2
* Licensed to the Apache Software Foundation (ASF) under one or more
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* contributor license agreements. See the NOTICE file distributed with
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* this work for additional information regarding copyright ownership.
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* The ASF licenses this file to You under the Apache License, Version 2.0
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* (the "License"); you may not use this file except in compliance with
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* the License. You may obtain a copy of the License at
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* http://www.apache.org/licenses/LICENSE-2.0
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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package org.apache.commons.math.stat.inference;
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import org.apache.commons.math.MathException;
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import org.apache.commons.math.stat.descriptive.StatisticalSummary;
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* An interface for Student's t-tests.
26
* <li>One-sample or two-sample</li>
27
* <li>One-sided or two-sided</li>
28
* <li>Paired or unpaired (for two-sample tests)</li>
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* <li>Homoscedastic (equal variance assumption) or heteroscedastic
30
* (for two sample tests)</li>
31
* <li>Fixed significance level (boolean-valued) or returning p-values.
34
* Test statistics are available for all tests. Methods including "Test" in
35
* in their names perform tests, all other methods return t-statistics. Among
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* the "Test" methods, <code>double-</code>valued methods return p-values;
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* <code>boolean-</code>valued methods perform fixed significance level tests.
38
* Significance levels are always specified as numbers between 0 and 0.5
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* (e.g. tests at the 95% level use <code>alpha=0.05</code>).</p>
41
* Input to tests can be either <code>double[]</code> arrays or
42
* {@link StatisticalSummary} instances.</p>
45
* @version $Revision: 617953 $ $Date: 2008-02-02 22:54:00 -0700 (Sat, 02 Feb 2008) $
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public interface TTest {
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* Computes a paired, 2-sample t-statistic based on the data in the input
50
* arrays. The t-statistic returned is equivalent to what would be returned by
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* computing the one-sample t-statistic {@link #t(double, double[])}, with
52
* <code>mu = 0</code> and the sample array consisting of the (signed)
53
* differences between corresponding entries in <code>sample1</code> and
54
* <code>sample2.</code>
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* <strong>Preconditions</strong>: <ul>
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* <li>The input arrays must have the same length and their common length
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* @param sample1 array of sample data values
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* @param sample2 array of sample data values
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* @throws IllegalArgumentException if the precondition is not met
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* @throws MathException if the statistic can not be computed do to a
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* convergence or other numerical error.
68
public abstract double pairedT(double[] sample1, double[] sample2)
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throws IllegalArgumentException, MathException;
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* Returns the <i>observed significance level</i>, or
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* <i> p-value</i>, associated with a paired, two-sample, two-tailed t-test
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* based on the data in the input arrays.
75
* The number returned is the smallest significance level
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* at which one can reject the null hypothesis that the mean of the paired
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* differences is 0 in favor of the two-sided alternative that the mean paired
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* difference is not equal to 0. For a one-sided test, divide the returned
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* This test is equivalent to a one-sample t-test computed using
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* {@link #tTest(double, double[])} with <code>mu = 0</code> and the sample
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* array consisting of the signed differences between corresponding elements of
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* <code>sample1</code> and <code>sample2.</code></p>
86
* <strong>Usage Note:</strong><br>
87
* The validity of the p-value depends on the assumptions of the parametric
88
* t-test procedure, as discussed
89
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
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* <strong>Preconditions</strong>: <ul>
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* <li>The input array lengths must be the same and their common length must
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* @param sample1 array of sample data values
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* @param sample2 array of sample data values
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* @return p-value for t-test
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* @throws IllegalArgumentException if the precondition is not met
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* @throws MathException if an error occurs computing the p-value
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public abstract double pairedTTest(double[] sample1, double[] sample2)
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throws IllegalArgumentException, MathException;
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* Performs a paired t-test evaluating the null hypothesis that the
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* mean of the paired differences between <code>sample1</code> and
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* <code>sample2</code> is 0 in favor of the two-sided alternative that the
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* mean paired difference is not equal to 0, with significance level
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* <code>alpha</code>.
