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Controlling the false discovery rate under a non-parametric graphical dependence model

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper establishes a finite-sample FDR guarantee for any multiple testing procedure that is monotone, self-consistent, and neighbor-blind under a declared dependency graph, and it constructs efficient procedures that meet these…

desk verdict Solid and useful FDR framework for dependency graphs, but the paper's stated 0/0=1 convention formally invalidates Theorem 1 as written; fix that and the central contribution stands. read the letter →

arxiv 2506.24126 v1 pith:IXVI4FIB submitted 2025-06-30 stat.ME math.STstat.TH

Drew T. Nguyen , William Fithian This is my paper · ORCID
classification stat.MEmath.STstat.TH MSC 62H1562J15
keywords multipletestingfalsediscoveryratedependencygraphlocaldependenceBenjamini-Hochbergprocedureindependentsetsp-valuesgenome-wideassociationstudies
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

The paper establishes finite-sample false discovery rate control under a non-parametric model of local dependence: the analyst declares a graph on the p-values, and only p-values outside each other's declared neighborhoods are assumed independent. Its main theorem shows that any D-adapted procedure—one that is monotone, self-consistent, and neighbor-blind—controls FDR at the nominal level when the declared graph is a true dependency graph. The paper then constructs a family of D-adapted procedures, IndBH and its iterated improvements, whose rejection set matches Benjamini–Hochberg (BH) whenever the BH rejections have no edges between them. Because the procedures ignore hypotheses outside the BH rejection set, they scale to large problems while retaining BH-like power on sparse graphs. The upshot is a rigorous, computationally practical middle ground between running BH uncorrected and paying the full conservative cost of the Benjamini–Yekutieli correction.

What carries the argument

The key machinery is the dependency graph D with neighborhoods $N_i$, where the only assumption is marginal independence between $p_i$ and all $p_j$ outside $N_i$. A D-adapted procedure is characterized by local thresholds: $i$ is rejected exactly when $p_i \le c_i(S_i)$ with $S_i = (p_j)_{j \notin N_i}$, so the threshold is independent of $p_i$. The procedures IndBH and IndBH(k) are built from independent sets: IndBH unions the BH rejection sets obtained after masking every independent set's complement, and the gap-chasing update, $R^{(k+1)}(p) = \{i : p_i \le \alpha|\{i\} \cup R^{(k)}(1_{N_i^\circ} p)|/m\}$, sharpens any monotone procedure toward the most liberal D-adapted procedure SUD while preserving D-adaptivity.

What would settle it

Simulate m null p-values from equicorrelated Gaussian blocks with edges exactly matching the blocks, run IndBH(0.1) on $10^4$ replicates, and compare the empirical FDR to 0.1; because the declared graph is a true dependency graph, Theorem 1 predicts FDR $\le 0.1$ (indeed $\le 0.1|H_0|/m$), so any systematic excess would refute the theorem.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: whenever D is a dependency graph for p, any D-adapted procedure controls FDR at level $\alpha|H_0|/m$, hence at the nominal level $\alpha$. The three defining conditions do the work: self-consistency (rejections require $p_i \le \alpha|R|/m$), monotonicity (smaller p-values only add rejections), and neighbor-blindness (whether $i$ is rejected is unchanged when all p-values in $i$'s punctured neighborhood are masked to 1). Neighbor-blindness makes the local threshold for each null $p_i$ depend only on p-values outside its neighborhood, which are independent of $p_i$, reducing each FDR contribution to a superuniformity bound. The paper's main procedures—IndBH, IndBH(k), and the optimal SUD—are all D-adapted; IndBH reruns BH on masked vectors for every independent set and unions the results, and its rejection set is exactly the BH rejection set when no BH rejections are connected.

Load-bearing premise

The analyst's graph must be a true dependency graph: each $p_i$ must be genuinely independent of every $p_j$ outside its declared neighborhood, and if real dependence leaks outside those neighborhoods, the finite-sample guarantee can fail.

Editorial extensions

If this is right

  • IndBH and IndBH(k) provide finite-sample FDR control for any true dependency graph, with no parametric model of the dependence inside neighborhoods.
  • On sparse graphs, IndBH(3) approaches BH's power; on block-dependent data, IndBH reduces to a simple threshold based on the BH rejection set of the per-block minima.
  • All of the proposed procedures reduce to BH on the empty graph and to Bonferroni on the complete graph, interpolating between independence and fully arbitrary dependence.
  • The methods are computationally safe to run on large data because they need only examine the BH rejection set; in simulations with $10^6$ hypotheses and a 1% non-null rate, IndBH finishes in about a second and IndBH(3) in about five seconds.

