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Left-truncated Cauchy combination test

Mathematical Methods AIMech:LeftTruncatedCauchyCombinationTest DRAFT

Aliases: Truncated Cauchy Combination Test, Truncated Cauchy Test, CA^tr, Left-Truncated Student-t_1 Combination Test

A statistical p-value combination method that aggregates m dependent hypothesis test p-values p_1, ..., p_m in (0, 1] by bounding the left tail of the standard Cauchy inverse-CDF transformation at a finite negative threshold—either via Winsorized p-value truncation p_i^tr = min(p_i, 1 - delta) producing a point mass delta at nu_delta = tan(pi * (delta - 0.5)) < 0 (Fang et al. 2023) or via conditional left-truncation F_trunc(x; c) = (F(x) - F(c)) / (1 - F(c)) I_{x >= c} with truncation probability p_0 = 1 - F(c) in (0.5, 1) producing the smooth quantile map tan(pi * (0.5 - p_0 * p_i)) >= tan(pi * (0.5 - p_0)) (Gui et al. 2023)—thereby eliminating unbounded negative score penalties as p_i -> 1 while preserving regularly varying right-tail index gamma = 1 and asymptotic tail robustness under arbitrary non-perfect correlation.

Identity Notes & Concept Scope No normalized identity match for the Left-Truncated Cauchy Combination Test across all 442 pinned AIO records and 567 maintained IDs, names, aliases and scoped synonyms. Fixed-string searches across data, sources, research, history, and reviews returned only neighbour references in AIMech:HalfCauchyCombinationTest. Distinct from AIMech:CauchyCombinationTest (Liu & Xie 2020), which uses the untruncated transformation tan(pi * (0.5 - p_i)) on (-infinity, +infinity) with symmetric tail balance p = q = 0.5 and suffers unbounded negative divergence as p_i -> 1. Distinct from AIMech:HalfCauchyCombinationTest (Liu, Meng, & Pillai 2025), which restricts support strictly to [0, +infinity) via tan(pi * (1 - p_i) / 2) (corresponding to the boundary case p_0 = 0.5, c = 0) and evaluates finite-sample null distributions via sine/cosine integral Laplace contour integration rather than negative-support truncation (p_0 in (0.5, 1), c < 0 or point-mass Winsorization at nu_delta < 0) with Generalized Central Limit Theorem / importance-sampling hybrid calibration.

Claims & Grounded Evidence

4 Claims

Mechanism #winsorized-and-conditional-left-truncated-cauchy-transformations

To prevent p-values near 1 from generating unbounded negative Cauchy scores tan(pi * (0.5 - p_i)) -> -infinity, the left-truncated Cauchy combination test bounds each transformed score below at a finite negative threshold: in the Winsorized point-mass formulation (Fang et al. 2023), p_i is truncated to p_i^tr = min(p_i, 1 - delta) (default delta = 0.01) so g^tr(p_i) = tan(pi * (0.5 - p_i^tr)) >= nu_delta = tan(pi * (delta - 0.5)); in the conditional left-truncated formulation (Gui et al. 2023), the Cauchy CDF F is conditioned on X >= c = Q_F(1 - p_0) for p_0 in (0.5, 1) (default p_0 = 0.9), yielding the smooth inverse-survival transformation Q_{F^trunc}(1 - p_i) = tan(pi * (0.5 - p_0 * p_i)) >= c.

supports Source: Heavy-tailed distribution for combining dependent p-values with asymptotic robustness · Locator: Section 2.2, lines 40–45; Appendix A, Remark A1.4, line 508
supports Source: Aggregating Dependent Signals with Heavy-Tailed Combination Tests · Locator: Section 2.2, lines 134–219, Equation 4 and Table 1

Mechanism #regularly-varying-index-one-and-asymptotic-dependence-robustness

Both left-truncated Cauchy transformations belong to the regularly varying tailed class R_{-1} with tail index gamma = 1 and one-sided tail balance parameters p = 1, q = 0, satisfying Condition (A2.2) (bounded below and strictly decreasing near 0); under bivariate normal test statistics with pairwise correlations |rho_{ij}| < 1 (Theorem 1) or partial perfect correlation (Theorem 2), the upper tail probability satisfies P(sum_{i=1}^n w_i U_i > t) ~ (sum_{i=1}^n w_i) P(U_1 > t) as t -> infinity and attains the Donoho-Jin sparse detection boundary sqrt(tau) + sqrt(beta) > 1 for 0 < beta < 1/4 under banded correlation (Theorem 3).

supports Source: Heavy-tailed distribution for combining dependent p-values with asymptotic robustness · Locator: Section 3.1–3.3, lines 66–101, Theorems 1–3; Appendix A, lines 449–453, 508–526
supports Source: Aggregating Dependent Signals with Heavy-Tailed Combination Tests · Locator: Section 3.1.2–3.1.3, lines 290–430, Proposition 3.2 and Remark 3.3