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* Returns <code>true</code> iff the null hypothesis can be rejected with
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* confidence <code>1 - alpha</code>. To perform a 1-sided test, use
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* <code>alpha * 2</code></p>
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* <strong>Usage Note:</strong><br>
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* The validity of the test depends on the assumptions of the parametric
118
* t-test procedure, as discussed
119
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
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* <strong>Preconditions</strong>: <ul>
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* <li>The input array lengths must be the same and their common length
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* must be at least 2.
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* <li> <code> 0 < alpha < 0.5 </code>
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* @param sample1 array of sample data values
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* @param sample2 array of sample data values
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* @param alpha significance level of the test
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* @return true if the null hypothesis can be rejected with
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* confidence 1 - alpha
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* @throws IllegalArgumentException if the preconditions are not met
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* @throws MathException if an error occurs performing the test
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public abstract boolean pairedTTest(
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throws IllegalArgumentException, MathException;
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* Computes a <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm#formula">
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* t statistic </a> given observed values and a comparison constant.
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* This statistic can be used to perform a one sample t-test for the mean.
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* <strong>Preconditions</strong>: <ul>
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* <li>The observed array length must be at least 2.
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* @param mu comparison constant
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* @param observed array of values
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* @return t statistic
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* @throws IllegalArgumentException if input array length is less than 2
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public abstract double t(double mu, double[] observed)
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throws IllegalArgumentException;
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* Computes a <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm#formula">
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* t statistic </a> to use in comparing the mean of the dataset described by
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* <code>sampleStats</code> to <code>mu</code>.
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* This statistic can be used to perform a one sample t-test for the mean.
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* <strong>Preconditions</strong>: <ul>
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* <li><code>observed.getN() > = 2</code>.
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* @param mu comparison constant
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* @param sampleStats DescriptiveStatistics holding sample summary statitstics
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* @return t statistic
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* @throws IllegalArgumentException if the precondition is not met
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public abstract double t(double mu, StatisticalSummary sampleStats)
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throws IllegalArgumentException;
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* Computes a 2-sample t statistic, under the hypothesis of equal
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* subpopulation variances. To compute a t-statistic without the
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* equal variances hypothesis, use {@link #t(double[], double[])}.
182
* This statistic can be used to perform a (homoscedastic) two-sample
183
* t-test to compare sample means.</p>
185
* The t-statisitc is</p>
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* <code> t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))</code>
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* where <strong><code>n1</code></strong> is the size of first sample;
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* <strong><code> n2</code></strong> is the size of second sample;
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* <strong><code> m1</code></strong> is the mean of first sample;
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* <strong><code> m2</code></strong> is the mean of second sample</li>
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* and <strong><code>var</code></strong> is the pooled variance estimate:
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* <code>var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))</code>
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* with <strong><code>var1<code></strong> the variance of the first sample and
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* <strong><code>var2</code></strong> the variance of the second sample.
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* <strong>Preconditions</strong>: <ul>
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* <li>The observed array lengths must both be at least 2.
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* @param sample1 array of sample data values
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* @param sample2 array of sample data values
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* @return t statistic
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* @throws IllegalArgumentException if the precondition is not met
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public abstract double homoscedasticT(double[] sample1, double[] sample2)
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throws IllegalArgumentException;
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* Computes a 2-sample t statistic, without the hypothesis of equal
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* subpopulation variances. To compute a t-statistic assuming equal
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* variances, use {@link #homoscedasticT(double[], double[])}.
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* This statistic can be used to perform a two-sample t-test to compare
220
* The t-statisitc is</p>
222
* <code> t = (m1 - m2) / sqrt(var1/n1 + var2/n2)</code>
224
* where <strong><code>n1</code></strong> is the size of the first sample
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* <strong><code> n2</code></strong> is the size of the second sample;
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* <strong><code> m1</code></strong> is the mean of the first sample;
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* <strong><code> m2</code></strong> is the mean of the second sample;
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* <strong><code> var1</code></strong> is the variance of the first sample;
229
* <strong><code> var2</code></strong> is the variance of the second sample;
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* <strong>Preconditions</strong>: <ul>
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* <li>The observed array lengths must both be at least 2.
