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

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. How to cite Please cite this deposit if you use the package or the method. Citing the concept DOI references the work in general and always resolves to the latest version; cite a specific version DOI to point at an exact snapshot. Dwyer, W. J. (2026). The deflated-Welch statistic: a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA — reproducibility deposit [Software]. Zenodo. https://doi.org/10.5281/zenodo.21908169 BibTeX: bibtex @software{dwyer_m01A_2026, author = {Dwyer, William J.}, title = {The deflated-Welch statistic: a closed-form, guaranteed-level test for heteroscedastic one-way ANOVA --- reproducibility deposit}, year = {2026}, publisher = {Zenodo}, doi = {10.5281/zenodo.21908169}, url = {https://doi.org/10.5281/zenodo.21908169}, orcid = {0009-0004-0855-7222} } The DOI above is the concept DOI (resolves to the latest version); to cite a specific release, use that version's DOI and add version = {v1.0.NN}. When the accompanying journal article is published, please cite it as the primary reference for the method and this deposit as the reproducibility archive. Version history (consolidated changelog) v1.0.83 ✅ 10.5281/zenodo.22653786 (2026-09-08; concept resolves to it) — figure-finalization fold (no simulation number, data CSV, table, or deployed radius changed). Figure 9 (the validity–power frontier facet grid) was corrected to the raw-power metric (mean power at α = 0.05; an earlier edit pass had silently regenerated it on the size-adjusted axis, relocating every point) and given a manuscript-readability pass: enlarged legend + title, single-spaced title/legend gutters, the abscissa now includes 0.00 with the right tail trimmed at >110, and per-panel data-label placements (W to the left of its point in panels 2–5, split combined labels, PF/WY*/AG/Z/J2/KLrepositioned). Figure 11 (the routing curve) label + legend placements: panel A "route to T_BB" moved left and its legend lifted above the α = 0.05 line; panel C λ(z) annotation moved up-left. Added mseries_labels.py — the label auto-placer (place_labels()) with the new per-panel pinned override mechanism, now a runtime dependency of make_m01A_frontier_figure.py — so Figure 9 is reproducible from the deposit. Dwyer_M01A + _anon docx/pdf and arXiv rebuilt to embed the finalized figures (clean-manuscript guard + org-map + math-render audits PASS). Deterministic m01A_anova_reproducibility_v1.0.83.zip, md5 ec4fa990f27917b19415b5cba8b5cb09, 391 files (member diff vs v1.0.82: +1 mseries_labels.py, 0 removed; rc_anova_frontier.csv and rc_anova_routing_curve.csv byte-identical). Forward change, not a retraction of v1.0.82. v1.0.82 ✅ 10.5281/zenodo.22653145 (2026-09-08) — organization-summary pivot (no simulation number, figure, table, or deployed radius changed). Removed the in-manuscript four-layer "Read at your depth" spine block from Dwyer_M01A + _anon — the manuscript is a clean scientific paper again — and added the separate canonical organization map M01A_ReadAtYourDepth_OrgMap.md (a senior-seminar deliverable that maps the four depths to the real files and points into the manuscript, deposit, companion, and demonstrator). teaching_layers_audit.pywas repurposed to a clean-manuscript guard + org-map validation (every file the map names must exist on disk). Dwyer_M01A + _anon + arXiv rebuilt from the clean manuscript (teaching-layers + math-render audits PASS). Deterministic m01A_anova_reproducibility_v1.0.82.zip, md5 e7c910c5eca8aed02bd2357881f48248, 390 files (member diff vs v1.0.81: +1 M01A_ReadAtYourDepth_OrgMap.md, 0 removed; result CSVs byte-identical). Forward change, not a retraction of v1.0.81. v1.0.81 ✅ 10.5281/zenodo.22652698 (2026-09-08; concept resolves to it) — presentation/honesty pass (no simulation number, figure, table, or deployed radius changed). Added the four-layer teaching-architecture front-matter spine (the "Read at your depth" L1–L4 map) to both manuscript variants, and reworded the guarantee language for honesty across Table 3, the abstract, §6, and the Figure 14 textbox — from "exact guarantee" to "empirical worst-case size ≤ 0.048 across the studied envelope" for the fixed/adaptive radius, with the proved-Gaussian R_eig distinguished as the one operating point carrying a proof. This upload also folds v1.0.80 (proof-tightening; below), which was cut locally but never published on its own. Dwyer_M01A + _anon + arXiv rebuilt (teaching-layers + math-render audits PASS). Deterministic m01A_anova_reproducibility_v1.0.81.zip, md5 051062cdc8a94a17e002f10633bc1906, 389 files (added teaching_layers_audit.py + the v1.0.80 proof-tightening files; chain verified — local v1.0.79 byte-identical to the published one, so v1.0.81 is a clean superset). v1.0.80 (2026-09-07, rolled into v1.0.81; never published on its own) — proof-tightening for the CSDA rejection: the non-Gaussian step of the guaranteed radius is stated at its true strength (documents + scripts; no simulation number, figure, table, or deployed radius changed). Derivations gain DA12.7 (an impossibility proposition — no finite-moment, distribution-free radius controls the level at the deployed small-n operating points, because the sharp J-moment tail bound on the required lower quantile of v_min = min_i s_i²/σ_i² is vacuous for every finite J; a moment-matched law on [0,∞) retains mass near zero) and DA12.8 (a conditional theorem — the bounded-kurtosis radius R_eig,κ is proved to control the level under one explicit, checkable lower-tail-domination regularity condition, which holds on every continuous parent tested — skewed, symmetric-leptokurtic, and platykurtic-bounded — and fails only for atomic parents at n ≤ 6). §4.4 is rewritten to state the guarantee at that strength rather than "empirically". New seeded scripts rerun_cochran/rc_anova_proof_tighten.py (four-radius comparison + the vacuous Cantelli bound), rc_anova_momentbound.py (the sharp J-moment Chebyshev–Markov linear program), rc_anova_da126_adversarial.py (platykurtic/discrete stress), the CSV proof_tighten.csv, and the deep-dive M01A_ProofTightening_BoundedKurtosis_DeepDive_2026-09-07.md. Dwyer_M01A + _anon + Derivations rebuilt (math-render PASS). v1.0.79 ✅ 10.5281/zenodo.22573646 (2026-09-07) — Reporting-standard consolidation. The honest-by-default reporting standard was folded into the methods manuscript as §6 (the three-regime routing rule with the per-family CV* thresholds, the two-estimand distinction, the receipt JSON schema, and one worked receipt; the abstract gained the reporting-standard clause); the standalone reporting-standard artifact is retained in the bundle as a citable document, so m01Ar is no longer a separate submission target. Both manuscript variants were rebuilt. The two bundled reproducibility scripts (mseries_docx_core.py, build_arxiv_latex.py) were synced to the canonical house versions. No simulation, number, figure, table, or word changed; every docx/p