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Compares two paired samples by the ranks of their differences.

public static TestResult Paired(ReadOnlySpan<double> x, ReadOnlySpan<double> y, ZeroMethod zeroMethod = ZeroMethod.Wilcox, Alternative alternative = Alternative.TwoSided, Continuity continuity = Continuity.None, ExactMethod method = ExactMethod.Auto)

Parameters — x is the first measurement of each pair. y is the second measurement of each pair, in the same order as x. zeroMethod says what to do with pairs whose difference is zero. alternative says which tail the p-value covers. continuity says whether the normal approximation gets the half-unit correction. method chooses the exact null distribution, the exhaustive permutation test, its normal approximation, or a choice between them.

Returns — TestResult: the smaller of the two signed-rank sums, and the p-value.

Exceptions — ArgumentException when the two samples differ in length, or are empty. ArgumentOutOfRangeException when method is ExactMethod.Exact and the ranked sample exceeds 500 values.

Example — seven pairs, two of them unchanged: what the three zero methods disagree about.

using Lodestar.Stats;

double[] before = [12.0, 9.0, 15.0, 11.0, 8.0, 14.0, 10.0, 13.0];
double[] after = [9.0, 6.0, 16.0, 11.0, 4.0, 15.0, 10.0, 17.0];

TestResult wilcox = Wilcoxon.Paired(before, after);
TestResult pratt = Wilcoxon.Paired(before, after, ZeroMethod.Pratt);
TestResult zsplit = Wilcoxon.Paired(before, after, ZeroMethod.ZSplit);

double wilcoxW = wilcox.Statistic;   // => 8.5
double prattW = pratt.Statistic;     // => 14.5
double zsplitW = zsplit.Statistic;   // => 16

Remarks — this delegates to OneSample on the pairwise differences x[i] - y[i]; Paired(x, y, ...) and OneSample(differences, ...) agree exactly wherever the differences agree, the same relationship TTest.Paired has to TTest.OneSample.

Two of the seven pairs above are unchanged, and the three zero methods read that differently: Wilcox drops both pairs before ranking the rest; Pratt ranks them alongside everything else but excludes their ranks from the sums that follow; ZSplit ranks them the same way and keeps them, splitting their rank sum evenly between the positive and negative totals. All three land on different statistics here — the ZeroMethod page has the full three-way comparison, including the p-values.

method here refuses past a lower bound than MannWhitney.Test's: ExactMethod.Exact is only honoured up to 500 ranked values, because the exact null distribution's total, 2^n, overflows a double to infinity past n = 1023 and every p-value it divides would silently become exactly zero rather than throwing. ExactMethod.Auto never reaches that bound on its own — it is unconditionally asymptotic above 50 values, exact only below that and free of both ties and zeros, and falls to an exhaustive permutation test at 13 values or fewer when ties or zeros rule out the plain exact table.

Applies to — net10.0, netstandard2.0.

See also — Wilcoxon.OneSample, TTest.Paired for the parametric counterpart, ZeroMethod, MannWhitney.Test, the Python equivalence table.