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arxiv: 2604.10986 · v1 · submitted 2026-04-13 · 📊 stat.ME · stat.CO

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Optimal multiple testing under family-wise error control: elementary symmetric polynomials and a scalable algorithm

Prasanjit Dubey , Xiaoming Huo
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Pith reviewed 2026-05-10 16:30 UTC · model grok-4.3

classification 📊 stat.ME stat.CO
keywords multiple testingfamily-wise error rateelementary symmetric polynomialsmonotonicity theoremcoordinate descentexchangeable hypotheseslikelihood ratiosoptimal testing
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The pith

Elementary symmetric polynomials of likelihood ratios yield closed-form FWER constraints and a scalable optimal test for exchangeable hypotheses.

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

The paper establishes that the coefficients appearing in the family-wise error rate constraints of the dual program for optimal multiple testing admit explicit closed-form expressions when the hypotheses are exchangeable. These expressions are built from elementary symmetric polynomials evaluated on the vector of likelihood ratios. The resulting algebraic structure produces a global monotonicity property: each target function decreases when any coordinate of the dual vector is increased. This monotonicity guarantees that a simple coordinate-wise bisection search locates the optimal dual vector and therefore the most powerful valid test. A reader would care because the procedure recovers noticeably more power than classical methods such as Hommel while still enforcing strict family-wise error control.

Core claim

The family-wise error rate constraint coefficients b_{l,k}(vec u) admit closed-form expressions through elementary symmetric polynomials of the likelihood-ratio values g(u_1),...,g(u_K). This algebraic structure implies a global monotonicity theorem: the target functions F_gamma(mu) = FWER_gamma(D^mu) are simultaneously non-increasing in every component of mu, for arbitrary K, which guarantees unique coordinate-wise roots and enables a bisection-based coordinate-descent algorithm with O(log ε^{-1}) convergence rate.

What carries the argument

Elementary symmetric polynomials of the likelihood-ratio values g(u_1) to g(u_K), which supply the closed-form expressions for the FWER constraint coefficients b_{l,k} and establish the monotonicity property.

If this is right

  • The optimal dual vector mu* can be recovered by a coordinate-descent procedure whose per-coordinate search converges at rate O(log epsilon inverse).
  • The resulting test produces relative power gains over Hommel's method that grow from 15 percent at K equals 3 to 84 percent at K equals 12.
  • Unique coordinate-wise roots exist for every equation because each target function is strictly monotone in its own variable.
  • The procedure applies directly to replication studies, clinical trials, and replicability assessments that satisfy the exchangeability condition.

Where Pith is reading between the lines

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

  • If the exchangeability assumption holds only approximately, the same algorithmic skeleton may still yield useful power improvements in practice.
  • Analogous closed-form reductions could be sought for other multiple-testing criteria whenever a comparable symmetry is present.
  • The monotonicity result supplies a design principle that may guide the construction of tests even when the hypotheses exhibit limited dependence.

Load-bearing premise

The K hypotheses are exchangeable, that is, identically distributed under the null hypothesis.

What would settle it

A concrete numerical counter-example with small K in which any of the target functions F_gamma(mu) increases when one coordinate of mu is increased while the others are held fixed would disprove the monotonicity theorem.

Figures

Figures reproduced from arXiv: 2604.10986 by Prasanjit Dubey, Xiaoming Huo.

Figure 1
Figure 1. Figure 1: Left: absolute power gain over standard procedures at [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Power curves for 𝐾=3 across four distributional models. Top left: truncated normal. Top right: mixture normal. Bottom left: Student-𝑡. Bottom right: beta. Left panels within each subplot show Π3; right panels show Πany. The algorithm (solid line) dominates all baselines across the signal range. Alt text: Four two-panel plots showing power versus signal strength for K equals 3. In each plot, the algorithm c… view at source ↗
Figure 3
Figure 3. Figure 3: Power curves for 𝐾=6 across four distributional models. Comparing with [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: plots the convergence trajectory ∥𝜇 (𝑡) − 𝜇 (𝑡−1) ∥2 on a logarithmic scale for several 𝐾 values under the truncated normal model at 𝜃 = −2·0. The approximately linear decrease on the log scale confirms geometric convergence, consistent with a contraction mapping. 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 Outer iteration t 10 2 10 1 10 0 (t) (t 1) 2 K = 3 K = 4 K = 6 K = 8 K = 12 [PITH_FULL_IMAGE:figures/full_… view at source ↗
read the original abstract

