Package org.apache.commons.statistics.inference

Class BinomialTest

java.lang.Object

org.apache.commons.statistics.inference.BinomialTest

public final class BinomialTest
extends Object

Implements binomial test statistics.

Performs an exact test for the statistical significance of deviations from a theoretically expected distribution of observations into two categories.

Since:

1.1

See Also:

Binomial test (Wikipedia)

Method Summary

Modifier and Type	Method	Description
SignificanceResult	test(int numberOfTrials, int numberOfSuccesses, double probability)	Performs a binomial test about the probability of success \( \pi \).
BinomialTest	with(AlternativeHypothesis v)	Return an instance with the configured alternative hypothesis.
static BinomialTest	withDefaults()	Return an instance using the default options.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

withDefaults

public static BinomialTest withDefaults()

Return an instance using the default options.

• AlternativeHypothesis.TWO_SIDED

Returns:

default instance

with

public BinomialTest with(AlternativeHypothesis v)

Return an instance with the configured alternative hypothesis.

Parameters:

v - Value.

Returns:

an instance

test

public SignificanceResult test(int numberOfTrials,
                               int numberOfSuccesses,
                               double probability)

Performs a binomial test about the probability of success \( \pi \).

The null hypothesis is \( H_0:\pi=\pi_0 \) where \( \pi_0 \) is between 0 and 1.

The probability of observing \( k \) successes from \( n \) trials with a given probability of success \( p \) is:

\[ \Pr(X=k)=\binom{n}{k}p^k(1-p)^{n-k} \]

The test is defined by the AlternativeHypothesis.

To test \( \pi < \pi_0 \) (less than):

\[ p = \sum_{i=0}^k\Pr(X=i)=\sum_{i=0}^k\binom{n}{i}\pi_0^i(1-\pi_0)^{n-i} \]

To test \( \pi > \pi_0 \) (greater than):

\[ p = \sum_{i=0}^k\Pr(X=i)=\sum_{i=k}^n\binom{n}{i}\pi_0^i(1-\pi_0)^{n-i} \]

To test \( \pi \ne \pi_0 \) (two-sided) requires finding all \( i \) such that \( \mathcal{I}=\{i:\Pr(X=i)\leq \Pr(X=k)\} \) and compute the sum:

\[ p = \sum_{i\in\mathcal{I}}\Pr(X=i)=\sum_{i\in\mathcal{I}}\binom{n}{i}\pi_0^i(1-\pi_0)^{n-i} \]

The two-sided p-value represents the likelihood of getting a result at least as extreme as the sample, given the provided probability of success on a single trial.

The test statistic is equal to the estimated proportion \( \frac{k}{n} \).

Parameters:

numberOfTrials - Number of trials performed.

numberOfSuccesses - Number of successes observed.

probability - Assumed probability of a single trial under the null hypothesis.

Returns:

test result

Throws:

IllegalArgumentException - if numberOfTrials or numberOfSuccesses is negative; probability is not between 0 and 1; or if numberOfTrials < numberOfSuccesses

See Also:

with(AlternativeHypothesis)