Rank Based Hypothesis Testing

Rank-Based Hypothesis Testing (cross-cutting/numerics/rank-based-hypothesis-testing)

Use when the task is a distribution-free location comparison that must stay valid when the normality assumption of the parametric sibling fails: the Wilcoxon rank-sum test (Mann-Whitney U) for two independent samples, the Wilcoxon signed-rank test and the sign test for paired data. This leaf ranks the observations, applies the normal approximation with the 0.5 continuity correction toward zero, and returns the statistic, z, two-sided p-value and the reject or accept verdict at the chosen alpha. It pairs with cross-cutting/numerics/hypothesis-testing, which owns the parametric family of location and scale tests, and with the distribution and descriptive-moment context leaves.

Domain quick reference

• Rank-sum (Mann-Whitney U): merge the samples, rank all observations with average ranks for ties, R1 = sum of ranks of sample 1. U = R1 - n1 (n1 + 1) / 2 (the other orientation is n1 n2 + n1 (n1 + 1) / 2 - R1; sample 1 is the x sample here). Mean mu_U = n1 n2 / 2, sd_U = sqrt(n1 n2 (n1 + n2 + 1) / 12), z = (U - mu_U +/- 0.5) / sd_U with the correction toward zero.
• Signed-rank (paired): differences d_i = x_i - y_i, zeros dropped, absolute differences ranked with average ranks for ties. W = sum of positive-difference ranks minus sum of negative-difference ranks. mu_W = 0, sd_W = sqrt(n (n + 1) (2 n + 1) / 6), z = (W - sign(W) * 0.5) / sd_W.
• Sign test (paired): n+ = positive differences among n nonzero ones; z = (n+ - n/2 +/- 0.5) / sqrt(n/4) with the correction toward zero.
• Two-sided p = 2 (1 - Phi(|z|)) with Phi(z) = 0.5 (1 + erf(z / sqrt(2))) from math.erf; verdict reject when p <= alpha.
• Default alpha 0.05; pure stdlib, deterministic, offline.

Workflow

1. Decide the design: two independent samples (wilcoxon_rank_sum) or paired measurements (wilcoxon_signed_rank for magnitudes with signs, sign_test for signs only).
2. Pick alpha (default 0.05, must lie in (0, 1)).
3. Call the chosen function on the raw samples; every function returns a dict with the exact keys of the spec.
4. Read z and p_value for the effect direction and size, reject for the verdict.
5. For a uniform report use rank_test_summary("rank-sum" | "signed-rank" | "sign", x, y, alpha), which dispatches to the same three functions.
6. Confirm the deterministic checks with the contract test scripts/test_rank_based_hypothesis_testing.py.

Worked example

Finish-roughness batches x = [0.82, 0.79, 0.85, 0.80, 0.83] vs y = [0.96, 1.02, 0.94, 0.98, 0.99], alpha 0.05: the x values occupy the five lowest merged ranks, so wilcoxon_rank_sum returns r1 = 15.0, u = 0.0, mu_u = 12.5, sd_u = 4.7871, z = -2.5067, p_value = 0.01219, reject True.

Paired series (1.0, 1.1) to (6.0, 6.6) with all differences negative: wilcoxon_signed_rank returns n = 6, w = -21.0, sd_w = 9.5394 (sqrt of 6 * 7 * 13 / 6), z = -2.1490, p_value = 0.03164, reject True at 0.05.

Sign test with 8 positive of 10 differences: sign_test returns n_pos = 8, n_neg = 2, n = 10, z = 1.5811, p_value = 0.11385, reject False. With 10 of 10 positive it returns z = 2.8461, p_value = 0.00443, reject True (the continuity-corrected approximation region hand checks to z near 3 and p near 0.003).

Pitfalls

• Using the rank-sum test on paired measurements: wilcoxon_rank_sum assumes two independent samples; paired designs belong in wilcoxon_signed_rank (magnitudes with signs) or sign_test (signs only), and zero differences are dropped from the paired statistics.
• Ignoring ties: tied values share average ranks in both the rank-sum and the signed-rank paths, so assigning plain integer ranks perturbs r1/u/w and the resulting p-value.
• Reading direction from the wrong sample order: swapping the two samples mirrors z (negative to positive) with the same p-value, and sample 1 is the x sample by convention.
• Picking alpha outside (0, 1) or passing undersized or mismatched inputs: fewer than 2 observations per rank-sum sample, paired length mismatch, fewer than 2 nonzero signed-rank differences, and no nonzero sign-test differences all raise ValueError.
• Treating the normal-approximation output as exact: z and p come from the normal approximation with the 0.5 continuity correction, so the 10-of-10 sign-test case returns z = 2.8461, p = 0.00443 while the region hand-checks to z near 3 and p near 0.003.
• Using the sign test where magnitudes matter: sign_test counts only the signs of the differences, while the signed-rank test ranks their magnitudes; the two can disagree on the same pair series.

Verification

• Confirm wilcoxon_rank_sum(x, y) above returns r1 15.0, u 0.0, z -2.5067, p 0.01219 and reject True; swapping the samples mirrors z positive with the same p.
• Confirm identical samples return u = n1 n2 / 2, z = 0.0 and p = 1.0 within 1e-6, reject False; symmetric paired data give w = 0.0, p = 1.0.
• Confirm tied values share average ranks: [1, 2, 3] vs [1, 2, 3] and the tied-magnitude signed-rank case both return the averaged statistics.
• Confirm every non-physical input raises ValueError: fewer than 2 observations per rank-sum sample, alpha outside (0, 1), paired length mismatch, fewer than 2 nonzero signed-rank differences, no nonzero sign-test differences, and an unknown test name in rank_test_summary.
• Run the contract test offline: python3 scripts/test_rank_based_hypothesis_testing.py (33 tests, deterministic).

Related leaves

• cross-cutting/numerics/hypothesis-testing: the parametric sibling (boundary: parametric location tests against this leaf's distribution-free ranks).
• cross-cutting/numerics/descriptive-statistics: moments and ordering context for the samples being compared.
• cross-cutting/numerics/probability-distributions: normal CDF and quantile context for the approximation used here.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_rank_based_hypothesis_testing.py

The test covers the worked-example anchors for all three tests (rank-sum r1 15.0 / u 0.0 / z -2.5067 / p 0.01219; signed-rank w -21.0 / z -2.1490 / p 0.03164; sign test z 1.5811 / p 0.11385), reversed and all-positive/all-negative rejection directions, the identical-sample and symmetric-pair identities (z 0, p 1), average-rank tie handling, zero-difference dropping, exact dict keys, run-to-run determinism, dispatch via rank_test_summary, and ValueError rejection of every non-physical input listed in Verification.

Compliance

• Standards referenced, not reproduced: NACA TR-824 is named as the numerics-pack reference; the rank procedures above are standard statistical methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.