Simultaneously testing $K$ hypotheses while controlling the family-wise error rate is a fundamental problem in statistics. Existing procedures (Bonferroni, Holm, Hochberg, Hommel) provide valid control but sacrifice power, increasingly so as $K$ grows, because they base decisions on marginal $p$-value ranks rather than the joint likelihood. Rosset et al. (2022) formulated the most powerful family-wise-error-rate-controlling test as a dual program and proved the existence of an optimal dual vector $\mu^*$, but left its computation as an open problem. We solve this problem for $K$ exchangeable hypotheses. The key insight is that the family-wise error rate constraint coefficients $b_{l,k}(\vec{u})$ admit closed-form expressions through elementary symmetric polynomials of the likelihood-ratio values $g(u_1), \ldots, g(u_K)$. This algebraic structure implies a global monotonicity theorem: the target functions $F_\gamma(\mu) = {\rm FWER}_\gamma(\vec{D}^\mu)$ are simultaneously non-increasing in every component of $\mu$, for arbitrary $K$, which guarantees unique coordinate-wise roots and enables a bisection-based coordinate-descent algorithm with $O(\log \varepsilon^{-1})$ convergence rate. The relative power gain over Hommel's method grows from 15\% at $K{=}3$ to 84\% at $K{=}12$. Applications to replication studies, a clinical trial, and a replicability assessment illustrate both the power gains and the role of the exchangeability assumption.

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 / 3 minor

Summary. The paper claims to resolve the open computational problem posed by Rosset et al. (2022) for finding the optimal dual vector μ* that yields the most powerful FWER-controlling test. For exchangeable hypotheses, the authors derive closed-form expressions for the FWER constraint coefficients b_{l,k}(u) in terms of elementary symmetric polynomials of the likelihood-ratio values g(u_i), establish a global monotonicity theorem showing that F_γ(μ) is simultaneously non-increasing in each coordinate of μ for any K, and use this to construct a coordinate-wise bisection algorithm with O(log ε^{-1}) convergence. Numerical results report relative power gains over Hommel's procedure ranging from 15% at K=3 to 84% at K=12, with illustrations on replication studies, a clinical trial, and replicability assessment.

Significance. If the algebraic reduction to symmetric polynomials and the accompanying monotonicity theorem are correct, the work supplies a practical, scalable algorithm that closes the computational gap left by the dual-program formulation of Rosset et al. and delivers measurable power improvements for moderate K under the stated exchangeability assumption. The approach is parameter-free once the g(u_i) are fixed, directly leverages the symmetry to avoid exponential complexity, and is supported by reproducible applications; these features make the contribution substantive for multiple-testing methodology in statistics.

major comments (1)
  1. [§4, Theorem 2] §4, Theorem 2 (monotonicity): the claim that F_γ(μ) is simultaneously non-increasing in every coordinate of μ for arbitrary K rests on the closed-form b_{l,k} via symmetric polynomials; the provided sketch should be expanded to an explicit inductive or combinatorial argument showing that the partial derivatives remain non-positive under exchangeability, as this is the load-bearing step enabling the bisection procedure.
minor comments (3)
  1. [§5.2, Table 2] §5.2, Table 2: the reported power gains are given as relative percentages; adding absolute power values (or the underlying rejection counts) for each method and K would allow readers to assess the practical magnitude of the improvement.
  2. [Notation] Notation: the definition of the likelihood-ratio function g(u_i) and its relation to the dual variables μ should be restated once in the main text (currently only in the appendix), as it is used repeatedly in the symmetric-polynomial expressions.
  3. [§6] §6 (applications): the replicability-assessment example would benefit from an explicit statement of how the exchangeability assumption is justified or checked for the selected studies.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and the specific suggestion regarding the proof of Theorem 2. We address the comment below and will incorporate the requested expansion in the revised manuscript.

