/*
* Copyright 2004-2005 The Apache Software Foundation.
*
* Licensed 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.math.stat.inference;
import org.apache.commons.math.MathException;
import org.apache.commons.math.distribution.DistributionFactory;
import org.apache.commons.math.distribution.TDistribution;
import org.apache.commons.math.stat.StatUtils;
import org.apache.commons.math.stat.descriptive.StatisticalSummary;
/**
* Implements t-test statistics defined in the {@link TTest} interface.
* <p>
* Uses commons-math {@link org.apache.commons.math.distribution.TDistribution}
* implementation to estimate exact p-values.
*
* @version $Revision: 165583 $ $Date: 2005-05-01 22:14:49 -0700 (Sun, 01 May 2005) $
*/
public class TTestImpl implements TTest {
/** Cached DistributionFactory used to create TDistribution instances */
private DistributionFactory distributionFactory = null;
/**
* Default constructor.
*/
public TTestImpl() {
super();
}
/**
* 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</code> and the sample array consisting of the (signed)
* differences between corresponding entries in <code>sample1</code> and
* <code>sample2.</code>
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>The input arrays must have the same length and their common length
* must be at least 2.
* </li></ul>
*
* @param sample1 array of sample data values
* @param sample2 array of sample data values
* @return t statistic
* @throws IllegalArgumentException if the precondition is not met
* @throws MathException if the statistic can not be computed do to a
* convergence or other numerical error.
*/
public double pairedT(double[] sample1, double[] sample2)
throws IllegalArgumentException, MathException {
if ((sample1 == null) || (sample2 == null ||
Math.min(sample1.length, sample2.length) < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
double meanDifference = StatUtils.meanDifference(sample1, sample2);
return t(meanDifference, 0,
StatUtils.varianceDifference(sample1, sample2, meanDifference),
(double) sample1.length);
}
/**
* Returns the <i>observed significance level</i>, or
* <i> p-value</i>, 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>
* This test is equivalent to a one-sample t-test computed using
* {@link #tTest(double, double[])} with <code>mu = 0</code> and the sample
* array consisting of the signed differences between corresponding elements of
* <code>sample1</code> and <code>sample2.</code>
* <p>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The input array lengths must be the same and their common length must
* be at least 2.
* </li></ul>
*
* @param sample1 array of sample data values
* @param sample2 array of sample data values
* @return p-value for t-test
* @throws IllegalArgumentException if the precondition is not met
* @throws MathException if an error occurs computing the p-value
*/
public double pairedTTest(double[] sample1, double[] sample2)
throws IllegalArgumentException, MathException {
double meanDifference = StatUtils.meanDifference(sample1, sample2);
return tTest(meanDifference, 0,
StatUtils.varianceDifference(sample1, sample2, meanDifference),
(double) sample1.length);
}
/**
* Performs a paired t-test evaluating the null hypothesis that the
* mean of the paired differences between <code>sample1</code> and
* <code>sample2</code> is 0 in favor of the two-sided alternative that the
* mean paired difference is not equal to 0, with significance level
* <code>alpha</code>.
* <p>
* Returns <code>true</code> iff the null hypothesis can be rejected with
* confidence <code>1 - alpha</code>. To perform a 1-sided test, use
* <code>alpha * 2</code>
* <p>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The input array lengths must be the same and their common length
* must be at least 2.
* </li>
* <li> <code> 0 < alpha < 0.5 </code>
* </li></ul>
*
* @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 IllegalArgumentException if the preconditions are not met
* @throws MathException if an error occurs performing the test
*/
public boolean pairedTTest(double[] sample1, double[] sample2, double alpha)
throws IllegalArgumentException, MathException {
if ((alpha <= 0) || (alpha > 0.5)) {
throw new IllegalArgumentException("bad significance level: " + 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</strong>: <ul>
* <li>The observed array length must be at least 2.
* </li></ul>
*
* @param mu comparison constant
* @param observed array of values
* @return t statistic
* @throws IllegalArgumentException if input array length is less than 2
*/
public double t(double mu, double[] observed)
throws IllegalArgumentException {
if ((observed == null) || (observed.length < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
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</code> to <code>mu</code>.
