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Bayesian power and type-I error for Bayes-factor phase II designs can be computed exactly, with no simulation, by solving for a critical response count.

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2026-07-31 23:03 UTC pith:BKEXAMXF

load-bearing objection Solid tutorial, not a methods paper: useful examples and code, but the monotonicity assumption behind the central root-finding step needs to be stated and the 80/90 hybrid typo fixed. the 2 major comments →

arxiv 2607.24084 v1 pith:BKEXAMXF submitted 2026-07-27 stat.ME stat.AP

Simulation-Free Bayesian Power and Sample Size Calculations for Bayes Factors in Single-Arm Phase II Trials with Binary Endpoints

Riko Kelter , Kathrin M\"ollenhoff This is my paper
classification stat.ME stat.AP MSC 62F1562C1062P10
keywords Bayes factorphase II trialbinary endpointsample size calculationBayesian powertype-I errorpredictive probabilitysimulation-free calibration
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Single-arm phase II trials with binary endpoints usually get their sample size from frequentist power calculations or from Monte-Carlo-calibrated Bayesian designs. This paper argues that when the decision rule is a Bayes factor, the entire calibration can be done exactly, without Monte Carlo simulation: Bayesian power, Bayesian type-I error, and the probability of compelling evidence for the null are all sums of prior-predictive probabilities over response counts beyond a critical threshold. The threshold is found by solving BF01(y)=k for the evidence threshold k. The paper demonstrates the workflow on two real oncology trials and shows how design priors and calibration mode (Bayesian, frequentist, hybrid) change the required sample size. The takeaway is that Bayes-factor trial design no longer needs simulation at all.

Core claim

The paper's central claim is that for a single-arm phase II trial with Y ~ Binomial(n, p), a Bayes-factor decision rule BF01(y) < k declaring efficacy is exactly equivalent to observing more than a critical number of responses y_crit, where y_crit is the unique root of BF01(y) - k = 0. Because of this equivalence, the Bayesian power P(BF01<k|H1) = P(y>y_crit|H1), the Bayesian type-I error P(BF01<k|H0) = P(y>y_crit|H0), and the probability of compelling evidence for the null P(BF01>k_f|H0) can each be evaluated by summing closed-form prior-predictive densities f(y|H0) and f(y|H1) over the tail y > y_crit. No Monte Carlo simulation is needed. The paper shows that this root-finding calibration

What carries the argument

The central object is the Bayes factor BF01(y), the ratio of predictive probabilities of observing y responses under the null and alternative hypotheses, together with its monotonicity in y. Under beta priors the predictive densities f(y|H0) and f(y|H1) are available in closed form, so the equation BF01(y) - k = 0 has a unique root y_crit. The root defines the rejection region {y > y_crit}, and all operating characteristics reduce to tail sums of the predictive distributions. The identity BF01(y) < k ⇔ y > y_crit is the load-bearing equivalence that converts design calibration into one-dimensional root-finding followed by summation.

Load-bearing premise

The whole calculation assumes that the Bayes factor BF01(y) is strictly decreasing in the number of responses y, so that BF01(y) < k is exactly equivalent to y > y_crit, and that the closed-form Bayes-factor and predictive-density formulas quoted from earlier work are correct, since they are not derived in this paper.

What would settle it

Compute BF01(y) for every y from 0 to n under the chosen beta priors and a fixed n; if the sequence is not strictly decreasing, the equivalence BF01(y) < k ⇔ y > y_crit fails and the computed power and type-I error are wrong. Also, for a small n, compare the exact tail-sum power with a brute-force Monte Carlo estimate; a mismatch beyond Monte Carlo error would indicate an error in the quoted closed-form expressions.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Bayesian power, Bayesian type-I error, and the probability of compelling evidence for the null can be reported as exact numbers instead of Monte Carlo averages, eliminating Monte Carlo error from design calibration.
  • Sample size determination becomes a one-dimensional search over n: pick the smallest n for which P(y > y_crit | H1) exceeds 1 - beta (and type-I error is below alpha).
  • Design priors under H1 directly control the calibrated sample size: more concentrated H1 priors yield smaller n at similar operating characteristics.
  • Frequentist calibration at a point alternative p1 is a limiting case of Bayesian calibration with an extremely concentrated design prior, making the implicit prior assumptions of classical power calculations explicit.
  • The probability of compelling evidence for the null can be added as a constraint, allowing designs to guarantee that an ineffective treatment is recognised as such with strong evidence.