read point-by-point responses

  1. Referee: [§4, Theorem 2] §4, Theorem 2 (monotonicity): the claim that F_γ(μ) is simultaneously non-increasing in every coordinate of μ for arbitrary K rests on the closed-form b_{l,k} via symmetric polynomials; the provided sketch should be expanded to an explicit inductive or combinatorial argument showing that the partial derivatives remain non-positive under exchangeability, as this is the load-bearing step enabling the bisection procedure.

    Authors: We agree that a fully explicit proof strengthens the paper. The manuscript currently presents a sketch that derives the closed-form b_{l,k}(u) via elementary symmetric polynomials and invokes their properties to conclude monotonicity. In the revision we will replace the sketch with a complete inductive argument on K. The base case K=1 is immediate. For the inductive step we fix all but one coordinate of μ, increase the remaining coordinate, and track the resulting change in each FWER coefficient b_{l,k} through the recurrence relations satisfied by the elementary symmetric polynomials. Because the likelihood ratios g(u_i) are non-negative and the hypotheses are exchangeable, each incremental change in a symmetric polynomial term is non-positive, which propagates to show that every partial derivative of F_γ(μ) is non-positive. This explicit induction makes the load-bearing step self-contained and directly justifies the coordinate-wise bisection procedure. No other changes to the results or algorithm are required. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins from the external dual-program formulation of Rosset et al. (2022) and introduces independent algebraic closed forms for the b_{l,k} coefficients via elementary symmetric polynomials of the g(u_i) under the stated exchangeability assumption. The global monotonicity theorem for F_gamma(mu) is proved directly from those closed forms, enabling the coordinate-wise bisection algorithm. No equation reduces by construction to a fitted input, self-referential definition, or load-bearing self-citation; the cited prior work supplies only the optimization setup while the paper supplies the explicit solution and convergence guarantee.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central construction rests on the domain assumption of exchangeability; no free parameters or new invented entities are introduced.

axioms (1)
  • domain assumption The K hypotheses are exchangeable (identically distributed under the null).
    Required to obtain the elementary-symmetric-polynomial closed forms for the FWER coefficients and the global monotonicity theorem.

pith-pipeline@v0.9.0 · 5584 in / 1306 out tokens · 37666 ms · 2026-05-10T16:30:04.459269+00:00 · methodology

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    Yoav Benjamini and Yosef Hochberg

    doi: 10.1126/science.1211180. Yoav Benjamini and Yosef Hochberg. Controlling the false discovery rate: A practical and powerful approach to multiple testing.Journal of the Royal Statistical Society. Series B (Methodological), 57(1):289–300,

    work page doi:10.1126/science.1211180

  2. [2]

    Controlling the false discovery rate: a practical and powerful approach to multiple testing.Journal of the Royal Statistical Society: Series B (Methodological), 57(1):289–300, 1995

    URLhttp://www.jstor.org/stable/2346101. Carlo Emilio Bonferroni. Teoria statistica delle classi e calcolo delle probabilit `a. InPubblicazioni del R Istituto Superiore di Scienze Economiche e Commerciali di Firenze, volume 8, pages 3–62. Libreria Internazionale Seeber, Firenze, Italy,

    work page arXiv

  3. [3]

    doi: 10.1002/sim.3495. Colin F. Camerer, Anna Dreber, Felix Holzmeister, Teck-Hua Ho, J¨ urgen Huber, Magnus Johannesson, Michael Kirchler, Gideon Nave, Brian A. Nosek, Thomas Pfeiffer, et al. Evaluating the replicability of 20 social science experiments in Nature and Science between 2010 and 2015.Nature Human Behaviour, 2: 637–644,

    work page doi:10.1002/sim.3495 2010

  4. [4]

