1. A Bayesian Design with Conditional Borrowing of Historical Data in a Rare Disease Setting
Peng Sun*
July 30, 2019
*Joint work with Ming-Hui Chen, Yiwei Zhang, John Zhong, Charlie Cao, Guochen Song, and Zhenxun Wang

2. Outline
Introduction
Bayesian design using power prior with conditional borrowing of historical data
Comparison to the Bayesian hierarchical modelling approach
Discussion

3. Introduction (1)
Rare disease setting: An efficacious standard of care (S) is already on the market
A new modality (N) for the treatment of the same disease: Synergistic effect of N and S is expected
Efficacy endpoint: Change from baseline in HINE* total milestones score at Month 10
Despite strong efficacy of S, the gap remains:
With the treatment of S, the observed mean change from baseline in Month 10 HINE total milestones score was 14.9 with a standard deviation of 4.12 (N=15)
For healthy subjects, the expected change from baseline in Month 10 HINE total milestones score is 20
*HINE: Hammersmith Infant Neurological Examination

4. Introduction (2)
Hypothesis of interest: The population mean change from baseline in Month 10 HINE total milestones score is greater for subjects treated with N+S comparing to subjects treated with S alone
Historical data: 15 subjects treated with the standard of care (S)
Study design:
Randomized control trial with a total of 27 subjects and a 2:1 randomization ratio (N+S vs. S)
Conditional borrowing through power prior to augment current control with historical data
Without borrowing historical data, assuming an effect size of 4.5, the study has power 77% power with one-sided alpha of 0.025

5. Bayesian Design Notations

6. Bayesian Design Power Prior and Conditional Borrowing

7. Bayesian Design Simulation Set up
Borrowing region: [13.9, 15.9], which is approximately one stand error around the historical mean of 14.9
a0 ranges from 0.2 to 0.7
Number of simulations: 5,000
Bayesian modeling: 10,000 posterior draws with 3000 as burn-ins
Assumed standard deviation for Month 10 HINE change from baseline: 4.12 for the monotherapy arm (S) and 2.8 for the combination arm
All simulations were conducted in R with the rstan package for Bayesian modeling

8. Bayesian Design Choice of Gamma
Evaluated via numerical integration and simulation under no borrowing assuming a common mean of 15. Hence gamma=0.972 is selected for subsequent simulations.

9. Bayesian Design Type I Error under Fixed Borrowing

10. Bayesian Design Posterior Probability of Control Mean Exceeding a Reference Value Under Historical Data

11. Bayesian Design Type I Error under Conditional Borrowing
a0=0.3 was selected: Maximum inflation is below 5% at a population mean with low posterior probability of exceeding it

12. Bayesian Design Power Gain under Fixed Effect Size of 4.5 (a0=0.3)

13. Bayesian Design Power Gain Assuming Pop
Bayesian Design Power Gain Assuming Pop. Mean Change from Baseline of 19.5 for Combination (a0=0.3)

14. Bayesian Hierarchical Model

15. Comparison of Type I Error
a0=0.3 for conditional borrowing

16. Comparison of Power
a0=0.3 for conditional borrowing

17. Bias of Effect Size Estimate (True Effect Size = 0)
a0=0.3 for conditional borrowing; bias is defined as the deviation of posterior mean estimate from the population parameter

18. Bayesian Design Discussion
With conditional borrowing, the pre-specified borrowing region prevents borrowing if the observed historical data and the current control data drastically differ in efficacy
With conditional borrowing, the Type I error is bounded within the range of population control mean. This is in contrast to the fixed borrowing approach with power prior
The Bayesian estimator of treatment difference based on conditional borrowing is unbiased. This is in contrast to the Bayesian hierarchical modelling approach, where moderate bias* is observed.
* Bias is defined as the deviation of posterior mean estimate from the population parameter

19. Reference
Ibrahim J, Chen M. Power prior distributions for regressive models. Statistical Science. 2000;15
Allocco D et al. A prospective evaluation of the safety and efficacy of the TAXUS Element paclitaxel-eluting coronary stent system for the treatment of de novo coronary artery lesions: Design and statistical methods of the PERSEUS clinical program. Trials. 2010; 11:1
Stan Development Team Stan Modeling Language Users Guide and Reference Manual, Version