E-Values For Multiplicity Control In Multiverse Analysis

David Rossell; Paul Rognon-Vael

REVIEW 4 major objections 6 minor 24 references

A simple p-to-e calibration rule can outperform generic e-value strategies for false-discovery control in multiverse analysis, and reveals previously missed associations between teen technology use and mental well-being.

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2026-08-01 17:32 UTC pith:RA5MRP4T

load-bearing objection Useful comparison paper; the mixture p-to-e calibrator recommendation is plausible but oversold by 'significant' without error bars, and finite-n e-variable validity is asserted rather than checked. the 4 major comments →

arxiv 2607.17596 v1 pith:RA5MRP4T submitted 2026-07-20 stat.ME

E-Values For Multiplicity Control In Multiverse Analysis

Paul Rognon-Vael , David Rossell This is my paper

classification stat.ME MSC 62F0362J1262H15

keywords multiverse analysise-valuesfalse discovery ratemultiple testinggeneralized linear modelsp-to-e calibrationteenager well-being

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.

This paper argues that multiverse analyses—testing many treatment–outcome pairs at once—can be made statistically honest at modest cost by using e-values instead of p-values. It shows that a simple conversion of ordinary likelihood-ratio p-values into e-values, called p-to-e calibration, delivers higher statistical power than generic universal or soft-rank e-value constructions in generalized-linear-model settings. The paper's core claim is that this calibration rule, combined with the e-BH procedure, controls the false discovery rate regardless of how the individual tests depend on each other. Applied to a large survey of teenagers, the method finds four significant technology–wellbeing associations that a previous aggregate specification-curve analysis had obscured. The sympathetic reader should take the central message to be that e-values can make multiverse analysis inferentially valid without needing unrealistic assumptions about test dependence.

Core claim

The paper claims that, for GLM-based multiverse analyses, p-to-e calibration—especially the mixture calibrator Ec1—has higher statistical power than universal mixture e-values and soft-rank e-values, while retaining the e-value property that guarantees FDR control under arbitrary dependence via the e-BH procedure. In a cohort of 11,884 UK teenagers, applying Ec1 with refinement to e-BH rejects four outcome–treatment pairs: internet use with low self-esteem, internet use with depressive symptoms, internet use with high peer problems, and social media use with depressive symptoms, with estimated odds ratios between roughly 2 and 4. The paper interprets these as practically relevant association

What carries the argument

The central object is the mixture p-to-e calibrator Ec1(π) = (1 − π + π log π) / [π (log π)²], which maps a p-value π into an e-value—a test statistic whose expectation under the null is at most 1. The calibrator does the power work by blending a family of simple calibrator functions. With e-values in hand, the e-BH procedure rejects hypotheses with the largest e-values, up to a threshold that uses the order statistics of the e-values, and this controls the FDR without requiring independence, positive dependence, or estimation of nuisance parameters.

Load-bearing premise

The FDR guarantee rests on the calibrated e-values being true e-values, which holds only if the likelihood-ratio p-values fed into the calibrator are super-uniform under the null—an assumption the paper verifies by simulation for n ≥ 100 but does not prove for finite samples in non-Gaussian models.

What would settle it

Run the paper's logistic-regression setting with n = 50 and a rare binary outcome under the null, applying Ec1 to the asymptotic likelihood-ratio p-value; if the fraction of rejected hypotheses at level α exceeds α over many replications, the p-value is anti-conservative and the e-variable guarantee fails in that regime.

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

If this is right

Multiverse analyses can achieve finite-sample FDR control without assuming independence or positive dependence among the multiple tests.
The mixture calibrator Ec1 is preferable to universal and soft-rank e-values for well-specified GLM-based multiverse testing.
E-values remain a conservative tool: statistical power may be moderate unless sample sizes and effect sizes are sufficiently large.
The teenager well-being application indicates that averaging technology effects across treatments, as specification-curve analysis does, can hide strong and practically relevant associations.