    Maxime Derex, Marie-Pauline Beugin, Bernard Godelle, and Michel Raymond

    doi: 10.1038/s41562-018-0399-z. Maxime Derex, Marie-Pauline Beugin, Bernard Godelle, and Michel Raymond. Experimental evidence for the influence of group size on cultural complexity.Nature, 503:389–391,

    work page doi:10.1038/s41562-018-0399-z

  5. [5]

    Edgar Dobriban, Kristen Fortney, Stuart K

    doi: 10.1038/nature12774. Edgar Dobriban, Kristen Fortney, Stuart K. Kim, and Art B. Owen. Optimal multiple testing under a gaussian prior on the effect sizes.Biometrika, 102(4):753–766, 11

    work page doi:10.1038/nature12774

  6. [6]

    URLhttps://doi.org/10.1093/biomet/asv050

    doi: 10.1093/biomet/asv050. URLhttps://doi.org/10.1093/biomet/asv050. Prasanjit Dubey and Xiaoming Huo. Most Powerful Test with Exact Family-Wise Error Rate Control: Necessary Conditions and a Path to Fast Computing.arXiv preprint arXiv:2512.14131,

    work page doi:10.1093/biomet/asv050

  7. [7]

    Katherine Duncan, Arjun Sadanand, and Lila Davachi

    URL https://arxiv.org/abs/2512.14131. Katherine Duncan, Arjun Sadanand, and Lila Davachi. Memory’s penumbra: episodic memory decisions induce lingering mnemonic biases.Science, 337:485–488,

    work page arXiv

  8. [8]

    Yosef Hochberg

    doi: 10.1126/science.1221936. Yosef Hochberg. A sharper Bonferroni procedure for multiple tests of significance.Biometrika, 75(4): 800–802,

    work page doi:10.1126/science.1221936

  9. [9]

    URLhttp://www.jstor.org/stable/2336325

    doi: 10.2307/2336325. URLhttp://www.jstor.org/stable/2336325. Sture Holm. A simple sequentially rejective multiple test procedure.Scandinavian Journal of Statistics, 6 (2):65–70,

    work page doi:10.2307/2336325

  10. [10]

    URLhttp://www.jstor.org/stable/4615733. G. Hommel. A stagewise rejective multiple test procedure based on a modified bonferroni test.Biometrika, 75(2):383–386,

    work page arXiv

  11. [11]

    Ramesh Johari, Pete Koomen, Leonid Pekelis, and David Walsh

    URLhttp://www.jstor.org/stable/2336190. Ramesh Johari, Pete Koomen, Leonid Pekelis, and David Walsh. Always valid inference: Continuous monitoring of A/B tests.Operations Research, 70(3):1806–1821,

    work page arXiv

  12. [12]

    ´Agnes Melinda Kov´acs, Ern˝o T´egl´as, and Ansgar Denis Endress

    doi: 10.1287/opre.2021.2135. ´Agnes Melinda Kov´acs, Ern˝o T´egl´as, and Ansgar Denis Endress. The social sense: Susceptibility to others’ beliefs in human infants and adults.Science, 330:1830–1834,

    work page doi:10.1287/opre.2021.2135 2021

  13. [13]

    doi: 10.1126/science.1190792. E. L. Lehmann, Joseph P. Romano, and Juliet Popper Shaffer. On optimality of stepdown and stepup multiple test procedures.The Annals of Statistics, 33(3):1084 – 1108,

    work page doi:10.1126/science.1190792

  14. [14]

    URLhttps://doi.org/10.1214/009053605000000066

    doi: 10.1214/009053605000000066. URLhttps://doi.org/10.1214/009053605000000066. John A List, Azeem M Shaikh, and Yang Xu. Multiple hypothesis testing in experimental economics. Working Paper 21875, National Bureau of Economic Research, January

    work page doi:10.1214/009053605000000066

  15. [15]

