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arXiv · 2608.07979

Sparse departures from independence in two-way tables: a heteroscedasticity profile and detection boundary, an adaptive higher-criticism gate, and an assumption-lean exact anchor

William J. Dwyer
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Abstract

Two-way contingency tables are tested for independence throughout applied statistics (genomics, network and text co-occurrence, pharmacovigilance, ecology, survey cross-tabulation), and the routine test reads asymptotic Gaussian tail probabilities off the table cell by cell. On the tables people actually analyze this is badly miscalibrated: across 5,543 real public two-way tables, two-thirds have counts small enough or margins heterogeneous enough that the asymptotic maximum-cell independence scan false-positives at a mean 33% against a 0.05 target, while an exact margin-conditional anchor holds at about 1%. The failure is sharpest when the departure is sparse, the association concentrated in a few cells, where the omnibus chi-square is inefficient and the sharp object is the detection boundary. In three parts we answer where a sparse signal can be seen, which combiner attains that limit, and how to calibrate at small counts. Part I specializes the sparse-detection theory of Ingster (1997), Donoho and Jin (2004), and Chhor, Mukherjee and Sen (2024) to independence: the table is a heteroscedastic Gaussian sequence whose variance profile is the expected-count table, fixed by the margins via CVe^2 = (1 + CVr^2)(1 + CVc^2) - 1, giving an exact signal map and a closed-form separation radius valid under a moderate-deviation growing-count condition. Part II shows higher criticism, with its closed-form Jaeschke-Eicker null, attains that boundary adaptively over unknown sparsity. Part III measures the small-count calibration failure (size 0.3 to 0.6 even under uniform margins, so the driver is the per-cell tail), removes it with exact margin-conditional laws (size 0.003 to 0.009), and gives a routing rule sending large-count tables to the asymptotic gate and small-count tables to the exact anchor. Every claim is reproduced from openly deposited, deterministically seeded code.

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William J. Dwyer. 2026-08-08. Sparse departures from independence in two-way tables: a heteroscedasticity profile and detection boundary, an adaptive higher-criticism gate, and an assumption-lean exact anchor. https://arxiv.org/abs/2608.07979

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