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REVIEW 2 major objections 5 minor 62 references

Mediation hypotheses can be screened in a first stage and tested in a second stage using the same sample, with asymptotically independent p-values, so standard FDR control applies to the survivors.

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2026-08-01 17:34 UTC pith:QWFO7ZK2

load-bearing objection Genuinely useful transformation pair for two-stage mediation testing, but the FDR guarantee is formally only under cross-hypothesis independence, which the real-data regime violates. the 2 major comments →

arxiv 2607.17579 v1 pith:QWFO7ZK2 submitted 2026-07-20 stat.ME

Two-stage Adaptive Testing of Large-scale Mediation Hypotheses

Yueqi Xu , Kwun Chuen Gary Chan This is my paper
classification stat.ME MSC 62F0362F0762P10
keywords mediation analysistwo-stage testingfalse discovery ratemultiple testingp-value transformationcomposite null hypothesisasymptotic independenceepigenome-wide association
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.

The paper claims that large-scale mediation testing can be run in two stages without paying the usual price for selection: screening and testing can use the same full sample, yet the p-value used to decide whether a hypothesis survives screening is asymptotically independent of the p-value used to decide whether it is a discovery. The key is a pair of transformations of the two path-specific p-values—a transformed minimum for screening and a rescaled p-value for testing. Under a composite null in which either the exposure–mediator path or the mediator–outcome path is null, the rescaled p-value is exactly uniform and independent of the screening input, so a standard step-up FDR procedure can be applied to the survivors. The authors establish asymptotic FDR control and demonstrate in simulations and an epigenome-wide mediation analysis that screening sharply reduces the multiplicity burden, recovering all findings of existing methods plus additional mediators.

Core claim

The paper's central claim is that each mediation hypothesis's two path-specific p-values can be transformed into a screening p-value and a testing p-value that are asymptotically independent even though both come from the same data. Screening uses the transformed minimum to eliminate likely global nulls, while testing uses the rescaled p-value on survivors. Because the selection and testing inputs are asymptotically independent, no sample splitting or selection-dependent adjustment is needed, and a standard step-up FDR procedure controls the false discovery rate asymptotically. The paper implements this with an adaptive signal-proportion estimator for the screening threshold and a conservati

What carries the argument

The central objects are two transformed p-values built from the order statistics of the two path-specific p-values: the transformed minimum, p*_1/2 = 1 - (1 - p_(1))^2, and the rescaled p-value, p*_2/2 = 1 - (1 - p_(2))/(1 - p_(1)). Under the global null both are asymptotically independent standard uniforms; under a partial null the rescaled p-value remains uniform and independent of the screening input; under a full alternative both degenerate to zero. This asymptotic independence lets the first-stage screen be data-adaptive and arbitrary without invalidating the second-stage FDR procedure, and the proof uses a classical representation of ordered uniform variables to establish the separatio

Load-bearing premise

The formal FDR guarantee assumes that the null p-values across different candidate mediators are mutually independent asymptotically; the paper only shows simulation robustness for one block-exchangeable dependence pattern, so under stronger or more complex cross-hypothesis correlation the guarantee is unproven.

What would settle it

Simulate a mediation study with, say, J = 50,000 null mediators whose p-values are positively correlated beyond the paper's block design—for instance, a single-factor correlation with rho = 0.6 or an autoregressive order-1 structure—and run the procedure at a nominal FDR of 0.2. If the empirical FDR systematically exceeds 0.2, the asymptotic FDR control claim is contradicted.

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

If this is right

  • Screening can remove the majority of global-null hypotheses, reducing the multiplicity burden in Stage 2 and increasing power, especially when signals are sparse.
  • The Stage 1 threshold can be chosen data-adaptively, since FDR control is asymptotically unaffected by the choice of screening threshold.
  • Any step-up FDR-controlling procedure can be used in Stage 2; substituting an adaptive Bonferroni correction yields family-wise error rate control without changing Stage 1.
  • In the paper's data application, screening retained 3,556 of 484,613 candidates and the procedure identified 23 mediators at FDR 0.05, compared to 6 without screening while covering all findings of prior approaches.
  • The two-stage procedure preserves the strongest signals; the screening step primarily removes unpromising hypotheses rather than sacrificing tail discoveries.

Where Pith is reading between the lines

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

  • The two-p-value transformation may generalize to other composite-null problems with paired p-values, such as replicability analysis, but each setting must independently justify the asymptotic independence; the paper only proves it for the mediation structure.
  • If cross-hypothesis dependence in real data is stronger or more long-range than the block-exchangeable pattern simulated (e.g., factor-model or autoregressive correlations), the asymptotic FDR guarantee has not been proven and could break, so practitioners should verify empirically before trusting nominal levels.
  • The asymptotic nature of the result suggests finite-sample performance may degrade with small sample sizes, so users with modest n might need calibration studies before applying the procedure.

