org.apache.commons.math.stat.inference.TTestImpl Maven / Gradle / Ivy
Show all versions of commons-math Show documentation
/*
* 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.MathRuntimeException;
import org.apache.commons.math.distribution.TDistribution;
import org.apache.commons.math.distribution.TDistributionImpl;
import org.apache.commons.math.exception.util.LocalizedFormats;
import org.apache.commons.math.stat.StatUtils;
import org.apache.commons.math.stat.descriptive.StatisticalSummary;
import org.apache.commons.math.util.FastMath;
/**
* Implements t-test statistics defined in the {@link TTest} interface.
*
* Uses commons-math {@link org.apache.commons.math.distribution.TDistributionImpl}
* implementation to estimate exact p-values.
*
* @version $Revision: 1042336 $ $Date: 2010-12-05 13:40:48 +0100 (dim. 05 déc. 2010) $
*/
public class TTestImpl implements TTest {
/** Distribution used to compute inference statistics.
* @deprecated in 2.2 (to be removed in 3.0).
*/
@Deprecated
private TDistribution distribution;
/**
* Default constructor.
*/
public TTestImpl() {
this(new TDistributionImpl(1.0));
}
/**
* Create a test instance using the given distribution for computing
* inference statistics.
* @param t distribution used to compute inference statistics.
* @since 1.2
* @deprecated in 2.2 (to be removed in 3.0).
*/
@Deprecated
public TTestImpl(TDistribution t) {
super();
setDistribution(t);
}
/**
* 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
* mu = 0 and the sample array consisting of the (signed)
* differences between corresponding entries in sample1 and
* sample2.
*
* Preconditions:
* - The input arrays must have the same length and their common length
* must be at least 2.
*
*
* @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 {
checkSampleData(sample1);
checkSampleData(sample2);
double meanDifference = StatUtils.meanDifference(sample1, sample2);
return t(meanDifference, 0,
StatUtils.varianceDifference(sample1, sample2, meanDifference),
sample1.length);
}
/**
* Returns the observed significance level, or
* p-value, associated with a paired, two-sample, two-tailed t-test
* based on the data in the input arrays.
*
* 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.
*
* This test is equivalent to a one-sample t-test computed using
* {@link #tTest(double, double[])} with mu = 0 and the sample
* array consisting of the signed differences between corresponding elements of
* sample1 and sample2.
*
* Usage Note:
* The validity of the p-value depends on the assumptions of the parametric
* t-test procedure, as discussed
*
* here
*
* Preconditions:
* - The input array lengths must be the same and their common length must
* be at least 2.
*
*
* @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),
sample1.length);
}
/**
* Performs a paired t-test evaluating the null hypothesis that the
* mean of the paired differences between sample1 and
* sample2 is 0 in favor of the two-sided alternative that the
* mean paired difference is not equal to 0, with significance level
* alpha.
*
* Returns true iff the null hypothesis can be rejected with
* confidence 1 - alpha. To perform a 1-sided test, use
* alpha * 2
*
* Usage Note:
* The validity of the test depends on the assumptions of the parametric
* t-test procedure, as discussed
*
* here
*
* Preconditions:
* - The input array lengths must be the same and their common length
* must be at least 2.
*
* -
0 < alpha < 0.5
*
*
* @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 {
checkSignificanceLevel(alpha);
return pairedTTest(sample1, sample2) < alpha;
}
/**
* Computes a
* t statistic given observed values and a comparison constant.
*
* This statistic can be used to perform a one sample t-test for the mean.
*
* Preconditions:
* - The observed array length must be at least 2.
*
*
* @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 {
checkSampleData(observed);
return t(StatUtils.mean(observed), mu, StatUtils.variance(observed),
observed.length);
}
/**
* Computes a
* t statistic to use in comparing the mean of the dataset described by
* sampleStats to mu.
*
* This statistic can be used to perform a one sample t-test for the mean.
*
* Preconditions:
* observed.getN() > = 2.
*
*
* @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 {
checkSampleData(sampleStats);
return t(sampleStats.getMean(), mu, sampleStats.getVariance(),
sampleStats.getN());
}
/**
* Computes a 2-sample t statistic, under the hypothesis of equal
* subpopulation variances. To compute a t-statistic without the
* equal variances hypothesis, use {@link #t(double[], double[])}.
*
* This statistic can be used to perform a (homoscedastic) two-sample
* t-test to compare sample means.
*
* The t-statisitc is
*
* t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))
*
* where n1 is the size of first sample;
* n2 is the size of second sample;
* m1 is the mean of first sample;
* m2 is the mean of second sample
*
* and var is the pooled variance estimate:
*
* var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))
*
* with var1 the variance of the first sample and
* var2 the variance of the second sample.
