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Java example source code file (TTest.java)

This example Java source code file (TTest.java) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM.

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Java - Java tags/keywords

dimensionmismatchexception, mathillegalargumentexception, maxcountexceededexception, nodataexception, notstrictlypositiveexception, nullargumentexception, numberistoosmallexception, outofrangeexception, statisticalsummary, tdistribution, ttest

The TTest.java Java example source code

/*
 * 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
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package org.apache.commons.math3.stat.inference;

import org.apache.commons.math3.distribution.TDistribution;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.stat.StatUtils;
import org.apache.commons.math3.stat.descriptive.StatisticalSummary;
import org.apache.commons.math3.util.FastMath;

/**
 * An implementation for Student's t-tests.
 * <p>
 * Tests can be:<ul>
 * <li>One-sample or two-sample
 * <li>One-sided or two-sided
 * <li>Paired or unpaired (for two-sample tests)
 * <li>Homoscedastic (equal variance assumption) or heteroscedastic
 * (for two sample tests)</li>
 * <li>Fixed significance level (boolean-valued) or returning p-values.
 * </li>

* <p> * Test statistics are available for all tests. Methods including "Test" in * in their names perform tests, all other methods return t-statistics. Among * the "Test" methods, <code>double-valued methods return p-values; * <code>boolean-valued methods perform fixed significance level tests. * Significance levels are always specified as numbers between 0 and 0.5 * (e.g. tests at the 95% level use <code>alpha=0.05).

* <p> * Input to tests can be either <code>double[] arrays or * {@link StatisticalSummary} instances.</p>

* Uses commons-math {@link org.apache.commons.math3.distribution.TDistribution} * implementation to estimate exact p-values.</p> * */ public class TTest { /** * Computes a paired, 2-sample t-statistic based on the data in the input * arrays. The t-statistic returned is equivalent to what would be returned by * computing the one-sample t-statistic {@link #t(double, double[])}, with * <code>mu = 0 and the sample array consisting of the (signed) * differences between corresponding entries in <code>sample1 and * <code>sample2. * <p> * <strong>Preconditions:

    * <li>The input arrays must have the same length and their common length * must be at least 2. * </li>

* * @param sample1 array of sample data values * @param sample2 array of sample data values * @return t statistic * @throws NullArgumentException if the arrays are <code>null * @throws NoDataException if the arrays are empty * @throws DimensionMismatchException if the length of the arrays is not equal * @throws NumberIsTooSmallException if the length of the arrays is < 2 */ public double pairedT(final double[] sample1, final double[] sample2) throws NullArgumentException, NoDataException, DimensionMismatchException, NumberIsTooSmallException { checkSampleData(sample1); checkSampleData(sample2); double meanDifference = StatUtils.meanDifference(sample1, sample2); return t(meanDifference, 0, StatUtils.varianceDifference(sample1, sample2, meanDifference), sample1.length); } /** * Returns the <i>observed significance level, or * <i> p-value, associated with a paired, two-sample, two-tailed t-test * based on the data in the input arrays. * <p> * The number returned is the smallest significance level * at which one can reject the null hypothesis that the mean of the paired * differences is 0 in favor of the two-sided alternative that the mean paired * difference is not equal to 0. For a one-sided test, divide the returned * value by 2.</p> * <p> * This test is equivalent to a one-sample t-test computed using * {@link #tTest(double, double[])} with <code>mu = 0 and the sample * array consisting of the signed differences between corresponding elements of * <code>sample1 and sample2.

* <p> * <strong>Usage Note:
* The validity of the p-value depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"> * here</a>

* <p> * <strong>Preconditions:
    * <li>The input array lengths must be the same and their common length must * be at least 2. * </li>

* * @param sample1 array of sample data values * @param sample2 array of sample data values * @return p-value for t-test * @throws NullArgumentException if the arrays are <code>null * @throws NoDataException if the arrays are empty * @throws DimensionMismatchException if the length of the arrays is not equal * @throws NumberIsTooSmallException if the length of the arrays is < 2 * @throws MaxCountExceededException if an error occurs computing the p-value */ public double pairedTTest(final double[] sample1, final double[] sample2) throws NullArgumentException, NoDataException, DimensionMismatchException, NumberIsTooSmallException, MaxCountExceededException { double meanDifference = StatUtils.meanDifference(sample1, sample2); return tTest(meanDifference, 0, StatUtils.varianceDifference(sample1, sample2, meanDifference), sample1.length); } /** * Performs a paired t-test evaluating the null hypothesis that the * mean of the paired differences between <code>sample1 and * <code>sample2 is 0 in favor of the two-sided alternative that the * mean paired difference is not equal to 0, with significance level * <code>alpha. * <p> * Returns <code>true iff the null hypothesis can be rejected with * confidence <code>1 - alpha. To perform a 1-sided test, use * <code>alpha * 2

