The deflated-Welch statistic: a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA. (T_BB, m01a): Methods, confidence-set-geometry development, derivations, novelty, and reporting standard): reproducibility deposit

Abstract

The deflated-Welch statistic: a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA William J. Dwyer, MD, MPH, FAAP — Department of Mathematics and Statistics, University of Massachusetts Lowell. ORCID 0009-0004-0855-7222. Concept DOI (always resolves to the latest version): 10.5281/zenodo.21908169. What this is The reproducibility deposit for the deflated-Welch statistic T_BB, a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA (the Behrens–Fisher problem for k ≥ 3 groups). Welch's test becomes liberal under skew and unstable variance weights at small samples; T_BB = Q(s²)·exp(−R) keeps the ordinary group means and buys a guaranteed level by deflating the Welch quadratic by a Berger–Boos scale-inflation radius R. Three operating points are provided: a fixedcalibrated radius (κ_s), a design-adaptive near-guarantee radius (closed-form polygamma Cornish–Fisher with a finite-nkurtosis guard), and a fully proved smallest-eigenvalue radius R_eig (Gaussian, extended under bounded kurtosis). What the deposit contains Manuscript (author + anonymized) and a derivations supplement (DA1–DA13) plus a long-form derivations companion, covering: why Welch fails under skew in closed form; the Berger–Boos deflation and its exact worst-case radius; the polygamma-cumulant Cornish–Fisher radius with saddlepoint-exact normal backbone; the excess-kurtosis tail term with its finite-n upper-confidence guard; the imbalance correction; the fully proved smallest-eigenvalue radius (with the k-group multiplicity fix, free-β optimization, and the proved-under-bounded-kurtosis widening); and the k-sample Behrens–Fisher null distribution. Interactive demonstrator rerun_cochran/honest_anova.html — computes raw-mean Welch, the fixed / adaptive / proved T_BB radii, the estimand-changing transform routes, and the full routing receipt in the browser, reproducing the deposited Python. Its engine is extracted as a standalone Node module (m01A_anova_engine.js) and checked cell-by-cell against Python across an 84-design taxonomy (verify_anova_engine_taxonomy.py/.js, max |Δp| = 0.00000). Reproducibility scripts (rerun_cochran/, rerun/) — every reported number traces to a named, deterministically-seeded script (size/power/surface, the calibration and information-limit decompositions, the proved-radius verification, the imbalance calibration, the skew-router branch, and the figures). Real-data evidence — anova_flip_scan.py scans 2,783 public one-way layouts (254 datasets): guaranteed T_BBwithholds ~41% of Welch-significant calls, concentrated where the weight-instability screen fires, and never manufactures significance (Table 7 / Figure 15). Figures and the deterministic deposit builder (fixed timestamps → stable md5). All evaluation is simulation-based; the one empirical component is the public-dataset scan, which uses only openly distributed data. Code is released under the MIT License; text and figures under CC BY 4.0. Version history (consolidated changelog) Published version DOIs are marked ✅; the concept DOI above always resolves to the latest. Staged versions were rolled into the next published one unless noted. v1.0.77 ✅ 10.5281/zenodo.22167690 (2026-08-30): CSDA guide-for-authors conformance — abstract trimmed to 247 words (from 284), keywords cut to 7 (from 11), the withholding highlight shortened to ≤85 characters, and the arXiv PDF/source regenerated. No change to methods, results, figures, or code. v1.0.76 ✅ 10.5281/zenodo.22167536 (2026-08-30) — AI-disclosure heading aligned to Elsevier. The manuscript's declaration heading is now "Declaration of generative AI and AI-assisted technologies in the manuscript preparation process" (was "Use of generative AI"); the disclosure body is unchanged. Prepared alongside an Elsevier-compliant cover-letter variant and an EM suggested-reviewer sheet (both kept outside the deposit). docx/pdf rebuilt; deterministic md5 refreshed. v1.0.75 ✅ 10.5281/zenodo.22167304 (2026-08-30) — Submission-sharpening pass. Graphical abstract + Elsevier Highlights; figures and tables renumbered into reading order with per-table Source clauses; the validity–power frontier (Figure 8) now carries the proved R_eig operating point (100% validity, size-adjusted power 0.613, merge_tbb_proved_frontier.py); new Section 7 "Recovering power by design" + Table 8 (rc_anova_power_by_design.py); and