*
* Preconditions:
* - The observed array lengths must both be at least 2.
*
*
* @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 {
checkSampleData(sample1);
checkSampleData(sample2);
return homoscedasticT(StatUtils.mean(sample1), StatUtils.mean(sample2),
StatUtils.variance(sample1), StatUtils.variance(sample2),
sample1.length, sample2.length);
}
/**
* Computes a 2-sample t statistic, without the hypothesis of equal
* subpopulation variances. To compute a t-statistic assuming equal
* variances, use {@link #homoscedasticT(double[], double[])}.
*
* This statistic can be used to perform a two-sample t-test to compare
* sample means.
*
* The t-statisitc is
*
* t = (m1 - m2) / sqrt(var1/n1 + var2/n2)
*
* where n1 is the size of the first sample
* n2 is the size of the second sample;
* m1 is the mean of the first sample;
* m2 is the mean of the second sample;
* var1 is the variance of the first sample;
* var2 is the variance of the second sample;
*
* Preconditions:
* - The observed array lengths must both be at least 2.
*
*
* @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 {
checkSampleData(sample1);
checkSampleData(sample2);
return t(StatUtils.mean(sample1), StatUtils.mean(sample2),
StatUtils.variance(sample1), StatUtils.variance(sample2),
sample1.length, sample2.length);
}
/**
* Computes a 2-sample t statistic , 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.
*
* This statistic can be used to perform a two-sample t-test to compare
* sample means.
*
* The returned t-statisitc is
*
* t = (m1 - m2) / sqrt(var1/n1 + var2/n2)
*
* where n1 is the size of the first sample;
* n2 is the size of the second sample;
* m1 is the mean of the first sample;
* m2 is the mean of the second sample
* var1 is the variance of the first sample;
* var2 is the variance of the second sample
*
* Preconditions:
* - The datasets described by the two Univariates must each contain
* at least 2 observations.
*
*
* @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 {
checkSampleData(sampleStats1);
checkSampleData(sampleStats2);
return t(sampleStats1.getMean(), sampleStats2.getMean(),
sampleStats1.getVariance(), sampleStats2.getVariance(),
sampleStats1.getN(), sampleStats2.getN());
}
/**
* Computes a 2-sample t statistic, comparing the means of the datasets
* described by two {@link StatisticalSummary} instances, under the
* assumption of equal subpopulation variances. To compute a t-statistic
* without the equal variances assumption, use
* {@link #t(StatisticalSummary, StatisticalSummary)}.
*
* This statistic can be used to perform a (homoscedastic) two-sample
* t-test to compare sample means.
*
* The t-statisitc returned is
*
* t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))
*
* where n1 is the size of first sample;
* n2 is the size of second sample;
* m1 is the mean of first sample;
* m2 is the mean of second sample
* and var is the pooled variance estimate:
*
* var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))
*
* with var1 the variance of the first sample and
* var2 the variance of the second sample.
*
* Preconditions:
* - The datasets described by the two Univariates must each contain
* at least 2 observations.
*
*
* @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 {
checkSampleData(sampleStats1);
checkSampleData(sampleStats2);
return homoscedasticT(sampleStats1.getMean(), sampleStats2.getMean(),
sampleStats1.getVariance(), sampleStats2.getVariance(),
sampleStats1.getN(), sampleStats2.getN());
}
/**
* Returns the observed significance level, or
* p-value, associated with a one-sample, two-tailed t-test
* comparing the mean of the input array with the constant mu.
*
* The number returned is the smallest significance level
* at which one can reject the null hypothesis that the mean equals
* mu in favor of the two-sided alternative that the mean
* is different from mu. For a one-sided test, divide the
* returned value by 2.
*
* Usage Note:
* The validity of the test depends on the assumptions of the parametric
* t-test procedure, as discussed
* here
*
* Preconditions:
* - The observed array length must be at least 2.
*
*
* @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 {
checkSampleData(sample);
return tTest( StatUtils.mean(sample), mu, StatUtils.variance(sample),
sample.length);
}
/**
* Performs a
* two-sided t-test evaluating the null hypothesis that the mean of the population from
* which sample is drawn equals mu.
*
* Returns true iff the null hypothesis can be
* rejected with confidence 1 - alpha. To
* perform a 1-sided test, use alpha * 2
*
* Examples:
* - To test the (2-sided) hypothesis
sample mean = mu at
* the 95% level, use
tTest(mu, sample, 0.05)
*
* - To test the (one-sided) hypothesis
sample mean < mu
* at the 99% level, first verify that the measured sample mean is less
* than mu and then use
*
tTest(mu, sample, 0.02)
*
*
* Usage Note:
* The validity of the test depends on the assumptions of the one-sample
* parametric t-test procedure, as discussed
* here
*
* Preconditions:
* - The observed array length must be at least 2.
