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REVIEW 1 major objections 5 minor 79 references

False Discovery Control in Multiple Testing: A Brief Overview of Theories and Methodologies

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This review claims that the vast majority of offline FDR control methods fit a three-step pattern — rank, estimate the false discovery proportion, then threshold — and organizes the literature around that pattern.

desk verdict A solid, useful FDR-control review with one clear typo in the adaptive-BH threshold range; publish after a minor fix. read the letter →

arxiv 2411.10647 v1 pith:VSGKN3IJ submitted 2024-11-16 stat.ME math.STstat.APstat.TH

Jianliang He , Bowen Gang , Luella Fu This is my paper · ORCID
classification stat.MEmath.STstat.APstat.TH MSC 62H1562J1562F03
keywords falsediscoveryratemultipletestingcomparisonstype1errorBenjamini-Hochbergprocedurelocale-valuemirrorsequence
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

This paper is a review of methods that control the false discovery rate (FDR), the expected proportion of false positives among rejected hypotheses. Its central organizing claim is that almost all offline FDR methods fit a three-step pattern: rank the hypotheses with a summary statistic, estimate the false discovery proportion (FDP) at each possible threshold, then reject everything whose statistic falls below the largest threshold with estimated FDP at or below the target level $\alpha$. The authors demonstrate the pattern on the Benjamini-Hochberg procedure, null-proportion adaptive versions, mirror-sequence and knockoff methods, weighted p-values, the Bayesian Sun-Cai local-FDR procedure, and dependence-robust methods including e-BH. A reader who follows the review gets a common vocabulary for comparing methods, because most innovation in the field happens in either the ranking step or the FDP-estimation step.

What carries the argument

The load-bearing object is the three-step framework, with the FDP estimators that make it concrete. For any threshold $t$, an FDR method must produce a ranking statistic $T_i$, an estimator $\widehat{\mathrm{FDP}}(t)$ for the proportion of false rejections among tests with $T_i \le t$, and a threshold $t_\alpha = \sup\{t : \widehat{\mathrm{FDP}}(t) \le \alpha\}$. The paper supplies explicit estimators that all turn on the same identity: the expected number of null p-values below $t$ divided by the observed number of rejections. The BH estimator $mt/\sum_i \mathbf{1}(P_i \le t)$, the Storey estimator $\hat\pi_0(\lambda) mt / \sum_i \mathbf{1}(P_i \le t)$, the mirror-sequence estimator $(1 + \sum_i \mathbf{1}(1-P_i \le t))/\sum_i \mathbf{1}(P_i \le t)$, and the weighted-BH estimator $\sum_i w_i t / \sum_i \mathbf{1}(P_i/w_i \le t)$ are all variants of this comparison, as is the Sun-Cai estimator that replaces counts by sums of local FDR values. The framework does the work of unifying frequentist and Bayesian methods under one decision rule.

What would settle it

One concrete test is a literature count: collect the offline FDR-controlling procedures from a fixed set of recent papers and textbooks and check whether each one explicitly ranks hypotheses and defines an estimated FDP function before thresholding. If a substantial share, say a quarter or more, do not, then the 'vast majority' claim in Section 2.1 fails.

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Extended reading notes

Core claim

The paper's central claim, stated in Section 2.1, is that the vast majority of methodologies developed to control FDR in offline analyses adhere to the three-step framework of ranking, FDP estimation, and thresholding. It then recasts the field's main procedures as instances of that recipe: the BH procedure uses p-values for ranking and the plug-in estimator $\widehat{\mathrm{FDP}}(t) = mt/\sum_i \mathbf{1}(P_i \le t)$; Storey's adaptive version multiplies that estimator by an estimated null proportion $\hat\pi_0(\lambda)$; mirror-sequence methods use the null symmetry of $P_i$ and $1-P_i$ to estimate FDP; weighted BH uses weights in both ranking and FDP estimation; and the Sun-Cai procedure replaces p-values by the local false discovery rate $\mathrm{Lfdr}(X_i) = P(\theta_i=0 \mid X_i)$, the posterior probability that a hypothesis is null given its statistic, which the paper argues carries more information about the alternative distribution. The paper also reviews dependence-robust methods, including the Benjamini-Yekutieli correction, conditional calibration, and e-BH, framed as ways to keep FDP estimation valid when independence fails. The claim is that the differences among methods are largely differences in Step 1 and Step 2, not in the overall decision structure.

