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Variance

Whether an independent-samples t-test pools the two variances.

public enum Variance { Equal, Welch }

Members — Equal pools the two variances — Student's t; scipy's equal_var=True. Welch does not pool, using the Welch-Satterthwaite degrees of freedom instead; scipy's equal_var=False.

Example — the same two samples, tested both ways.

using Lodestar.Stats;

double[] before = [102.0, 98.0, 110.0, 105.0, 99.0];
double[] after = [95.0, 92.0, 99.0, 91.0, 97.0];

TTestResult welch = TTest.Independent(before, after, Alternative.TwoSided, Variance.Welch);
TTestResult student = TTest.Independent(before, after, Alternative.TwoSided, Variance.Equal);

double welchDf = Math.Round(welch.Df, 4);     // => 7.0904
double studentDf = student.Df;                // => 8

Remarks — this package's one deliberate divergence from scipy. TTest.Independent defaults to Welch; scipy.stats.ttest_ind defaults to Equal (equal_var=True). Pooling is only correct when the two populations really do share a variance, which is an assumption a caller rarely has grounds to make going in — Welch is the safer choice when it is wrong and loses nothing when it happens to be right, so the safer default costs one word at the call site here rather than a wrong answer by default. Equal's degrees of freedom are always n + m - 2, a whole number; Welch's Satterthwaite denominator is not a count of anything, which is why TTestResult.Df is a double rather than an int — 7.0904 above is not a rounding artefact.

On these two samples the t statistic itself does not move — both give 3.028 — because the two sample sizes are equal; pooling and not pooling only disagree on the standard error, and hence the degrees of freedom, once the samples are unequal in size or spread.

Applies to — net10.0, netstandard2.0.

See also — TTest.Independent, TTestResult, the Python equivalence table.