*
*
* @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 {
checkSignificanceLevel(alpha);
return tTest(mu, sample) < alpha;
}
/**
* Returns the observed significance level, or
* p-value, associated with a one-sample, two-tailed t-test
* comparing the mean of the dataset described by sampleStats
* with the constant mu.
*
* The number returned is the smallest significance level
* at which one can reject the null hypothesis that the mean equals
* mu in favor of the two-sided alternative that the mean
* is different from mu. For a one-sided test, divide the
* returned value by 2.
*
* Usage Note:
* The validity of the test depends on the assumptions of the parametric
* t-test procedure, as discussed
*
* here
*
* Preconditions:
* - The sample must contain at least 2 observations.
*
*
* @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 {
checkSampleData(sampleStats);
return tTest(sampleStats.getMean(), mu, sampleStats.getVariance(),
sampleStats.getN());
}
/**
* Performs a
* two-sided t-test evaluating the null hypothesis that the mean of the
* population from which the dataset described by stats is
* drawn equals mu.
*
* Returns true iff the null hypothesis can be rejected with
* confidence 1 - alpha. To perform a 1-sided test, use
* alpha * 2.
*
* Examples:
* - To test the (2-sided) hypothesis
sample mean = mu at
* the 95% level, use
tTest(mu, sampleStats, 0.05)
*
* - To test the (one-sided) hypothesis
sample mean < mu
* at the 99% level, first verify that the measured sample mean is less
* than mu and then use
*
tTest(mu, sampleStats, 0.02)
*
*
* Usage Note:
* The validity of the test depends on the assumptions of the one-sample
* parametric t-test procedure, as discussed
* here
*
* Preconditions:
* - The sample must include at least 2 observations.
*
*
* @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 {
checkSignificanceLevel(alpha);
return tTest(mu, sampleStats) < alpha;
}
/**
* Returns the observed significance level, or
* p-value, associated with a two-sample, two-tailed t-test
* comparing the means of the input arrays.
*
* 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.
*
* 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
*
* here. To perform the test under the assumption of equal subpopulation
* variances, use {@link #homoscedasticTTest(double[], double[])}.
*
* Usage Note:
* The validity of the p-value depends on the assumptions of the parametric
* t-test procedure, as discussed
*
* here
*
* Preconditions:
* - The observed array lengths must both be at least 2.
*
*
* @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 {
checkSampleData(sample1);
checkSampleData(sample2);
return tTest(StatUtils.mean(sample1), StatUtils.mean(sample2),
StatUtils.variance(sample1), StatUtils.variance(sample2),
sample1.length, sample2.length);
}
/**
* Returns the observed significance level, or
* p-value, associated with a two-sample, two-tailed t-test
* comparing the means of the input arrays, under the assumption that
* the two samples are drawn from subpopulations with equal variances.
* To perform the test without the equal variances assumption, use
* {@link #tTest(double[], double[])}.
*
* 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.
*
* 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.
*
* Usage Note:
* The validity of the p-value depends on the assumptions of the parametric
* t-test procedure, as discussed
*
* here
*
* Preconditions:
* - The observed array lengths must both be at least 2.
*
*
* @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 {
checkSampleData(sample1);
checkSampleData(sample2);
return homoscedasticTTest(StatUtils.mean(sample1),
StatUtils.mean(sample2), StatUtils.variance(sample1),
StatUtils.variance(sample2), sample1.length,
sample2.length);
}
/**
* Performs a
*
* two-sided t-test evaluating the null hypothesis that sample1
* and sample2 are drawn from populations with the same mean,
* with significance level alpha. This test does not assume
* that the subpopulation variances are equal. To perform the test assuming
* equal variances, use
* {@link #homoscedasticTTest(double[], double[], double)}.
*
* Returns true iff the null hypothesis that the means are
* equal can be rejected with confidence 1 - alpha. To
* perform a 1-sided test, use alpha / 2
*
* See {@link #t(double[], double[])} for the formula used to compute the
* t-statistic. Degrees of freedom are approximated using the
*
* Welch-Satterthwaite approximation.
*
* Examples:
* - To test the (2-sided) hypothesis
mean 1 = mean 2 at
* the 95% level, use
*
tTest(sample1, sample2, 0.05).
