Power and Sample Size Calculations for Bayes Factors in two-arm clinical Phase II Trials with binary Endpoints

TL;DR

The approach allows for a Bayes-frequentist compromise by providing a Bayesian analogue to a frequentist power analysis for various Bayes factors in the two-arm binomial setting of a phase II clinical trial.

Abstract

Bayesian sample size calculations in clinical trials usually rely on complex Monte Carlo simulations in practice. Obtaining bounds on Bayesian notions of the false-positive rate and power often lack closed-form or approximate numerical solutions. In this paper, we focus on power and sample size calculations for Bayes factors in the two-arm binomial setting of phase II trials. We cover point-null versus composite and directional hypothesis tests, derive the corresponding Bayes factors, and discuss relevant aspects to consider when pursuing Bayesian design of experiments with the introduced approach. Based on these Bayes factors, we propose a numerical approach which allows to determine the necessary sample size to obtain prespecified bounds of Bayesian power and type-I-error rate in a computationally efficient way. Our method does not rely on Monte Carlo simulations and instead solely relies on standard numerical methods. Real-world examples of phase II trials from oncology and autoimmune diseases illustrate the advantage of the proposed calibration method. In summary, our approach allows for a Bayes-frequentist compromise by providing a Bayesian analogue to a frequentist power analysis for various Bayes factors in the two-arm binomial setting of a phase II clinical trial. The methods are implemented in our R package bfbin2arm.

Figures (10)

• Figure 1: Overview of the methodology underlying Bayesian power and sample size calculations for Bayes factors in the two-arm binomial setting, modified and adapted from the one-arm setting in KelterPawel2025
• Figure 2: Illustration of the computation of critical values for the Bayes factor power calculation, based on $n_1=n_2=5$ and flat analysis priors with $a_i^a=b_i^a=1$ for $i=1,2$.
• Figure 3: Untruncated Beta$(a_1^d,a_2^d)$ design (or analysis) prior (right) and truncated Beta$(a_1^d,a_2^d)$ prior (left) for $a_1^d=2$, $b_1^d=5$, $a_2^d=3$ and $b_2^d=4$, where the truncation is to the set $\iint_{A} \pi_{\text{untr}}(p_1,p_2)\,dp_1\,dp_2 = P(p_2>p_1)$.
• Figure 4: Bayesian power and sample size calculations for the riociguat trial, where $H_0:p_1=p_2$ versus $H_1:p_2>p_1$ is tested. Flat design and analysis priors are used with moderate evidence thresholds $k=1/3$, $k_f=3$. The calibration shows the results for 80% power, 5% type-I-error rate and 80% probability of compelling evidence under $H_0$. Frequentist power was obtained under $p_1=0.4$ and $p_2=0.6$.
• Figure 5: Bayesian power and sample size calculations for the riociguat trial, where $H_0:p_1=p_2$ versus $H_1:p_2>p_1$ is tested. Informative $B(1,2)$ and $B(2,1)$ design and flat analysis priors are used with strong evidence threshold $k=1/10$ and evidence threshold for compelling evidence under $H_0$ of $k_f=3$. The calibration shows the results for 80% power, 5% type-I-error rate and 80% probability of compelling evidence under $H_0$. Frequentist power was obtained under $p_1=0.4$ and $p_2=0.6$.

Theorems & Definitions (5)

• proof : Prior-predictive under $H_1:\eta \neq 0$
• proof : Prior-predictive under $H_0:\eta=0$
• proof : Finite-sum form for $C$ (integer shapes)
• proof : Predictive density under $H_+:\eta >0$
• proof : Finite-sum form for $I(y_1,y_2)$ (integer shapes)