235
* @param sample1 array of sample data values
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* @param sample2 array of sample data values
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* @return t statistic
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* @throws IllegalArgumentException if the precondition is not met
240
public abstract double t(double[] sample1, double[] sample2)
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throws IllegalArgumentException;
243
* Computes a 2-sample t statistic </a>, comparing the means of the datasets
244
* described by two {@link StatisticalSummary} instances, without the
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* assumption of equal subpopulation variances. Use
246
* {@link #homoscedasticT(StatisticalSummary, StatisticalSummary)} to
247
* compute a t-statistic under the equal variances assumption.
249
* This statistic can be used to perform a two-sample t-test to compare
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* The returned t-statisitc is</p>
254
* <code> t = (m1 - m2) / sqrt(var1/n1 + var2/n2)</code>
256
* where <strong><code>n1</code></strong> is the size of the first sample;
257
* <strong><code> n2</code></strong> is the size of the second sample;
258
* <strong><code> m1</code></strong> is the mean of the first sample;
259
* <strong><code> m2</code></strong> is the mean of the second sample
260
* <strong><code> var1</code></strong> is the variance of the first sample;
261
* <strong><code> var2</code></strong> is the variance of the second sample
263
* <strong>Preconditions</strong>: <ul>
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* <li>The datasets described by the two Univariates must each contain
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* at least 2 observations.
268
* @param sampleStats1 StatisticalSummary describing data from the first sample
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* @param sampleStats2 StatisticalSummary describing data from the second sample
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* @return t statistic
271
* @throws IllegalArgumentException if the precondition is not met
273
public abstract double t(
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StatisticalSummary sampleStats1,
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StatisticalSummary sampleStats2)
276
throws IllegalArgumentException;
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* Computes a 2-sample t statistic, comparing the means of the datasets
279
* described by two {@link StatisticalSummary} instances, under the
280
* assumption of equal subpopulation variances. To compute a t-statistic
281
* without the equal variances assumption, use
282
* {@link #t(StatisticalSummary, StatisticalSummary)}.
284
* This statistic can be used to perform a (homoscedastic) two-sample
285
* t-test to compare sample means.</p>
287
* The t-statisitc returned is</p>
289
* <code> t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))</code>
291
* where <strong><code>n1</code></strong> is the size of first sample;
292
* <strong><code> n2</code></strong> is the size of second sample;
293
* <strong><code> m1</code></strong> is the mean of first sample;
294
* <strong><code> m2</code></strong> is the mean of second sample
295
* and <strong><code>var</code></strong> is the pooled variance estimate:
297
* <code>var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))</code>
299
* with <strong><code>var1<code></strong> the variance of the first sample and
300
* <strong><code>var2</code></strong> the variance of the second sample.
302
* <strong>Preconditions</strong>: <ul>
303
* <li>The datasets described by the two Univariates must each contain
304
* at least 2 observations.
307
* @param sampleStats1 StatisticalSummary describing data from the first sample
308
* @param sampleStats2 StatisticalSummary describing data from the second sample
309
* @return t statistic
310
* @throws IllegalArgumentException if the precondition is not met
312
public abstract double homoscedasticT(
313
StatisticalSummary sampleStats1,
314
StatisticalSummary sampleStats2)
315
throws IllegalArgumentException;
317
* Returns the <i>observed significance level</i>, or
318
* <i>p-value</i>, associated with a one-sample, two-tailed t-test
319
* comparing the mean of the input array with the constant <code>mu</code>.
321
* The number returned is the smallest significance level
322
* at which one can reject the null hypothesis that the mean equals
323
* <code>mu</code> in favor of the two-sided alternative that the mean
324
* is different from <code>mu</code>. For a one-sided test, divide the
325
* returned value by 2.</p>
327
* <strong>Usage Note:</strong><br>
328
* The validity of the test depends on the assumptions of the parametric
329
* t-test procedure, as discussed
330
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">here</a>
332
* <strong>Preconditions</strong>: <ul>
333
* <li>The observed array length must be at least 2.
336
* @param mu constant value to compare sample mean against
337
* @param sample array of sample data values
339
* @throws IllegalArgumentException if the precondition is not met
340
* @throws MathException if an error occurs computing the p-value
342
public abstract double tTest(double mu, double[] sample)
343
throws IllegalArgumentException, MathException;
345
* Performs a <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm">
346
* two-sided t-test</a> evaluating the null hypothesis that the mean of the population from
347
* which <code>sample</code> is drawn equals <code>mu</code>.
