Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Bayesian optimal two-stage design for clinical phase II trials based on Bayes factors

Riko Kelter, Samuel Pawel

TL;DR

The paper tackles the calibration challenge of Bayesian two-stage phase II trials using Bayes factors with binary endpoints. It introduces a trinomial-tree framework to represent interim and final decision branches, derives corrections for power and type-I-error introduced by an interim analysis, and presents practical calibration algorithms to minimize the expected sample size under the null. The authors demonstrate that the proposed design can recover Simon's two-stage design as a special case, yields faster calibration without Monte Carlo simulations, and allows a minimum probability of compelling evidence for the null hypothesis. They illustrate with examples that sequential Bayes-factor designs can reduce expected sample sizes and provide a fully frequentist interpretation under certain priors, while offering greater flexibility through prior choice and evidence thresholds. The approach is generalizable beyond binary endpoints and Bayes factors, offering a practical, reproducible framework for Bayesian adaptive trial designs in clinical research.

Abstract

Sequential trial design is an important statistical approach to increase the efficiency of clinical trials. Bayesian sequential trial design relies primarily on conducting a Monte Carlo simulation under the hypotheses of interest and investigating the resulting design characteristics via Monte Carlo estimates. This approach has several drawbacks, namely that replicating the calibration of a Bayesian design requires repeating a possibly complex Monte Carlo simulation. Furthermore, Monte Carlo standard errors are required to judge the reliability of the simulation. All of this is due to a lack of closed-form or numerical approaches to calibrate a Bayesian design which uses Bayes factors. In this paper, we propose the Bayesian optimal two-stage design for clinical phase II trials based on Bayes factors. The optimal two-stage Bayes factor design is a sequential clinical trial design that is built on the idea of trinomial tree branching, a method we propose to correct the resulting design characteristics for introducing a single interim analysis. We build upon this idea to invent a calibration algorithm which yields the optimal Bayesian design that minimizes the expected sample size under the null hypothesis. Examples show that our design recovers Simon's two-stage optimal design as a special case, improves upon non-sequential Bayesian design based on Bayes factors, and can be calibrated quickly, as it makes use only of standard numerical techniques instead of time-consuming Monte Carlo simulations. Furthermore, the design allows to ensure a minimum probability on compelling evidence in favour of the null hypothesis, which is not possible with other designs. As the idea of trinomial tree branching is neither dependent on the endpoint, nor on the use of Bayes factors, the design can therefore be generalized to other settings, too.

The Bayesian optimal two-stage design for clinical phase II trials based on Bayes factors

TL;DR

The paper tackles the calibration challenge of Bayesian two-stage phase II trials using Bayes factors with binary endpoints. It introduces a trinomial-tree framework to represent interim and final decision branches, derives corrections for power and type-I-error introduced by an interim analysis, and presents practical calibration algorithms to minimize the expected sample size under the null. The authors demonstrate that the proposed design can recover Simon's two-stage design as a special case, yields faster calibration without Monte Carlo simulations, and allows a minimum probability of compelling evidence for the null hypothesis. They illustrate with examples that sequential Bayes-factor designs can reduce expected sample sizes and provide a fully frequentist interpretation under certain priors, while offering greater flexibility through prior choice and evidence thresholds. The approach is generalizable beyond binary endpoints and Bayes factors, offering a practical, reproducible framework for Bayesian adaptive trial designs in clinical research.

Abstract

Sequential trial design is an important statistical approach to increase the efficiency of clinical trials. Bayesian sequential trial design relies primarily on conducting a Monte Carlo simulation under the hypotheses of interest and investigating the resulting design characteristics via Monte Carlo estimates. This approach has several drawbacks, namely that replicating the calibration of a Bayesian design requires repeating a possibly complex Monte Carlo simulation. Furthermore, Monte Carlo standard errors are required to judge the reliability of the simulation. All of this is due to a lack of closed-form or numerical approaches to calibrate a Bayesian design which uses Bayes factors. In this paper, we propose the Bayesian optimal two-stage design for clinical phase II trials based on Bayes factors. The optimal two-stage Bayes factor design is a sequential clinical trial design that is built on the idea of trinomial tree branching, a method we propose to correct the resulting design characteristics for introducing a single interim analysis. We build upon this idea to invent a calibration algorithm which yields the optimal Bayesian design that minimizes the expected sample size under the null hypothesis. Examples show that our design recovers Simon's two-stage optimal design as a special case, improves upon non-sequential Bayesian design based on Bayes factors, and can be calibrated quickly, as it makes use only of standard numerical techniques instead of time-consuming Monte Carlo simulations. Furthermore, the design allows to ensure a minimum probability on compelling evidence in favour of the null hypothesis, which is not possible with other designs. As the idea of trinomial tree branching is neither dependent on the endpoint, nor on the use of Bayes factors, the design can therefore be generalized to other settings, too.

Paper Structure

This paper contains 21 sections, 2 theorems, 23 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Outline
  3. The Bayesian two-stage design for phase II clinical trials with binary endpoints
  4. Operating characteristics
  5. Trinomial tree branching and the problems with power and error rates
  6. Graphical structure of the design: The trinomial tree
  7. Problem I - Calculating the correct power
  8. Problem II - Calculating the type-I-error rate
  9. Solving both problems
  10. Computing the correct power for fixed sample sizes $n_1$ and $n_2$
  11. Computing the correct type-I-error rate for fixed sample sizes $n_1$ and $n_2$
  12. Calibrating the two-stage design
  13. Calibration algorithm
  14. Possible issues with the calibration algorithm
  15. The price of introducing an interim analysis
  16. ...and 6 more sections

Key Result

Theorem 1

Let $n_1$ and $n_2$ be the sample sizes of the interim and final analysis in the Bayesian two-stage design. Let the data $Y_i \mid \theta \sim \mathrm{Bin}(n_i, \theta)$ for $i\in\{1,2\}$ and assume the success probability $\theta$ has a truncated Beta prior, $\theta \sim \mathrm{Beta}(a,b)_{[l,u]}$

Figures (6)

  • Figure 1: Overview of the Bayesian two-stage design for phase IIa clinical trials
  • Figure 2: A trinomial tree illustrating the three outcomes the Bayes factor (or a statistical test, in general) can yield during the interim analysis and the final analysis of the Bayesian two-stage design.
  • Figure 3: A trinomial tree illustrating the three outcomes the Bayes factor (or a statistical test, in general) can yield during the interim analysis and the final analysis of the Bayesian two-stage design. In addition to \ref{['fig:treePower']}, the probabilities of each branch are added.
  • Figure 4: Design characteristics for the first example, showing how oscillations occur due to the discreteness of the binomially distributed data. Vertical line shows the value $n_1=10$ which is the optimal value that, together with $n_2=29$ yields a calibrated sequential Bayes factor design based on frequentist power computed at $p_1=0.3$, and which is optimal in the sense that it minimizes the expected sample size $E[N|H_0]$.
  • Figure 5: Informative $\mathrm{Beta}_{[0.1,1]}(a_d,b_d)$ priors used in the first example. A prior with larger sum of $a_d+b_d$ corresponds to a more informative prior, where $a_d$ can be interpreted as already observed failures and $b_d$ as already observed successes.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 1: Optimal two-stage Bayes factor design
  • proof : Proof of \ref{['theorem:1']}