* <p>
* This statistic can be used to perform a one sample t-test for the mean.
* <p>
* <strong>Preconditions</strong>: <ul>
* <li><code>observed.getN() > = 2</code>.
* </li></ul>
*
* @param mu comparison constant
* @param sampleStats DescriptiveStatistics holding sample summary statitstics
* @return t statistic
* @throws IllegalArgumentException if the precondition is not met
*/
public double t(double mu, StatisticalSummary sampleStats)
throws IllegalArgumentException {
if ((sampleStats == null) || (sampleStats.getN() < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
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>
* The t-statisitc is
* <p>
* <code> t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))</code>
* <p>
* where <strong><code>n1</code></strong> is the size of first sample;
* <strong><code> n2</code></strong> is the size of second sample;
* <strong><code> m1</code></strong> is the mean of first sample;
* <strong><code> m2</code></strong> is the mean of second sample</li>
* </ul>
* and <strong><code>var</code></strong> is the pooled variance estimate:
* <p>
* <code>var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))</code>
* <p>
* with <strong><code>var1<code></strong> the variance of the first sample and
* <strong><code>var2</code></strong> the variance of the second sample.
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>The observed array lengths must both be at least 2.
* </li></ul>
*
* @param sample1 array of sample data values
* @param sample2 array of sample data values
* @return t statistic
* @throws IllegalArgumentException if the precondition is not met
*/
public double homoscedasticT(double[] sample1, double[] sample2)
throws IllegalArgumentException {
if ((sample1 == null) || (sample2 == null ||
Math.min(sample1.length, sample2.length) < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
return homoscedasticT(StatUtils.mean(sample1), StatUtils.mean(sample2),
StatUtils.variance(sample1), StatUtils.variance(sample2),
(double) sample1.length, (double) 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>
* The t-statisitc is
* <p>
* <code> t = (m1 - m2) / sqrt(var1/n1 + var2/n2)</code>
* <p>
* where <strong><code>n1</code></strong> is the size of the first sample
* <strong><code> n2</code></strong> is the size of the second sample;
* <strong><code> m1</code></strong> is the mean of the first sample;
* <strong><code> m2</code></strong> is the mean of the second sample;
* <strong><code> var1</code></strong> is the variance of the first sample;
* <strong><code> var2</code></strong> is the variance of the second sample;
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>The observed array lengths must both be at least 2.
* </li></ul>
*
* @param sample1 array of sample data values
* @param sample2 array of sample data values
* @return t statistic
* @throws IllegalArgumentException if the precondition is not met
*/
public double t(double[] sample1, double[] sample2)
throws IllegalArgumentException {
if ((sample1 == null) || (sample2 == null ||
Math.min(sample1.length, sample2.length) < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
return t(StatUtils.mean(sample1), StatUtils.mean(sample2),
StatUtils.variance(sample1), StatUtils.variance(sample2),
(double) sample1.length, (double) 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>
* The returned t-statisitc is
* <p>
* <code> t = (m1 - m2) / sqrt(var1/n1 + var2/n2)</code>
* <p>
* where <strong><code>n1</code></strong> is the size of the first sample;
* <strong><code> n2</code></strong> is the size of the second sample;
* <strong><code> m1</code></strong> is the mean of the first sample;
* <strong><code> m2</code></strong> is the mean of the second sample
* <strong><code> var1</code></strong> is the variance of the first sample;
* <strong><code> var2</code></strong> is the variance of the second sample
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>The datasets described by the two Univariates must each contain
* at least 2 observations.