Reading between the lines

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

  • Because the proof only uses the local-threshold structure, any future procedure that enforces neighbor-blindness while preserving monotonicity and self-consistency will inherit the same FDR guarantee, including weighted or covariate-adaptive variants that choose independent sets adaptively.
  • The same decomposition of FDR into per-hypothesis superuniformity bounds could be exported to decision problems beyond reject/accept, such as constructing confidence sets for selected parameters under local dependence.
  • The paper's bounds on BH's worst-case FDR under a dependency graph suggest a practical diagnostic: for block dependence, the inflation cost of running BH is roughly a factor of $\log(\text{block size})$, so one could report BH results alongside a graph-based inflation check.
  • A data-driven version that estimates the dependency graph from the same p-values rather than from an external reference would need to account for estimation error, since the current guarantee holds only for a pre-specified true graph.
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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

3 major / 4 minor

Summary. The paper proposes a nonparametric dependency-graph model for p-values, defines D-adapted multiple testing procedures via three conditions (self-consistency, monotonicity, neighbor-blindness), and proves finite-sample FDR control at level α|H0|/m whenever the input graph is a true dependency graph (Theorem 1) and under a partial positive regression dependence extension (Theorem 2). It then introduces practical procedures: IndBH, its iterated improvement IndBH(k), and the optimal-but-hard-to-compute SUD, with a 'gap chasing' framework (Theorem 3) and computational reductions based on the BH rejection set and connected components. Simulations and a schizophrenia GWAS example illustrate near-BH power in sparse dependence settings.

Significance. If valid, the results give a clean middle ground between full independence and arbitrary dependence: any procedure satisfying the three local conditions inherits a finite-sample FDR bound, and the proposed IndBH procedures interpolate between BH and Bonferroni while remaining computationally tractable when the BH rejection set is small. The core proof of Theorem 1 is self-contained and elegant, the gap-chasing iteration is a useful algorithmic idea, and the authors ship an R package and reproducible code, which strengthens the paper's practical value. The main formal issues (the 0/0 convention in the FDR definition, and the unjustified PRD statement in Theorem 2's proof) are localized and fixable, but they do need correction before the claims as stated are supported.

major comments (3)
  1. [Section 2.1 and Theorem 1] The FDR definition in Section 2.1 states the convention 0/0=1 inside the expectation. Under this convention, the empty procedure R_α(p)=∅ has FDP=0/0=1 for every p, hence FDR=1, while it is trivially D-adapted for any dependency graph; this directly contradicts Theorem 1, which would require FDR ≤ α|H0|/m < 1 for α<1. The proof of Theorem 1 uses the identity FDR = Σ_{i∈H0} E[1{i∈R(p)}/|R(p)|], which is only valid when terms with zero numerator are evaluated as 0, i.e., 0/0=0. The convention should be changed to 0/0=0 (or the proof should handle the empty-rejection event separately), and the statement of Theorem 1 should be aligned with that correction.
  2. [Appendix A, proof of Theorem 2] The proof asserts that 'Since p is PPRD on H0 w.r.t. D, we have that q is PRD on H0.' This is not implied by PPRD: PPRD only gives monotonicity of P(p_{N_i^c}∈K | p_i=t) for each i and increasing sets in p_{N_i^c}, whereas PRD of q would require monotonicity for all increasing sets in the full q-space, including coordinates that are unmasked neighbors of other indices. For the fixed i in the FDR contribution, the needed conditional monotonicity does follow from PPRD, so the proof can be repaired, but as written the application of Lemma 1 is not justified. In the same chain, the equality labelled (P1) should be an inequality, since P1 gives only one direction of the indicator inclusion.
  3. [Section 5.4] The real-data dependency graph is built by drawing edges between SNPs on the same chromosome with LD r^2>0.2. Definition 2.1 requires that each p_i be independent of all p_j with j outside N_i; the threshold r^2>0.2 does not guarantee independence for pairs with r^2≤0.2, and the same-chromosome rule is a proxy. Consequently, the finite-sample FDR guarantee of Theorem 1 does not formally transfer to the GWAS analysis as presented. The authors should either state this limitation explicitly or provide additional justification that the resulting graph is a true dependency graph for the SNP p-values.
minor comments (4)
  1. [Appendix A, Lemma 2 proof] The symbol i is used both as the running index in the set comprehension and as the fixed index being masked; the final expression R^{IndBH(k+1)}_{α',-i}(p_{-i}) and the reference to R^{IndBH(k+1)}_α(1_{\{i\}}p) should use the fixed index j to avoid confusion.
  2. [Appendix A, Lemma 1] Lemma 1 is stated with the convention 0/0=1, which interacts with the FDR definition issue raised in the major comments; if the convention is corrected to 0/0=0 in Section 2.1, Lemma 1 and its use in Theorem 2 should be restated consistently.
  3. [Appendix D, display (23)] The upper-bound notation (|N_i| − (|N_i| − 1)m^{−1/(|N_i|−1)}}?) is typeset in a way that is hard to parse; please clarify the exponent and whether the intended expression matches Proposition 11's L_{D,i}.
  4. [Section 4.2.2] The inequalities β+ ≥ β^{IndBH}_{α,i}(p_{−i}) and β− ≤ β^{IndBH}_{α,i}(p_{−i}) are stated without proof in the main text; a short justification or an explicit pointer to the appendix derivation would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the FDR-control theorem is derived from the paper's own D-adaptivity conditions, with self-citations used only as supporting tools.