349
* Returns <code>true</code> iff the null hypothesis can be
350
* rejected with confidence <code>1 - alpha</code>. To
351
* perform a 1-sided test, use <code>alpha * 2</code></p>
353
* <strong>Examples:</strong><br><ol>
354
* <li>To test the (2-sided) hypothesis <code>sample mean = mu </code> at
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* the 95% level, use <br><code>tTest(mu, sample, 0.05) </code>
357
* <li>To test the (one-sided) hypothesis <code> sample mean < mu </code>
358
* at the 99% level, first verify that the measured sample mean is less
359
* than <code>mu</code> and then use
360
* <br><code>tTest(mu, sample, 0.02) </code>
363
* <strong>Usage Note:</strong><br>
364
* The validity of the test depends on the assumptions of the one-sample
365
* parametric t-test procedure, as discussed
366
* <a href="http://www.basic.nwu.edu/statguidefiles/sg_glos.html#one-sample">here</a>
368
* <strong>Preconditions</strong>: <ul>
369
* <li>The observed array length must be at least 2.
372
* @param mu constant value to compare sample mean against
373
* @param sample array of sample data values
374
* @param alpha significance level of the test
376
* @throws IllegalArgumentException if the precondition is not met
377
* @throws MathException if an error computing the p-value
379
public abstract boolean tTest(double mu, double[] sample, double alpha)
380
throws IllegalArgumentException, MathException;
382
* Returns the <i>observed significance level</i>, or
383
* <i>p-value</i>, associated with a one-sample, two-tailed t-test
384
* comparing the mean of the dataset described by <code>sampleStats</code>
385
* with the constant <code>mu</code>.
387
* The number returned is the smallest significance level
388
* at which one can reject the null hypothesis that the mean equals
389
* <code>mu</code> in favor of the two-sided alternative that the mean
390
* is different from <code>mu</code>. For a one-sided test, divide the
391
* returned value by 2.</p>
393
* <strong>Usage Note:</strong><br>
394
* The validity of the test depends on the assumptions of the parametric
395
* t-test procedure, as discussed
396
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
399
* <strong>Preconditions</strong>: <ul>
400
* <li>The sample must contain at least 2 observations.
403
* @param mu constant value to compare sample mean against
404
* @param sampleStats StatisticalSummary describing sample data
406
* @throws IllegalArgumentException if the precondition is not met
407
* @throws MathException if an error occurs computing the p-value
409
public abstract double tTest(double mu, StatisticalSummary sampleStats)
410
throws IllegalArgumentException, MathException;
412
* Performs a <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm">
413
* two-sided t-test</a> evaluating the null hypothesis that the mean of the
414
* population from which the dataset described by <code>stats</code> is
415
* drawn equals <code>mu</code>.
417
* Returns <code>true</code> iff the null hypothesis can be rejected with
418
* confidence <code>1 - alpha</code>. To perform a 1-sided test, use
419
* <code>alpha * 2.</code></p>
421
* <strong>Examples:</strong><br><ol>
422
* <li>To test the (2-sided) hypothesis <code>sample mean = mu </code> at
423
* the 95% level, use <br><code>tTest(mu, sampleStats, 0.05) </code>
425
* <li>To test the (one-sided) hypothesis <code> sample mean < mu </code>
426
* at the 99% level, first verify that the measured sample mean is less
427
* than <code>mu</code> and then use
428
* <br><code>tTest(mu, sampleStats, 0.02) </code>
431
* <strong>Usage Note:</strong><br>
432
* The validity of the test depends on the assumptions of the one-sample
433
* parametric t-test procedure, as discussed
434
* <a href="http://www.basic.nwu.edu/statguidefiles/sg_glos.html#one-sample">here</a>
436
* <strong>Preconditions</strong>: <ul>
437
* <li>The sample must include at least 2 observations.
440
* @param mu constant value to compare sample mean against
441
* @param sampleStats StatisticalSummary describing sample data values
442
* @param alpha significance level of the test
444
* @throws IllegalArgumentException if the precondition is not met
445
* @throws MathException if an error occurs computing the p-value
447
public abstract boolean tTest(
449
StatisticalSummary sampleStats,
451
throws IllegalArgumentException, MathException;
453
* Returns the <i>observed significance level</i>, or
454
* <i>p-value</i>, associated with a two-sample, two-tailed t-test
455
* comparing the means of the input arrays.
