Principles in harmony: Closed testing meets the partitioning principle for computational efficiency

Liane Kluge; Werner Brannath

Principles in harmony: Closed testing meets the partitioning principle for computational efficiency

T0 review · 0 major / 5 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read Closed testing and partitioning share the same projection algorithms once FWER control is read as a simultaneous interval for binary truth indicators.

desk verdict Solid unification of closed testing and partitioning that yields usable projection algorithms for informative SCIs; the math checks out and the open problems are honestly flagged. read the letter →

arxiv 2607.04707 v1 pith:I7UPKNUM submitted 2026-07-06 stat.ME stat.CO

Werner Brannath , Liane Kluge This is my paper · ORCID

classification stat.MEstat.CO MSC 62J1562F25

keywords closedtestingpartitioningprincipleFWERsimultaneousconfidenceintervalsconsonanceshort-cutalgorithmsweightedBonferroniclinicaltrials

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that any multiple test controlling the family-wise error rate is formally identical to a one-sided simultaneous confidence interval for a binary vector that simply marks which nulls are true. Because of that identity, the closed-testing and partitioning principles use exactly the same computational projection step. The authors therefore transplant the classical idea of consonance from closed tests into the partitioning setting, introducing weaker and stronger versions that still guarantee correct projection. Under those properties they obtain linear-cost algorithms that compute the lower simultaneous confidence bounds for both discrete and continuous parameters, recovering and extending known constructions for weighted Bonferroni and continuously weighted Holm procedures. The practical payoff is that informative simultaneous intervals become as easy to compute as the familiar step-down multiple tests that inspired them.

What carries the argument

Weak (uniform) consonance: rejection of an intersection at θ (or δ) forces the existence of a coordinate direction i such that the entire “half-space” G(i)θ or D(i)θ is also rejected; this property licenses the coordinate-wise stepping of Algorithms 1–3.

What would settle it

Exhibit a continuous weighted-Bonferroni family whose weights violate the stated monotonicity, then check whether Algorithm 3 still produces lower bounds that stay below the true projection L and converge to it; any systematic overshoot falsifies the claim.

Watch

Extended reading notes

Core claim

A multiple test with strong FWER control is precisely a one-sided simultaneous confidence interval for the binary indicator vector δ of true versus false nulls; consequently the closed-testing and partitioning principles share identical projection algorithms, and the newly defined weak (uniform) consonance properties are sufficient for linear-cost computation of the projected SCI bounds.

Load-bearing premise

The local tests and their weights must keep the monotonicity that makes weak uniform-consonance hold at every point and in every coordinate; without it the stepping rules lose their correctness guarantees.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The punchline is straightforward: once you treat an FWER-controlling multiple test as a one-sided SCI for the binary indicator vector δ of true/false nulls, closed testing and the partitioning principle become the same projection problem. Brannath and Kluge then introduce weak and weak-uniform consonance, prove they are enough for linear-cost algorithms, and give three concrete projection procedures (discrete, continuous grid+bisection, and the monotone weighted-Bonferroni iteration). That is the real contribution.

What works well is the formal hygiene. The equivalence of (5) and (7) is immediate from the definitions; Lemmas 1 and 4 establish correctness of the discrete and continuous grid algorithms by induction; Theorem 1 plus Lemmas 2–3 give an explicit maximizer for the continuously weighted Holm case; and Theorem 2 settles monotonicity and uniqueness of the fixed-point for Algorithm 3 under the usual continuity and monotonicity assumptions on p-values and weights. The continuously weighted Holm example is cleanly worked out, and the initialization arguments for unbounded parameters are practical. Self-citations to their earlier SCI papers supply concrete weight functions but are not load-bearing for the new consonance notions.

Soft spots are minor and mostly already acknowledged. The algorithms require the monotonicity conditions that guarantee weak uniform-consonance; without them the coordinate-wise updates lose their guarantees. Mixed discrete/continuous parameters, gatekeeping, and restricted parameter spaces (all-pairwise comparisons) are left open, which is honest rather than evasive. There are no numerical experiments, but this is a methods paper whose claims are mathematical, not empirical.

This is for people who already care about informative simultaneous confidence intervals and graphical or weighted closed tests in clinical-trial settings. The reader who wants a clean algorithmic bridge between two classical principles and ready-to-implement projection routines will get value. The mathematics is self-contained once the classical definitions are granted, the citation pattern is appropriate, and nothing looks circular. I would send it to peer review without hesitation; a serious referee will find it solid methodological progress for a well-defined niche.

