coverage probability across various scenarios. An application to a dose-response study illustrates its importance in identifying the minimum effective dose in Phase II clinical trials. Software implementation of the proposed methods is available in an accompanying R package.

Problem

• Bayesian simultaneous credible band
• dose-response study
• polynomial regression
• uncertainty quantification
• simultaneous inference

Innovation

• Bayesian simultaneous credible band
• polynomial regression
• dose-response modeling
• asymptotic equivalence
• posterior coverage probability