Where Pith is reading between the lines

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

The paper does not say this, but the same calibration recipe would extend to permutation-based p-values, which are super-uniform by construction and would remove the asymptotic caveat outside Gaussian models.
Because e-BH is dependence-robust, the framework could be used to compare results across different control-covariate subsets without modeling their correlation; whether the four findings survive such specification choices remains open.
E-values compose across independent datasets, so the approach could be extended to anytime-valid pooling of subpopulation or cohort analyses, which would be useful for ongoing surveillance of technology–wellbeing associations.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Useful comparison paper; the mixture p-to-e calibrator recommendation is plausible but oversold by 'significant' without error bars, and finite-n e-variable validity is asserted rather than checked.

read the letter

Short version: this is a competent, useful comparison paper, not a breakthrough. The genuinely new bit is the simulation-based comparison showing p-to-e calibration (especially the mixture calibrator Ec1) beats universal mixture and soft-rank e-values for GLM multiverse testing, plus the MCS application that flips the earlier 'negligible' finding. The math for e-value validity is standard and correct, and the authors explicitly flag the finite-sample caveat for LRT p-values. Worth a proper look, but the authors oversell 'significantly outperforms' relative to the evidence, and the FDR guarantee rests on a property the simulations don't actually check.

What's good: the framing is honest, the comparison is fair, and the practical recommendation (Ec1 + e-BH for multiverse GLM analyses) is concrete. The application is a nice demonstration that e-value FDR control can change conclusions from SCA. The authors also clearly state that e-values are conservative and that power can be moderate—rare candor.

Soft spots, in order of severity. First, the stress-test concern lands: Table 1 reports rejection proportions at alpha=1% and 5%, but the e-BH guarantee requires E[E]<=1, not just tail control. A p-value can be slightly anti-conservative and still give rejection rates below alpha, while E[Ec1(P)]>1. So the finite-n validity of Ec1/Ec2 is not demonstrated; the authors' acknowledgment is honest but the simulation doesn't close the gap. For the MCS application n is large, so this is minor for the headline application, but for the general recommendation it matters. Second, 'significantly outperforms' is not supported: Figures 1-2 are averages without error bars and there are no MC intervals in Table 1. The dominance is visually clear in many scenarios, so the conclusion is probably right, but the wording is stronger than the numbers. Third, the code/data statement says 'dedicated Github repository' with no URL or commit—unverifiable as is. Fourth, the pre-screening before e-BH in Section 4 is ambiguous: six hypotheses with 1/Ec1>20, then refined e-BH; it's likely fine but should be spelled out.

All fixable. The central comparative finding holds up for the investigated regimes. This deserves a serious referee; I'd want a revision that adds proper Monte Carlo error bars or a direct check of e-variable validity, and a real code link. For readers in multiverse analysis or applied e-value methodology, this is a useful addition. I'd cite the mixture-calibrator recommendation if I worked in that area.

Referee Report

4 major / 6 minor

Summary. The paper proposes using e-values to control the false discovery rate in GLM-based multiverse analyses. It reviews three e-value constructions: universal mixture e-variables (Eq. 3), soft-rank e-variables (Eq. 5), and p-to-e calibrators (Eqs. 7-8), proves the validity of the universal mixture, and compares power via simulation. Based on average log e-values in linear and logistic regression, it recommends the mixture calibrator Ec1 and applies it with e-BH to the Millennium Cohort Study teenage well-being data, reporting four significant outcome-treatment associations. The central claim is that simple p-to-e calibration dominates generic e-value strategies in this setting.

Significance. If established, the paper offers a practical, dependence-robust FDR control method for multiverse analyses, avoiding independence or positive-dependence assumptions and nuisance parameter estimation. Strengths include a correct e-variable validity proof in Section 3.1, explicit discussion of the finite-sample caveat, and reproducible code/data availability. The substantive application is a useful counterpoint to specification-curve averaging. However, the empirical support for the headline power comparison is thin, and the finite-sample validity of the recommended calibrator is not verified at the level needed for strict FDR guarantees.

major comments (4)