    Science , volume =

    doi: 10.1126/science.aac4716. Stuart J. Pocock, Nancy L. Geller, and Anastasios A. Tsiatis. The analysis of multiple endpoints in clinical trials.Biometrics, 43(3):487–498,

    work page doi:10.1126/science.aac4716

  16. [16]

    21 David G

    URLhttp://www.jstor.org/stable/2531989. 21 David G. Rand, Joshua D. Greene, and Martin A. Nowak. Spontaneous giving and calculated greed.Nature, 489:427–430,

    work page arXiv

  17. [17]

    Joseph P

    doi: 10.1038/nature11467. Joseph P. Romano and Michael Wolf. Stepwise multiple testing as formalized data snooping.Econometrica, 73(4):1237–1282,

    work page doi:10.1038/nature11467

  18. [18]

    Joseph P

    doi: 10.1111/j.1468-0262.2005.00615.x. Joseph P. Romano and Michael Wolf. Control of generalized error rates in multiple testing.The Annals of Statistics, 35(4):1378 – 1408,

    work page doi:10.1111/j.1468-0262.2005.00615.x 2005

  19. [19]

    URLhttps://doi.org/10

    doi: 10.1214/009053606000001622. URLhttps://doi.org/10. 1214/009053606000001622. Michael Rosenblum, Han Liu, and En-Hsu Yen. Optimal tests of treatment effects for the overall population and two subpopulations in randomized trials, using sparse linear programming.Journal of the American Statistical Association, 109(507):1216–1228,

    work page doi:10.1214/009053606000001622

  20. [20]

    Journal of the American Statistical Association 108(503), 1062–1074 (2013) https://doi.org/10.1080/01621459.2013.820134

    doi: 10.1080/01621459.2013.879063. URLhttps: //doi.org/10.1080/01621459.2013.879063. Saharon Rosset, Ruth Heller, Amichai Painsky, and Ehud Aharoni. Optimal and maximin procedures for multiple testing problems.Journal of the Royal Statistical Society Series B, 84(4):1105–1128, September

    work page doi:10.1080/01621459.2013.879063 2013

  21. [21]

    URLhttps://ideas.repec.org/a/bla/jorssb/ v84y2022i4p1105-1128.html

    doi: 10.1111/rssb.12507. URLhttps://ideas.repec.org/a/bla/jorssb/ v84y2022i4p1105-1128.html. Betsy Sparrow, Jenny Liu, and Daniel M. Wegner. Google effects on memory: Cognitive consequences of having information at our fingertips.Science, 333:776–778,

    work page doi:10.1111/rssb.12507

  22. [22]

    Emil Spjøtvoll

    doi: 10.1126/science.1207745. Emil Spjøtvoll. On the optimality of some multiple comparison procedures.Annals of Mathematical Statistics, 43:398–411,

    work page doi:10.1126/science.1207745

  23. [23]

    Victoria Vickerstaff, Rumana Omar, and Gareth Ambler

    doi: 10.1056/NEJMoa1511939. Victoria Vickerstaff, Rumana Omar, and Gareth Ambler. Methods to adjust for multiple comparisons in the analysis and sample size calculation of randomised controlled trials with multiple primary outcomes. BMC Medical Research Methodology, 19, 06

    work page doi:10.1056/nejmoa1511939

  24. [24]

    Davide Viviano, Kaspar Wuthrich, and Paul Niehaus

    doi: 10.1186/s12874-019-0754-4. Davide Viviano, Kaspar Wuthrich, and Paul Niehaus. A model of multiple hypothesis testing,

    work page doi:10.1186/s12874-019-0754-4

  25. [25]

    URL https://arxiv.org/abs/2104.13367. Peter H. Westfall, Alok Krishen, and S. Stanley Young. Using prior information to allocate significance levels for multiple endpoints.Statistics in Medicine, 17(18):2107–2119,

    work page arXiv