* </li></ul>
*
* @param sampleStats1 StatisticalSummary describing data from the first sample
* @param sampleStats2 StatisticalSummary describing data from the second sample
* @return t statistic
* @throws IllegalArgumentException if the precondition is not met
*/
public double t(StatisticalSummary sampleStats1,
StatisticalSummary sampleStats2)
throws IllegalArgumentException {
if ((sampleStats1 == null) ||
(sampleStats2 == null ||
Math.min(sampleStats1.getN(), sampleStats2.getN()) < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
return t(sampleStats1.getMean(), sampleStats2.getMean(),
sampleStats1.getVariance(), sampleStats2.getVariance(),
(double) sampleStats1.getN(), (double) 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>
* The t-statisitc returned is
* <p>
* <code> t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))</code>
* <p>
* where <strong><code>n1</code></strong> is the size of first sample;
* <strong><code> n2</code></strong> is the size of second sample;
* <strong><code> m1</code></strong> is the mean of first sample;
* <strong><code> m2</code></strong> is the mean of second sample
* and <strong><code>var</code></strong> is the pooled variance estimate:
* <p>
* <code>var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))</code>
* <p>
* with <strong><code>var1<code></strong> the variance of the first sample and
* <strong><code>var2</code></strong> the variance of the second sample.
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>The datasets described by the two Univariates must each contain
* at least 2 observations.
* </li></ul>
*
* @param sampleStats1 StatisticalSummary describing data from the first sample
* @param sampleStats2 StatisticalSummary describing data from the second sample
* @return t statistic
* @throws IllegalArgumentException if the precondition is not met
*/
public double homoscedasticT(StatisticalSummary sampleStats1,
StatisticalSummary sampleStats2)
throws IllegalArgumentException {
if ((sampleStats1 == null) ||
(sampleStats2 == null ||
Math.min(sampleStats1.getN(), sampleStats2.getN()) < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
return homoscedasticT(sampleStats1.getMean(), sampleStats2.getMean(),
sampleStats1.getVariance(), sampleStats2.getVariance(),
(double) sampleStats1.getN(), (double) sampleStats2.getN());
}
/**
* Returns the <i>observed significance level</i>, or
* <i>p-value</i>, associated with a one-sample, two-tailed t-test
* comparing the mean of the input array with the constant <code>mu</code>.
* <p>
* The number returned is the smallest significance level
* at which one can reject the null hypothesis that the mean equals
* <code>mu</code> in favor of the two-sided alternative that the mean
* is different from <code>mu</code>. For a one-sided test, divide the
* returned value by 2.
* <p>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The observed array length must be at least 2.
* </li></ul>
*
* @param mu constant value to compare sample mean against
* @param sample array of sample data values
* @return p-value
* @throws IllegalArgumentException if the precondition is not met
* @throws MathException if an error occurs computing the p-value
*/
public double tTest(double mu, double[] sample)
throws IllegalArgumentException, MathException {
if ((sample == null) || (sample.length < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
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</code> is drawn equals <code>mu</code>.
* <p>
* Returns <code>true</code> iff the null hypothesis can be
* rejected with confidence <code>1 - alpha</code>. To
* perform a 1-sided test, use <code>alpha * 2</code>
* <p>
* <strong>Examples:</strong><br><ol>
* <li>To test the (2-sided) hypothesis <code>sample mean = mu </code> at
* the 95% level, use <br><code>tTest(mu, sample, 0.05) </code>
* </li>
* <li>To test the (one-sided) hypothesis <code> sample mean < mu </code>
* at the 99% level, first verify that the measured sample mean is less
* than <code>mu</code> and then use
* <br><code>tTest(mu, sample, 0.02) </code>
* </li></ol>
* <p>
* <strong>Usage Note:</strong><br>
* 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</a>
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>The observed array length must be at least 2.
* </li></ul>
*
* @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 IllegalArgumentException if the precondition is not met
* @throws MathException if an error computing the p-value
*/
public boolean tTest(double mu, double[] sample, double alpha)
throws IllegalArgumentException, MathException {
if ((alpha <= 0) || (alpha > 0.5)) {
throw new IllegalArgumentException("bad significance level: " + alpha);
}
return (tTest(mu, sample) < alpha);
}
/**
* Returns the <i>observed significance level</i>, or
* <i>p-value</i>, associated with a one-sample, two-tailed t-test
* comparing the mean of the dataset described by <code>sampleStats</code>
* with the constant <code>mu</code>.