Where Pith is reading between the lines

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

  • The same root-finding machinery should extend to other one-parameter exponential families wherever the Bayes factor is monotone in the sufficient statistic, suggesting a general simulation-free calibration recipe beyond binomial endpoints.
  • Because the calibration is exact, trialists can in principle exhaustively audit the entire design space of priors, thresholds, and sample sizes, which could support regulatory review of Bayesian designs.
  • The critical-count equivalence could be tested empirically in a re-analysis: compute BF01(y) for all y in an observed trial and verify that the decision boundary is a single threshold; any non-monotonicity would break the method.
  • The paper's contrast between Bayesian and frequentist calibration suggests a practical diagnostic: report the effective prior sample size implied by a frequentist power calculation, so investigators see how strong the prior assumption is.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper presents a tutorial on simulation-free Bayesian power and sample-size calculations for single-arm Phase II trials with binary endpoints, using Bayes-factor decision rules. The central procedure, summarized in §1.3 and Figure 1, is to find the critical response count y_crit by solving BF01(y) - k = 0 and then to compute Bayesian and frequentist power, type-I error, and probability of compelling evidence for the null by summing prior-predictive (or binomial) probabilities over the region y > y_crit. Two worked examples (DESTINY-Gastric02 and LCMC3) compare Bayesian, frequentist, and hybrid calibration modes, with implementations in the bfbin2arm R package and code provided on OSF.

Significance. If the computational claims are correct, the paper offers a useful practical bridge between theoretical Bayes-factor power work and clinical trial planning. Its strengths include open-source R software, reproducible OSF code, clearly worked oncology examples, and a helpful pedagogical framing of the differences between Bayesian and frequentist calibration. The value is primarily expository and software-oriented: the underlying formulas and root-finding device are attributed to prior (unpublished) work by the same authors, and the tutorial does not add new methodology. However, the central equivalence BF01(y)<k iff y>y_crit is asserted without proof or a statement of the required monotonicity condition. That gap is load-bearing, so the paper should not be accepted until it is closed.

major comments (2)
  1. [Section 1.3, Eqs. (2)-(3), Figures 1-2] The entire algorithm rests on the equivalence BF01(y)<k ⇔ y>y_crit, where y_crit solves BF01(y)-k=0. This equivalence is valid only if BF01(y) is strictly decreasing in y, or otherwise the set {y: BF01(y)<k} is not a single upper tail. The manuscript nowhere states or proves this monotonicity. For beta-binomial predictive ratios under truncated-beta priors, monotonicity is not automatic; it requires a likelihood-ratio ordering condition on the priors. If the Bayes factor crosses the threshold more than once, the tail sums in (2)-(3) and all results in Tables 3 and 5 are incorrect. The same issue affects the CE(k0) events in §3.3 and PCE(H0), where BF01(y)>k0 must correspond to a lower tail. Please add an explicit theorem or assumption giving conditions under which monotonicity holds, and verify it numerically across the entire n search grid used for calibration.
  2. [Section 2 (and Section 1.3)] The paper refers to Kelter & Pawel (2025b) for the closed-form Bayes-factor and prior-predictive expressions but does not reproduce them. Since the tutorial's central claim is that these quantities can be evaluated without simulation, and since the monotonicity issue above cannot be checked without the formula, the key expressions for f(y|H0), f(y|H1), and BF01(y) under truncated beta priors should be included in an appendix or stated explicitly. This is not merely a citation preference: a reader of this paper alone cannot verify that the root-finding calibration is correct, and the R package's behavior is not transparent without these formulas.
minor comments (5)
  1. [Section 4.1.5, Table 3, Appendix A code] The text says hybrid calibration 'enforces 80% Bayesian power,' but Table 3 reports Bayesian power 0.901 and the Appendix code uses target_power = 0.90. This inconsistency should be corrected (likely to 90%).
  2. [Figure 2] The threshold y_crit = 37.92274 is non-integer; the text should state explicitly that the event is Y ≥ 38, to avoid ambiguity in equations (2)-(3).
  3. [Sections 2.3 vs 3.3] The threshold for compelling evidence for H0 is denoted k0 in §3.3 but k_f or k_ce elsewhere. Unify notation for consistency.
  4. [Table 3 footnote] The footnote for the frequentist calibration row is confusing: design priors are said to be irrelevant for frequentist calibration, yet the note discusses a trunc.Beta(2250,2750) prior. Reword to distinguish the computational irrelevance of design priors from the interpretive argument that point-mass power calculations correspond to an extreme Bayesian design prior.
  5. [Figure 5 caption] The caption almost duplicates Figure 4's caption and should instead emphasize the added PCE(H0) constraint and its effect (or non-effect) on the selected sample size.

Circularity Check

1 steps flagged

Load-bearing self-citation for the closed-form Bayes-factor machinery; no circular fitting in the worked examples.

specific steps
  1. self citation load bearing [Section 1.3, Figure 1, Eqs. (2)-(3); also Section 2.2]
    "Kelter and Pawel (2025b) derived Bayes factors and the prior-predictive densities f(y|H0) and f(y|H1) (step 3. and 4. in Figure 1) for point-null and composite hypotheses in the binomial setting and showed that Bayesian power, Bayesian type-I-error, and related operating characteristics can be computed numerically without simulation"

    The paper's simulation-free power/type-I-error calculation depends on closed-form BF01(y) and predictive densities f(y|H0), f(y|H1), and on the threshold equivalence BF01(y)<k <=> y>y_crit. None of these are derived or proved here; they are imported from Kelter & Pawel (2025b), an arXiv preprint by the same authors that the present draft itself does not verify. Every numerical design in Tables 3 and 5 therefore rests, at the base of the derivation chain, on accepting the authors' own prior work as an input. This is load-bearing self-citation rather than an independent derivation. It is not a fitted-input-called-prediction: the examples fix priors and thresholds a priori and compute exact tail sums, so no result is reverse-engineered from the reported targets.