Load-bearing premise

The claim stands or falls on the accuracy of the paper's restatements of other people's theorems and on whether its selection of methods fairly represents the offline FDR literature.

Editorial extensions

If this is right

  • Because the BH procedure controls FDR at $\alpha |H_0|/m$ under independence and PRDS dependence, practitioners can expect it to be conservative when many nulls are true, and can sharpen it by estimating the null proportion.
  • Ranking by local FDR instead of p-value can improve power, because the local FDR uses information about the alternative distribution; the paper states this is why the Sun-Cai procedure can beat optimal p-value procedures.
  • Mirror-sequence and mirror-statistic constructions, including knockoffs, control FDR in finite samples by exploiting symmetry under the null, without requiring a user-chosen tuning parameter like $\lambda$.
  • Under arbitrary dependence, e-BH controls FDR by turning e-values into super-uniform p-values, and e-values remain valid under optional stopping, making them attractive for sequential and aggregate evidence.
  • The three-step framework gives practitioners a direct way to compare methods: any new offline procedure can be described by what statistic it ranks on and how it estimates FDP before thresholding.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The three-step decomposition is implicitly a design recipe: a methodologist can invent a new FDR procedure by swapping in a better ranking statistic or a sharper FDP estimator while leaving the thresholding rule untouched.
  • Because the review is explicitly limited to offline testing, a natural extension of the framework is to online FDR procedures; the paper's discussion of e-values and martingales suggests the same ranking-estimation-thresholding logic may carry over in a time-indexed form.
  • The paper's framing predicts that methods will be compared mainly by which step they improve, so a reader can classify any newly published FDR method by asking whether it contributes a new statistic, a new FDP estimator, or a new thresholding rule.
  • A testable consequence of the e-value material is that in strongly dependent settings e-BH will typically reject less than BH or adaptive BH; the paper cites comparisons but does not itself run one, so a simulation study could quantify the power gap.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. This manuscript is a review article on false discovery rate (FDR) control in offline multiple testing. It proposes a three-step framework (ranking, FDP estimation, thresholding) and uses it to organize a large literature: the Benjamini-Hochberg procedure and its variants (adaptive/π0 procedures, mirror sequences, weighted p-values), the Sun-Cai local FDR procedure and its side-information extensions, FDR control under dependence (PRDS, weak dependence, general dependence via the BY correction, dBH, and e-values), and pointers to online testing. The paper is explicitly non-exhaustive and directs readers to original sources; its own contribution is an expository conceptual map rather than new theorems.

Significance. If corrected, the survey would be a useful and generally accurate compact map of a large and rapidly evolving literature. Its strengths are the clear three-step organizing principle, the broadly correct statements of the central results (BH, PRDS, e-BH), and the up-to-date coverage of mirror sequences, knockoffs, conformal p-values, and generalized e-values. The paper is also commendably explicit about its selective scope and cites accessible proofs for several key results. However, because the value of the paper is precisely that readers can rely on its formulas and conditions, a correctness error in the adaptive-BH threshold interval and its propagation into the mirror-sequence derivation are not merely cosmetic; they need to be fixed before the review can serve as a dependable reference.

major comments (1)
  1. [Section 3.2 and Section 3.3 (Eq. (7)-(9))] The definition of t_λ_α as sup{t ∈ (0, 1−λ] : \hat FDP_λ(t) ≤ α} is incorrect as printed and should be sup{t ∈ (0, λ] : ...}. Since \hat π0(λ) is estimated from the upper tail {P_i ≥ λ}, the conservative interpretation of \hat FDP_λ(t) requires the rejection set {P_i ≤ t} and the estimation set {P_i ≥ λ} to be disjoint, which is guaranteed by t ≤ λ. The printed range permits t > λ, where the estimator can be anti-conservative. For example, under the global null with m = 2, λ = 0.2, and t = 0.8, the event that both p-values are below 0.5 gives \hat FDP_λ(t) = 0.5 while the true FDP is 1. This error propagates to Section 3.3: substituting λ = 1 − t into the corrected inequality t ≤ λ yields t ≤ 0.5, which is exactly Eq. (9), whereas substituting into the printed inequality t ≤ 1 − λ gives only the tautology t ≤ t and cannot justify the 0.5 bound. Theorem 2 itself appears correct under the corrected range, but the derivation as printed is incoherent.
minor comments (5)
  1. [Section 3.2, Eq. (5)-(6)] The sentence 'The ≥ in (6) is sharper when λ is bigger' is unclear and seems backwards, since the lower bound involves P(P_i ≥ λ), which decreases as λ increases; please rephrase or correct.
  2. [Section 3.3, Theorem 2] Unlike Theorem 1, Theorem 2 states no explicit assumptions on the p-values; please state the required conditions (e.g., independence of null p-values, or the exact conditions in Leung and Sun 2022) so that the theorem statement is self-contained.
  3. [Section 3.4] There is a notational slip in the sentence defining the weights: 'independent of P_1, ..., P_n' should be 'independent of P_1, ..., P_m'.
  4. [Section 5.1, Example 1] The PRDS characterization for the multivariate normal is garbled: it should be stated in terms of the index set I0 (not H0) and, in the standard form, nonnegative covariances Σij ≥ 0 rather than strict positivity.
  5. [Throughout] Please copyedit for typos such as 'acros s', 'Univeristy', 'is is', 'weighed' in Section 3.4, and 'a overview' in Section 5.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a survey whose descriptions are attributed to external sources, and its own organizational framework is not used to derive any of the cited results.

