Stats binomialresult proportionconfidenceinterval

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BinomialResult.ProportionConfidenceInterval

The confidence interval for the proportion.

public (double Low, double High) ProportionConfidenceInterval(double level = 0.95, ProportionInterval method = ProportionInterval.Exact)

Parameters — level is the confidence level, strictly between 0 and 1. method chooses which interval to compute, defaulting to ProportionInterval.Exact as scipy's proportion_ci does.

Returns — (double Low, double High): the interval around the observed proportion. A one-sided test's interval is half-open, and the far bound is the proportion's own limit, 0 or 1, rather than a wider number — a proportion cannot leave that range. So is a two-sided interval's bound when every trial succeeded or none did.

Exceptions — ArgumentOutOfRangeException when level is NaN or outside (0, 1), or method is not one of the three.

Example — the same seven of twenty, under two of the three methods.

using Lodestar.Stats;

BinomialResult result = Binomial.Test(7, 20, 0.5);
(double exactLow, double exactHigh) = result.ProportionConfidenceInterval();
(double wilsonLow, double wilsonHigh) =
    result.ProportionConfidenceInterval(0.95, ProportionInterval.Wilson);

double clopperPearson = Math.Round(exactLow, 4);   // => 0.1539
double wilson = Math.Round(wilsonLow, 4);          // => 0.1812

Remarks — "exact" names the derivation, not the coverage. Clopper-Pearson inverts the binomial tails themselves, so it is guaranteed never to cover less than the level asked for — and because the binomial is discrete, it usually covers more, which makes it wider than it needs to be. Wilson inverts the normal approximation instead: it is close to the level on average and can fall below it for particular counts. Neither is wrong; they answer "at least this much coverage" and "about this much coverage", and which one a reader wants depends on what a wrong interval costs.

A bound the data pins exactly is reported as the limit, not computed. With every trial a success there is no upper bound to estimate, and the interval runs to 1.

using Lodestar.Stats;

(double low, double high) = Binomial.Test(20, 20, 0.5).ProportionConfidenceInterval();

double lower = Math.Round(low, 4);   // => 0.8316
double upper = high;                 // => 1

Applies to — net10.0, netstandard2.0.

See also — Binomial.Test, ProportionInterval, PearsonResult.ConfidenceInterval, the Python equivalence table.