* <p>
* The number returned is the smallest significance level
* at which one can reject the null hypothesis that the mean equals
* <code>mu</code> in favor of the two-sided alternative that the mean
* is different from <code>mu</code>. For a one-sided test, divide the
* returned value by 2.
* <p>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The sample must contain at least 2 observations.
* </li></ul>
*
* @param mu constant value to compare sample mean against
* @param sampleStats StatisticalSummary describing sample data
* @return p-value
* @throws IllegalArgumentException if the precondition is not met
* @throws MathException if an error occurs computing the p-value
*/
public double tTest(double mu, StatisticalSummary sampleStats)
throws IllegalArgumentException, MathException {
if ((sampleStats == null) || (sampleStats.getN() < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
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</code> is
* drawn equals <code>mu</code>.
* <p>
* Returns <code>true</code> iff the null hypothesis can be rejected with
* confidence <code>1 - alpha</code>. To perform a 1-sided test, use
* <code>alpha * 2.</code>
* <p>
* <strong>Examples:</strong><br><ol>
* <li>To test the (2-sided) hypothesis <code>sample mean = mu </code> at
* the 95% level, use <br><code>tTest(mu, sampleStats, 0.05) </code>
* </li>
* <li>To test the (one-sided) hypothesis <code> sample mean < mu </code>
* at the 99% level, first verify that the measured sample mean is less
* than <code>mu</code> and then use
* <br><code>tTest(mu, sampleStats, 0.02) </code>
* </li></ol>
* <p>
* <strong>Usage Note:</strong><br>
* 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</a>
* <p>
* <strong>Preconditions</strong>: <ul>
* <li>The sample must include at least 2 observations.
* </li></ul>
*
* @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 IllegalArgumentException if the precondition is not met
* @throws MathException if an error occurs computing the p-value
*/
public boolean tTest( double mu, StatisticalSummary sampleStats,
double alpha)
throws IllegalArgumentException, MathException {
if ((alpha <= 0) || (alpha > 0.5)) {
throw new IllegalArgumentException("bad significance level: " + alpha);
}
return (tTest(mu, sampleStats) < alpha);
}
/**
* Returns the <i>observed significance level</i>, or
* <i>p-value</i>, 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>
* 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>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The observed array lengths must both be at least 2.
* </li></ul>
*
* @param sample1 array of sample data values
* @param sample2 array of sample data values
* @return p-value for t-test
* @throws IllegalArgumentException if the precondition is not met
* @throws MathException if an error occurs computing the p-value
*/
public double tTest(double[] sample1, double[] sample2)
throws IllegalArgumentException, MathException {
if ((sample1 == null) || (sample2 == null ||
Math.min(sample1.length, sample2.length) < 2)) {
throw new IllegalArgumentException("insufficient data");
}
return tTest(StatUtils.mean(sample1), StatUtils.mean(sample2),
StatUtils.variance(sample1), StatUtils.variance(sample2),
(double) sample1.length, (double) sample2.length);
}
/**
* Returns the <i>observed significance level</i>, or
* <i>p-value</i>, 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>
* 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>
* 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>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The observed array lengths must both be at least 2.
* </li></ul>
*
* @param sample1 array of sample data values
* @param sample2 array of sample data values
* @return p-value for t-test
* @throws IllegalArgumentException if the precondition is not met
* @throws MathException if an error occurs computing the p-value
*/
public double homoscedasticTTest(double[] sample1, double[] sample2)
throws IllegalArgumentException, MathException {
if ((sample1 == null) || (sample2 == null ||
Math.min(sample1.length, sample2.length) < 2)) {
throw new IllegalArgumentException("insufficient data");
}
return homoscedasticTTest(StatUtils.mean(sample1),
StatUtils.mean(sample2), StatUtils.variance(sample1),
StatUtils.variance(sample2), (double) sample1.length,
(double) 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 <code>sample1</code>
* and <code>sample2</code> are drawn from populations with the same mean,
* with significance level <code>alpha</code>. 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</code> iff the null hypothesis that the means are
* equal can be rejected with confidence <code>1 - alpha</code>. To
* perform a 1-sided test, use <code>alpha / 2</code>
* <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:</strong><br><ol>
* <li>To test the (2-sided) hypothesis <code>mean 1 = mean 2 </code> at
* the 95% level, use
* <br><code>tTest(sample1, sample2, 0.05). </code>
* </li>
* <li>To test the (one-sided) hypothesis <code> mean 1 < mean 2 </code> at
* the 99% level, first verify that the measured mean of <code>sample 1</code>
* is less than the mean of <code>sample 2</code> and then use
* <br><code>tTest(sample1, sample2, 0.02) </code>
* </li></ol>
* <p>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The observed array lengths must both be at least 2.