*
* - To test the (one-sided) hypothesis
mean 1 < mean 2 at
* the 99% level, first verify that the measured mean of sample 1
* is less than the mean of sample 2 and then use
*
tTest(sample1, sample2, 0.02)
*
*
* Usage Note:
* The validity of the test depends on the assumptions of the parametric
* t-test procedure, as discussed
*
* here
*
* Preconditions:
* - The observed array lengths must both be at least 2.
*
* -
0 < alpha < 0.5
*
*
* @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 {
checkSignificanceLevel(alpha);
return tTest(sample1, sample2) < alpha;
}
/**
* Performs a
*
* two-sided t-test evaluating the null hypothesis that sample1
* and sample2 are drawn from populations with the same mean,
* with significance level alpha, assuming that the
* subpopulation variances are equal. Use
* {@link #tTest(double[], double[], double)} to perform the test without
* the assumption of equal variances.
*
* Returns true iff the null hypothesis that the means are
* equal can be rejected with confidence 1 - alpha. To
* perform a 1-sided test, use alpha * 2. To perform the test
* without the assumption of equal subpopulation variances, use
* {@link #tTest(double[], double[], double)}.
*
* 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.
*
* Examples:
* - To test the (2-sided) hypothesis
mean 1 = mean 2 at
* the 95% level, use
tTest(sample1, sample2, 0.05).
*
* - To test the (one-sided) hypothesis
mean 1 < mean 2,
* at the 99% level, first verify that the measured mean of
* sample 1 is less than the mean of sample 2
* and then use
*
tTest(sample1, sample2, 0.02)
*
*
* Usage Note:
* The validity of the test depends on the assumptions of the parametric
* t-test procedure, as discussed
*
* here
*
* Preconditions:
* - The observed array lengths must both be at least 2.
*
* -
0 < alpha < 0.5
*
*
* @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 {
checkSignificanceLevel(alpha);
return homoscedasticTTest(sample1, sample2) < alpha;
}
/**
* Returns the observed significance level, or
* p-value, associated with a two-sample, two-tailed t-test
* comparing the means of the datasets described by two StatisticalSummary
* instances.
*
* 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.
*
* 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)}.
*
* Usage Note:
* The validity of the p-value depends on the assumptions of the parametric
* t-test procedure, as discussed
*
* here
*
* Preconditions:
* - The datasets described by the two Univariates must each contain
* at least 2 observations.
*
*
* @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 {
checkSampleData(sampleStats1);
checkSampleData(sampleStats2);
return tTest(sampleStats1.getMean(), sampleStats2.getMean(), sampleStats1.getVariance(),
sampleStats2.getVariance(), sampleStats1.getN(),
sampleStats2.getN());
}
/**
* Returns the observed significance level, or
* p-value, associated with a two-sample, two-tailed t-test
* comparing the means of the datasets described by two StatisticalSummary
* instances, under the hypothesis of equal subpopulation variances. To
* perform a test without the equal variances assumption, use
* {@link #tTest(StatisticalSummary, StatisticalSummary)}.
*
* 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.
*
* 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.
*
* Usage Note:
* The validity of the p-value depends on the assumptions of the parametric
* t-test procedure, as discussed
* here
*
* Preconditions:
* - The datasets described by the two Univariates must each contain
* at least 2 observations.
*
*
* @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 {
checkSampleData(sampleStats1);
checkSampleData(sampleStats2);
return homoscedasticTTest(sampleStats1.getMean(),
sampleStats2.getMean(), sampleStats1.getVariance(),
sampleStats2.getVariance(), sampleStats1.getN(),
sampleStats2.getN());
}
/**
* Performs a
*
* two-sided t-test evaluating the null hypothesis that
* sampleStats1 and sampleStats2 describe
* datasets drawn from populations with the same mean, with significance
* level alpha. This test does not assume that the
* subpopulation variances are equal. To perform the test under the equal
* variances assumption, use
* {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}.
*
* Returns true iff the null hypothesis that the means are
* equal can be rejected with confidence 1 - alpha. To
* perform a 1-sided test, use alpha * 2
*
* See {@link #t(double[], double[])} for the formula used to compute the
* t-statistic. Degrees of freedom are approximated using the
*
* Welch-Satterthwaite approximation.
*
* Examples:
* - To test the (2-sided) hypothesis
mean 1 = mean 2 at
* the 95%, use
*
tTest(sampleStats1, sampleStats2, 0.05)
*
* - To test the (one-sided) hypothesis
mean 1 < mean 2
* at the 99% level, first verify that the measured mean of
* sample 1 is less than the mean of sample 2
* and then use
*
tTest(sampleStats1, sampleStats2, 0.02)
*
*
* Usage Note:
* The validity of the test depends on the assumptions of the parametric
* t-test procedure, as discussed
*
* here
*
* Preconditions:
* - The datasets described by the two Univariates must each contain
* at least 2 observations.