full rationale

The central claim (Theorem 1) is derived in the paper from the three D-adaptivity conditions (P1)-(P3), with a self-contained proof that uses only superuniformity of null p-values, the graph-masking argument, and the tower rule; no parameter is fitted to data and no external result containing the target FDR bound is assumed. IndBH and IndBH(k) are shown to satisfy (P1)-(P3) by direct arguments in Propositions 3-5 and Theorem 3, so the inherited guarantees do not reduce to the definitions of the procedures. Citations to Blanchard and Roquain (2008) and Ramdas et al. (2019) supply a standard superuniformity lemma, and the citation to Fithian and Lei (2020) provides a conditional-calibration perspective in Section 6.1.2 and a pruning idea in Appendix C; these are supporting tools or framing, not load-bearing premises of the main theorem. Two non-circular caveats are flagged for completeness: (i) Section 2.1 defines FDR with the convention 0/0=1, while the proof of Theorem 1 uses the decomposition FDR = sum_i E[1{i in R}/|R|], which requires 0/0=0; as written, the empty, D-adapted procedure would have FDR 1, contradicting Theorem 1 for alpha<1. This is an internal convention inconsistency, not an equivalence-to-inputs circularity, and it does not raise the circularity score. (ii) The GWAS example builds the dependency graph from LD r2>0.2 (Section 5.4), which is an approximation that may not strictly satisfy Definition 2.1; this is a validity caveat about the supplied graph, not a circular step. The paper is otherwise self-contained, including simulation comparisons to BH and BY as external benchmarks.

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

The method has no fitted parameters; alpha and the graph D are user inputs. The load-bearing assumptions are superuniform null p-values and correctness of the dependency graph. The paper also imports the superuniformity lemma from prior work and, for the extended theorem, assumes PPRD. No new entities are postulated.

assumptions (4)
  • domain assumption Null p-values are super-uniform, meaning stochastically larger than Uniform[0,1].
    Standard in multiple testing; used in the Theorem 1 proof through the tower rule and superuniformity.
  • domain assumption The user-specified graph D is a correct dependency graph: for every i, p_i is independent of (p_j) for j outside N_i.
    Load-bearing; if the graph is too sparse, FDR control is not guaranteed. Stated in Definition 2.1.
  • domain assumption For Theorem 2, partial positive regression dependence (PPRD) on H0 with respect to D.
    Weaker than independence and used to extend FDR control; Proposition 1 verifies it for multivariate t and linear model p-values.
  • standard math The superuniformity lemma of Blanchard-Roquain and Ramdas et al. (Lemma 1).
    Imported from prior literature; used in the proof of Theorem 2.

