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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
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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.
 * 
 * Exact test for the statistical significance of deviations from a
 * theoretically expected distribution of observations into two categories.
 *
 * @see Binomial test (Wikipedia)
 * @since 3.3
 */
public class BinomialTest {

    /**
     * Returns whether the null hypothesis can be rejected with the given confidence level.
     * 

     * Preconditions:
     * 

     * Number of trials must be ≥ 0.
     * Number of successes must be ≥ 0.
     * Number of successes must be ≤ number of trials.
     * Probability must be ≥ 0 and ≤ 1.
     * 
     *
     * @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 observed significance level, or
     * p-value,
     * associated with a  Binomial test.
     * 
     * 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}.
     * 
     * 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.
     * 
     * Preconditions:
     * 

     * Number of trials must be ≥ 0.
     * Number of successes must be ≥ 0.
     * Number of successes must be ≤ number of trials.
     * Probability must be ≥ 0 and ≤ 1.
     * 

     *
     * @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);
        }
    }
}