* </li>
* <li> <code> 0 < alpha < 0.5 </code>
* </li></ul>
*
* @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 IllegalArgumentException if the preconditions are not met
* @throws MathException if an error occurs performing the test
*/
public boolean tTest(double[] sample1, double[] sample2,
double alpha)
throws IllegalArgumentException, MathException {
if ((alpha <= 0) || (alpha > 0.5)) {
throw new IllegalArgumentException("bad significance level: " + 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 <code>sample1</code>
* and <code>sample2</code> are drawn from populations with the same mean,
* with significance level <code>alpha</code>, 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</code> iff the null hypothesis that the means are
* equal can be rejected with confidence <code>1 - alpha</code>. To
* perform a 1-sided test, use <code>alpha * 2.</code> To perform the test
* without the assumption of equal subpopulation variances, use
* {@link #tTest(double[], double[], double)}.
* <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>
* <strong>Examples:</strong><br><ol>
* <li>To test the (2-sided) hypothesis <code>mean 1 = mean 2 </code> at
* the 95% level, use <br><code>tTest(sample1, sample2, 0.05). </code>
* </li>
* <li>To test the (one-sided) hypothesis <code> mean 1 < mean 2, </code>
* at the 99% level, first verify that the measured mean of
* <code>sample 1</code> is less than the mean of <code>sample 2</code>
* and then use
* <br><code>tTest(sample1, sample2, 0.02) </code>
* </li></ol>
* <p>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The observed array lengths must both be at least 2.
* </li>
* <li> <code> 0 < alpha < 0.5 </code>
* </li></ul>
*
* @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 IllegalArgumentException if the preconditions are not met
* @throws MathException if an error occurs performing the test
*/
public boolean homoscedasticTTest(double[] sample1, double[] sample2,
double alpha)
throws IllegalArgumentException, MathException {
if ((alpha <= 0) || (alpha > 0.5)) {
throw new IllegalArgumentException("bad significance level: " + alpha);
}
return (homoscedasticTTest(sample1, sample2) < alpha);
}
/**
* Returns the <i>observed significance level</i>, or
* <i>p-value</i>, 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>
* 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. To perform the test assuming
* equal variances, use
* {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}.
* <p>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The datasets described by the two Univariates must each contain
* at least 2 observations.
* </li></ul>
*
* @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 IllegalArgumentException if the precondition is not met
* @throws MathException if an error occurs computing the p-value
*/
public double tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
throws IllegalArgumentException, MathException {
if ((sampleStats1 == null) || (sampleStats2 == null ||
Math.min(sampleStats1.getN(), sampleStats2.getN()) < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
return tTest(sampleStats1.getMean(), sampleStats2.getMean(), sampleStats1.getVariance(),
sampleStats2.getVariance(), (double) sampleStats1.getN(),
(double) sampleStats2.getN());
}
/**
* Returns the <i>observed significance level</i>, or
* <i>p-value</i>, 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>
* 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>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The datasets described by the two Univariates must each contain
* at least 2 observations.