*
* -
0 < alpha < 0.5
*
*
* @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 {
checkSignificanceLevel(alpha);
return tTest(sampleStats1, sampleStats2) < alpha;
}
//----------------------------------------------- Protected methods
/**
* Computes approximate degrees of freedom for 2-sample t-test.
*
* @param v1 first sample variance
* @param v2 second sample variance
* @param n1 first sample n
* @param n2 second sample n
* @return approximate degrees of freedom
*/
protected double df(double v1, double v2, double n1, double n2) {
return (((v1 / n1) + (v2 / n2)) * ((v1 / n1) + (v2 / n2))) /
((v1 * v1) / (n1 * n1 * (n1 - 1d)) + (v2 * v2) /
(n2 * n2 * (n2 - 1d)));
}
/**
* Computes t test statistic for 1-sample t-test.
*
* @param m sample mean
* @param mu constant to test against
* @param v sample variance
* @param n sample n
* @return t test statistic
*/
protected double t(double m, double mu, double v, double n) {
return (m - mu) / FastMath.sqrt(v / n);
}
/**
* Computes t test statistic for 2-sample t-test.
*
* 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) / FastMath.sqrt((v1 / n1) + (v2 / n2));
}
/**
* Computes t test statistic for 2-sample t-test under the hypothesis
* of equal subpopulation variances.
*
* @param m1 first sample mean
* @param m2 second sample mean
* @param v1 first sample variance
* @param v2 second sample variance
* @param n1 first sample n
* @param n2 second sample n
* @return t test statistic
*/
protected double homoscedasticT(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) / FastMath.sqrt(pooledVariance * (1d / n1 + 1d / n2));
}
/**
* Computes p-value for 2-sided, 1-sample t-test.
*
* @param m sample mean
* @param mu constant to test against
* @param v sample variance
* @param n sample n
* @return p-value
* @throws MathException if an error occurs computing the p-value
*/
protected double tTest(double m, double mu, double v, double n)
throws MathException {
double t = FastMath.abs(t(m, mu, v, n));
distribution.setDegreesOfFreedom(n - 1);
return 2.0 * distribution.cumulativeProbability(-t);
}
/**
* Computes p-value for 2-sided, 2-sample t-test.
*
* 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 = FastMath.abs(t(m1, m2, v1, v2, n1, n2));
double degreesOfFreedom = 0;
degreesOfFreedom = df(v1, v2, n1, n2);
distribution.setDegreesOfFreedom(degreesOfFreedom);
return 2.0 * distribution.cumulativeProbability(-t);
}
/**
* Computes p-value for 2-sided, 2-sample t-test, under the assumption
* of equal subpopulation variances.
*
* 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 = FastMath.abs(homoscedasticT(m1, m2, v1, v2, n1, n2));
double degreesOfFreedom = n1 + n2 - 2;
distribution.setDegreesOfFreedom(degreesOfFreedom);
return 2.0 * distribution.cumulativeProbability(-t);
}
/**
* Modify the distribution used to compute inference statistics.
* @param value the new distribution
* @since 1.2
* @deprecated in 2.2 (to be removed in 3.0).
*/
@Deprecated
public void setDistribution(TDistribution value) {
distribution = value;
}
/** Check significance level.
* @param alpha significance level
* @exception IllegalArgumentException if significance level is out of bounds
*/
private void checkSignificanceLevel(final double alpha)
throws IllegalArgumentException {
if ((alpha <= 0) || (alpha > 0.5)) {
throw MathRuntimeException.createIllegalArgumentException(
LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL,
alpha, 0.0, 0.5);
}
}
/** Check sample data.
* @param data sample data
* @exception IllegalArgumentException if there is not enough sample data
*/
private void checkSampleData(final double[] data)
throws IllegalArgumentException {
if ((data == null) || (data.length < 2)) {
throw MathRuntimeException.createIllegalArgumentException(
LocalizedFormats.INSUFFICIENT_DATA_FOR_T_STATISTIC,
(data == null) ? 0 : data.length);
}
}
/** Check sample data.
* @param stat statistical summary
* @exception IllegalArgumentException if there is not enough sample data
*/
private void checkSampleData(final StatisticalSummary stat)
throws IllegalArgumentException {
if ((stat == null) || (stat.getN() < 2)) {
throw MathRuntimeException.createIllegalArgumentException(
LocalizedFormats.INSUFFICIENT_DATA_FOR_T_STATISTIC,
(stat == null) ? 0 : stat.getN());
}
}
}