multiple-testing-correctionlisted

# 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`