Zhang–Xu k-sample Scheffé transformation

Status: primary-paper transformation, both base tests, dimension, scale, and boundary gates pass. Row-pairing sensitivity is intrinsic and documented.

After ordering groups by sample size, Zhang and Xu transform the unequal- covariance k-sample problem into a one-sample problem with n_min rows and (k - 1) p columns. zx_ksamp then applies either the Bai–Saranadasa test or, when dimension permits, the exact Gaussian Hotelling test.

Section 2 of the primary paper (pp. 4–5) gives the transformation and the resulting independent Gaussian sample in Equations (2.2)–(2.3), after assuming n_1 <= ... <= n_k. pySHT evaluates that display literally.

For a larger group l, the transformation explicitly uses its first n_min observations both in the partial mean and paired row term. Therefore, permuting rows in that group can change the realized covariance and test statistic. This is part of the published Scheffé construction, whose null distribution remains valid for arbitrarily ordered i.i.d. rows; a row-order invariance gate is consequently not applicable. The unit ledger deliberately demonstrates this sensitivity. Users should ensure row order is unrelated to the measurements and should not sort rows by an outcome before testing. When several groups tie for the minimum sample size, their input order resolves the paper’s otherwise unspecified reference tie. Relabeling such tied groups can therefore change the realized statistic. A data-dependent tie-break would compromise the fixed-design Gaussian transformation, so pySHT does not silently reorder tied groups by their observed values.

One shared original-coordinate feature anchor is removed from all groups before constructing the transformation. This changes neither the Scheffé blocks nor their units and prevents a huge common location from driving the working scale.

Null calibration gate for the default base test

The advertised scenario uses three independent balanced Gaussian groups with n_i=30, p=500, and identity covariance. After the published transformation the default Bai–Saranadasa branch has 30 rows and 1000 columns. The independent batched implementation evaluates 20,000 complete null transformations per seed.

Seed	alpha=0.01	alpha=0.05	alpha=0.10
20260810	0.01235	0.05105	0.10125
20260811	0.01100	0.05035	0.09935

Both rows pass the release tolerance at every level. The optional Hotelling base has exact Gaussian F calibration when its dimension requirement holds and is covered by formula fixtures rather than this asymptotic gate.

Targeted alternative-power gate for the default base test

Seed 2026090318 initializes a SeedSequence; child 0 drives one persistent PCG64 data stream and child 1 is a separate persistent PCG64 auxiliary stream, unused here. In each of 1,000 replications, child 0 draws standard-normal matrices with (n,p)=(25,30), (30,30), and (35,30) in call order. Their columns are multiplied respectively by linspace(0.8,1.2,30), linspace(1.0,1.5,30), and linspace(0.7,1.1,30); the second and third group means are then shifted by \(+0.45\mathbf 1\) and \(-0.45\mathbf 1\). The public call is mean.zx_ksamp(x, y, z), exercising the default Bai–Saranadasa base. Streams advance in replication order. For pvalue < alpha:

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

The intentionally strong dense alternative checks response under unequal sample sizes and unequal diagonal covariance; it is not a broad power claim. python -m tools.mean_power_audits reproduces the result 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.-T. Zhang and J. Xu, On the k-Sample Behrens–Fisher Problem for High-Dimensional Data, Science in China Series A 52 (2009), 1285–1304, doi:10.1007/s11425-009-0091-x.