Referee Report

0 major / 5 minor

Summary. The paper establishes a formal equivalence between FWER-controlling multiple tests and one-sided simultaneous confidence intervals for the binary indicator vector δ of true/false nulls (eqs. 2–3). From this it shows that the closed-testing and partitioning principles share the same projection algorithms (eqs. 5 and 7). It then introduces weaker notions of consonance (weak consonance, Definitions 1–2; weak uniform-consonance, Definition 3) that are sufficient for linear-cost projection algorithms computing the SCI bounds for discrete parameters (Algorithm 1) and continuous parameters (Algorithms 2–3). The continuously weighted Holm procedure is shown to satisfy weak uniform-consonance via an explicit maximizer argument (Theorem 1, Lemmas 2–3), and convergence of the coordinate-wise updates of Algorithm 3 is proved under standard monotonicity and continuity assumptions on weights and p-values (Theorem 2). The development is illustrated by simple examples that recover and extend earlier informative-SCI constructions.

Significance. If the results hold, the paper supplies a clean conceptual unification of two classical multiple-inference principles and, more importantly, a set of practically usable algorithms for computing simultaneous confidence intervals that inherit the structure of familiar closed tests (weighted Bonferroni/Holm, graphical procedures). The new weak-(uniform)-consonance notions are weaker than classical consonance and therefore easier to verify, yet still guarantee correctness of the projection steps (Lemmas 1 and 4). The continuous-parameter algorithms improve on earlier work by the same authors by providing both lower and upper approximations with a pre-specified precision. The proofs are self-contained once the classical definitions are granted, and the monotonicity conditions required for weak uniform-consonance are standard for the weighted-Bonferroni family. This is a solid methodological contribution for clinical-trial and multiple-testing practice.

minor comments (5)

The abstract and introduction repeatedly use the future tense (“We will then utilise…”). Convert to present tense for consistency with the rest of the manuscript.
Figure 1 caption and surrounding text refer to “red squares” and “blue ellipses/circles”; ensure the published figure actually uses those colours (or replace colour references by shape/line-style descriptions).
Section 4.3 and the initialization paragraphs of Algorithms 2–3 leave the construction of a data-driven starting vector θ0 for unbounded parameters as a case-by-case exercise. A short remark that the weighted-Bonferroni family admits a simple construction (already given later) would help the reader.
A few typographical slips remain: “inlnon-increasing” (p. 5), “Gird Traversal” (Appendix C), and occasional missing spaces after punctuation. A careful proof-reading pass is recommended.
The discussion of gatekeeping and mixed discrete/continuous parameters (Section 6.1 and final section) correctly flags open questions; a one-sentence pointer to the concrete algorithms already available for graphical tests (Brannath et al. 2026, Kluge & Brannath 2026) would orient the reader more clearly.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the formal equivalence of FWER tests to SCIs for δ and the weak-consonance algorithms are self-contained once classical definitions are granted; self-citations supply examples, not load-bearing uniqueness.

full rationale

The paper's central chain is definitional and algorithmic, not predictive. Equations (2)–(3) simply rewrite strong FWER control as coverage for the binary indicator δ; equations (5) and (7) then make the projection steps of closed testing and partitioning identical by construction. Weak (uniform) consonance is introduced as a new, weaker sufficient condition (Definitions 1–3) and is proved to guarantee correctness of Algorithms 1–3 via Lemmas 1 and 4 and Theorems 1–2; the proofs rely only on the monotonicity of the weights and p-values that the paper itself assumes for the weighted-Bonferroni family. Self-citations to Brannath & Schmidt (2014), Schmidt & Brannath (2014, 2015), Brannath et al. (2026) and Kluge & Brannath (2026) appear only as concrete instances of weight functions that already satisfy those monotonicity conditions; they are not invoked as uniqueness theorems that force the present constructions. Consequently the new algorithms properly generalize the earlier ones rather than reducing to them by algebraic identity. Score 1 reflects a single non-load-bearing self-citation pattern that is normal and expected.

