Accuracy and certification

Accuracy is measured as a Log Relative Error (LRE):

LRE = −log₁₀( |computed − certified| / |certified| )

The condition “|a − b| < 1e-10” measures nothing on its own: near 1e-15 it is trivially satisfied and near 1e12 it is impossible — it measures the magnitude of the operands, not the quality of the computation. LRE is scale-free and reads directly as “how many digits are correct”. It is also the metric NIST itself publishes, which keeps our figures directly comparable with its certification.

Float64 carries about 15.95 decimal digits, so reported values are capped at 15; a value marked “exact” matched identically.

The minimum LRE permitted for each quantity. These are the project's published commitment: a change that drops below one of them is not accepted at all.

Each dataset is solved with the model NIST declares in its own header. κ is the condition number: the larger it is, the harder the problem. The Coef. and SE columns report the worst entry of the coefficient vector and of its standard errors.

Every dataset is solved by QR (Householder) decomposition, and none is found rank-deficient. Forming XᵀX squares the condition number — which is precisely why commercial packages returned zero correct digits on Filip.

Wampler4 and Wampler5 are deliberately constructed: the certified solution stays exactly (1, 1, 1, 1, 1, 1) while a very large noise term is added on top, which is what the certified R² reveals. Least-squares error is bounded by a κ²·ε·tan θ term, and with a residual that large it is that term which dominates. This is therefore the float64 ceiling for the problem, not a defect in the solver. Loosening the solver to “improve” these figures would be the wrong fix.

NIST built these datasets to test the mean and the standard deviation. The right-hand column matters: in NumAcc3 and NumAcc4 the decimal inputs do not round-trip through float64, so the stored data genuinely has a different standard deviation from the one NIST certifies. Reading the two columns side by side separates algorithm error from input-representation error.

A reference value of unknown origin is a superstition. Every number here is either certified by NIST or produced by the versions recorded below.

Every analysis is checked against reference values, but the strength of the evidence differs — and pretending otherwise would be dishonest. Each analysis states which source backs it.

SciPy — independent implementation

An implementation of the same test in a different language and of a different lineage. The strongest routine check available here.

statsmodels — independent OLS

An independent OLS implementation for coefficients, standard errors, confidence intervals and VIF.

SciPy + a documented formula

The statistic and p-value are cross-verified against SciPy; a quantity SciPy does not expose (a rank-biserial correlation, ε²) comes from a documented closed form. That is formula-verified, not cross-library verified.

A documented formula

A closed form with no second implementation in this toolchain. Verified by derivation and by algebraic identities asserted in the tests.

An accuracy report may only claim what it measured. The following sits outside it:

• The confidence interval of the ε² effect size is a seeded bootstrap (BCa). No fixture can certify it, because a second implementation would use a different random stream. The fixture carries the point estimate only; the interval is verified by property instead — determinism under a fixed seed, containment of the point estimate, and one-sidedness.
• The studentized range distribution (used for Tukey comparisons) is computed at 60 points through SciPy QUADPACK in float64, not in arbitrary precision, because no library in this toolchain ships ptukey. The k = 2 case is pinned to an exact closed form, which validates the entire integration path at full float64 precision.
• Accuracy does not guarantee that a result is interpreted correctly. Choosing the test, framing the hypothesis and drawing the conclusion remain the researcher's.
• The figures on this page belong to this version of the code. If a reference-library version is deliberately upgraded, that lands as its own change with the LRE deltas reported.