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* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.stat.inference;
import org.apache.commons.math3.distribution.BinomialDistribution;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.MathInternalError;
import org.apache.commons.math3.exception.NotPositiveException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
/**
* Implements binomial test statistics.
* <p>
* Exact test for the statistical significance of deviations from a
* theoretically expected distribution of observations into two categories.
*
* @see <a href="http://en.wikipedia.org/wiki/Binomial_test">Binomial test (Wikipedia)</a>
* @since 3.3
*/
public class BinomialTest {
/**
* Returns whether the null hypothesis can be rejected with the given confidence level.
* <p>
* <strong>Preconditions</strong>:
* <ul>
* <li>Number of trials must be ≥ 0.</li>
* <li>Number of successes must be ≥ 0.</li>
* <li>Number of successes must be ≤ number of trials.</li>
* <li>Probability must be ≥ 0 and ≤ 1.</li>
* </ul>
*
* @param numberOfTrials number of trials performed
* @param numberOfSuccesses number of successes observed
* @param probability assumed probability of a single trial under the null hypothesis
* @param alternativeHypothesis type of hypothesis being evaluated (one- or two-sided)
* @param alpha significance level of the test
* @return true if the null hypothesis can be rejected with confidence {@code 1 - alpha}
* @throws NotPositiveException if {@code numberOfTrials} or {@code numberOfSuccesses} is negative
* @throws OutOfRangeException if {@code probability} is not between 0 and 1
* @throws MathIllegalArgumentException if {@code numberOfTrials} < {@code numberOfSuccesses} or
* if {@code alternateHypothesis} is null.
* @see AlternativeHypothesis
*/
public boolean binomialTest(int numberOfTrials, int numberOfSuccesses, double probability,
AlternativeHypothesis alternativeHypothesis, double alpha) {
double pValue = binomialTest(numberOfTrials, numberOfSuccesses, probability, alternativeHypothesis);
return pValue < alpha;
}
/**
* Returns the <i>observed significance level</i>, or
* <a href="http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue">p-value</a>,
* associated with a <a href="http://en.wikipedia.org/wiki/Binomial_test"> Binomial test</a>.
* <p>
* The number returned is the smallest significance level at which one can reject the null hypothesis.
* The form of the hypothesis depends on {@code alternativeHypothesis}.</p>
* <p>
* The p-Value represents the likelihood of getting a result at least as extreme as the sample,
* given the provided {@code probability} of success on a single trial. For single-sided tests,
* this value can be directly derived from the Binomial distribution. For the two-sided test,
* the implementation works as follows: we start by looking at the most extreme cases
* (0 success and n success where n is the number of trials from the sample) and determine their likelihood.
* The lower value is added to the p-Value (if both values are equal, both are added). Then we continue with
* the next extreme value, until we added the value for the actual observed sample.</p>
* <p>
* <strong>Preconditions</strong>:
* <ul>
* <li>Number of trials must be ≥ 0.</li>
* <li>Number of successes must be ≥ 0.</li>
* <li>Number of successes must be ≤ number of trials.</li>
* <li>Probability must be ≥ 0 and ≤ 1.</li>
* </ul></p>
*
* @param numberOfTrials number of trials performed
* @param numberOfSuccesses number of successes observed
* @param probability assumed probability of a single trial under the null hypothesis
* @param alternativeHypothesis type of hypothesis being evaluated (one- or two-sided)
* @return p-value
* @throws NotPositiveException if {@code numberOfTrials} or {@code numberOfSuccesses} is negative
* @throws OutOfRangeException if {@code probability} is not between 0 and 1
* @throws MathIllegalArgumentException if {@code numberOfTrials} < {@code numberOfSuccesses} or
* if {@code alternateHypothesis} is null.
* @see AlternativeHypothesis
*/
public double binomialTest(int numberOfTrials, int numberOfSuccesses, double probability,
AlternativeHypothesis alternativeHypothesis) {
if (numberOfTrials < 0) {
throw new NotPositiveException(numberOfTrials);
}
if (numberOfSuccesses < 0) {
throw new NotPositiveException(numberOfSuccesses);
}
if (probability < 0 || probability > 1) {
throw new OutOfRangeException(probability, 0, 1);
}
if (numberOfTrials < numberOfSuccesses) {
throw new MathIllegalArgumentException(
LocalizedFormats.BINOMIAL_INVALID_PARAMETERS_ORDER,
numberOfTrials, numberOfSuccesses);
}
if (alternativeHypothesis == null) {
throw new NullArgumentException();
}
// pass a null rng to avoid unneeded overhead as we will not sample from this distribution
final BinomialDistribution distribution = new BinomialDistribution(null, numberOfTrials, probability);
switch (alternativeHypothesis) {
case GREATER_THAN:
return 1 - distribution.cumulativeProbability(numberOfSuccesses - 1);
case LESS_THAN:
return distribution.cumulativeProbability(numberOfSuccesses);
case TWO_SIDED:
int criticalValueLow = 0;
int criticalValueHigh = numberOfTrials;
double pTotal = 0;
while (true) {
double pLow = distribution.probability(criticalValueLow);
double pHigh = distribution.probability(criticalValueHigh);
if (pLow == pHigh) {
pTotal += 2 * pLow;
criticalValueLow++;
criticalValueHigh--;
} else if (pLow < pHigh) {
pTotal += pLow;
criticalValueLow++;
} else {
pTotal += pHigh;
criticalValueHigh--;
}
if (criticalValueLow > numberOfSuccesses || criticalValueHigh < numberOfSuccesses) {
break;
}
}
return pTotal;
default:
throw new MathInternalError(LocalizedFormats. OUT_OF_RANGE_SIMPLE, alternativeHypothesis,
AlternativeHypothesis.TWO_SIDED, AlternativeHypothesis.LESS_THAN);
}
}
}