* <p> * <strong>Usage Note:
* The validity of the test depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"> * here</a>

* <p> * <strong>Preconditions:
    * <li>The input array lengths must be the same and their common length * must be at least 2. * </li> * <li> 0 < alpha < 0.5 * </li>

* * @param sample1 array of sample data values * @param sample2 array of sample data values * @param alpha significance level of the test * @return true if the null hypothesis can be rejected with * confidence 1 - alpha * @throws NullArgumentException if the arrays are <code>null * @throws NoDataException if the arrays are empty * @throws DimensionMismatchException if the length of the arrays is not equal * @throws NumberIsTooSmallException if the length of the arrays is < 2 * @throws OutOfRangeException if <code>alpha is not in the range (0, 0.5] * @throws MaxCountExceededException if an error occurs computing the p-value */ public boolean pairedTTest(final double[] sample1, final double[] sample2, final double alpha) throws NullArgumentException, NoDataException, DimensionMismatchException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException { checkSignificanceLevel(alpha); return pairedTTest(sample1, sample2) < alpha; } /** * Computes a <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm#formula"> * t statistic </a> given observed values and a comparison constant. * <p> * This statistic can be used to perform a one sample t-test for the mean. * </p>

* <strong>Preconditions:

    * <li>The observed array length must be at least 2. * </li>

* * @param mu comparison constant * @param observed array of values * @return t statistic * @throws NullArgumentException if <code>observed is null * @throws NumberIsTooSmallException if the length of <code>observed is < 2 */ public double t(final double mu, final double[] observed) throws NullArgumentException, NumberIsTooSmallException { checkSampleData(observed); // No try-catch or advertised exception because args have just been checked return t(StatUtils.mean(observed), mu, StatUtils.variance(observed), observed.length); } /** * Computes a <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm#formula"> * t statistic </a> to use in comparing the mean of the dataset described by * <code>sampleStats to mu. * <p> * This statistic can be used to perform a one sample t-test for the mean. * </p>

* <strong>Preconditions:

    * <li>observed.getN() ≥ 2. * </li>

* * @param mu comparison constant * @param sampleStats DescriptiveStatistics holding sample summary statitstics * @return t statistic * @throws NullArgumentException if <code>sampleStats is null * @throws NumberIsTooSmallException if the number of samples is < 2 */ public double t(final double mu, final StatisticalSummary sampleStats) throws NullArgumentException, NumberIsTooSmallException { checkSampleData(sampleStats); return t(sampleStats.getMean(), mu, sampleStats.getVariance(), sampleStats.getN()); } /** * Computes a 2-sample t statistic, under the hypothesis of equal * subpopulation variances. To compute a t-statistic without the * equal variances hypothesis, use {@link #t(double[], double[])}. * <p> * This statistic can be used to perform a (homoscedastic) two-sample * t-test to compare sample means.</p> * <p> * The t-statistic is</p> * <p> *   <code> t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var)) * </p>

* where <strong>n1 is the size of first sample; * <strong> n2 is the size of second sample; * <strong> m1 is the mean of first sample; * <strong> m2 is the mean of second sample * </ul> * and <strong>var is the pooled variance estimate: * </p>

* <code>var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1))) * </p>

* with <strong>var1 the variance of the first sample and * <strong>var2 the variance of the second sample. * </p>

* <strong>Preconditions:

    * <li>The observed array lengths must both be at least 2. * </li>

* * @param sample1 array of sample data values * @param sample2 array of sample data values * @return t statistic * @throws NullArgumentException if the arrays are <code>null * @throws NumberIsTooSmallException if the length of the arrays is < 2 */ public double homoscedasticT(final double[] sample1, final double[] sample2) throws NullArgumentException, NumberIsTooSmallException { checkSampleData(sample1); checkSampleData(sample2); // No try-catch or advertised exception because args have just been checked return homoscedasticT(StatUtils.mean(sample1), StatUtils.mean(sample2), StatUtils.variance(sample1), StatUtils.variance(sample2), sample1.length, sample2.length); } /** * Computes a 2-sample t statistic, without the hypothesis of equal * subpopulation variances. To compute a t-statistic assuming equal * variances, use {@link #homoscedasticT(double[], double[])}. * <p> * This statistic can be used to perform a two-sample t-test to compare * sample means.</p> * <p> * The t-statistic is</p> * <p> *    <code> t = (m1 - m2) / sqrt(var1/n1 + var2/n2) * </p>

* where <strong>n1 is the size of the first sample * <strong> n2 is the size of the second sample; * <strong> m1 is the mean of the first sample; * <strong> m2 is the mean of the second sample; * <strong> var1 is the variance of the first sample; * <strong> var2 is the variance of the second sample; * </p>

