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Half-Cauchy combination test

Mathematical Methods AIMech:HalfCauchyCombinationTest DRAFT

Aliases: HCCT, Heavily right Half-Cauchy p-value combination test

A statistical p-value combination method that transforms m individual hypothesis test p-values p_1, ..., p_m in (0, 1] into non-negative standard Half-Cauchy random variables X_j = tan((1 - p_j) * pi / 2), forms the non-negatively weighted sum T_HC(p) = sum_{j=1}^m w_j tan((1 - p_j) * pi / 2) with sum_{j=1}^m w_j = 1, and evaluates global significance using the exact finite-sample Half-Cauchy convolution null distribution (or its location-shifted Landau domain-of-attraction approximation as m -> infinity), thereby remaining insensitive to large p-values near 1, guaranteeing strictly convex confidence regions under linear-Gaussian test inversion, and preserving asymptotic valid tail control under pairwise asymptotically independent or bivariate normal test statistics.

Identity Notes & Concept Scope No normalized identity match for the Half-Cauchy combination test across all 442 pinned AIO records and 565 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only frontier-tracking references in prior discovery reports noting Liu, Meng, & Pillai (2025) as a pending concept. Distinct from the standard Cauchy Combination Test (CCT) of Liu & Xie (2020), which maps p_j to full-support Cauchy variables tan((0.5 - p_j) * pi) that diverge to -infinity as p_j -> 1, allowing a single large p-value to cancel a small p-value and producing disconnected confidence sets when inverted. Distinct from the Harmonic Mean P-Value (HMP) and Exact Harmonic Mean P-Value (EHMP), which transform p_j via reciprocal Pareto variables 1/p_j rather than the Half-Cauchy tangent map tan((1 - p_j) * pi / 2).

Claims & Grounded Evidence

4 Claims

Mechanism #half-cauchy-tangent-map-and-large-p-insensitivity

For individual p-values p_j in (0, 1] and non-negative weights w_j summing to 1, HCCT defines T_HC(p) = sum_{j=1}^m w_j tan((1 - p_j) * pi / 2), where each X_j = tan((1 - p_j) * pi / 2) follows a standard Half-Cauchy distribution HC(0, 1) on [0, infinity) under Uniform(0, 1); because tan((1 - p_j) * pi / 2) -> 0 as p_j -> 1, p-values near 1 contribute negligible non-negative mass and cannot cancel strong evidence from small p-values, unlike the two-sided Cauchy transformation tan((0.5 - p_j) * pi) -> -infinity.

supports Source: A Heavily Right Strategy for Integrating Dependent Studies in Any Dimension · Locator: Section 1.1, Equations 1.2–1.4, lines 109–166; Section 1.2, lines 167–205

Mechanism #exact-convolution-and-landau-domain-of-attraction-null

Under independent Uniform(0, 1) p-values, T_HC(p) has an exact finite-sample null distribution given by the m-fold weighted Half-Cauchy convolution (with closed-form density f_{m=2}(t) = (4 / (pi^2 * (t^2 + 4))) * [pi/2 + (t/2) log(t^2 / 4) + ((t^2 - 4)/(4t)) * arctan(t/2)] for m = 2 equal weights in Theorem 2.3), and as m -> infinity with max_j w_j -> 0, (pi / 2) * (T_HC(p) - A_m) converges in distribution to the standard Landau law Stable(1, 1, 1, 0), where A_m = (2 / pi) * (1 - gamma - sum_{j=1}^m w_j log w_j) and gamma is the Euler-Mascheroni constant.

supports Source: A Heavily Right Strategy for Integrating Dependent Studies in Any Dimension · Locator: Section 2, Theorem 2.1, Theorem 2.3, and Equations 2.1–2.10, lines 302–385; Appendix B.1–B.2, lines 1681–1785

Mechanism #convex-confidence-regions-and-inconsistency-lower-bound

When combining two-sided normal test p-values p_j(theta) = 2 * Phi(-||A_j(hat_theta_j - theta)||_2) for a shared parameter vector theta in R^d, each composite function tan((1 - p_j(theta)) * pi / 2) is strictly convex on R^d, so inverting T_HC(p(theta)) <= c_{m, 1 - alpha} yields a bounded, connected, convex confidence set I_{1 - alpha}^HC and a unique minimizer hat_theta_HC = arg min_theta T_HC(p(theta)) whose minimum value satisfies T_HC(p(hat_theta_HC)) >= (4 / (m * pi)) * sum_{1 <= k < j <= m} ||Sigma_{kj}^(-1/2)(hat_theta_k - hat_theta_j)||_2 for equal weights, providing an intrinsic inter-study inconsistency check that rejects all theta (producing an empty confidence set) when between-study conflict is severe.