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Cite this review

Pith. "Pith review of Controlling the false discovery rate under a non-parametric graphical dependence model." pith.science (2026). https://pith.science/paper/IXVI4FIB

@misc{pith2026250624126,
  author       = {Pith},
  title        = {Pith review of: Controlling the false discovery rate under a non-parametric graphical dependence model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IXVI4FIB}},
  note         = {Machine review of arXiv:2506.24126}
}
abstract

We propose sufficient conditions and computationally efficient procedures for false discovery rate control in multiple testing when the $p$-values are related by a known \emph{dependency graph} -- meaning that we assume independence of $p$-values that are not within each other's neighborhoods, but otherwise leave the dependence unspecified. Our methods' rejection sets coincide with that of the Benjamini--Hochberg (BH) procedure whenever there are no edges between BH rejections, and we find in simulations and a genomics data example that their power approaches that of the BH procedure when there are few such edges, as is commonly the case. Because our methods ignore all hypotheses not in the BH rejection set, they are computationally efficient whenever that set is small. Our fastest method, the IndBH procedure, typically finishes within seconds even in simulations with up to one million hypotheses.

Figures

Figures reproduced from arXiv: 2506.24126 by the authors.

Figure 1
Figure 1. The expected true positive ratio, relative to BH, of our [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Some notation depicted on a dependency graph with [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. An example realization of m = 5 p-values on the graph from [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: The same graph and p-values from [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Results for block dependence with scattered signals. FDR control level set to [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Results for banded dependence with clustered signals. FDR control level set to [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: A rug plot of one generation of the non-nulls [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Results with distributions designed to inflate FDR. FDR control level set to [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Top: A Manhattan plot of the SNP p-values from the schizophrenia GWAS of Ripke et al. (2014), after filtering to about 140,000 p-values. Bottom left, middle: Dependency matrices for SNP p-values within a single chromosome. (SNPs on different chromosomes are assumed ind…
Figure 10
Figure 10. Figure 10: When α = 0.05, the IndBH(∞) (α) procedure gives the same rejection set as IndBH(2)(α)—the fixed point is reached after only one iteration. This rejection set is H3, H4, H5, but SU(α) rejects everything. Monotonicity holds because Rα is monotone. Finally, neighbor-blin…
Figure 11
Figure 11. Figure 11: Results for block dependence with scattered signals. FDR control level set to [PITH_FULL_IMAGE:figures/full_fig_p043_11.png]
Figure 12
Figure 12. Figure 12: Results for banded dependence with clustered signals. FDR control level set to [PITH_FULL_IMAGE:figures/full_fig_p043_12.png]
Figure 13
Figure 13. Figure 13: Results for block dependence with scattered signals. FDR control level set to [PITH_FULL_IMAGE:figures/full_fig_p044_13.png]
Figure 14
Figure 14. Figure 14: Results for banded dependence with clustered signals. FDR control level set [PITH_FULL_IMAGE:figures/full_fig_p044_14.png]
Figure 15
Figure 15. Figure 15: Results for block dependence with scattered signals. FDR control level set to [PITH_FULL_IMAGE:figures/full_fig_p045_15.png]
Figure 16
Figure 16. Figure 16: Results for banded dependence with clustered signals. FDR control level set to [PITH_FULL_IMAGE:figures/full_fig_p045_16.png]
Figure 17
Figure 17. Figure 17: Results for block dependence with scattered signals. FDR control level set to [PITH_FULL_IMAGE:figures/full_fig_p046_17.png]
Figure 18
Figure 18. Figure 18: Results for banded dependence with clustered signals. FDR control level set [PITH_FULL_IMAGE:figures/full_fig_p046_18.png]

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Works this paper leans on

38 extracted references · 23 canonical work pages

  1. [1]

    Controlling the False Discovery Rate : A Practical and Powerful Approach to Multiple Testing

    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 0 (1): 0 289--300, 1995. ISSN 0035-9246. URL https://www.jstor.org/stable/2346101. Publisher: [Royal Statistical Society, Wiley]

    arXiv 1995

  2. [2]

    The control of the false discovery rate in multiple testing under dependency

    Yoav Benjamini and Daniel Yekutieli. The control of the false discovery rate in multiple testing under dependency. The Annals of Statistics, 29 0 (4): 0 1165--1188, August 2001. ISSN 0090-5364, 2168-8966. doi:10.1214/aos/1013699998. URL https://projecteuclid.org/journals/annals-of-statistics/volume-29/issue-4/The-control-of-the-false-discovery-rate-in-mul...