* <strong>Preconditions:

    * <li>The observed array lengths must both be at least 2. * </li>

* * @param sample1 array of sample data values * @param sample2 array of sample data values * @return t statistic * @throws NullArgumentException if the arrays are <code>null * @throws NumberIsTooSmallException if the length of the arrays is < 2 */ public double t(final double[] sample1, final double[] sample2) throws NullArgumentException, NumberIsTooSmallException { checkSampleData(sample1); checkSampleData(sample2); // No try-catch or advertised exception because args have just been checked return t(StatUtils.mean(sample1), StatUtils.mean(sample2), StatUtils.variance(sample1), StatUtils.variance(sample2), sample1.length, sample2.length); } /** * Computes a 2-sample t statistic </a>, comparing the means of the datasets * described by two {@link StatisticalSummary} instances, without the * assumption of equal subpopulation variances. Use * {@link #homoscedasticT(StatisticalSummary, StatisticalSummary)} to * compute a t-statistic under the equal variances assumption. * <p> * This statistic can be used to perform a two-sample t-test to compare * sample means.</p> * <p> * The returned t-statistic is</p> * <p> *    <code> t = (m1 - m2) / sqrt(var1/n1 + var2/n2) * </p>

* where <strong>n1 is the size of the first sample; * <strong> n2 is the size of the second sample; * <strong> m1 is the mean of the first sample; * <strong> m2 is the mean of the second sample * <strong> var1 is the variance of the first sample; * <strong> var2 is the variance of the second sample * </p>

* <strong>Preconditions:

    * <li>The datasets described by the two Univariates must each contain * at least 2 observations. * </li>

* * @param sampleStats1 StatisticalSummary describing data from the first sample * @param sampleStats2 StatisticalSummary describing data from the second sample * @return t statistic * @throws NullArgumentException if the sample statistics are <code>null * @throws NumberIsTooSmallException if the number of samples is < 2 */ public double t(final StatisticalSummary sampleStats1, final StatisticalSummary sampleStats2) throws NullArgumentException, NumberIsTooSmallException { checkSampleData(sampleStats1); checkSampleData(sampleStats2); return t(sampleStats1.getMean(), sampleStats2.getMean(), sampleStats1.getVariance(), sampleStats2.getVariance(), sampleStats1.getN(), sampleStats2.getN()); } /** * Computes a 2-sample t statistic, comparing the means of the datasets * described by two {@link StatisticalSummary} instances, under the * assumption of equal subpopulation variances. To compute a t-statistic * without the equal variances assumption, use * {@link #t(StatisticalSummary, StatisticalSummary)}. * <p> * This statistic can be used to perform a (homoscedastic) two-sample * t-test to compare sample means.</p> * <p> * The t-statistic returned is</p> * <p> *   <code> t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var)) * </p>

* where <strong>n1 is the size of first sample; * <strong> n2 is the size of second sample; * <strong> m1 is the mean of first sample; * <strong> m2 is the mean of second sample * and <strong>var is the pooled variance estimate: * </p>

* <code>var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1))) * </p>

* with <strong>var1 the variance of the first sample and * <strong>var2 the variance of the second sample. * </p>

* <strong>Preconditions:

    * <li>The datasets described by the two Univariates must each contain * at least 2 observations. * </li>

* * @param sampleStats1 StatisticalSummary describing data from the first sample * @param sampleStats2 StatisticalSummary describing data from the second sample * @return t statistic * @throws NullArgumentException if the sample statistics are <code>null * @throws NumberIsTooSmallException if the number of samples is < 2 */ public double homoscedasticT(final StatisticalSummary sampleStats1, final StatisticalSummary sampleStats2) throws NullArgumentException, NumberIsTooSmallException { checkSampleData(sampleStats1); checkSampleData(sampleStats2); return homoscedasticT(sampleStats1.getMean(), sampleStats2.getMean(), sampleStats1.getVariance(), sampleStats2.getVariance(), sampleStats1.getN(), sampleStats2.getN()); } /** * Returns the <i>observed significance level, or * <i>p-value, associated with a one-sample, two-tailed t-test * comparing the mean of the input array with the constant <code>mu. * <p> * The number returned is the smallest significance level * at which one can reject the null hypothesis that the mean equals * <code>mu in favor of the two-sided alternative that the mean * is different from <code>mu. For a one-sided test, divide the * returned value by 2.</p> * <p> * <strong>Usage Note:
* The validity of the test depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">here * </p>

* <strong>Preconditions:

    * <li>The observed array length must be at least 2. * </li>

* * @param mu constant value to compare sample mean against * @param sample array of sample data values * @return p-value * @throws NullArgumentException if the sample array is <code>null * @throws NumberIsTooSmallException if the length of the array is < 2 * @throws MaxCountExceededException if an error occurs computing the p-value */ public double tTest(final double mu, final double[] sample) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException { checkSampleData(sample); // No try-catch or advertised exception because args have just been checked return tTest(StatUtils.mean(sample), mu, StatUtils.variance(sample), sample.length); } /** * Performs a <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"> * two-sided t-test</a> evaluating the null hypothesis that the mean of the population from * which <code>sample is drawn equals mu. * <p> * Returns <code>true iff the null hypothesis can be * rejected with confidence <code>1 - alpha. To * perform a 1-sided test, use <code>alpha * 2

* <p> * <strong>Examples:
    * <li>To test the (2-sided) hypothesis sample mean = mu at * the 95% level, use <br>tTest(mu, sample, 0.05) * </li> * <li>To test the (one-sided) hypothesis sample mean < mu * at the 99% level, first verify that the measured sample mean is less * than <code>mu and then use * <br>tTest(mu, sample, 0.02) * </li>

* <p> * <strong>Usage Note:
* The validity of the test depends on the assumptions of the one-sample * parametric t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/sg_glos.html#one-sample">here * </p>

* <strong>Preconditions:

    * <li>The observed array length must be at least 2. * </li>

* * @param mu constant value to compare sample mean against * @param sample array of sample data values * @param alpha significance level of the test * @return p-value * @throws NullArgumentException if the sample array is <code>null * @throws NumberIsTooSmallException if the length of the array is < 2 * @throws OutOfRangeException if <code>alpha is not in the range (0, 0.5] * @throws MaxCountExceededException if an error computing the p-value */ public boolean tTest(final double mu, final double[] sample, final double alpha) throws NullArgumentException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException { checkSignificanceLevel(alpha); return tTest(mu, sample) < alpha; } /** * Returns the <i>observed significance level, or * <i>p-value, associated with a one-sample, two-tailed t-test * comparing the mean of the dataset described by <code>sampleStats * with the constant <code>mu. * <p> * The number returned is the smallest significance level * at which one can reject the null hypothesis that the mean equals * <code>mu in favor of the two-sided alternative that the mean * is different from <code>mu. For a one-sided test, divide the * returned value by 2.</p> * <p> * <strong>Usage Note:
* The validity of the test depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"> * here</a>

* <p> * <strong>Preconditions:
    * <li>The sample must contain at least 2 observations. * </li>

* * @param mu constant value to compare sample mean against * @param sampleStats StatisticalSummary describing sample data * @return p-value * @throws NullArgumentException if <code>sampleStats is null * @throws NumberIsTooSmallException if the number of samples is < 2 * @throws MaxCountExceededException if an error occurs computing the p-value */ public double tTest(final double mu, final StatisticalSummary sampleStats) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException { checkSampleData(sampleStats); return tTest(sampleStats.getMean(), mu, sampleStats.getVariance(), sampleStats.getN()); } /** * Performs a <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"> * two-sided t-test</a> evaluating the null hypothesis that the mean of the * population from which the dataset described by <code>stats is * drawn equals <code>mu. * <p> * Returns <code>true iff the null hypothesis can be rejected with * confidence <code>1 - alpha. To perform a 1-sided test, use * <code>alpha * 2.

* <p> * <strong>Examples:
    * <li>To test the (2-sided) hypothesis sample mean = mu at * the 95% level, use <br>tTest(mu, sampleStats, 0.05) * </li> * <li>To test the (one-sided) hypothesis sample mean < mu * at the 99% level, first verify that the measured sample mean is less * than <code>mu and then use * <br>tTest(mu, sampleStats, 0.02) * </li>

* <p> * <strong>Usage Note:
* The validity of the test depends on the assumptions of the one-sample * parametric t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/sg_glos.html#one-sample">here * </p>

* <strong>Preconditions:

