Stats kendallvariant

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KendallVariant

Which normalisation Kendall's tau divides the concordance excess by.

public enum KendallVariant { TauB, TauC }

Members — TauB divides by the tied-pair counts of each sample separately; scipy's 'b', and the default there too. TauC is Stuart's tau-c, scaled by the smaller number of distinct values, scipy's 'c'.

Example — two judges ranking eight entries, with ties on both cards: the same count, normalised twice.

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];

TestResult b = KendallTau.Test(judgeA, judgeB);
TestResult c = KendallTau.Test(judgeA, judgeB, variant: KendallVariant.TauC);

double tauB = Math.Round(b.Statistic, 4);   // => 0.8573
double tauC = Math.Round(c.Statistic, 4);   // => 0.875

Remarks — the p-value is the same either way. Both variants normalise the same quantity, the excess of concordant pairs over discordant ones, and that excess is what the null distribution is read against; only the divisor differs. The two statistics above sit either side of each other, and both come with 0.006153.

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];

TestResult b = KendallTau.Test(judgeA, judgeB);
TestResult c = KendallTau.Test(judgeA, judgeB, variant: KendallVariant.TauC);

double shared = Math.Round(c.PValue - b.PValue, 12);   // => 0

Which to report is a question about the table, not about the data. Tau-b cannot reach 1 when the two samples hold different numbers of distinct values, because its divisor counts the ties it could not have ordered; tau-c rescales by the smaller of those counts so that a table with unequal margins can still reach the ends of the range. On an untied sample the two are equal, and both equal Kendall's original tau-a — which is why no third member exists here, and none does in scipy either.

Applies to — net10.0, netstandard2.0.

See also — KendallTau.Test, ExactMethod, the Python equivalence table.