    arXiv 2001

  3. [3]

    Two simple sufficient conditions for FDR control

    Gilles Blanchard and Etienne Roquain. Two simple sufficient conditions for FDR control. Electronic Journal of Statistics, 2, October 2008. doi:10.1214/08-EJS180

    doi:10.1214/08-ejs180 2008

  4. [4]

    Louis H. Y. Chen and Qi-Man Shao. Normal approximation under local dependence. The Annals of Probability, 32 0 (3), July 2004. ISSN 0091-1798. doi:10.1214/009117904000000450

    doi:10.1214/009117904000000450 2004

  5. [5]

    Multiple testing under negative dependence

    Ziyu Chi, Aaditya Ramdas, and Ruodu Wang. Multiple testing under negative dependence. arXiv preprint arXiv:2212.09706, 2022

    arXiv 2022

  6. [6]

    Igraph for R : R interface of the igraph library for graph theory and network analysis

    G \'a bor Cs \'a rdi, Tam \'a s Nepusz, Kirill M \"u ller, Szabolcs Horv \'a t, Vincent Traag, Fabio Zanini, and Daniel Noom. Igraph for R : R interface of the igraph library for graph theory and network analysis. Zenodo, January 2025

    work page 2025

  7. [7]

    The International Genome Sample Resource ( IGSR ) collection of open human genomic variation resources

    Susan Fairley, Ernesto Lowy-Gallego , Emily Perry, and Paul Flicek. The International Genome Sample Resource ( IGSR ) collection of open human genomic variation resources. Nucleic Acids Research, 48 0 (D1): 0 D941--D947, January 2020. ISSN 0305-1048. doi:10.1093/nar/gkz836

    doi:10.1093/nar/gkz836 2020

  8. [8]

    Some Results on the Control of the False Discovery Rate under Dependence

    Alessio Farcomeni. Some Results on the Control of the False Discovery Rate under Dependence . Scandinavian Journal of Statistics, 34 0 (2): 0 275--297, 2007. ISSN 1467-9469. doi:10.1111/j.1467-9469.2006.00530.x. URL https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9469.2006.00530.x. \_eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-94...

    arXiv 2007

Show all 38 references
  1. [9]

    Helmut Finner and M. Roters. On the False Discovery Rate and Expected Type I Errors . Biometrical Journal, 43 0 (8): 0 985--1005, 2001. ISSN 1521-4036. doi:10.1002/1521-4036(200112)43:8<985::AID-BIMJ985>3.0.CO;2-4

    2001 doi

  2. [10]

    Dependency and false discovery rate: Asymptotics

    Helmut Finner, Thorsten Dickhaus, and Markus Roters. Dependency and false discovery rate: Asymptotics . The Annals of Statistics, 35 0 (4): 0 1432--1455, August 2007. ISSN 0090-5364, 2168-8966. doi:10.1214/009053607000000046. URL https://projecteuclid.org/journals/annals-of-st...

    2007 doi

  3. [11]

    Conditional calibration for false discovery rate control under dependence, July 2020

    William Fithian and Lihua Lei. Conditional calibration for false discovery rate control under dependence, July 2020. URL http://arxiv.org/abs/2007.10438. arXiv:2007.10438 [math, stat]

    2020 arXiv

  4. [12]

    Goeman and Aldo Solari

    Jelle J. Goeman and Aldo Solari. Multiple hypothesis testing in genomics. Statistics in Medicine, 33 0 (11): 0 1946--1978, 2014. ISSN 1097-0258. doi:10.1002/sim.6082. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/sim.6082. \_eprint: https://onlinelibrary.wiley.com/doi/pd...

    1946 doi

  5. [13]

    Adaptive controls of FWER and FDR under block dependence

    Wenge Guo and Sanat Sarkar. Adaptive controls of FWER and FDR under block dependence. Journal of Statistical Planning and Inference, 208: 0 13--24, September 2020. ISSN 03783758. doi:10.1016/j.jspi.2018.03.008. URL https://linkinghub.elsevier.com/retrieve/pii/S0378375819301181

    2020 doi

  6. [14]

    Covariate powered cross-weighted multiple testing

    Nikolaos Ignatiadis and Wolfgang Huber. Covariate powered cross-weighted multiple testing. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 83 0 (4): 0 720--751, 2021. ISSN 1467-9868. doi:10.1111/rssb.12411. URL https://onlinelibrary.wiley.com/doi/...