    * <li>The sample must include at least 2 observations. * </li>

* * @param mu constant value to compare sample mean against * @param sampleStats StatisticalSummary describing sample data values * @param alpha significance level of the test * @return p-value * @throws NullArgumentException if <code>sampleStats is null * @throws NumberIsTooSmallException if the number of samples is < 2 * @throws OutOfRangeException if <code>alpha is not in the range (0, 0.5] * @throws MaxCountExceededException if an error occurs computing the p-value */ public boolean tTest(final double mu, final StatisticalSummary sampleStats, final double alpha) throws NullArgumentException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException { checkSignificanceLevel(alpha); return tTest(mu, sampleStats) < alpha; } /** * Returns the <i>observed significance level, or * <i>p-value, associated with a two-sample, two-tailed t-test * comparing the means of the input arrays. * <p> * The number returned is the smallest significance level * at which one can reject the null hypothesis that the two means are * equal in favor of the two-sided alternative that they are different. * For a one-sided test, divide the returned value by 2.</p> * <p> * The test does not assume that the underlying popuation variances are * equal and it uses approximated degrees of freedom computed from the * sample data to compute the p-value. The t-statistic used is as defined in * {@link #t(double[], double[])} and the Welch-Satterthwaite approximation * to the degrees of freedom is used, * as described * <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm"> * here.</a> To perform the test under the assumption of equal subpopulation * variances, use {@link #homoscedasticTTest(double[], double[])}.</p> * <p> * <strong>Usage Note:
* The validity of the p-value depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"> * here</a>

* <p> * <strong>Preconditions:
    * <li>The observed array lengths must both be at least 2. * </li>

* * @param sample1 array of sample data values * @param sample2 array of sample data values * @return p-value for t-test * @throws NullArgumentException if the arrays are <code>null * @throws NumberIsTooSmallException if the length of the arrays is < 2 * @throws MaxCountExceededException if an error occurs computing the p-value */ public double tTest(final double[] sample1, final double[] sample2) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException { checkSampleData(sample1); checkSampleData(sample2); // No try-catch or advertised exception because args have just been checked return tTest(StatUtils.mean(sample1), StatUtils.mean(sample2), StatUtils.variance(sample1), StatUtils.variance(sample2), sample1.length, sample2.length); } /** * Returns the <i>observed significance level, or * <i>p-value, associated with a two-sample, two-tailed t-test * comparing the means of the input arrays, under the assumption that * the two samples are drawn from subpopulations with equal variances. * To perform the test without the equal variances assumption, use * {@link #tTest(double[], double[])}.</p> * <p> * The number returned is the smallest significance level * at which one can reject the null hypothesis that the two means are * equal in favor of the two-sided alternative that they are different. * For a one-sided test, divide the returned value by 2.</p> * <p> * A pooled variance estimate is used to compute the t-statistic. See * {@link #homoscedasticT(double[], double[])}. The sum of the sample sizes * minus 2 is used as the degrees of freedom.</p> * <p> * <strong>Usage Note:
* The validity of the p-value depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"> * here</a>

* <p> * <strong>Preconditions:
    * <li>The observed array lengths must both be at least 2. * </li>

* * @param sample1 array of sample data values * @param sample2 array of sample data values * @return p-value for t-test * @throws NullArgumentException if the arrays are <code>null * @throws NumberIsTooSmallException if the length of the arrays is < 2 * @throws MaxCountExceededException if an error occurs computing the p-value */ public double homoscedasticTTest(final double[] sample1, final double[] sample2) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException { checkSampleData(sample1); checkSampleData(sample2); // No try-catch or advertised exception because args have just been checked return homoscedasticTTest(StatUtils.mean(sample1), StatUtils.mean(sample2), StatUtils.variance(sample1), StatUtils.variance(sample2), sample1.length, sample2.length); } /** * Performs a * <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"> * two-sided t-test</a> evaluating the null hypothesis that sample1 * and <code>sample2 are drawn from populations with the same mean, * with significance level <code>alpha. This test does not assume * that the subpopulation variances are equal. To perform the test assuming * equal variances, use * {@link #homoscedasticTTest(double[], double[], double)}. * <p> * Returns <code>true iff the null hypothesis that the means are * equal can be rejected with confidence <code>1 - alpha. To * perform a 1-sided test, use <code>alpha * 2

* <p> * See {@link #t(double[], double[])} for the formula used to compute the * t-statistic. Degrees of freedom are approximated using the * <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm"> * Welch-Satterthwaite approximation.</a>

* <p> * <strong>Examples:
    * <li>To test the (2-sided) hypothesis mean 1 = mean 2 at * the 95% level, use * <br>tTest(sample1, sample2, 0.05). * </li> * <li>To test the (one-sided) hypothesis mean 1 < mean 2 , * at the 99% level, first verify that the measured mean of <code>sample 1 * is less than the mean of <code>sample 2 and then use * <br>tTest(sample1, sample2, 0.02) * </li>

* <p> * <strong>Usage Note:
* The validity of the test depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"> * here</a>

* <p> * <strong>Preconditions:
    * <li>The observed array lengths must both be at least 2. * </li> * <li> 0 < alpha < 0.5 * </li>

