Batch aliased with condition (confounding / non-identifiability / correct-then-test)

Stub. Corpus-first: this page starts minimal and grows only when a real case demands it (content/meta/guiding-principles.md). Below is the one-line phenomenon, the two regimes that share the mechanism, and the primary-source outline to summarize next. Each concrete signature is added only when its primary is read into a content/research/papers/<id>/ note. Frontmatter contract for patterns is deferred to repo standup (AGENTS.md).

Phenomenon (one line): a technical/batch factor co-varies with the biological contrast. When they are perfectly aliased, the design matrix is rank-deficient and the batch and biological effects are non-identifiable — no correction can separate them. When partially confounded, the common “correct-then-test” move (remove the batch effect, then test as if it were never there) inflates significance. No error is thrown; the effect estimate or its p/q-value simply lies.

Why two regimes, not one. Same confounding, two clothes — and they earn opposite verdicts, which is exactly why this pattern is the audit-method-validity referee’s ESCALATE anchor (double-dipping is its REVISE anchor — fixable; this one has an unfixable extreme):

• Perfect aliasing → ESCALATE (unfixable). Every case in batch 1, every control in batch 2: the contrast column and the batch column span the same subspace, the design is rank-deficient, the coefficients are non-identifiable. No method — ComBat, covariate adjustment, surrogate variables — can recover the separation; the only fix is a redesign. [corpus-observed anchor: Leek 2010’s bladder/sTCC case — biology (CIS status) confounded with processing date; even the control samples clustered perfectly by date, proving the split was batch, and “statistical adjustment cannot recover it.” The strict “rank-deficient ⇒ non-identifiable” restatement is a linear-algebra fact — see the convention flag below.]
• Partial confounding → “correct-then-test” inflation. Two-step batch adjustment (e.g. ComBat to “clean” the data, then a naive test that ignores the adjustment) inflates false-positive and false-discovery rates, catastrophically under unbalanced group-batch designs. The remedy is a one-step model that carries batch as a covariate (~ batch + condition), not correct-then-test. [corpus-observed: Nygaard et al. 2016 — reanalyses 2011 vs 11 and 1003 vs 377 DE genes (correct-then-test vs batch-blocked); inflation is a fixed factor set by design imbalance, not sample size; corroborated by limma/DESeq2 tool docs]

Primary sources to summarize (→ content/research/papers/<id>/)

Sourcing pre-verified in content/research/05-skill-backing-references.md §3 ([VS] = search-confirmed venue/DOI). ✓ ingested = note exists at content/research/papers/<slug>/ and wiki-links back here.

Spine / validity axis — the correct-then-test cardinal sin (highest recoverability value)

• Nygaard, Rødland & Hovig 2016, Biostatistics 17(1):29–39 (DOI 10.1093/biostatistics/kxv027; PMID 26272994; CC BY 4.0) — “Methods that remove batch effects while retaining group differences may lead to exaggerated confidence in downstream analyses.” Grounds the correct-then-test signature + the unbalanced-design inflation + the one-step-model remedy (~ batch + condition). The portable null detector: ComBat + F-test on N(0,1) data with the real design retained yields a non-uniform p-value distribution / inflated F. Silent on RNA-seq counts — do not assume the NB case transfers. → ✓ ingested nygaard-2016

Overview / awareness + the confounded-with-outcome example

• Leek, Scharpf, Corrada Bravo, Simcha, Langmead, Johnson, Geman, Baggerly & Irizarry 2010, Nature Reviews Genetics 11(10):733–739 (DOI 10.1038/nrg2825) — “Tackling the widespread and critical impact of batch effects in high-throughput data.” Grounds prevalence (32.1–99.5% of features batch-associated; 12.2–100% confounding R² across 9 datasets) + the unfixable confounded-design anchor (bladder/sTCC: controls cluster by date ⇒ batch-driven; adjustment cannot recover). Paywalled publisher version; read via the open NIH author manuscript (PMC3880143), all-rights-reserved → own-words note. → ✓ ingested leek-2010

Convention — flag, don’t cite (no clean primary)

• “Complete (perfect) confounding is uncorrectable.” This is a design-of-experiments / linear-algebra fact (rank-deficient design ⇒ non-identifiable coefficients). Both ingested notes speak to it qualitatively — Leek’s bladder/sTCC anchor + own-words “adjustment cannot recover it”; Nygaard’s balance condition (no inflation iff each batch reproduces the overall group ratio, ν₀ = ν) — but no single crisp citable “complete confounding is uncorrectable” sentence exists in either (content/research/05-skill-backing-references.md §3). Label it as convention; cite Nygaard + Leek jointly for the surrounding claim, never as the source of this exact sentence.

Tool-doc corroboration (community guidance, not primaries)

• limma removeBatchEffect doc: use for plots/EDA only, not before linear modelling.
• DESeq2 vignette: model batch as ~ batch + condition; don’t correct-then-test.

Adjacent (belongs to sibling leaves, cross-link — do not absorb here)

• The diagnostic that detects aliasing (cross-tabulate batch × condition; check design-matrix rank) is a protocol-altitude referee → [[assess-batch-effects-and-confounding]], not this note.
• Surrogate-variable / unwanted-variation methods (sva, RUVSeq) target unknown/unmeasured confounders — a different regime than a measured batch aliased with the contrast. Cross-link, don’t merge.