arxiv: 2604.20016 · v1 · submitted 2026-04-21 · 📊 stat.ME · math.ST· stat.TH

Recognition: unknown

Weighted Holm Procedures: Theory, Properties, and Recommendations

Beibei Li, Wenge Guo

Authors on Pith no claims yet

Pith reviewed 2026-05-10 01:30 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH

keywords weighted multiple testingHolm procedurefamilywise error rateclosed testing procedureadjusted p-valuespower comparisongraphical representation

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The pith

Ordering weighted p-values first yields a uniformly more powerful Holm procedure than ordering raw p-values while controlling familywise error.

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

This paper compares two common weighted versions of the Holm method for multiple testing when hypotheses carry different importance weights. The weighted Holm procedure orders by weighted p-values and is shown via closed testing to reject at least as many false nulls as the alternative that orders by raw p-values, without inflating the familywise error rate. The authors also supply graphical displays using one starting graph and two updating rules, compute adjusted p-values for both methods, and prove that the weighted version is optimal in the sense that no critical value can be relaxed without breaking error control. Simulations illustrate the power gain in practice.

Core claim

The paper establishes that the weighted Holm procedure (WHP), which orders hypotheses by their weighted p-values, is uniformly more powerful than the weighted alternative Holm procedure (WAP) that orders by raw p-values. This follows from direct comparison of their closed testing procedures. Additional results include that WAP is consonant but not monotone, graphical representations that share an initial graph yet use distinct updating strategies, derivation of both adjusted p-values and adjusted weighted p-values, and an optimality theorem stating that WHP cannot have any critical value enlarged without violating FWER control whereas WAP achieves optimality only under specific conditions.

What carries the argument

Closed testing procedures attached to each weighted Holm variant, which determine rejections by checking adjusted thresholds on all supersets of a given hypothesis.

If this is right

Practitioners should prefer ordering by weighted p-values when weights are available to increase the chance of rejecting important false nulls.
The shared-graph graphical representations allow non-statisticians to follow and verify the rejection steps for either procedure.
Adjusted p-values and adjusted weighted p-values make both methods directly usable in reporting without needing to recompute thresholds.
The optimality result for WHP means its critical values are tight for the class of procedures that use the same weighted p-value ordering.

Where Pith is reading between the lines

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

Incorporating weights at the ordering step rather than only at the threshold step appears to be the source of the power advantage.
Similar ordering comparisons could be applied to other weighted step-down or step-up procedures to check for parallel gains.
In clinical trial design the result suggests assigning higher weights to primary endpoints can be paired with the WHP rule to improve overall study power without extra sample size.

Load-bearing premise

The p-values are valid and uniformly distributed under each null, and the weights are fixed in advance without depending on the observed data.

What would settle it

A concrete counter-example or simulation in which the weighted alternative procedure rejects strictly more hypotheses than the weighted Holm procedure while keeping the familywise error rate at or below the target alpha level.

Figures

Figures reproduced from arXiv: 2604.20016 by Beibei Li, Wenge Guo.

**Figure 1.** Figure 1: Graphical representation of WAP with α = 0.05, weights wi = i for i = 1, 2, 3, and initial allocation α = (α/6, α/3, α/2). Given the raw p-values p1 = 0.01, p2 = 0.014, and p3 = 0.3, no hypotheses are rejected under WAP. Rejected nodes are shown in yellow, and nodes that remain non-rejected at the final stage are shown in red. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

**Figure 2.** Figure 2: Graphical representation of WHP under the same setting as Figure [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: Simulated performance under weight setting [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: Average power under two weight settings: strongly informative (high separation, [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: Average power of WHP, WAP, and Holm under the weight setting [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

read the original abstract

In many statistical applications, particularly in clinical studies, hypotheses may carry different levels of importance, motivating the use of weighted multiple testing procedures (wMTPs) to control the familywise error rate (FWER). Among these approaches, two weighted Holm procedures are commonly used: the weighted Holm procedure (WHP), which is based on ordered weighted $p$-values, and the weighted alternative Holm procedure (WAP), which relies on ordered raw $p$-values. This paper provides a systematic comparison of these two procedures, along with practical recommendations for their use. We first examine their corresponding closed testing procedures (CTPs) and show that WHP is uniformly more powerful than WAP. We further investigate their structural properties, demonstrating that WAP, while consonant, lacks monotonicity. To facilitate communication with non-statisticians, we introduce graphical representations of both procedures using a common initial graph and distinct updating strategies. In addition, we derive adjusted $p$-values and adjusted weighted $p$-values for both methods. Finally, we establish an optimality result: WHP cannot be improved by enlarging any of its critical values without violating FWER control, whereas WAP is optimal only under specific conditions. Simulation studies support these theoretical findings and highlight the superior FWER control and average power of WHP.

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.

