test: migrate stats/ttest2 to ULP-based assertions

[kgryte]

Resolves a part of #11352.

Description

What is the purpose of this pull request?

This pull request:

• migrates the tests for stats/ttest2 from relative tolerance testing to ULP difference testing.

Specifically, in test/test.js, each of the 16 relative tolerance assertions of the form

expected = unequal.pValue;
delta = abs( out.pValue - expected );
tol = 18.0 * EPS * abs( expected );
t.ok( delta <= tol, 'within tolerance. a: ['+a+']. b: ['+b+']. pValue: '+out.pValue+'. E: '+expected+'. Δ: '+delta+'. tol: '+tol );

is replaced by

expected = unequal.pValue;
t.strictEqual( isAlmostSameValue( out.pValue, expected, 17 ), true, 'returns expected value' );

along with the corresponding require changes (@stdlib/math/base/special/abs and @stdlib/constants/float64/eps dropped in favor of @stdlib/assert/is-almost-same-value) and the removal of the now-unused delta/tol locals.

ULP bounds (all measured minimums). Because each assertion compares a distinct output quantity against its own R-derived fixture value, the bounds are per-assertion rather than a single package-wide constant:

Test case	Quantity	ULP
Welch, unequal variances (default)	pValue	17
	statistic	0
	ci[0]	0
	ci[1]	1
variance: 'equal'	pValue	1
	statistic	0
	ci[0]	0
	ci[1]	0
difference: -2.0	ci[0]	0
	ci[1]	0
alternative: 'less'	pValue	17
	statistic	0
	ci[1]	0
alternative: 'greater'	pValue	0
	statistic	0
	ci[0]	0

Each bound was determined empirically by computing, for every (actual, expected) pair, the smallest integer N for which isAlmostSameValue( actual, expected, N ) returns true. The three non-zero bounds were additionally confirmed to fail at N-1, so none can be tightened further:

Assertion	N	N-1
Welch pValue	17 → true	16 → false
Welch ci[1]	1 → true	0 → false
variance: 'equal' pValue	1 → true	0 → false
alternative: 'less' pValue	17 → true	16 → false

The remaining twelve comparisons are bit-exact against the R fixtures and therefore use 0. The suite was run twice at the final bounds to confirm determinism (identical results, no FMA/arch variation).

The two loosest bounds (17 for the Welch and one-sided p-values) correspond to the pre-existing loosest relative tolerance in the file (18.0 * EPS); these p-values are ~1e-5 tail probabilities computed via t/cdf, where the accumulated error is largest.

Notes:

• The package has no test/test.native.js, so test/test.js is the only file changed.
• Exact assertions elsewhere in the file were left untouched, including t.strictEqual( out.pValue, diff.pValue, ... ) and t.strictEqual( out.statistic, diff.statistic, ... ) in the difference test case (already bit-exact), t.strictEqual( out.ci[0], NINF, ... ), t.strictEqual( out.ci[1], PINF, ... ), and the .print() and argument-validation tests.
• test/test.validate.js contains no relative tolerance assertions and is unchanged.

Verification:

• make test TESTS_FILTER=".*/stats/ttest2/.*" — 46/46 and 89/89 passing (run twice, identical results).
• make eslint-tests TESTS_FILTER=".*/stats/ttest2/.*" — clean.

Related Issues

Does this pull request have any related issues?

This pull request has the following related issues:

• [RFC]: Migrate math/base/special packages from relative tolerance testing to ULP difference testing (tracking issue) #11352

Questions

Any questions for reviewers of this pull request?

No.

Other

Any other information relevant to this pull request? This may include screenshots, references, and/or implementation notes.

The change mirrors the idiom established in stats/ttest (#15147), the closest prior conversion in the same family, which has the same per-quantity fixture structure.

Two environment notes, neither affecting the diff:

• make install-node-modules failed in this environment because the npm registry resolved for this session has no es-object-atoms@^1.1.2 (a transitive dependency); dev dependencies were installed with that dependency pinned to 1.1.1 so that the test and lint targets could run. package.json is unchanged in this PR.
• The pre-commit hook could not run lint-editorconfig, as it downloads the editorconfig-checker binary from GitHub releases and network access to that repository is not available in this session. The changed file was verified manually to be tab-indented, LF-terminated, free of trailing whitespace, ASCII-only, and to end with a final newline.

Checklist

Please ensure the following tasks are completed before submitting this pull request.

• Read, understood, and followed the contributing guidelines.

AI Assistance

When authoring the changes proposed in this PR, did you use any kind of AI assistance?

• Yes
• No

If you answered "yes" above, how did you use AI assistance?

• Code generation (e.g., when writing an implementation or fixing a bug)
• Test/benchmark generation
• Documentation (including examples)
• Research and understanding

Disclosure

If you answered "yes" to using AI assistance, please provide a short disclosure indicating how you used AI assistance. This helps reviewers determine how much scrutiny to apply when reviewing your contribution. Example disclosures: "This PR was written primarily by Claude Code." or "I consulted ChatGPT to understand the codebase, but the proposed changes were fully authored manually by myself.".

This PR was authored by Claude Code, running unattended as a scheduled task. It reviewed prior converted packages to match the established idiom, applied the conversion, and empirically determined the minimum passing ULP bound for each assertion by measuring the ULP difference against the R fixtures, confirming each non-zero bound fails at N-1, and re-running the suite to confirm determinism.

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https://claude.ai/code/session_01RokNCyA4ZqnTxGzuyCAzqH

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Coverage Report

Package	Statements	Branches	Functions	Lines
stats/ttest2	$\\color{green}416/416$ $\\color{green}+100.00\\%$	$\\color{green}71/71$ $\\color{green}+100.00\\%$	$\\color{green}3/3$ $\\color{green}+100.00\\%$	$\\color{green}416/416$ $\\color{green}+100.00\\%$

The above coverage report was generated for the changes in this PR.