[Section 3.2, Figures 1-2, Abstract] The statement that p-to-e calibration 'significantly outperforms' the other approaches in statistical power is not supported by the reported evidence. The figures plot average log e-values from 100 repetitions with no error bars, confidence intervals, or paired comparisons, and an average log e-value is not a power curve. There are no simulations of e-BH FDR control in a multiverse setting with multiple correlated outcomes/treatments, which is the paper's stated target. Please add Monte Carlo intervals, formal comparisons, and FDR/power simulations with realistic multiverse dimensions, or weaken the claim.
[Section 3.1, Eqs. (7)-(8), Table 1] p-to-e calibrators are e-variables only when input p-values are super-uniform under the null. For logistic regression, the LRT p-values are asymptotic only; the paper acknowledges this, yet relies on the e-BH finite-sample FDR guarantee. Table 1 reports only rejection proportions at alpha=0.01 and 0.05, which does not validate E[Ec1(P)] <= 1. Since Ec1(p) ~ 1/(p log^2 p) as p goes to 0, its expectation is sensitive to small-p behavior even if tail rejection rates are controlled. This is load-bearing for Section 4. Please prove or empirically validate finite-sample super-uniformity (e.g., report null means of calibrated e-values with Monte Carlo intervals), use exact/permutation p-values, or label the FDR control as asymptotic.
[Section 2, Eq. (2)] The displayed null H_lj includes 'beta_j' != 0 for j' != j', contradicting the following sentence that no restrictions are placed on remaining treatment effects. The global H_l (all effects nonzero) is not the relevant alternative for a single test H_lj. This affects the definition of q_l in Eq. (3) and the interpretation of all simulation and application results. Please clarify the hypothesis space and the alternative under which q_l is specified.
[Section 4, Table 2] The sentence 'there were six hypotheses for which 1/Ec1 > 20' appears inverted; the individual-test threshold at level 0.05 is Ec1 > 20 (equivalently 1/Ec1 < 0.05). In addition, the application's four rejections disappear under universal and soft-rank e-values; this is informative, but it means the headline result hinges entirely on the asymptotic validity of Ec1 and should be presented together with the caveat in the second major comment.

minor comments (6)

[Abstract, Section 1] Typos: 'self-steem' should be 'self-esteem'; 'sensitvy' should be 'sensitivity'.
[Section 3.1] The inverse-gamma notation IG(a/2,l/2) is confusing because l is also used to index outcomes. Use a different symbol such as b/2.
[Table 1] The caption and text do not specify the null coefficient values used for the type I error rows. State explicitly which beta* are set to zero.
[Figures 1-2] Captions say 'average over 100 repetitions' but no uncertainty is displayed. Add error bars or confidence intervals, or state that none are shown.
[Section 3.1] The closed-form marginal likelihood uses inconsistent notation for sigma^2_l and sigma-hat^2_l. Clarify the parameterization of the inverse-gamma prior.
[Section 4] The claim that odds ratios in [2,4] are 'practically relevant' could use a sentence on effect-size calibration or external benchmarks, since practical relevance is a subjective judgment.

Circularity Check

0 steps flagged

No significant circularity; the only self-citation ([2]) is minor and not load-bearing.

full rationale

The paper's central derivation is self-contained. Section 3.1 proves the universal mixture e-variable property in-line (E[E^um] <= 1 follows from the pointwise bound p(y) <= \hat p_lj(y) and \int q_l = 1). The p-to-e calibrators (7)-(8) are standard transforms whose e-variable property is inherited from super-uniformity of the input p-values, with the finite-n caveat for LRT p-values stated explicitly in Section 3.1. Soft-rank e-values are justified by exchangeability ([18]) and e-BH FDR control by [5]; neither is imported from the authors' prior work. The simulation comparison of average log e-values and type I error rates is an empirical evaluation, not a fitted quantity renamed as a prediction. The MCS application applies the pre-specified calibrator to LRT p-values and reports e-BH rejections; no parameter fitted to the data is called a prediction. The only self-citation is [2], co-authored by Rossell, used for data preparation and for motivating the critique of SCA; that citation is not the load-bearing step for any e-value proof or for the comparative power claim. The paper's explicit finite-sample caveat ('Since such a p-value is not exact for finite n, neither is the resulting calibrated e-value') is a validity limitation to verify in applications, not a circularity. Hence score 2 only for the minor, non-load-bearing self-citation.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central claim rests on the GLM being correct, on the asymptotic validity of LRT p-values used by the recommended calibrator, and on the external e-BH theorem. The universal-mixture prior choice and permutation exchangeability matter only for the two non-recommended methods. No invented entities are introduced.