Mechanism #gclt-stable-approximation-and-finite-sample-tail-bounds

Because left truncation breaks exact finite-sample Cauchy stability, Fang et al. (2023) derive analytical finite-n lower and upper tail bounds P(X_1 >= t) <= P(n^(-1) T_{CA^tr} >= t) <= sum_{j=0}^{n-1} C(n, j) delta^j P(X_1 >= (n*t - j*nu_delta)/(n - j)) <= ((1 + delta)^n - delta^n) P(X_1 >= t) (Proposition 4) and a Generalized Central Limit Theorem approximation n^(-1) T_{CA^tr} - n * theta_n ->_d S(1, 1, 1/2, 0) (Proposition 3), using GCLT when alpha < 5e-3 or n >= 25 and cross-entropy importance sampling within the Proposition 4 bounds when n < 25 and alpha >= 5e-3.

supports Source: Heavy-tailed distribution for combining dependent p-values with asymptotic robustness · Locator: Section 2.2, lines 41–55, Propositions 3–4; Appendix A, Proof of Proposition 4, lines 528–535

Application #power-recovery-under-discrete-and-negatively-correlated-p-values

In discrete contingency-table Fisher exact tests, misspecified or negatively correlated one-sided tests, and gene-level GWAS aggregation (such as genes SLC2A9 and PCSK6 in neuroticism GWAS containing both SNPs with p < 1e-4 and SNPs with p > 0.99), left-truncated Cauchy combination prevents p-values near 1 from cancelling true signals and recovers statistical power comparable to harmonic mean combination.

supports Source: Heavy-tailed distribution for combining dependent p-values with asymptotic robustness · Locator: Section 4.3, lines 14–16 (ContentOffset 46080); Section 5, lines 18–20 (ContentOffset 46080)
supports Source: Aggregating Dependent Signals with Heavy-Tailed Combination Tests · Locator: Section 3.1.3, lines 382–430; Section 4, Simulation Results

⚠️ Curated Limitations & Negative Results

Left truncation destroys the exact finite-sample stability under addition of the standard Cauchy distribution (where the average of n i.i.d. standard Cauchy variables is exactly standard Cauchy for all t in R); consequently, exact finite-sample calibration at moderate significance levels alpha >= 5e-3 and small n < 25 requires importance sampling or numerical convolution rather than a closed-form arctangent formula.
Because the right tail is one-sided (balance parameters p = 1, q = 0) with infinite first moment, the Generalized Central Limit Theorem limit S(1, 1, 1/2, 0) introduces a sample-size-dependent location shift n * theta_n that grows logarithmically with n, and using the unshifted asymptotic single-big-jump tail approximation at large m and moderate alpha (such as alpha = 0.05) inflates Type I error under dependence.
The choice of truncation threshold (delta = 0.01 in Fang et al. 2023 or p_0 = 0.9 in Gui et al. 2023) is a tuning hyperparameter trading off negative-score cancellation immunity against retention of weak negative-evidence cancellation near p = 1.
All 546 lines of Fang et al. (2023, arXiv:2103.12967) and all 1,071 lines of Gui, Jiang, & Wang (2023, arXiv:2310.20460v1) were read. No author R package or neuroticism GWAS pipeline was executed; constructed numerical checks in research/discovery/left-truncated-cauchy-combination-boundaries.py verify the p_0 = 0.5 and p_0 = 1.0 boundary identities, cancellation immunity on (p_1 = 1e-4, p_2 = 1 - 1e-6), and Proposition 3/4 GCLT and binomial tail bounds only.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:DataDredgingBias` Data Dredging Bias	related_match	The left-truncated Cauchy combination test aggregates dependent hypothesis test p-values for global Type I error control when screening multiple correlated hypotheses, mitigating false-positive inflation from selective reporting (AIO:DataDredgingBias). AIO:DataDredgingBias denotes the multiple-testing bias rather than the statistical combination procedure.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`fang-chang-park-tseng2023`	Heavy-tailed distribution for combining dependent p-values with asymptotic robustness Yusi Fang, Chung Chang, Yongseok Park, George C. Tseng	primary paper full_text	ID: https://doi.org/10.5705/ss.202021.0078 sha256:432b18ec119b…
`gui-jiang-wang2023`	Aggregating Dependent Signals with Heavy-Tailed Combination Tests Lin Gui, Yuchao Jiang, Jingshu Wang	primary paper full_text	ID: https://doi.org/10.48550/arXiv.2310.20460 sha256:3f487fc6bca4…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini 3

Created: 2026-10-17T10:20:00Z

Updated: 2026-10-17T10:20:00Z

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

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