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Correlates two paired samples by counting concordant and discordant pairs.

public static TestResult Test(ReadOnlySpan<double> x, ReadOnlySpan<double> y, Alternative alternative = Alternative.TwoSided, KendallVariant variant = KendallVariant.TauB, ExactMethod method = ExactMethod.Auto, NanPolicy nanPolicy = NanPolicy.Propagate)

Parameters — x and y are the two samples, of the same length; both spans are read, never modified. alternative says which tail the p-value covers. variant chooses which normalisation the statistic carries, and changes nothing else — see KendallVariant. method chooses the exact null distribution, its normal approximation, or a choice between them by sample size and ties. nanPolicy says what to do with a NaN; scipy's nan_policy, defaulting to NanPolicy.Propagate.

Returns — TestResult: tau, and the p-value. Fewer than two pairs, or a sample in which every value is tied, leaves tau undefined and both numbers are NaN.

Exceptions — ArgumentException when the samples differ in length, when method is ExactMethod.Exact and either sample holds a tie, or when nanPolicy is NanPolicy.Raise and either sample holds a NaN. ArgumentOutOfRangeException when method is ExactMethod.Exact and the null distribution's table would exceed twenty million cells.

Example — the same eight tasting panels the other two measure.

using Lodestar.Stats;

double[] sugar = [4.0, 6.5, 5.0, 9.0, 7.5, 3.0, 8.0, 6.0];
double[] sweetness = [3.1, 5.9, 4.0, 7.4, 6.8, 2.2, 6.1, 5.2];

TestResult result = KendallTau.Test(sugar, sweetness);

double tau = Math.Round(result.Statistic, 4);   // => 0.9286
double p = Math.Round(result.PValue, 6);        // => 0.000397

Remarks — the exact method is refused on a tied sample, where MannWhitney.Test answers. The two families differ because their references do: mannwhitneyu builds its table and reads it whatever the data, while kendalltau raises ValueError: Ties found; exact method cannot be used. Each is matched rather than reconciled, and the Python equivalence table carries both rows so the difference is met in the mapping rather than in a stack trace.

using Lodestar.Stats;

double[] judgeA = [1.0, 1.0, 2.0, 2.0, 3.0, 3.0, 4.0, 4.0];
double[] judgeB = [1.0, 2.0, 2.0, 3.0, 3.0, 4.0, 4.0, 5.0];

string message = "nothing was thrown";
try
{
    KendallTau.Test(judgeA, judgeB, method: ExactMethod.Exact);
}
catch (ArgumentException error)
{
    message = error.Message;
}

string what = message;   // => The exact Kendall distribution is defined over untied…

ExactMethod.Auto never throws for it: a tie sends it to the asymptotic route, and so does a table past the twenty-million-cell ceiling, because nothing the caller wrote asked for an exact answer. Untied, Auto is exact up to and including thirty-three observations, or whenever one tail holds at most a single pair — the two routes disagree on the number, which is why the choice is a parameter rather than an optimisation.

using Lodestar.Stats;

double[] sugar = [4.0, 6.5, 5.0, 9.0, 7.5, 3.0, 8.0, 6.0];
double[] sweetness = [3.1, 5.9, 4.0, 7.4, 6.8, 2.2, 6.1, 5.2];

TestResult auto = KendallTau.Test(sugar, sweetness);
TestResult asymptotic = KendallTau.Test(sugar, sweetness, method: ExactMethod.Asymptotic);

double exactP = Math.Round(auto.PValue, 6);             // => 0.000397
double approximateP = Math.Round(asymptotic.PValue, 6); // => 0.001297

The discordant pairs are counted in O(n log n), not by the double loop the definition suggests. Ordering the pairs by the first sample turns the count into an inversion count of the second, which a merge sort answers; at ten thousand pairs that is a hundred and thirty thousand comparisons against the definition's fifty million. scipy reaches the same order through a Fenwick tree.

Applies to — net10.0, netstandard2.0.

See also — Pearson.Test, Spearman.Test, KendallVariant, the Python equivalence table.