    2021 doi

  7. [15]

    Large deviations for sums of partly dependent random variables

    Svante Janson. Large deviations for sums of partly dependent random variables. Random Structures & Algorithms, 24 0 (3): 0 234--248, May 2004. ISSN 1042-9832, 1098-2418. doi:10.1002/rsa.20008

    2004 doi

  8. [16]

    van de Wiel

    Kyung In Kim and Mark A. van de Wiel. Effects of dependence in high-dimensional multiple testing problems. BMC bioinformatics, 9: 0 114, February 2008. ISSN 1471-2105. doi:10.1186/1471-2105-9-114

    2008 doi

  9. [17]

    Kluger and Art B

    Dan M. Kluger and Art B. Owen. A central limit theorem for the Benjamini-Hochberg false discovery proportion under a factor model. Bernoulli, 30 0 (1), February 2024. ISSN 1350-7265. doi:10.3150/23-BEJ1615

    2024 doi

  10. [18]

    A Short Review on Novel Approaches for Maximum Clique Problem : From Classical algorithms to Graph Neural Networks and Quantum algorithms, March 2024

    Raffaele Marino, Lorenzo Buffoni, and Bogdan Zavalnij. A Short Review on Novel Approaches for Maximum Clique Problem : From Classical algorithms to Graph Neural Networks and Quantum algorithms, March 2024

    2024

  11. [19]

    Shaun Purcell, Benjamin Neale, Kathe Todd-Brown , Lori Thomas, Manuel A. R. Ferreira, David Bender, Julian Maller, Pamela Sklar, Paul I. W. de Bakker, Mark J. Daly, and Pak C. Sham. PLINK : A Tool Set for Whole-Genome Association and Population-Based Linkage Analyses . The Ame...

    2007 doi

  12. [20]

    Ramdas, Rina F

    Aaditya K. Ramdas, Rina F. Barber, Martin J. Wainwright, and Michael I. Jordan. A unified treatment of multiple testing with prior knowledge using the p-filter. The Annals of Statistics, 47 0 (5): 0 2790--2821, October 2019. ISSN 0090-5364, 2168-8966. doi:10.1214/18-AOS1765. U...

    2019 doi

  13. [21]

    FDR Control by the BH Procedure for Two - Sided Correlated Tests with Implications to Gene Expression Data Analysis

    Anat Reiner-Benaim. FDR Control by the BH Procedure for Two - Sided Correlated Tests with Implications to Gene Expression Data Analysis . Biometrical Journal, 49 0 (1): 0 107--126, 2007. ISSN 1521-4036. doi:10.1002/bimj.200510313. URL https://onlinelibrary.wiley.com/doi/abs/10...

    2007 doi

  14. [22]

    Biological Insights From 108 Schizophrenia-Associated Genetic Loci

    Stephan Ripke et al. Biological Insights From 108 Schizophrenia-Associated Genetic Loci . Nature, 511 0 (7510): 0 421--427, July 2014. ISSN 0028-0836. doi:10.1038/nature13595

    2014 doi

  15. [23]

    Romano and Azeem M

    Joseph P. Romano and Azeem M. Shaikh. Stepup Procedures for Control of Generalizations of the Familywise Error Rate . The Annals of Statistics, 34 0 (4): 0 1850--1873, 2006. ISSN 0090-5364

    2006

  16. [24]

    Inf \'e rence de graphes par une proc \'e dure de test multiple avec application en Neuroimagerie

    Marine Roux. Inf \'e rence de graphes par une proc \'e dure de test multiple avec application en Neuroimagerie . thesis , Universit \'e Grenoble Alpes (ComUE), September 2018

    2018

  17. [25]

    Sanat K. Sarkar. Some Results on False Discovery Rate in Stepwise multiple testing procedures. The Annals of Statistics, 30 0 (1), February 2002. ISSN 0090-5364. doi:10.1214/aos/1015362192. URL https://projecteuclid.org/journals/annals-of-statistics/volume-30/issue-1/Some-Resu...

