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ProportionInterval

Which interval a binomial proportion is reported with.

public enum ProportionInterval { Exact, Wilson, WilsonCorrected }

Members — Exact is Clopper-Pearson, inverted from the binomial tails; scipy's 'exact', and the default on both sides. Wilson is Wilson's score interval, scipy's 'wilson'. WilsonCorrected is Wilson's with the half-unit continuity correction, scipy's 'wilsoncc'.

Example — the three on the same seven of twenty, widest last.

using Lodestar.Stats;

BinomialResult result = Binomial.Test(7, 20, 0.5);

double exact = Math.Round(result.ProportionConfidenceInterval().Low, 4);
double wilson = Math.Round(
    result.ProportionConfidenceInterval(0.95, ProportionInterval.Wilson).Low, 4);

double clopperPearson = exact;   // => 0.1539
double score = wilson;           // => 0.1812

Remarks — the three differ only where the question is hard, which is small samples. At a thousand trials they agree to three decimals and the choice does not matter. At twenty they disagree by two percentage points, and at five they disagree by more than the estimate is worth. Clopper-Pearson is conservative by construction — it inverts a discrete distribution, so its coverage is at least the level and usually above it. Wilson is calibrated on average and can dip below. The continuity correction pushes Wilson back toward Clopper-Pearson, buying coverage at the cost of width.

Applies to — net10.0, netstandard2.0.

See also — BinomialResult.ProportionConfidenceInterval, Binomial.Test, the Python equivalence table.