* * @param sample1 array of sample data values * @param sample2 array of sample data values * @param alpha significance level of the test * @return true if the null hypothesis can be rejected with * confidence 1 - alpha * @throws NullArgumentException if the arrays are <code>null * @throws NumberIsTooSmallException if the length of the arrays is < 2 * @throws OutOfRangeException if <code>alpha is not in the range (0, 0.5] * @throws MaxCountExceededException if an error occurs computing the p-value */ public boolean tTest(final double[] sample1, final double[] sample2, final double alpha) throws NullArgumentException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException { checkSignificanceLevel(alpha); return tTest(sample1, sample2) < alpha; } /** * Performs a * <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"> * two-sided t-test</a> evaluating the null hypothesis that sample1 * and <code>sample2 are drawn from populations with the same mean, * with significance level <code>alpha, assuming that the * subpopulation variances are equal. Use * {@link #tTest(double[], double[], double)} to perform the test without * the assumption of equal variances. * <p> * Returns <code>true iff the null hypothesis that the means are * equal can be rejected with confidence <code>1 - alpha. To * perform a 1-sided test, use <code>alpha * 2. To perform the test * without the assumption of equal subpopulation variances, use * {@link #tTest(double[], double[], double)}.</p> * <p> * A pooled variance estimate is used to compute the t-statistic. See * {@link #t(double[], double[])} for the formula. The sum of the sample * sizes minus 2 is used as the degrees of freedom.</p> * <p> * <strong>Examples:
    * <li>To test the (2-sided) hypothesis mean 1 = mean 2 at * the 95% level, use <br>tTest(sample1, sample2, 0.05). * </li> * <li>To test the (one-sided) hypothesis mean 1 < mean 2, * at the 99% level, first verify that the measured mean of * <code>sample 1 is less than the mean of sample 2 * and then use * <br>tTest(sample1, sample2, 0.02) * </li>

* <p> * <strong>Usage Note:
* The validity of the test depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"> * here</a>

* <p> * <strong>Preconditions:
    * <li>The observed array lengths must both be at least 2. * </li> * <li> 0 < alpha < 0.5 * </li>

* * @param sample1 array of sample data values * @param sample2 array of sample data values * @param alpha significance level of the test * @return true if the null hypothesis can be rejected with * confidence 1 - alpha * @throws NullArgumentException if the arrays are <code>null * @throws NumberIsTooSmallException if the length of the arrays is < 2 * @throws OutOfRangeException if <code>alpha is not in the range (0, 0.5] * @throws MaxCountExceededException if an error occurs computing the p-value */ public boolean homoscedasticTTest(final double[] sample1, final double[] sample2, final double alpha) throws NullArgumentException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException { checkSignificanceLevel(alpha); return homoscedasticTTest(sample1, sample2) < alpha; } /** * Returns the <i>observed significance level, or * <i>p-value, associated with a two-sample, two-tailed t-test * comparing the means of the datasets described by two StatisticalSummary * instances. * <p> * The number returned is the smallest significance level * at which one can reject the null hypothesis that the two means are * equal in favor of the two-sided alternative that they are different. * For a one-sided test, divide the returned value by 2.</p> * <p> * The test does not assume that the underlying population variances are * equal and it uses approximated degrees of freedom computed from the * sample data to compute the p-value. To perform the test assuming * equal variances, use * {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}.</p> * <p> * <strong>Usage Note:
* The validity of the p-value depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"> * here</a>

* <p> * <strong>Preconditions:
    * <li>The datasets described by the two Univariates must each contain * at least 2 observations. * </li>

* * @param sampleStats1 StatisticalSummary describing data from the first sample * @param sampleStats2 StatisticalSummary describing data from the second sample * @return p-value for t-test * @throws NullArgumentException if the sample statistics are <code>null * @throws NumberIsTooSmallException if the number of samples is < 2 * @throws MaxCountExceededException if an error occurs computing the p-value */ public double tTest(final StatisticalSummary sampleStats1, final StatisticalSummary sampleStats2) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException { checkSampleData(sampleStats1); checkSampleData(sampleStats2); return tTest(sampleStats1.getMean(), sampleStats2.getMean(), sampleStats1.getVariance(), sampleStats2.getVariance(), sampleStats1.getN(), sampleStats2.getN()); } /** * Returns the <i>observed significance level, or * <i>p-value, associated with a two-sample, two-tailed t-test * comparing the means of the datasets described by two StatisticalSummary * instances, under the hypothesis of equal subpopulation variances. To * perform a test without the equal variances assumption, use * {@link #tTest(StatisticalSummary, StatisticalSummary)}. * <p> * The number returned is the smallest significance level * at which one can reject the null hypothesis that the two means are * equal in favor of the two-sided alternative that they are different. * For a one-sided test, divide the returned value by 2.</p> * <p> * See {@link #homoscedasticT(double[], double[])} for the formula used to * compute the t-statistic. The sum of the sample sizes minus 2 is used as * the degrees of freedom.</p> * <p> * <strong>Usage Note:
* The validity of the p-value depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">here * </p>

