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arxiv: 2606.02410 · v1 · pith:U7GEZPZ7new · submitted 2026-06-01 · 📊 stat.ME · stat.AP

Optimal sequential two-stage Bayes Factor Design for two-arm clinical Phase II Trials with binary Endpoints

Riko Kelter This is my paper

Pith reviewed 2026-06-28 13:08 UTC · model grok-4.3

classification 📊 stat.ME stat.AP
keywords Bayes factortwo-stage designphase II trialbinary endpointBayesian designfutility interimoperating characteristicssimulation-free
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The pith

Exact correction formulas allow simulation-free calibration of optimal two-stage Bayes factor designs for two-arm phase II trials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a numerical procedure to find the best two-stage design for two-arm trials with binary outcomes that uses Bayes factors for evidence and includes a single futility stop. It starts from exact fixed-sample calculations and adjusts them to account for sample paths eliminated by early stopping. This produces precise values for Bayesian power, type-I error, and the chance of strong evidence for the null without any Monte Carlo runs. The search then picks the design with the smallest expected sample size under the null while meeting pre-set targets on those quantities. The method is demonstrated on standard phase II settings and on a re-analysis of an existing trial.

Core claim

The central claim is that Bayesian power and type-I error for a two-stage two-arm design with a single futility interim equal the corresponding fixed-sample quantities minus the contributions from trajectories removed at the interim, and that these exact corrections enable a fully numerical search over admissible interim and final sample sizes to minimize expected sample size under the null subject to constraints on power, type-I error, and the probability of compelling evidence for the null.

What carries the argument

The correction formulas that adjust fixed-sample Bayes factor operating characteristics for trajectories removed by early stopping at the futility interim.

If this is right

  • The procedure yields designs whose operating characteristics are known exactly rather than estimated with simulation error.
  • The search identifies the admissible design that minimizes expected sample size under the null while satisfying the target constraints.
  • The same numerical calibration applies directly to re-analysis of completed two-arm phase II trials such as the riociguat trial.
  • Sensitivity checks on prior choices or target thresholds become straightforward because each candidate design is evaluated without Monte Carlo variability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The correction approach could be tested on designs that allow early stopping for superiority as well as futility.
  • Extension to three or more stages would require deriving analogous but more involved trajectory corrections.
  • The method might be applied to non-binary endpoints once fixed-sample Bayes factor formulas for those endpoints are available.
  • Trial planners in other disease areas could adopt the same search to reduce average patient exposure under the null hypothesis.

Load-bearing premise

The correction formulas that work for one-arm two-stage designs remain valid when extended to the two-arm binary-endpoint case with one futility interim.

What would settle it

Compute the Bayesian power and type-I error of the derived optimal design both with the exact correction formulas and with an independent Monte Carlo simulation of the same design; mismatch between the two values would falsify the claim.

Figures

Figures reproduced from arXiv: 2606.02410 by Riko Kelter.

Figure 1
Figure 1. Figure 1: Overview of Bayesian power and sample size calculations for the case of a single-arm [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Trinomial tree underlying the two-stage Bayesian Bayes-factor design. At the interim [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Matrix-search procedure for two-arm Bayes factor power calculation ( [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the calibration algorithm for finding an optimal Bayesian two-arm two [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Calibrated one-stage Bayes factor design for the riociguat example. The figure il [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Optimal two-stage Bayes factor design for the riociguat example using mildly in [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Optimal two-stage Bayes factor design for the riociguat example using more informa [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Calibrated one-stage Bayes factor design for the riociguat example. The figure illus [PITH_FULL_IMAGE:figures/full_fig_p033_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Optimal two-stage Bayes factor design for the riociguat example using more informa [PITH_FULL_IMAGE:figures/full_fig_p034_9.png] view at source ↗
read the original abstract

