Multiple Testing Correction

Multiple Testing Correction

Every hypothesis test carries a false-positive rate at its stated confidence level — run enough of them and false positives arrive on schedule, not by bad luck. Screen 30 factors at the conventional |t| >= 2 (~95% single-test confidence) and roughly 1-2 will look "significant" even if every one of them is pure noise, purely from running 30 tests. Nobody needs to be dishonest for this to happen; the batch size does it on its own. The same arithmetic error shows up one level up: claim N "independent" signals or symbols and the Fundamental Law of Active Management (Grinold, 1989) says skill scales with sqrt(N) — but if those N things actually share high pairwise correlation (same model family, same data source, same market regime), the effective N is far smaller, and the implied Information Ratio was never really there.

The two failure modes this skill catches

1. Uncorrected batch significance. A naive per-test threshold (|t| >= 2, p < 0.05) does not control the batch's error rate. Bonferroni controls the probability of any false positive in the batch (conservative); Benjamini-Hochberg (BH/FDR) controls the expected proportion of false positives among what gets called significant (the more common choice in modern quant research — the same logic behind the Deflated Sharpe Ratio).
2. Breadth illusion. A count of "independent" bets that are actually correlated inflates the Fundamental Law's IR estimate. N nominal bets at average pairwise correlation rho behave like N / (1 + (N-1)*rho) effective bets — at rho=0.7 and N=20, that's ~1.4 effective bets, not 20.

Workflow

1. Collect every test actually run in the batch — not just the ones that looked interesting. The batch size (n) is the correction; leaving out the tests that failed and only reporting the survivors defeats the whole point, the same way undisclosed trials defeat the Deflated Sharpe Ratio in backtest-validation.
2. For each test, get the t-statistic and the sample size (or degrees of freedom) that produced it.
3. Run the gate: python scripts/multiplicity_gate.py --tests tests.json --alpha 0.05 where tests.json is {"name": {"t_stat": float, "n_periods": int}, ...} — one entry per hypothesis actually tested. With no data at hand, show the mechanics with --demo.
4. If a breadth/IR claim is also being made, add --ic <avg information coefficient> --n-bets <N> --avg-corr <rho> to the same call (or call it standalone with just those three flags) to check the Fundamental-Law breadth illusion independently of the correction above.
5. Report the BH-FDR survivor set, not the naive-significant set, as "what passed." Bonferroni's stricter survivor set is worth stating alongside it when the user needs the conservative answer (e.g. before capital deployment). Never report a factor as significant because it cleared the naive threshold alone.
6. Completion gate: a factor/variant is only "significant" in the final writeup if it is in the BH-FDR (or stricter) survivor set. On a significance_evaporates FAIL — naive flagged some, corrected keeps none — the correct deliverable is "this batch shows no evidence of a real effect at this batch size," which is a valid, useful research result, not a failure to find something.
7. If breadth was checked and flagged breadth_illusion, report the effective-breadth Information Ratio, not the nominal one, in any forward-looking claim.

Guardrails

• No "N of M factors were significant" claim using the naive per-test threshold when M > 1. Correct first.
• No citing a Fundamental-Law IR without disclosing the correlation assumption behind the breadth number that produced it.
• Both checks require honest inputs the skill cannot verify on its own: the full test batch (not a cherry-picked subset) and a real, not assumed-away, correlation estimate for the breadth check. Say so when either is unavailable rather than defaulting silently to "independent."
• Missing scipy degrades to a normal approximation for the p-value (accurate once n_periods is reasonably large) rather than failing — but the CLI says so; don't silently treat it as exact for small samples.

Bundled resources

• scripts/multiplicity_gate.py — Bonferroni + BH-FDR correction and the breadth-illusion check, in one gate with a PASS/WARN/FAIL verdict. --demo runs both scenarios with synthetic data and no dependencies beyond numpy/scipy (skips cleanly if absent).
• references/methodology.md — the exact formulas, when to prefer Bonferroni over BH-FDR, and how this relates to backtest-validation's Deflated Sharpe Ratio and risk-assessment's effective-N diversification check.