Schott high-dimensional one-way MANOVA

Status: primary-paper formula, independent fixture, exchange, scaling, and boundary gates pass in the advertised Gaussian asymptotic regime.

The MANOVA error sum-of-products matrix is

E = sum_i (n_i - 1) S_i,

with error degrees of freedom N - k. The raw Tnp statistic is returned; its standardized normal statistic is included in diagnostics.

The pinned SHT implementation used sum_i n_i S_i while retaining N-k as the error degrees of freedom. That is not the defining error SSP. pySHT uses the paper formula. A literal NumPy implementation independently checks E, the hypothesis SSP, variance estimator, standardized statistic, and upper normal tail. Group exchange leaves the result unchanged; multiplying every observation by c multiplies raw Tnp by c² and leaves its p-value unchanged.

All groups are first expressed relative to one deterministic feature-wise anchor. This preserves the raw statistic’s squared measurement units while avoiding location-driven cancellation; regressions cover common locations through 1e14.

Null calibration gate

The advertised Gaussian scenario has three independent groups of 20 rows, p=500, and identity covariance. The audit uses exact normal/Wishart sufficient-statistic algebra, independently matched to the full-data formula fixture, for 20,000 null datasets per seed.

Seed	alpha=0.01	alpha=0.05	alpha=0.10
20260810	0.01285	0.05335	0.10265
20260811	0.01215	0.05275	0.10060

Both rows pass the release tolerance at all three levels. The claim is scoped to the stated balanced Gaussian high-dimensional regime.

Targeted alternative-power gate

Seed 2026090316 initializes a SeedSequence; child 0 drives one persistent PCG64 data stream and child 1 drives a separate persistent PCG64 auxiliary stream, unused by this deterministic test. In each of 1,000 replications, child 0 draws, in call order, three 20-by-40 independent Gaussian matrices with identity covariance and mean vectors \(0\), \(0.35\mathbf 1\), and \(-0.35\mathbf 1\). The public call is mean.schott_ksamp(x, y, z). Streams advance in replication order. The counts for pvalue < alpha are:

alpha	0.01	0.05	0.10
Rejections / 1,000	1,000	1,000	1,000
Rate	1.000	1.000	1.000

This deliberately dense and strong alternative checks response direction in the p>n_i path; it is not a general power guarantee. It is reproduced by python -m tools.mean_power_audits under Python 3.12.13, NumPy 2.5.1, and SciPy 1.18.0. The worst-case binomial standard error at 1,000 outer replications is 0.0159.

Primary reference: J. R. Schott, Some High-Dimensional Tests for a One-Way MANOVA, Journal of Multivariate Analysis 98 (2007), 1825–1839, doi:10.1016/j.jmva.2006.11.007.