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. The paper proposes a two-stage procedure for large-scale mediation testing. For each mediator j, the two asymptotically independent path-specific p-values (p1j, p2j) are ordered and transformed into p*_{1/2}=1-(1-p_(1))^2 and p*_{2/2}=1-(1-p_(2))/(1-p_(1)). Proposition 1 shows that, under the composite mediation null, the screening p-value and the rescaled test p-value are asymptotically independent and the latter is uniform; under partial and full alternatives the screening p-value collapses to zero. Stage 1 uses an adaptive signal-missing-rate rule (adSMR) on p*_{1/2} to filter out global nulls; Stage 2 applies Storey's adaptive step-up procedure to p*_{2/2} among the survivors. Theorem 2 claims asymptotic FDR control under mutual independence of the null p-values. The paper compares the method with DACT, HDMT, and M-DACT in simulations under sparse, moderate, and dense signals, including block dependence, and applies it to the Normative Aging Study, recovering previously reported CpG sites and adding two further discoveries.

Significance. The core idea is attractive and potentially useful: exploit the composite-null structure of mediation to obtain within-hypothesis asymptotic independence between screening and testing p-values, thereby avoiding sample splitting and complicated selection-adjusted FDR procedures. The supplementary proofs are self-contained, and the order-statistic argument in Proposition 1 is transparent. The simulation study is broader than in many competing papers and covers both FDR and FWER endpoints. If the independence-based guarantee is treated as the formal result, the contribution is solid. The two concerns below concern how far that guarantee extends to the dependent, large-J settings that motivate the paper.

major comments (2)
  1. [Theorem 2 and S2; Remark after Theorem 2] The theorem is proved only under mutual independence of the null p-values across J. In the motivating NAS application (Section 4), the 484,613 CpG p-values are correlated within genomic regions, and the simulations in Section 3 cover only a block-exchangeable model with ρ=0.3 and block sizes 10–100. The remark that 'standard step-up FDR procedures are robust to positive regression dependency' does not transfer automatically to the selected set I(tγ): the Stage-1 event {p*_{1/2,j}≤tγ} is a complex, data-adaptive selection event that can destroy PRDS among the survivors even if the original p-values are PRDS. Because the abstract and introduction present FDR control without qualification, the manuscript should either prove a version of Theorem 2 under a stated dependence model (e.g., block dependence with explicit conditions) or delimit the formal claim to independent hypotheses and descri
  2. [Theorem 2 / S2 (asymptotic regime)] The proof in S2 treats J as fixed and lets n→∞: it assumes convergence of the J-dimensional vector P*(n) to P*(∞) and applies the continuous mapping theorem. The paper's stated domain is 'large-scale' testing with J on the order of 10^5, and the simulations use J=10,000. If J=J_n grows with n, the convergence and continuity argument for the FDP mapping is not defined, and no uniform-in-J control is given. The theorem should state its asymptotic regime explicitly. If the claim is only for fixed J, the 'large-scale' framing needs adjustment; if J is meant to diverge, a triangular-array or uniform convergence argument is needed.
minor comments (5)
  1. [Section 2, after definition of p*] Typo: 'at least at large as' should read 'as large as'; the following sentence about p*_{2/2,j} being 'neither linear nor monotone' is also awkwardly phrased.
  2. [Algorithm 2, step 3] \hat{s}=J\hat{\pi}_{MR} is not necessarily an integer, yet k* is defined using p*_{1/2,(\hat{s}+j)}. Please specify whether \hat{s} is rounded, or define order statistics with a real index convention.
  3. [Figure 1 caption] The caption states (π00, π11)=(0.9, 0), but the simulation setting described in the text uses (π00, π11)=(0.9, 0.04) with π10=π01=0.05. Please correct this apparent inconsistency.
  4. [Section 4, NAS analysis] The simulation-based bounding sequence c_sim_J used for the NAS data is not fully described. Since the calibration is meant to account for dependence, the block structure and simulation protocol should be specified for reproducibility.
  5. [Section 2, Stage 2 estimator] Storey's estimator depends on a user-chosen λ, but the paper gives no guidance on choosing λ or a sensitivity analysis for it. A brief remark or a small simulation would help practitioners.