full rationale

This is a review article rather than a derivation paper: the formal claims it reports are explicitly attributed to external literature (e.g., Theorem 1 to Benjamini and Hochberg 1995, Theorem 2 to Leung and Sun 2022, the adaptive-BH result to Storey et al. 2004, the e-BH guarantee to Wang and Ramdas 2022). The paper advances no new theorem whose conclusion depends on accepting the authors' own prior findings. The self-citations that do appear, such as Fu et al. (2022) in Section 4.2 and Banerjee et al. (2023) in Section 5.3, are used as literature pointers alongside independent external references, and they do not supply the premises of any load-bearing argument in this paper. Section 2.1's 'ranking, FDP estimation, thresholding' schema is an expository organizational device, not a result derived from or equivalent to the methods it later describes; the subsequent sections demonstrate how published procedures fit that schema, but the schema itself imposes no mathematical constraint on those procedures. The skeptical observation about the printed range t ∈ (0,1−λ] in Section 3.2, and the substitution λ = 1−t in Section 3.3, is best understood as an internal correctness or typographical issue in the exposition: the displayed FDP estimator and the attributed FDR-control theorem are not defined in terms of the conclusion that the survey wants to draw, so the error does not make the derivation circular. No step in the paper reduces, by construction or by self-citation, to its own inputs, so the appropriate circularity score is 0.

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

This review introduces no new parameters or entities. The axioms listed are standard assumptions of the methods being surveyed; the review reports these assumptions but does not derive them. All theorems are attributed to the cited literature.

assumptions (4)
  • standard math FDR is defined as the expectation of the false discovery proportion (Section 2, Eq. 1).
    Definition used throughout the review; accepted in the multiple testing literature.
  • domain assumption Under the null hypothesis, p-values are uniformly distributed on [0,1] (Section 2, paragraph 1).
    Underpins the BH procedure and related estimators of the false discovery proportion.
  • domain assumption Independence of p-values under the null for the BH theorem (Theorem 1).
    Condition stated in Theorem 1 for the BH procedure's FDR control.
  • domain assumption For Theorem 3, p-values satisfy positive regression dependence (PRDS) on the set of true nulls.
    Condition under which the BH procedure controls FDR, as defined in Section 5.1.

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Cite this review

Pith. "Pith review of False Discovery Control in Multiple Testing: A Brief Overview of Theories and Methodologies." pith.science (2026). https://pith.science/paper/VSGKN3IJ

@misc{pith2026241110647,
  author       = {Pith},
  title        = {Pith review of: False Discovery Control in Multiple Testing: A Brief Overview of Theories and Methodologies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VSGKN3IJ}},
  note         = {Machine review of arXiv:2411.10647}
}
read the original abstract

As the volume and complexity of data continue to expand across various scientific disciplines, the need for robust methods to account for the multiplicity of comparisons has grown widespread. A popular measure of type 1 error rate in multiple testing literature is the false discovery rate (FDR). The FDR provides a powerful and practical approach to large-scale multiple testing and has been successfully used in a wide range of applications. The concept of FDR has gained wide acceptance in the statistical community and various methods has been proposed to control the FDR. In this work, we review the latest developments in FDR control methodologies. We also develop a conceptual framework to better describe this vast literature; understand its intuition and key ideas; and provide guidance for the researcher interested in both the application and development of the methodology.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.