457
* The number returned is the smallest significance level
458
* at which one can reject the null hypothesis that the two means are
459
* equal in favor of the two-sided alternative that they are different.
460
* For a one-sided test, divide the returned value by 2.</p>
462
* The test does not assume that the underlying popuation variances are
463
* equal and it uses approximated degrees of freedom computed from the
464
* sample data to compute the p-value. The t-statistic used is as defined in
465
* {@link #t(double[], double[])} and the Welch-Satterthwaite approximation
466
* to the degrees of freedom is used,
468
* <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm">
469
* here.</a> To perform the test under the assumption of equal subpopulation
470
* variances, use {@link #homoscedasticTTest(double[], double[])}.</p>
472
* <strong>Usage Note:</strong><br>
473
* The validity of the p-value depends on the assumptions of the parametric
474
* t-test procedure, as discussed
475
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
478
* <strong>Preconditions</strong>: <ul>
479
* <li>The observed array lengths must both be at least 2.
482
* @param sample1 array of sample data values
483
* @param sample2 array of sample data values
484
* @return p-value for t-test
485
* @throws IllegalArgumentException if the precondition is not met
486
* @throws MathException if an error occurs computing the p-value
488
public abstract double tTest(double[] sample1, double[] sample2)
489
throws IllegalArgumentException, MathException;
491
* Returns the <i>observed significance level</i>, or
492
* <i>p-value</i>, associated with a two-sample, two-tailed t-test
493
* comparing the means of the input arrays, under the assumption that
494
* the two samples are drawn from subpopulations with equal variances.
495
* To perform the test without the equal variances assumption, use
496
* {@link #tTest(double[], double[])}.</p>
498
* The number returned is the smallest significance level
499
* at which one can reject the null hypothesis that the two means are
500
* equal in favor of the two-sided alternative that they are different.
501
* For a one-sided test, divide the returned value by 2.</p>
503
* A pooled variance estimate is used to compute the t-statistic. See
504
* {@link #homoscedasticT(double[], double[])}. The sum of the sample sizes
505
* minus 2 is used as the degrees of freedom.</p>
507
* <strong>Usage Note:</strong><br>
508
* The validity of the p-value depends on the assumptions of the parametric
509
* t-test procedure, as discussed
510
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
513
* <strong>Preconditions</strong>: <ul>
514
* <li>The observed array lengths must both be at least 2.
517
* @param sample1 array of sample data values
518
* @param sample2 array of sample data values
519
* @return p-value for t-test
520
* @throws IllegalArgumentException if the precondition is not met
521
* @throws MathException if an error occurs computing the p-value
523
public abstract double homoscedasticTTest(
526
throws IllegalArgumentException, MathException;
529
* <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm">
530
* two-sided t-test</a> evaluating the null hypothesis that <code>sample1</code>
531
* and <code>sample2</code> are drawn from populations with the same mean,
532
* with significance level <code>alpha</code>. This test does not assume
533
* that the subpopulation variances are equal. To perform the test assuming
534
* equal variances, use
535
* {@link #homoscedasticTTest(double[], double[], double)}.
537
* Returns <code>true</code> iff the null hypothesis that the means are
538
* equal can be rejected with confidence <code>1 - alpha</code>. To
539
* perform a 1-sided test, use <code>alpha * 2</code></p>
541
* See {@link #t(double[], double[])} for the formula used to compute the
542
* t-statistic. Degrees of freedom are approximated using the
543
* <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm">
544
* Welch-Satterthwaite approximation.</a></p>
546
* <strong>Examples:</strong><br><ol>
547
* <li>To test the (2-sided) hypothesis <code>mean 1 = mean 2 </code> at
549
* <br><code>tTest(sample1, sample2, 0.05). </code>
551
* <li>To test the (one-sided) hypothesis <code> mean 1 < mean 2 </code>,
552
* at the 99% level, first verify that the measured mean of <code>sample 1</code>
553
* is less than the mean of <code>sample 2</code> and then use
554
* <br><code>tTest(sample1, sample2, 0.02) </code>
557
* <strong>Usage Note:</strong><br>
558
* The validity of the test depends on the assumptions of the parametric
559
* t-test procedure, as discussed
560
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
563
* <strong>Preconditions</strong>: <ul>
564
* <li>The observed array lengths must both be at least 2.
