Stats binomial test - CyrilB1531/lodestar GitHub Wiki

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Binomial.Test

Tests a count of successes against a stated probability.

public static BinomialResult Test(int successes, int trials, double probability = 0.5, Alternative alternative = Alternative.TwoSided)

Parameters — successes is the count observed, between zero and trials. trials is how many there were, zero or more; zero trials answer NaN throughout, as scipy's binomtest does. probability is the probability of success under the null, in [0, 1], defaulting to 0.5 as scipy.stats.binomtest does. alternative says which tail the p-value covers.

Returns — BinomialResult: the observed proportion, the p-value, and a ProportionConfidenceInterval the result can be asked for afterwards.

Exceptions — ArgumentOutOfRangeException when trials is negative, successes lies outside [0, trials], or probability lies outside [0, 1]. ArgumentOutOfRangeException when an option is outside its enumeration.

Example — seven heads in twenty tosses of a coin assumed fair.

using Lodestar.Stats;

BinomialResult result = Binomial.Test(successes: 7, trials: 20, probability: 0.5);

double proportion = Math.Round(result.Statistic, 4);   // => 0.35
double p = Math.Round(result.PValue, 6);               // => 0.263176

Seven of twenty is well inside what a fair coin produces; nothing here is evidence against it.

Remarks — exact means exact at any size, which is the whole reason this exists beside ChiSquared.GoodnessOfFit. The p-value is a sum of binomial probabilities rather than a chi-squared approximation of one, so it stays right where the approximation is worst: few trials, or a probability near zero or one. The cost is that the two-sided p-value has to decide which outcomes on the far side are "at least as extreme", and two outcomes whose probabilities differ only in the last bits are counted as equally extreme — within the 1 + 1e-7 relative margin scipy's binomtest uses. Each probability is computed in Loader's saddle-point form rather than from three log-gammas, whose rounding past 1e8 trials would already exceed that margin.

The two-sided p-value is not twice the one-sided one. The binomial is discrete and, unless probability is 0.5, asymmetric; doubling a tail would be wrong in both directions at once.

using Lodestar.Stats;

BinomialResult twoSided = Binomial.Test(7, 20, 0.5);
BinomialResult greater = Binomial.Test(7, 20, 0.5, Alternative.Greater);

double both = Math.Round(twoSided.PValue, 6);    // => 0.263176
double upper = Math.Round(greater.PValue, 6);    // => 0.942341

Applies to — net10.0, netstandard2.0.