    2002

  18. [26]

    Sanat K. Sarkar. On Controlling the False Discovery Rate in Multiple Testing of the Means of Correlated Normals Against Two-Sided Alternatives , April 2023

    2023

  19. [27]

    Linkage disequilibrium --- understanding the evolutionary past and mapping the medical future

    Montgomery Slatkin. Linkage disequilibrium --- understanding the evolutionary past and mapping the medical future. Nature Reviews Genetics, 9 0 (6): 0 477--485, June 2008. ISSN 1471-0064. doi:10.1038/nrg2361

    2008 doi

  20. [28]

    Aldo Solari and Jelle J. Goeman. Minimally adaptive BH : A tiny but uniform improvement of the procedure of Benjamini and Hochberg . Biometrical Journal. Biometrische Zeitschrift, 59 0 (4): 0 776--780, July 2017. ISSN 1521-4036. doi:10.1002/bimj.201500253

    2017 doi

  21. [29]

    Weijie J. Su. The FDR - Linking Theorem , December 2018. URL http://arxiv.org/abs/1812.08965. arXiv:1812.08965 [math, stat]

    2018 arXiv

  22. [30]

    On weighted hochberg procedures

    Ajit C Tamhane and Lingyun Liu. On weighted hochberg procedures. Biometrika, 95 0 (2): 0 279--294, 2008

    2008

  23. [31]

    Tamhane, Wei Liu, and Charles W

    Ajit C. Tamhane, Wei Liu, and Charles W. Dunnett. A generalized step-up-down multiple test procedure. Canadian Journal of Statistics, 26 0 (2): 0 353--363, June 1998. ISSN 0319-5724, 1708-945X. doi:10.2307/3315516

    1998 doi

  24. [32]

    A New Algorithm for Generating All the Maximal Independent Sets

    Shuji Tsukiyama, Mikio Ide, Hiromu Ariyoshi, and Isao Shirakawa. A New Algorithm for Generating All the Maximal Independent Sets . SIAM J. Comput., 6 0 (3): 0 505--517, September 1977. ISSN 0097-5397. doi:10.1137/0206036

    1977 doi

  25. [33]

    Wall and Jonathan K

    Jeffrey D. Wall and Jonathan K. Pritchard. Haplotype blocks and linkage disequilibrium in the human genome. Nature Reviews Genetics, 4 0 (8): 0 587--597, August 2003. ISSN 1471-0064. doi:10.1038/nrg1123

    2003 doi

  26. [34]

    False discovery rate control with e-values, December 2021

    Ruodu Wang and Aaditya Ramdas. False discovery rate control with e-values, December 2021. URL http://arxiv.org/abs/2009.02824. arXiv:2009.02824 [math, stat]

    2021 arXiv

  27. [35]

    Werling, Sirisha Pochareddy, Jinmyung Choi, Joon-Yong An, Brooke Sheppard, Minshi Peng, Zhen Li, Claudia Dastmalchi, Gabriel Santpere, André M

    Donna M. Werling, Sirisha Pochareddy, Jinmyung Choi, Joon-Yong An, Brooke Sheppard, Minshi Peng, Zhen Li, Claudia Dastmalchi, Gabriel Santpere, André M. M. Sousa, Andrew T. N. Tebbenkamp, Navjot Kaur, Forrest O. Gulden, Michael S. Breen, Lindsay Liang, Michael C. Gilson, Xuefa...

    2020

  28. [36]

    Thorsten Wiegand and Kirk A. Moloney. Handbook of Spatial Point-Pattern Analysis in Ecology . Chapman and Hall/CRC , New York, December 2013. ISBN 978-0-429-13811-9. doi:10.1201/b16195

    2013 doi

  29. [37]

    A review on algorithms for maximum clique problems

    Qinghua Wu and Jin-Kao Hao. A review on algorithms for maximum clique problems. European Journal of Operational Research, 242 0 (3): 0 693--709, May 2015. ISSN 03772217. doi:10.1016/j.ejor.2014.09.064

    2015 doi

  30. [38]

    A selective inference approach for false discovery rate control using multiomics covariates yields insights into disease risk

    Ronald Yurko, Max G’Sell, Kathryn Roeder, and Bernie Devlin. A selective inference approach for false discovery rate control using multiomics covariates yields insights into disease risk. Proceedings of the National Academy of Sciences, 117 0 (26): 0 15028--15035, June 2020. d...

    2020 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.