566
* <li> <code> 0 < alpha < 0.5 </code>
569
* @param sample1 array of sample data values
570
* @param sample2 array of sample data values
571
* @param alpha significance level of the test
572
* @return true if the null hypothesis can be rejected with
573
* confidence 1 - alpha
574
* @throws IllegalArgumentException if the preconditions are not met
575
* @throws MathException if an error occurs performing the test
577
public abstract boolean tTest(
581
throws IllegalArgumentException, MathException;
584
* <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm">
585
* two-sided t-test</a> evaluating the null hypothesis that <code>sample1</code>
586
* and <code>sample2</code> are drawn from populations with the same mean,
587
* with significance level <code>alpha</code>, assuming that the
588
* subpopulation variances are equal. Use
589
* {@link #tTest(double[], double[], double)} to perform the test without
590
* the assumption of equal variances.
592
* Returns <code>true</code> iff the null hypothesis that the means are
593
* equal can be rejected with confidence <code>1 - alpha</code>. To
594
* perform a 1-sided test, use <code>alpha * 2.</code> To perform the test
595
* without the assumption of equal subpopulation variances, use
596
* {@link #tTest(double[], double[], double)}.</p>
598
* A pooled variance estimate is used to compute the t-statistic. See
599
* {@link #t(double[], double[])} for the formula. The sum of the sample
600
* sizes minus 2 is used as the degrees of freedom.</p>
602
* <strong>Examples:</strong><br><ol>
603
* <li>To test the (2-sided) hypothesis <code>mean 1 = mean 2 </code> at
604
* the 95% level, use <br><code>tTest(sample1, sample2, 0.05). </code>
606
* <li>To test the (one-sided) hypothesis <code> mean 1 < mean 2, </code>
607
* at the 99% level, first verify that the measured mean of
608
* <code>sample 1</code> is less than the mean of <code>sample 2</code>
610
* <br><code>tTest(sample1, sample2, 0.02) </code>
613
* <strong>Usage Note:</strong><br>
614
* The validity of the test depends on the assumptions of the parametric
615
* t-test procedure, as discussed
616
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
619
* <strong>Preconditions</strong>: <ul>
620
* <li>The observed array lengths must both be at least 2.
622
* <li> <code> 0 < alpha < 0.5 </code>
625
* @param sample1 array of sample data values
626
* @param sample2 array of sample data values
627
* @param alpha significance level of the test
628
* @return true if the null hypothesis can be rejected with
629
* confidence 1 - alpha
630
* @throws IllegalArgumentException if the preconditions are not met
631
* @throws MathException if an error occurs performing the test
633
public abstract boolean homoscedasticTTest(
637
throws IllegalArgumentException, MathException;
639
* Returns the <i>observed significance level</i>, or
640
* <i>p-value</i>, associated with a two-sample, two-tailed t-test
641
* comparing the means of the datasets described by two StatisticalSummary
644
* The number returned is the smallest significance level
645
* at which one can reject the null hypothesis that the two means are
646
* equal in favor of the two-sided alternative that they are different.
647
* For a one-sided test, divide the returned value by 2.</p>
649
* The test does not assume that the underlying popuation variances are
650
* equal and it uses approximated degrees of freedom computed from the
651
* sample data to compute the p-value. To perform the test assuming
652
* equal variances, use
653
* {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}.</p>
655
* <strong>Usage Note:</strong><br>
656
* The validity of the p-value depends on the assumptions of the parametric
657
* t-test procedure, as discussed
658
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
661
* <strong>Preconditions</strong>: <ul>
662
* <li>The datasets described by the two Univariates must each contain
663
* at least 2 observations.