Desk Editor's Note private letter to a colleague

WHP is uniformly more powerful than WAP with a stronger optimality result, backed by closed testing arguments and some practical graphs.

read the letter

The main thing to know is that this paper establishes uniform power superiority for the weighted Holm procedure over the weighted alternative Holm procedure, along with an optimality result that holds for WHP but only conditionally for WAP. The comparison rests on their closed testing versions, where the local tests for WHP produce at least as large critical values on every intersection hypothesis, so the rejection sets nest directly. This holds with valid p-values and fixed weights, independent of dependence structure among the tests. They also show WAP is consonant but fails monotonicity, which clarifies when each method can be expected to behave predictably as more data or hypotheses are added. The graphical representations start from the same initial graph but use different updating rules, which should help explain the procedures to non-statisticians. They derive adjusted p-values and adjusted weighted p-values for both, and the simulations are presented as confirming the power and FWER advantages. The optimality claim for WHP—that no critical value can be enlarged without losing FWER control—is a clean theoretical point that strengthens the recommendation to prefer it in practice. The work assumes standard conditions: p-values uniform under the null and weights chosen in advance. That is the usual setup for these methods, but it does mean the results do not cover data-dependent weighting schemes. The simulation details are referenced rather than fully expanded in the abstract, though the theory appears to cover the main cases without obvious gaps. This paper is for people working on weighted multiple testing in clinical trials or similar applications where hypotheses have unequal importance. It deserves a serious referee because the core claims follow from established closed testing properties and the new comparison plus optimality result are clearly stated and supported.

Referee Report

2 major / 3 minor

Summary. The manuscript compares two weighted multiple testing procedures for FWER control: the weighted Holm procedure (WHP), based on ordered weighted p-values, and the weighted alternative Holm procedure (WAP), based on ordered raw p-values. It establishes via their closed testing procedures that WHP is uniformly more powerful than WAP, shows that WAP is consonant but not monotone, introduces graphical representations with a shared initial graph but different updating rules, derives adjusted p-values and adjusted weighted p-values, and proves an optimality result that WHP's critical values cannot be enlarged without violating FWER control while WAP is optimal only under specific conditions. Simulation studies are cited in support of the theoretical findings and practical recommendations.

Significance. If the derivations hold, the work supplies clear, actionable guidance for clinical-trial settings where hypotheses carry unequal importance. The uniform power superiority, optimality characterization, and graphical tools are concrete contributions that can improve procedure selection and communication with non-statisticians.

major comments (2)

[Closed testing procedures] The section deriving the closed testing procedures: the uniform power superiority of WHP over WAP is asserted to follow from the local tests on intersection hypotheses; the manuscript should explicitly display the critical-value expressions for both procedures on a generic intersection hypothesis to make the nesting of rejection regions transparent.
[Optimality result] The optimality theorem: the statement that WHP cannot be improved by enlarging any critical value is load-bearing for the final recommendation; the proof should clarify whether this optimality is with respect to the class of all consonant procedures or only within weighted Holm-type procedures, and should contrast the specific conditions under which WAP is optimal.

minor comments (3)

[Simulation studies] The simulation section should report the exact number of hypotheses, the dependence structures examined, and the range of weight configurations; without these details it is difficult to assess how broadly the reported power and FWER advantages generalize.
[Adjusted p-values] Notation for adjusted weighted p-values is introduced but not contrasted with ordinary adjusted p-values in a single table; adding such a comparison would improve readability.
[Graphical representations] The graphical representations are described as using a common initial graph; a small numerical example showing the updating steps for both WHP and WAP on the same set of weighted p-values would make the distinction concrete.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and will incorporate revisions to improve clarity as suggested.

read point-by-point responses

Referee: [Closed testing procedures] The section deriving the closed testing procedures: the uniform power superiority of WHP over WAP is asserted to follow from the local tests on intersection hypotheses; the manuscript should explicitly display the critical-value expressions for both procedures on a generic intersection hypothesis to make the nesting of rejection regions transparent.

Authors: We agree that explicitly displaying the critical-value expressions for a generic intersection hypothesis will make the argument more transparent. In the revised manuscript we will add the explicit formulas for the local-test critical values under both WHP and WAP for an arbitrary intersection hypothesis H_I. These expressions will directly illustrate the nesting of the rejection regions and thereby confirm the uniform power superiority of WHP over WAP. revision: yes
Referee: [Optimality result] The optimality theorem: the statement that WHP cannot be improved by enlarging any critical value is load-bearing for the final recommendation; the proof should clarify whether this optimality is with respect to the class of all consonant procedures or only within weighted Holm-type procedures, and should contrast the specific conditions under which WAP is optimal.

Authors: The optimality result shows that no critical value of WHP can be enlarged without violating FWER control. We will revise the theorem statement and its proof to make explicit that the result holds within the class of weighted Holm-type procedures that employ the same weighting scheme. We will also expand the contrast with the more restrictive conditions under which WAP is optimal, as already derived in the manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central derivations rest on standard closed testing theory (Marcus et al.) and FWER definitions applied to fixed-weight local tests. The uniform power superiority of WHP over WAP follows from the nested rejection regions induced by comparing local critical values on every intersection hypothesis, which is a direct consequence of the CTP construction and does not reduce to any fitted parameter, self-definition, or self-citation chain. Graphical representations, adjusted p-values, and optimality results are likewise obtained by algebraic manipulation of the same standard framework without circular reduction. The paper is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard probability axioms for p-value validity and the closed testing principle; no free parameters or invented entities are introduced.

axioms (2)

domain assumption P-values are valid and stochastically larger than uniform under the null hypothesis
Standard assumption invoked for all FWER-controlling procedures in the abstract.
domain assumption Weights are fixed and known in advance
Required for the definition of both WHP and WAP.

pith-pipeline@v0.9.0 · 5529 in / 1241 out tokens · 48300 ms · 2026-05-10T01:30:27.547816+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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