Circularity Check

0 steps flagged

No circularity: the derivation uses external distributional facts, standard estimators, and an explicit independence assumption; the simulation-calibrated screening constant does not enter the FDR-control proof.

full rationale

The central derivation chain is self-contained. Proposition 1 is proved from the asymptotic distribution of the path-specific p-values, the joint density of order statistics, and Rényi's representation, not from the conclusion it supports. Theorem 2 is then proved by conditioning on the Stage-1 filtration and using a reverse-martingale/optional-stopping argument with Storey's estimator; the binomial step in the proof is exactly the stated mutual-independence assumption, which is an explicit hypothesis of the theorem rather than a fitted input. The simulation-calibrated bounding sequence cJ in adSMR is used only to set the Stage-1 screening threshold, and the paper explicitly states that 'the FDR or FWER guarantee is provided solely by the Stage-2 testing procedure and holds for any data-adaptive choice of Stage 1 screening threshold tγ due to martingale properties of Stage-2 procedures.' Thus the FDR conclusion does not reduce to the calibration constant. There are no self-citations that carry load-bearing weight: the cited results (Storey et al. 2004; Meinshausen and Rice 2006; Jeng et al. 2019; Liu et al. 2022; Dai et al. 2022) are external, and the transformations are attributed to Tippett (1931) and Rényi (1953). The paper openly flags its main limitation: 'the FDR guarantee in this work is asymptotic and relies on the asymptotic independence of the two transformed p-values,' and 'the theoretical results rely on mutual independence across j.' Whether cross-hypothesis dependence invalidates the real-data application is a correctness/robustness concern, not a circularity. No step in the derivation is equivalent to its own input by construction, and no fitted parameter is renamed as a prediction.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The method's central claims rest on: (1) the standard asymptotic behavior of the path-specific p-values (uniform under the corresponding null, degenerate at 0 under the alternative), (2) the asymptotic independence of the two base p-values within a hypothesis (score orthogonality, cited from Liu et al. 2022 and Dai et al. 2022), and (3) mutual independence of p-values across hypotheses, which is the most fragile assumption since real CpG data are correlated. The bound sequence cJ is a calibration constant chosen by simulation or the Gumbel approximation; it alters power but not the FDR guarantee. No invented entities.

free parameters (2)
  • λ (Storey estimator tuning parameter)
    User-prespecified threshold in the Storey FDR estimator; the proof in Theorem 2 holds for any λ∈(0,1), so it does not affect the FDR control claim, but the paper never states the value used in simulations or the NAS analysis.
  • cJ (bounding sequence in Meinshausen-Rice / adSMR) = csim_J (simulation) or cind_J (Gumbel approximation)
    Calibrated to satisfy P(VJ > cJ) → 0 under the null. It influences the Stage 1 threshold and hence power, but not the asymptotic FDR guarantee. Not fitted to observed data.
axioms (5)
  • domain assumption The base p-values p1j and p2j are asymptotically independent under the relevant null hypotheses (score orthogonality).
    Assumed in Proposition 1 and inherited from Liu et al. (2022)/Dai et al. (2022); the paper does not re-derive it. It underpins the independence of the transformed pair.
  • domain assumption For each base hypothesis, the p-value converges in distribution to Unif(0,1) under the null and to a point mass at 0 under the alternative.
    Standard asymptotic behavior of consistent tests; stated as the proposition's assumption in Section 2.
  • domain assumption Null p-values are mutually independent across hypotheses j (for Theorem 2).
    Stated in Theorem 2. Not satisfied in real genomic applications; the paper covers dependence only through simulation (Section 3, block-exchangeable correlation).
  • standard math Rényi's representation / order-statistics distributional facts for i.i.d. uniforms; continuous mapping theorem; optional stopping theorem.
    Used in the proofs of Proposition 1 (S1) and Theorem 2 (S2/S3).
  • domain assumption The linear mediation models in (1) are correctly specified with independent errors, so the path-specific p-values are valid.
    Underlies all inference; standard in the mediation literature.

pith-pipeline@v1.3.0-alltime-deepseek · 18506 in / 17872 out tokens · 144542 ms · 2026-08-01T17:34:29.321921+00:00 · methodology

0 comments
read the original abstract

In testing large-scale mediation hypotheses, the exposure-mediator and mediator-outcome path-specific p-values obtained for each hypothesis from the same sample can be combined into a pair of asymptotically independent p-values, which can then be used to test a global null hypothesis and a composite mediation null hypothesis, respectively. A first-stage screening procedure targeting the more stringent global null can effectively reduce the number of mediation hypotheses to be tested in the second stage, which in turn reduces the conservativeness of multiple comparison. The framework can incorporate any data-adaptive choice of screening threshold in Stage 1, and any step-up false discovery rate control method in Stage 2. The proposed procedure controls the false discovery rate asymptotically while being consistently well powered across a wide range of scenarios.

Figures

Figures reproduced from arXiv: 2607.17579 by Kwun Chuen Gary Chan, Yueqi Xu.

Figure 1
Figure 1. Figure 1: Q-Q plot comparing the original ordered p-values [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Empirical power (top row), FDR (middle row), and FWER (bottom row) of five mediation [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Empirical power (top row), FDR (middle row), and FWER (bottom row) of five mediation [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Q-Q plots comparing the rescaled p-values in the NAS data application against the [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

discussion (0)

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