supports Source: A Heavily Right Strategy for Integrating Dependent Studies in Any Dimension · Locator: Section 1.2, Example 1 and Figure 2, lines 206–299; Section 3, Theorems 3.1–3.2 and Equation 3.7, lines 387–485; Appendix C, lines 1786–1985

Application #asymptotic-validity-under-dependent-test-statistics

For fixed m as threshold t -> infinity, P(T_HC(p) > t) / P(HC(0, 1) > t) -> 1 whenever the transformed variables X_j are pairwise quasi-asymptotically independent or when underlying test statistics follow a multivariate normal distribution with any non-unit pairwise correlations, enabling valid global combination of dependent conflict or association p-values across high-dimensional studies and Bayesian evidence-synthesis submodels.

supports Source: A Heavily Right Strategy for Integrating Dependent Studies in Any Dimension · Locator: Section 4, Theorems 4.1–4.5, lines 486–575; Appendix D, lines 1986–2147
supports Source: Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies · Locator: Section 4.1, PDF p. 25

⚠️ Curated Limitations & Negative Results

Because the individual Half-Cauchy terms X_j = tan((1 - p_j) * pi / 2) are strictly positive with infinite mean, the finite-sample null distribution of T_HC(p) shifts upward at rate (2 / pi) * log m as the number of combined tests m grows; applying the unadjusted single HC(0, 1) tail 1 - (2 / pi) * arctan(t) at moderate thresholds t without the finite-m convolution or Landau location shift A_m inflates Type I error when m is large.
Asymptotic tail equivalence under arbitrary dependence holds in the extreme right tail (t -> infinity, or significance level alpha -> 0); at conventional significance levels (such as alpha = 0.05) and large dimension m, strong positive dependence or heavy-tailed copulas outside pairwise asymptotic independence can cause finite-threshold size distortion.
When inverting T_HC(p(theta)) <= c_{m, 1 - alpha} to form a confidence set for a shared parameter theta, severe inter-study conflict where T_HC(p(hat_theta_HC)) > c_{m, 1 - alpha} yields an empty confidence set, signaling violation of the common-parameter assumption rather than providing a pooled interval estimate.
All 2,200 lines of Liu, Meng, & Pillai (2025, arXiv:2501.01065v1) and all 52 pages of Yang, Nott, & Presanis (2026, arXiv:2511.02977v2) were read. Note that the first author name in arXiv:2501.01065v1 is Kairong Liu; both are noted in provenance. No author R package or empirical GWAS/meta-analysis pipeline was executed; constructed numerical checks in research/discovery/half-cauchy-combination-test-boundaries.py verify large-p insensitivity, exact m = 2 convolution, Example 1 confidence-set convexity, and Landau location shifts only.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	The Half-Cauchy combination test aggregates multiple hypothesis test p-values across dependent studies, features, or submodels to control global Type I error and prevent false-positive inflation from multiple testing (AIO:DataDredgingBias). AIO:DataDredgingBias denotes the use-and-interpretation bias rather than the statistical combination method.
`AIO:BayesianNetwork` Bayesian Network	related_match	In Bayesian directed acyclic graphical evidence synthesis (Yang, Nott, & Presanis, 2026), heavy-tailed Cauchy and Half-Cauchy p-value combination rules aggregate dependent node-level conflict p-values into a global model-checking statistic without requiring an explicit joint correlation matrix.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`liu-meng-pillai2025`	A Heavily Right Strategy for Integrating Dependent Studies in Any Dimension Kairong Liu, Xiao-Li Meng, Natesh S. Pillai	primary paper full_text	ID: https://doi.org/10.48550/arXiv.2501.01065 sha256:2fe912b58a74…
`yang2026`	Detecting Conflicts in Evidence Synthesis Models Using Score Discrepancies Fuming Yang, David J. Nott, Anne M. Presanis	primary paper full_text	ID: https://doi.org/10.48550/arXiv.2511.02977 sha256:4829e0d63d56…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T09:35:00Z

Updated: 2026-10-17T09:35:00Z

Source File: data/curated/HalfCauchyCombinationTest.yaml · Schema: curated_concept.yaml

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