Two-arm phase II clinical trials often benefit from an interim analysis that allows early stopping for futility, but Bayesian calibration of such designs is usually based on computationally intensive Monte Carlo simulation. In this work, a simulation-free methodology is developed to obtain Bayesian optimal two-stage designs in two-arm phase II trials with binary endpoints using Bayes factors as the primary measure of evidence. Building on recent matrix-search methods for fixed-sample two-arm Bayes factor designs and earlier correction formulas for one-arm two-stage designs, the proposed approach derives exact expressions for the operating characteristics of a two-stage two-arm design with a single futility interim. Bayesian power and type-I error are obtained by correcting the corresponding fixed-sample quantities for trajectories that would have been removed by early stopping, yielding a fully numerical calibration procedure that avoids Monte Carlo error entirely. The resulting method searches over admissible interim and final sample sizes to identify the optimal design that satisfies target constraints on Bayesian power, type-I error, and the probability of compelling evidence in favour of the null hypothesis, while minimizing the expected sample size under the null hypothesis. The methodology is illustrated in realistic phase II settings, including a detailed re-analysis of the riociguat trial in systemic sclerosis. Overall, the approach extends simulation-free Bayes factor design methodology to the practically important setting of two-arm two-stage phase II trials and provides a transparent basis for Bayesian design calibration and sensitivity analysis.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript develops a simulation-free methodology for Bayesian optimal two-stage two-arm phase II designs with binary endpoints and a single futility interim, using Bayes factors. It builds on matrix-search methods for fixed-sample two-arm designs and prior one-arm correction formulas to derive exact expressions for operating characteristics by subtracting probabilities of early-stopping trajectories from fixed-sample quantities, yielding a fully numerical calibration procedure that avoids Monte Carlo error. The approach searches admissible interim and final sample sizes to minimize expected sample size under the null subject to constraints on Bayesian power, type-I error, and probability of compelling evidence for the null, and illustrates the method on realistic settings including a re-analysis of the riociguat trial.

Significance. If the central extension holds, the work provides a clear advance by delivering exact, reproducible Bayesian design calibration for two-arm two-stage trials without simulation error. The numerical optimization under explicit constraints on power, type-I error, and null evidence probability, together with the focus on expected sample size under the null, supplies a transparent basis for sensitivity analysis and practical phase II planning that extends prior one-arm and fixed-sample results.

major comments (1)
  1. [Methodology (abstract description and §3)] The load-bearing step is the claim that the one-arm correction formulas extend exactly to the two-arm bivariate binomial setting (abstract, methodology paragraph). In the two-arm case the interim data form a pair of counts whose joint distribution is bivariate, the Bayes factor depends on both arms, and removable trajectories are defined on the product space; any mismatch in enumeration or re-weighting would make the reported Bayesian power and type-I error inexact. The manuscript must supply an explicit derivation or worked numerical example (e.g., in the methods section) showing how the joint-path corrections are constructed from the fixed-sample matrix-search quantities, rather than asserting direct inheritance from the one-arm case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for highlighting the need for greater transparency in the methodological extension. We address the single major comment below and will incorporate the requested material in the revision.

read point-by-point responses

  1. Referee: [Methodology (abstract description and §3)] The load-bearing step is the claim that the one-arm correction formulas extend exactly to the two-arm bivariate binomial setting (abstract, methodology paragraph). In the two-arm case the interim data form a pair of counts whose joint distribution is bivariate, the Bayes factor depends on both arms, and removable trajectories are defined on the product space; any mismatch in enumeration or re-weighting would make the reported Bayesian power and type-I error inexact. The manuscript must supply an explicit derivation or worked numerical example (e.g., in the methods section) showing how the joint-path corrections are constructed from the fixed-sample matrix-search quantities, rather than asserting direct inheritance from the one-arm case.

    Authors: We agree that an explicit derivation and a small-scale numerical illustration are necessary to make the bivariate extension fully transparent. The current manuscript states that exact expressions are obtained by subtracting the probabilities of early-stopping trajectories from the fixed-sample matrix-search quantities, but does not walk through the joint enumeration. In the revised version we will insert a new subsection (likely §3.2) that (i) defines the product-space trajectories for the pair of binomial counts, (ii) shows how the removable set is identified by evaluating the Bayes factor at the interim look, and (iii) provides a fully worked numerical example with n1=5 per arm, n=15 per arm, and explicit probability calculations for the four possible interim outcomes. This addition will confirm that the correction remains exact and that no Monte-Carlo error is introduced. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends prior independent methods without self-referential reduction

full rationale

The paper explicitly builds its two-stage correction formulas on cited prior matrix-search methods for fixed-sample designs and one-arm correction formulas. The central step—subtracting probabilities of early-stopping trajectories from fixed-sample Bayes factor operating characteristics to obtain exact two-arm quantities—is presented as a direct mathematical extension rather than a redefinition or fit of the target quantities themselves. No equation reduces the claimed exact Bayesian power or type-I error to a fitted parameter or self-citation chain by construction; the optimization over admissible sample sizes under explicit constraints is a standard numerical search independent of the paper's own outputs. The extension to bivariate binomial trajectories is the novel content and does not collapse to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; free parameters and axioms cannot be enumerated from the provided text. The method relies on target constraints (Bayesian power, type-I error, probability of compelling evidence for null) and the validity of the cited correction formulas, but no explicit free parameters or invented entities are described.

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discussion (0)

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