* <strong>Preconditions:

    * <li>The datasets described by the two Univariates must each contain * at least 2 observations. * </li>

* * @param sampleStats1 StatisticalSummary describing data from the first sample * @param sampleStats2 StatisticalSummary describing data from the second sample * @return p-value for t-test * @throws NullArgumentException if the sample statistics are <code>null * @throws NumberIsTooSmallException if the number of samples is < 2 * @throws MaxCountExceededException if an error occurs computing the p-value */ public double homoscedasticTTest(final StatisticalSummary sampleStats1, final StatisticalSummary sampleStats2) throws NullArgumentException, NumberIsTooSmallException, MaxCountExceededException { checkSampleData(sampleStats1); checkSampleData(sampleStats2); return homoscedasticTTest(sampleStats1.getMean(), sampleStats2.getMean(), sampleStats1.getVariance(), sampleStats2.getVariance(), sampleStats1.getN(), sampleStats2.getN()); } /** * Performs a * <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm"> * two-sided t-test</a> evaluating the null hypothesis that * <code>sampleStats1 and sampleStats2 describe * datasets drawn from populations with the same mean, with significance * level <code>alpha. This test does not assume that the * subpopulation variances are equal. To perform the test under the equal * variances assumption, use * {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}. * <p> * Returns <code>true iff the null hypothesis that the means are * equal can be rejected with confidence <code>1 - alpha. To * perform a 1-sided test, use <code>alpha * 2

* <p> * See {@link #t(double[], double[])} for the formula used to compute the * t-statistic. Degrees of freedom are approximated using the * <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm"> * Welch-Satterthwaite approximation.</a>

* <p> * <strong>Examples:
    * <li>To test the (2-sided) hypothesis mean 1 = mean 2 at * the 95%, use * <br>tTest(sampleStats1, sampleStats2, 0.05) * </li> * <li>To test the (one-sided) hypothesis mean 1 < mean 2 * at the 99% level, first verify that the measured mean of * <code>sample 1 is less than the mean of sample 2 * and then use * <br>tTest(sampleStats1, sampleStats2, 0.02) * </li>

* <p> * <strong>Usage Note:
* The validity of the test depends on the assumptions of the parametric * t-test procedure, as discussed * <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html"> * here</a>

* <p> * <strong>Preconditions:
    * <li>The datasets described by the two Univariates must each contain * at least 2 observations. * </li> * <li> 0 < alpha < 0.5 * </li>