* </li></ul>
*
* @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 IllegalArgumentException if the precondition is not met
* @throws MathException if an error occurs computing the p-value
*/
public double homoscedasticTTest(StatisticalSummary sampleStats1,
StatisticalSummary sampleStats2)
throws IllegalArgumentException, MathException {
if ((sampleStats1 == null) || (sampleStats2 == null ||
Math.min(sampleStats1.getN(), sampleStats2.getN()) < 2)) {
throw new IllegalArgumentException("insufficient data for t statistic");
}
return homoscedasticTTest(sampleStats1.getMean(),
sampleStats2.getMean(), sampleStats1.getVariance(),
sampleStats2.getVariance(), (double) sampleStats1.getN(),
(double) 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</code> and <code>sampleStats2</code> describe
* datasets drawn from populations with the same mean, with significance
* level <code>alpha</code>. 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</code> iff the null hypothesis that the means are
* equal can be rejected with confidence <code>1 - alpha</code>. To
* perform a 1-sided test, use <code>alpha * 2</code>
* <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:</strong><br><ol>
* <li>To test the (2-sided) hypothesis <code>mean 1 = mean 2 </code> at
* the 95%, use
* <br><code>tTest(sampleStats1, sampleStats2, 0.05) </code>
* </li>
* <li>To test the (one-sided) hypothesis <code> mean 1 < mean 2 </code>
* at the 99% level, first verify that the measured mean of
* <code>sample 1</code> is less than the mean of <code>sample 2</code>
* and then use
* <br><code>tTest(sampleStats1, sampleStats2, 0.02) </code>
* </li></ol>
* <p>
* <strong>Usage Note:</strong><br>
* 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</strong>: <ul>
* <li>The datasets described by the two Univariates must each contain
* at least 2 observations.
* </li>
* <li> <code> 0 < alpha < 0.5 </code>
* </li></ul>
*
* @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 IllegalArgumentException if the preconditions are not met
* @throws MathException if an error occurs performing the test
*/
public boolean tTest(StatisticalSummary sampleStats1,
StatisticalSummary sampleStats2, double alpha)
throws IllegalArgumentException, MathException {
if ((alpha <= 0) || (alpha > 0.5)) {
throw new IllegalArgumentException("bad significance level: " + alpha);
}
return (tTest(sampleStats1, sampleStats2) < alpha);
}
//----------------------------------------------- Protected methods
/**
* Gets a DistributionFactory to use in creating TDistribution instances.
* @return a distribution factory.
*/
protected DistributionFactory getDistributionFactory() {
if (distributionFactory == null) {
distributionFactory = DistributionFactory.newInstance();
}
return distributionFactory;
}
/**
* 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(double m, double mu, double v, double n) {
return (m - mu) / Math.sqrt(v / n);
}
/**
* Computes t test statistic for 2-sample t-test.
* <p>
* Does not assume that subpopulation variances are equal.
*
* @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(double m1, double m2, double v1, double v2, double n1,
double n2) {
return (m1 - m2) / Math.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(double m1, double m2, double v1,
double v2, double n1, double n2) {
double pooledVariance = ((n1 - 1) * v1 + (n2 -1) * v2 ) / (n1 + n2 - 2);
return (m1 - m2) / Math.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 MathException if an error occurs computing the p-value
*/
protected double tTest(double m, double mu, double v, double n)
throws MathException {
double t = Math.abs(t(m, mu, v, n));
TDistribution tDistribution =
getDistributionFactory().createTDistribution(n - 1);
return 1.0 - tDistribution.cumulativeProbability(-t, 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.
*
* @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 MathException if an error occurs computing the p-value
*/
protected double tTest(double m1, double m2, double v1, double v2,
double n1, double n2)
throws MathException {
double t = Math.abs(t(m1, m2, v1, v2, n1, n2));
double degreesOfFreedom = 0;
degreesOfFreedom= df(v1, v2, n1, n2);
TDistribution tDistribution =
getDistributionFactory().createTDistribution(degreesOfFreedom);
return 1.0 - tDistribution.cumulativeProbability(-t, 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.
*
* @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 MathException if an error occurs computing the p-value
*/
protected double homoscedasticTTest(double m1, double m2, double v1,
double v2, double n1, double n2)
throws MathException {
double t = Math.abs(homoscedasticT(m1, m2, v1, v2, n1, n2));
double degreesOfFreedom = 0;
degreesOfFreedom = (double) (n1 + n2 - 2);
TDistribution tDistribution =
getDistributionFactory().createTDistribution(degreesOfFreedom);
return 1.0 - tDistribution.cumulativeProbability(-t, t);
}
}