full rationale

No step in the worked examples fits a parameter to the reported power/type-I numbers; the priors and thresholds are specified in advance and the operating characteristics are exact sums over the event {BF01<k}. The tutorial's content is therefore not circular in the sense of Eq. X = Eq. Y by construction. The main circularity-adjacent weakness is that the closed-form Bayes factors, predictive densities, and the monotonicity justifying y>y_crit are all taken from an unpublished self-citation (Kelter & Pawel 2025b) rather than proved or reproduced here. I also flag, as a correctness risk rather than a circular step, the unstated monotonicity assumption: Section 1.3 asserts 'the relationship BF01(y)<k <=> y>ycrit holds' without proving BF01 is strictly decreasing; if that fails, equations (2)-(3) are not valid tail probabilities. This does not make the paper circular, but it makes the self-cited foundation more load-bearing. Score 4 reflects substantial self-citation; no prediction reduces by construction, so the score is not 6 or higher.

Axiom & Free-Parameter Ledger

8 free parameters · 4 axioms · 0 invented entities

No new entities are introduced; the method relies on standard beta-binomial Bayes factor machinery and on prior parameter choices.

free parameters (8)
  • H0 design prior (DESTINY) = truncated Beta(4,16) on [0,0.27]
    Chosen to reflect prior mean ~0.20; affects type-I error and sample size.
  • H1 design prior (DESTINY baseline) = truncated Beta(9,11) on (0.27,1]
    Chosen to center at 0.45; yields n=217 under Bayesian calibration.
  • H1 design prior (DESTINY concentrated) = truncated Beta(24,30) on (0.27,1]
    Same mean, higher concentration; reduces n to 125.
  • H0 design prior (LCMC3) = unspecified truncated beta on [0,0.15]
    Used to reflect null MPR rate at or below 0.15; exact parameters not given.
  • H1 design prior (LCMC3 baseline) = truncated Beta(5,15) on (0.15,1]
    Mean 0.25; yields n=295 under Bayesian calibration.
  • H1 design prior (LCMC3 concentrated) = truncated Beta(40,120) on (0.15,1]
    Same mean, tighter; yields n=175.
  • Evidence threshold k = 1/10 or 1/3
    Defines strong or moderate evidence for efficacy.
  • Futility threshold k_f = 3
    Defines compelling evidence for null.
axioms (4)
  • domain assumption Closed-form Bayes factor for binomial data under truncated beta priors (Kelter & Pawel 2025b)
    Section 2.2 and Figure 1 steps 3-4 rely on these formulas without derivation.
  • domain assumption BF01(y) is strictly decreasing in y, so the event BF01(y)<k is equivalent to y > y_crit
    Section 1.3 Step 6 and Figure 2 assume this monotonicity; no proof or reference is given.
  • domain assumption Predictive densities f(y|H0) and f(y|H1) are available in closed form for truncated-beta priors
    Section 1.3 Step 3; used for sums in Step 7.
  • domain assumption Design priors affect operating characteristics but not the Bayes factor computed at trial analysis
    Section 1.3 Step 1 vs 2; standard Bayesian design assumption.

pith-pipeline@v1.3.0-alltime-deepseek · 15853 in / 15103 out tokens · 121251 ms · 2026-07-31T23:03:36.966596+00:00 · methodology

0 comments
read the original abstract

Bayes factors provide a coherent Bayesian measure of evidence for competing hypotheses and have recently been used as the basis for single-arm phase II trial designs with binary endpoints. In contrast to classical power analyses based on test statistics and p-values, Bayes-factor based sample size calculations target high probabilities of obtaining compelling evidence either for a relevant treatment effect or for the null hypothesis, given pre-specified Bayes-factor thresholds. This paper explains how to design single-arm phase II binomial trials using Bayes factors with a focus on simulation-free calibration of Bayesian and frequentist power, type-I-error, and the probability of compelling evidence for the null. Two oncology-motivated examples illustrate the approach and are implemented in the bfbin2arm R package, with code provided in an appendix. The methodology fits naturally into current efforts to innovate and modernize clinical trial design through Bayesian and adaptive methods.

Figures

Figures reproduced from arXiv: 2607.24084 by Kathrin M\"ollenhoff, 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_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Relationship between the number of successes [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bayesian calibration for DESTINY-Gastric02 with baseline design priors under [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bayesian calibration for DESTINY-Gastric02 with a more concentrated design prior [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Bayesian calibration for DESTINY-Gastric02 with a more concentrated design prior [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Frequentist calibration for DESTINY-Gastric02 using the baseline design priors. The [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Hybrid calibration for DESTINY-Gastric02 with baseline design priors. The design [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Hybrid calibration for DESTINY-Gastric02 with extremely informative design prior [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

discussion (0)

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