free parameters (2)

Simulation design: effect-size grid, sample sizes, covariate count, pairwise covariance = beta* in {0, 0.15, 0.3, 0.5}; n in 100-10000; 10 covariates; pairwise covariance 0.5
Chosen by hand to mimic multiverse analyses in the social sciences. The ranking of e-value methods and the conclusion that Ec1 'significantly outperforms' could change outside this design region.
Inverse-gamma hyperparameters for sigma^2 in the universal mixture e-variable = not stated
The closed-form marginal likelihood q_l in Section 3.1 depends on these hyperparameters, which are not given in the text. They affect the reported E_um simulation results.

axioms (5)

domain assumption The GLM in Equation (1) with link function F is correctly specified for every outcome-treatment-subgroup combination, and the weighted sum-to-zero coding makes beta_j^(l) the average treatment effect.
Section 2 introduces Equation (1); all e-value strategies and the application rely on this model. The paper deliberately excludes covariate-subset specification curves.
domain assumption For p-to-e calibrators, the likelihood-ratio p-values are super-uniform under the null, or the chi-square approximation is adequate at the sample sizes used.
Section 3.1 states the caveat that outside the Gaussian family the LRT p-value is not exact for finite n, so calibrated e-values are only approximately valid. This is the load-bearing assumption behind the recommended Ec1.
standard math The e-BH procedure of Wang and Ramdas (2022) controls FDR at alpha K0/K <= alpha under arbitrary dependence.
Section 3.3 invokes this theorem to justify FDR control; it is not re-derived in the paper but is external established theory.
ad hoc to paper Zellner's unit information prior and the inverse-gamma prior on sigma^2 yield a proper marginal likelihood for the universal mixture e-variable.
Section 3.1 introduces these priors to avoid subjective choices, but the hyperparameters are not fully specified, so E_um results depend on an unstated prior choice.
domain assumption The permutation scheme (regressing the treatment on other covariates and permuting residuals) makes T0 exchangeable with T1,...,TB under the null.
Section 3.1 describes soft-rank e-variables; their validity relies on this exchangeability. This matters only if soft-rank e-values are used.

pith-pipeline@v1.3.0-alltime-deepseek · 11669 in / 17362 out tokens · 156847 ms · 2026-08-01T17:32:48.174096+00:00 · methodology

0 comments

read the original abstract

Multiverse analysis refers to a common situation where one wishes to assess the association between multiple possible treatment definitions and multiple possible outcome definitions, potentially within multiple sub-populations, among other possible analysis specifications. Multiverse analysis is a useful exploratory tool to assess heterogeneity across the considered specifications, but it is sometimes also used to assess statistical significance. In the latter case, it is critical to acknowledge that multiple comparisons are being performed, and to ensure a valid statistical control of false positive findings. We study the use of e-values within generalized linear models as a tool to control the false discovery rate regardless of the dependence structure of the multiple analyses being performed, while accounting for confounding covariates. We compare the performance of several approaches: universal e-values, soft-rank e-values, and p-to-e calibration. We find that, for problem characteristics typically encountered in multiverse analyses, p-to-e calibration significantly outperforms the other two approaches in terms of statistical power, but said power may be moderate unless the sample size or effect sizes are large enough. An application studying the association between teenager technology use and mental well-being reveals association between depression, low self-steem and peer problems with internet and social media usage.

discussion (0)

Reference graph

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Koning, N.W.: More power by using fewer permutations. Biometrika111(4), 1405–1412 (2024) https://doi.org/10. 1093/biomet/asae031 https://academic.oup.com/biomet/article- pdf/111/4/1405/60783709/asae031.pdf

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Ramdas, A., Manole, T.: Randomized and Exchangeable Improvements of Markov’s, Chebyshev’s and Chernoff’s Inequalities. Statistical Science41(1), 121–142 (2026) https://doi.org/10.1214/24-STS952

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2011