* * @param sampleStats1 StatisticalSummary describing sample data values * @param sampleStats2 StatisticalSummary describing sample data values * @param alpha significance level of the test * @return true if the null hypothesis can be rejected with * confidence 1 - alpha * @throws NullArgumentException if the sample statistics are <code>null * @throws NumberIsTooSmallException if the number of samples is < 2 * @throws OutOfRangeException if <code>alpha is not in the range (0, 0.5] * @throws MaxCountExceededException if an error occurs computing the p-value */ public boolean tTest(final StatisticalSummary sampleStats1, final StatisticalSummary sampleStats2, final double alpha) throws NullArgumentException, NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException { checkSignificanceLevel(alpha); return tTest(sampleStats1, sampleStats2) < alpha; } //----------------------------------------------- Protected methods /** * Computes approximate degrees of freedom for 2-sample t-test. * * @param v1 first sample variance * @param v2 second sample variance * @param n1 first sample n * @param n2 second sample n * @return approximate degrees of freedom */ protected double df(double v1, double v2, double n1, double n2) { return (((v1 / n1) + (v2 / n2)) * ((v1 / n1) + (v2 / n2))) / ((v1 * v1) / (n1 * n1 * (n1 - 1d)) + (v2 * v2) / (n2 * n2 * (n2 - 1d))); } /** * Computes t test statistic for 1-sample t-test. * * @param m sample mean * @param mu constant to test against * @param v sample variance * @param n sample n * @return t test statistic */ protected double t(final double m, final double mu, final double v, final double n) { return (m - mu) / FastMath.sqrt(v / n); } /** * Computes t test statistic for 2-sample t-test. * <p> * Does not assume that subpopulation variances are equal.</p> * * @param m1 first sample mean * @param m2 second sample mean * @param v1 first sample variance * @param v2 second sample variance * @param n1 first sample n * @param n2 second sample n * @return t test statistic */ protected double t(final double m1, final double m2, final double v1, final double v2, final double n1, final double n2) { return (m1 - m2) / FastMath.sqrt((v1 / n1) + (v2 / n2)); } /** * Computes t test statistic for 2-sample t-test under the hypothesis * of equal subpopulation variances. * * @param m1 first sample mean * @param m2 second sample mean * @param v1 first sample variance * @param v2 second sample variance * @param n1 first sample n * @param n2 second sample n * @return t test statistic */ protected double homoscedasticT(final double m1, final double m2, final double v1, final double v2, final double n1, final double n2) { final double pooledVariance = ((n1 - 1) * v1 + (n2 -1) * v2 ) / (n1 + n2 - 2); return (m1 - m2) / FastMath.sqrt(pooledVariance * (1d / n1 + 1d / n2)); } /** * Computes p-value for 2-sided, 1-sample t-test. * * @param m sample mean * @param mu constant to test against * @param v sample variance * @param n sample n * @return p-value * @throws MaxCountExceededException if an error occurs computing the p-value * @throws MathIllegalArgumentException if n is not greater than 1 */ protected double tTest(final double m, final double mu, final double v, final double n) throws MaxCountExceededException, MathIllegalArgumentException { final double t = FastMath.abs(t(m, mu, v, n)); // pass a null rng to avoid unneeded overhead as we will not sample from this distribution final TDistribution distribution = new TDistribution(null, n - 1); return 2.0 * distribution.cumulativeProbability(-t); } /** * Computes p-value for 2-sided, 2-sample t-test. * <p> * Does not assume subpopulation variances are equal. Degrees of freedom * are estimated from the data.</p> * * @param m1 first sample mean * @param m2 second sample mean * @param v1 first sample variance * @param v2 second sample variance * @param n1 first sample n * @param n2 second sample n * @return p-value * @throws MaxCountExceededException if an error occurs computing the p-value * @throws NotStrictlyPositiveException if the estimated degrees of freedom is not * strictly positive */ protected double tTest(final double m1, final double m2, final double v1, final double v2, final double n1, final double n2) throws MaxCountExceededException, NotStrictlyPositiveException { final double t = FastMath.abs(t(m1, m2, v1, v2, n1, n2)); final double degreesOfFreedom = df(v1, v2, n1, n2); // pass a null rng to avoid unneeded overhead as we will not sample from this distribution final TDistribution distribution = new TDistribution(null, degreesOfFreedom); return 2.0 * distribution.cumulativeProbability(-t); } /** * Computes p-value for 2-sided, 2-sample t-test, under the assumption * of equal subpopulation variances. * <p> * The sum of the sample sizes minus 2 is used as degrees of freedom.</p> * * @param m1 first sample mean * @param m2 second sample mean * @param v1 first sample variance * @param v2 second sample variance * @param n1 first sample n * @param n2 second sample n * @return p-value * @throws MaxCountExceededException if an error occurs computing the p-value * @throws NotStrictlyPositiveException if the estimated degrees of freedom is not * strictly positive */ protected double homoscedasticTTest(double m1, double m2, double v1, double v2, double n1, double n2) throws MaxCountExceededException, NotStrictlyPositiveException { final double t = FastMath.abs(homoscedasticT(m1, m2, v1, v2, n1, n2)); final double degreesOfFreedom = n1 + n2 - 2; // pass a null rng to avoid unneeded overhead as we will not sample from this distribution final TDistribution distribution = new TDistribution(null, degreesOfFreedom); return 2.0 * distribution.cumulativeProbability(-t); } /** * Check significance level. * * @param alpha significance level * @throws OutOfRangeException if the significance level is out of bounds. */ private void checkSignificanceLevel(final double alpha) throws OutOfRangeException { if (alpha <= 0 || alpha > 0.5) { throw new OutOfRangeException(LocalizedFormats.SIGNIFICANCE_LEVEL, alpha, 0.0, 0.5); } } /** * Check sample data. * * @param data Sample data. * @throws NullArgumentException if {@code data} is {@code null}. * @throws NumberIsTooSmallException if there is not enough sample data. */ private void checkSampleData(final double[] data) throws NullArgumentException, NumberIsTooSmallException { if (data == null) { throw new NullArgumentException(); } if (data.length < 2) { throw new NumberIsTooSmallException( LocalizedFormats.INSUFFICIENT_DATA_FOR_T_STATISTIC, data.length, 2, true); } } /** * Check sample data. * * @param stat Statistical summary. * @throws NullArgumentException if {@code data} is {@code null}. * @throws NumberIsTooSmallException if there is not enough sample data. */ private void checkSampleData(final StatisticalSummary stat) throws NullArgumentException, NumberIsTooSmallException { if (stat == null) { throw new NullArgumentException(); } if (stat.getN() < 2) { throw new NumberIsTooSmallException( LocalizedFormats.INSUFFICIENT_DATA_FOR_T_STATISTIC, stat.getN(), 2, true); } } }

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