| /* |
| * 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.math.stat.inference; |
| |
| import org.apache.commons.math.MathException; |
| import org.apache.commons.math.stat.descriptive.StatisticalSummary; |
| |
| /** |
| * An interface for Student's t-tests. |
| * <p> |
| * Tests can be:<ul> |
| * <li>One-sample or two-sample</li> |
| * <li>One-sided or two-sided</li> |
| * <li>Paired or unpaired (for two-sample tests)</li> |
| * <li>Homoscedastic (equal variance assumption) or heteroscedastic |
| * (for two sample tests)</li> |
| * <li>Fixed significance level (boolean-valued) or returning p-values. |
| * </li></ul></p> |
| * <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-</code>valued methods return p-values; |
| * <code>boolean-</code>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</code>).</p> |
| * <p> |
| * Input to tests can be either <code>double[]</code> arrays or |
| * {@link StatisticalSummary} instances.</p> |
| * |
| * |
| * @version $Revision: 811786 $ $Date: 2009-09-06 11:36:08 +0200 (dim. 06 sept. 2009) $ |
| */ |
| public interface 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</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></p> |
| * |
| * @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. |
| */ |
| double pairedT(double[] sample1, double[] sample2) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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> |
| * <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> |
| * <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></p> |
| * |
| * @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 |
| */ |
| double pairedTTest(double[] sample1, double[] sample2) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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> |
| * <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></p> |
| * |
| * @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 |
| */ |
| boolean pairedTTest( |
| double[] sample1, |
| double[] sample2, |
| double alpha) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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><p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The observed array length must be at least 2. |
| * </li></ul></p> |
| * |
| * @param mu comparison constant |
| * @param observed array of values |
| * @return t statistic |
| * @throws IllegalArgumentException if input array length is less than 2 |
| */ |
| double t(double mu, double[] observed) |
| throws IllegalArgumentException; |
| /** |
| * 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><p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li><code>observed.getN() > = 2</code>. |
| * </li></ul></p> |
| * |
| * @param mu comparison constant |
| * @param sampleStats DescriptiveStatistics holding sample summary statitstics |
| * @return t statistic |
| * @throws IllegalArgumentException if the precondition is not met |
| */ |
| double t(double mu, StatisticalSummary sampleStats) |
| throws IllegalArgumentException; |
| /** |
| * 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-statisitc is</p> |
| * <p> |
| * <code> t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))</code> |
| * </p><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><p> |
| * <code>var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))</code> |
| * </p><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><p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The observed array lengths must both be at least 2. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| double homoscedasticT(double[] sample1, double[] sample2) |
| throws IllegalArgumentException; |
| /** |
| * 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-statisitc is</p> |
| * <p> |
| * <code> t = (m1 - m2) / sqrt(var1/n1 + var2/n2)</code> |
| * </p><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><p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The observed array lengths must both be at least 2. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| double t(double[] sample1, double[] sample2) |
| throws IllegalArgumentException; |
| /** |
| * 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-statisitc is</p> |
| * <p> |
| * <code> t = (m1 - m2) / sqrt(var1/n1 + var2/n2)</code> |
| * </p><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><p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The datasets described by the two Univariates must each contain |
| * at least 2 observations. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| double t( |
| StatisticalSummary sampleStats1, |
| StatisticalSummary sampleStats2) |
| throws IllegalArgumentException; |
| /** |
| * 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-statisitc returned is</p> |
| * <p> |
| * <code> t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))</code> |
| * </p><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><p> |
| * <code>var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))</code> |
| * </p><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><p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The datasets described by the two Univariates must each contain |
| * at least 2 observations. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| double homoscedasticT( |
| StatisticalSummary sampleStats1, |
| StatisticalSummary sampleStats2) |
| throws IllegalArgumentException; |
| /** |
| * 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> |
| * <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><p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The observed array length must be at least 2. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| double tTest(double mu, double[] sample) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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> |
| * <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><p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The observed array length must be at least 2. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| boolean tTest(double mu, double[] sample, double alpha) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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> |
| * <p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The sample must contain at least 2 observations. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| double tTest(double mu, StatisticalSummary sampleStats) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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> |
| * <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><p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The sample must include at least 2 observations. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| boolean tTest( |
| double mu, |
| StatisticalSummary sampleStats, |
| double alpha) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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:</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> |
| * <p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The observed array lengths must both be at least 2. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| double tTest(double[] sample1, double[] sample2) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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:</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> |
| * <p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The observed array lengths must both be at least 2. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| double homoscedasticTTest( |
| double[] sample1, |
| double[] sample2) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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> |
| * <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> |
| * <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> |
| * <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></p> |
| * |
| * @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 |
| */ |
| boolean tTest( |
| double[] sample1, |
| double[] sample2, |
| double alpha) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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:</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> |
| * <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> |
| * <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></p> |
| * |
| * @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 |
| */ |
| boolean homoscedasticTTest( |
| double[] sample1, |
| double[] sample2, |
| double alpha) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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> |
| * <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> |
| * <p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The datasets described by the two Univariates must each contain |
| * at least 2 observations. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| double tTest( |
| StatisticalSummary sampleStats1, |
| StatisticalSummary sampleStats2) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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:</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><p> |
| * <strong>Preconditions</strong>: <ul> |
| * <li>The datasets described by the two Univariates must each contain |
| * at least 2 observations. |
| * </li></ul></p> |
| * |
| * @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 |
| */ |
| double homoscedasticTTest( |
| StatisticalSummary sampleStats1, |
| StatisticalSummary sampleStats2) |
| throws IllegalArgumentException, MathException; |
| /** |
| * 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> |
| * <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> |
| * <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> |
| * <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> |
| * <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></p> |
| * |
| * @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 |
| */ |
| boolean tTest( |
| StatisticalSummary sampleStats1, |
| StatisticalSummary sampleStats2, |
| double alpha) |
| throws IllegalArgumentException, MathException; |
| } |
