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The paired t-test: a one-sample test on the differences.

public static TTestResult Paired(ReadOnlySpan<double> a, ReadOnlySpan<double> b, Alternative alternative = Alternative.TwoSided, NanPolicy nanPolicy = NanPolicy.Propagate)

Parameters — a is the first measurement of each pair. b is the second measurement of each pair, in the same order as a. alternative says which tail the p-value covers. nanPolicy says what to do with a NaN; scipy's nan_policy, defaulting to NanPolicy.Propagate.

Returns — TTestResult: the t statistic of the differences a[i] - b[i], its p-value, and the degrees of freedom — one less than the number of pairs actually tested. That is a.Length - 1 under the default NanPolicy.Propagate, and one less than the aligned-filtered pair count under NanPolicy.Omit.

Exceptions — ArgumentException when the two samples differ in length, hold fewer than two pairs, or nanPolicy is NanPolicy.Raise and either sample holds a NaN.

Example — the same seven machines, measured before and after a configuration change.

using Lodestar.Stats;

double[] before = [102.0, 98.0, 110.0, 105.0, 99.0, 101.0, 108.0];
double[] after = [99.0, 96.0, 104.0, 103.0, 95.0, 99.0, 102.0];

TTestResult result = TTest.Paired(before, after);

double t = Math.Round(result.Statistic, 4);   // => 5.2129
double df = result.Df;                        // => 6

Remarks — this is TTest.OneSample on the pairwise differences against a population mean of 0.0: Paired(a, b, alternative) and OneSample(differences, 0.0, alternative) agree exactly, which is also how scipy.stats.ttest_rel is defined against scipy.stats.ttest_1samp. There is no Variance parameter here — pooling only means something when two separate samples each carry their own variance, and a paired test has one sample of differences.

Order matters, sign included. Paired(a, b) and Paired(b, a) report the same magnitude and the opposite sign, so a one-sided alternative answers a different question depending on which argument is a.

Under NanPolicy.Omit the two inputs are filtered together: an index is kept only when neither side holds a NaN, so a pair survives or neither value does.

Applies to — net10.0, netstandard2.0.

See also — TTest.Independent, TTest.OneSample, Wilcoxon.Paired for the rank-based counterpart, the Python equivalence table.