a live required-n calculator in honest_anova.html (per-group and total n for 80% power, "power now @ total n"), with a numeric-heading CSS fix and the engine re-verified against Python at 0.00000. v1.0.74 ✅ 10.5281/zenodo.22165892 (2026-08-29) — Proved-under-bounded-kurtosis radius (DA12.6). The proved non-normal widening now keys on excess kurtosis, √(1 + κ̂·(n−1)/(2n)), from the exact Var(s²/σ²) = 2/(n−1) + κ/n, so symmetric heavy tails (Student-t) are covered where the old skew form √(1 + 0.75·skew²) under-covered; tbbProvedswitched to the kurtosis form across the demonstrator, engine, and Python truth (re-verified JS-vs-Python at 0.00000); new rc_anova_kurtosis_proof.py + deep-dive. v1.0.73 ✅ 10.5281/zenodo.22165709 (2026-08-29) — Reconstructed & verified demonstrator engine (standalone Node module + taxonomy verifier, max |Δp| = 0.00000 across 84 designs; Yuen zero-variance fix; T_BB-routed presets both directions); series-impact deep-dive (the corrected R_eig k-group multiplicity gap also reaches m03 and m01t). v1.0.72 (2026-08-29) — Title set to "The deflated-Welch statistic…"; corrected + optimized proved radius R_eig (β/k multiplicity fix + β-optimization, DA12); real-data Welch-vs-T_BB flip scan (2,783 layouts; Table 7 / Figure 15) + demonstrator imbalance-factor fix; long-form derivations companion. v1.0.71 / v1.0.70 (2026-08-21) — Zhang normal-reference comparator benchmarked on the efficiency frontier (valid on only 24% of designs, in the calibrated-liberal cluster); k = 2 adaptive-radius case-study fold (design-scaling vs shape-keying distinction). v1.0.69 ✅ 10.5281/zenodo.22035826 (2026-08-20) — HTML R1/R2 presentation pass + Figure 9 adaptive per-cluster label merge. v1.0.68 ✅ 10.5281/zenodo.22033737 (2026-08-20) — Companion consolidation into a single six-column Table 6; Figures 11–14 harmonized into one story. v1.0.67 / v1.0.65 / v1.0.60 (2026-08-19/20) — Guarded-reference naming-collision fix; the 40,000-replication expanded-frontier pin (Table 3 + Figure 8) with the symmetric-heteroscedastic skew-router branch; the mean-preserving lightened-R_eig do-not-use fallback. v1.0.59 ✅ 10.5281/zenodo.21995320 (2026-08-18) — Reporting standard + honest_anova.html demonstrator re-aligned to the current T_BB methods paper. v1.0.57 ✅ 10.5281/zenodo.21986847 (2026-08-17) — Reviewer-comprehension pass (multi-paragraph abstract, contributions list, trimmed captions); proved radius R_eig added as a Table 3 scorecard row; corner tail-index correction (N−k)/2 (low-order moments exist in every deployed design). v1.0.56–v1.0.49 (2026-08-16) — The k-sample Behrens–Fisher corner-distribution program: two-moment scaled-χ² corner reference, derived corner cumulants, the secular-eigenvalue law + closed CGF + power-law tail, consolidated into derivations DA13 with a prior-art/novelty audit. v1.0.48 ✅ 10.5281/zenodo.21963458 (2026-08-16) — The unifying λ(z) correction (a smooth instability-keyed deflation strength). v1.0.45 ✅ 10.5281/zenodo.21962965 (2026-08-16) — Atomic sparsity index + bootstrap-t edge hardening + shape-aware pooled standardized-residual bootstrap (SA-PSRB); multivariate transfer to m03. v1.0.44–v1.0.41 (2026-08-16) — Shape-moment re-injection order (skew is the sweet spot), validated and hardened pooled standardized-residual bootstrap, atomic weight-noise probes. v1.0.40 ✅ 10.5281/zenodo.21961667 (2026-08-16) — Log-domain weight-stabilization probe (negative for stabilization; clarifies the size-adjusted oracle ceiling); includes the oracle-power gap decomposition (≈92% conservatism, ≈8% estimation). v1.0.37 ✅ 10.5281/zenodo.21961327 (2026-08-16) — Residual-bootstrap qualification of the shoot-out + the first proved Gaussian smallest-eigenvalue radius R_eig (DA12, the p = 1 specialization of the m03 theorem). v1.0.36 (2026-08-15) — Figure 11 T_BB-region colour fix (amber, matching the routing figures). v1.0.27 ✅ 10.5281/zenodo.21908170 — Earlier published baseline of the deposit. Provenance: every number traces to a named, deterministically-seeded script listed in the manuscript Declarations; the demonstrator engine reproduces the deposited Python to max |Δp| = 0.00000 across the taxonomy verification. License. Code and scripts in the deposit are released under the MIT License; text and figures under CC BY 4.0. Reuse is permitted with attribution to the author and citation of the concept DOI above. How to cite. Dwyer, W. J. The deflated-Welch statistic: a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA. Reproducibility deposit, Zenodo. https://doi.org/10.5281/zenodo.21908169