Assumptions & free parameters 0 free parameters · 4 assumptions · 3 invented entities

The work rests on standard multiple-testing axioms (FWER, free hypotheses, local level-α tests) plus the monotonicity and continuity conditions needed for the new consonance properties. No numerical free parameters are fitted; the invented entities are the weakened consonance notions and the projection algorithms themselves.

assumptions (4)

domain assumption Parameters are variationally independent (free hypotheses): every combination of component-wise values is possible, so intersection hypotheses are non-empty.
Stated in Section 2; required for the natural partition (Θ_δ) to cover the whole parameter space.
domain assumption Local tests ψ_θ satisfy P_θ(ψ_θ=1)≤α for every θ (or the corresponding supremum over the partition cell).
Definition of a valid partitioning procedure (eq. 6); inherited from Finner & Strassburger (2002).
domain assumption For the weighted Bonferroni family, each weight w_i(θ) is non-decreasing in θ_j (j≠i) and the marginal p-values are continuous and strictly increasing.
Used throughout Sections 4–6 and Appendices B–D to establish weak (uniform) consonance and convergence of the algorithms.
domain assumption An initial lower bound vector λ_0≤L can be obtained (either by boundedness of the parameter space or by a data-dependent initialization that uses a positive lower bound on the weights).
Required for Algorithms 1–3 to start; discussed in Sections 4.3 and 6.

invented entities (3)

weak consonance (Definition 1 / 2) independent evidence
purpose: Sufficient condition guaranteeing that the discrete projection algorithm advances correctly without enumerating the full power set.
New weakening of Gabriel’s classical consonance; independent evidence is the correctness proof of Algorithm 1.
weak uniform-consonance (Definition 3) independent evidence
purpose: Continuous-parameter analogue that prevents the algorithm from jumping over the true lower bound L_i.
Introduced to handle uncountable parameter spaces; verified for the continuous weighted Holm family.
Projection Algorithms 1–3 independent evidence
purpose: Linear-cost (or bisection-refined) computation of the projected simultaneous confidence bounds under the new consonance notions.
Core algorithmic contribution; correctness proved in the appendices.

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Pith. "Pith review of Principles in harmony: Closed testing meets the partitioning principle for computational efficiency." pith.science (2026). https://pith.science/paper/I7UPKNUM

@misc{pith2026260704707,
  author       = {Pith},
  title        = {Pith review of: Principles in harmony: Closed testing meets the partitioning principle for computational efficiency},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I7UPKNUM}},
  note         = {Machine review of arXiv:2607.04707}
}

read the original abstract

We explore and utilize the algorithmic relationship between the closed testing principle for multiple tests with family-wise error rate (FWER) control and the partitioning principle for the construction of simultaneous confidence intervals. Starting with the simple observation that a multiple test with FWER control is formally equivalent to a one-sided simultaneous confidence interval for the vector of binary parameter indicating whether the null or alternative hypothesis is true, we show that the closed testing and partitioning principles follow the same computational approach. We will then utilise this relationship to extend concepts of consonance for closed tests to the partitioning principle, with the aim of deriving computationally feasible and efficient algorithms for the calculation of simultaneous confidence intervals. We will also utilize the relationship between closed testing and partitioning principle to extend common closed testing procedures to simultaneous confidence intervals, referencing the existing literature on informative simultaneous confidence intervals. The relationships and extensions will be illustrated by simple, instructive examples.

Figures

Figures reproduced from arXiv: 2607.04707 by the authors.

**Figure 2.** Consonance of closed test (left) and weak consonance for partitioning principle (right). [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Projection algorithm for discrete parameter [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Grid Traversal (Step 2) of projection algorithm for continuous parameter [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.

[1] [1]

Statistical Methods in Medical Research , volume=

Informative simultaneous confidence intervals for graphical test procedures , author=. Statistical Methods in Medical Research , volume=. doi:10.1177/09622802251393666 , year=

work page doi:10.1177/09622802251393666

[2] [2]

and Bretz, F

Strassburger, K. and Bretz, F. , Journal =. Compatible simultaneous lower confidence bounds for the. 2008 , Number =. doi:10.1002/sim.3338 , Fjournal =

doi:10.1002/sim.3338 2008

[3] [3]

Statistics in Biopharmaceutical Research , volume=

Multiple testing in group sequential trials using graphical approaches , author=. Statistics in Biopharmaceutical Research , volume=. 2013 , publisher=

2013

[4] [4]

Statistics in Medicine , volume=

Powerful short-cuts for multiple testing procedures with special reference to gatekeeping strategies , author=. Statistics in Medicine , volume=. 2007 , publisher=

2007

[5] [5]

Biometrika , volume=

On closed testing procedures with special reference to ordered analysis of variance , author=. Biometrika , volume=. 1976 , publisher=

1976

[6] [6]

Biometrical Journal , volume=

General solutions to multiple testing problems , author=. Biometrical Journal , volume=. 2008 , publisher=

2008

[7] [7]

, author=

Allgemeine Lösungen multipler Testprobleme. , author=. EDV in Medizin und Biologie , volume=

[8] [8]

Annals of statistics , pages=

The partitioning principle: a powerful tool in multiple decision theory , author=. Annals of statistics , pages=. 2002 , publisher=

2002