666
* @param sampleStats1 StatisticalSummary describing data from the first sample
667
* @param sampleStats2 StatisticalSummary describing data from the second sample
668
* @return p-value for t-test
669
* @throws IllegalArgumentException if the precondition is not met
670
* @throws MathException if an error occurs computing the p-value
672
public abstract double tTest(
673
StatisticalSummary sampleStats1,
674
StatisticalSummary sampleStats2)
675
throws IllegalArgumentException, MathException;
677
* Returns the <i>observed significance level</i>, or
678
* <i>p-value</i>, associated with a two-sample, two-tailed t-test
679
* comparing the means of the datasets described by two StatisticalSummary
680
* instances, under the hypothesis of equal subpopulation variances. To
681
* perform a test without the equal variances assumption, use
682
* {@link #tTest(StatisticalSummary, StatisticalSummary)}.
684
* The number returned is the smallest significance level
685
* at which one can reject the null hypothesis that the two means are
686
* equal in favor of the two-sided alternative that they are different.
687
* For a one-sided test, divide the returned value by 2.</p>
689
* See {@link #homoscedasticT(double[], double[])} for the formula used to
690
* compute the t-statistic. The sum of the sample sizes minus 2 is used as
691
* the degrees of freedom.</p>
693
* <strong>Usage Note:</strong><br>
694
* The validity of the p-value depends on the assumptions of the parametric
695
* t-test procedure, as discussed
696
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">here</a>
698
* <strong>Preconditions</strong>: <ul>
699
* <li>The datasets described by the two Univariates must each contain
700
* at least 2 observations.
703
* @param sampleStats1 StatisticalSummary describing data from the first sample
704
* @param sampleStats2 StatisticalSummary describing data from the second sample
705
* @return p-value for t-test
706
* @throws IllegalArgumentException if the precondition is not met
707
* @throws MathException if an error occurs computing the p-value
709
public abstract double homoscedasticTTest(
710
StatisticalSummary sampleStats1,
711
StatisticalSummary sampleStats2)
712
throws IllegalArgumentException, MathException;
715
* <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda353.htm">
716
* two-sided t-test</a> evaluating the null hypothesis that
717
* <code>sampleStats1</code> and <code>sampleStats2</code> describe
718
* datasets drawn from populations with the same mean, with significance
719
* level <code>alpha</code>. This test does not assume that the
720
* subpopulation variances are equal. To perform the test under the equal
721
* variances assumption, use
722
* {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}.
724
* Returns <code>true</code> iff the null hypothesis that the means are
725
* equal can be rejected with confidence <code>1 - alpha</code>. To
726
* perform a 1-sided test, use <code>alpha * 2</code></p>
728
* See {@link #t(double[], double[])} for the formula used to compute the
729
* t-statistic. Degrees of freedom are approximated using the
730
* <a href="http://www.itl.nist.gov/div898/handbook/prc/section3/prc31.htm">
731
* Welch-Satterthwaite approximation.</a></p>
733
* <strong>Examples:</strong><br><ol>
734
* <li>To test the (2-sided) hypothesis <code>mean 1 = mean 2 </code> at
736
* <br><code>tTest(sampleStats1, sampleStats2, 0.05) </code>
738
* <li>To test the (one-sided) hypothesis <code> mean 1 < mean 2 </code>
739
* at the 99% level, first verify that the measured mean of
740
* <code>sample 1</code> is less than the mean of <code>sample 2</code>
742
* <br><code>tTest(sampleStats1, sampleStats2, 0.02) </code>
745
* <strong>Usage Note:</strong><br>
746
* The validity of the test depends on the assumptions of the parametric
747
* t-test procedure, as discussed
748
* <a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
751
* <strong>Preconditions</strong>: <ul>
752
* <li>The datasets described by the two Univariates must each contain
753
* at least 2 observations.
755
* <li> <code> 0 < alpha < 0.5 </code>
758
* @param sampleStats1 StatisticalSummary describing sample data values
759
* @param sampleStats2 StatisticalSummary describing sample data values
760
* @param alpha significance level of the test
761
* @return true if the null hypothesis can be rejected with
762
* confidence 1 - alpha
763
* @throws IllegalArgumentException if the preconditions are not met
764
* @throws MathException if an error occurs performing the test
766
public abstract boolean tTest(
767
StatisticalSummary sampleStats1,
768
StatisticalSummary sampleStats2,
770
throws IllegalArgumentException, MathException;
b'\\ No newline at end of file'
added
removed
src/java/org/apache/commons/math/stat/inference/TTest.java
