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* 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,
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* See the License for the specific language governing permissions and
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package org.apache.commons.math3.stat.inference;
import org.apache.commons.math3.distribution.TDistribution;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.MathIllegalArgumentException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.stat.StatUtils;
import org.apache.commons.math3.stat.descriptive.StatisticalSummary;
import org.apache.commons.math3.util.FastMath;
/**
* An implementation for Student's t-tests.
*
* Tests can be:
* - One-sample or two-sample
* - One-sided or two-sided
* - Paired or unpaired (for two-sample tests)
* - Homoscedastic (equal variance assumption) or heteroscedastic
* (for two sample tests)
* - Fixed significance level (boolean-valued) or returning p-values.
*
*
* 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, double-valued methods return p-values;
* boolean-valued methods perform fixed significance level tests.
* Significance levels are always specified as numbers between 0 and 0.5
* (e.g. tests at the 95% level use alpha=0.05).
*
* Input to tests can be either double[] arrays or
* {@link StatisticalSummary} instances.
* Uses commons-math {@link org.apache.commons.math3.distribution.TDistribution}
* implementation to estimate exact p-values.
*
*/
public class TTest {
/**
* Computes a paired, 2-sample t-statistic based on the data in the input
* arrays. The t-statistic returned is equivalent to what would be returned by
* computing the one-sample t-statistic {@link #t(double, double[])}, with
* 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 NullArgumentException if the arrays are null
* @throws NoDataException if the arrays are empty
* @throws DimensionMismatchException if the length of the arrays is not equal
* @throws NumberIsTooSmallException if the length of the arrays is < 2
*/
public double pairedT(final double[] sample1, final double[] sample2)
throws NullArgumentException, NoDataException,
DimensionMismatchException, NumberIsTooSmallException {
checkSampleData(sample1);
checkSampleData(sample2);
double meanDifference = StatUtils.meanDifference(sample1, sample2);
return t(meanDifference, 0,
StatUtils.varianceDifference(sample1, sample2, meanDifference),
sample1.length);
}
/**
* Returns the 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 NullArgumentException if the arrays are null
* @throws NoDataException if the arrays are empty
* @throws DimensionMismatchException if the length of the arrays is not equal
* @throws NumberIsTooSmallException if the length of the arrays is < 2
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public double pairedTTest(final double[] sample1, final double[] sample2)
throws NullArgumentException, NoDataException, DimensionMismatchException,
NumberIsTooSmallException, MaxCountExceededException {
double meanDifference = StatUtils.meanDifference(sample1, sample2);
return tTest(meanDifference, 0,
StatUtils.varianceDifference(sample1, sample2, meanDifference),
sample1.length);
}
/**
* Performs a paired t-test evaluating the null hypothesis that the
* mean of the paired differences between 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 NullArgumentException if the arrays are null
* @throws NoDataException if the arrays are empty
* @throws DimensionMismatchException if the length of the arrays is not equal
* @throws NumberIsTooSmallException if the length of the arrays is < 2
* @throws OutOfRangeException if alpha is not in the range (0, 0.5]
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public boolean pairedTTest(final double[] sample1, final double[] sample2,
final double alpha)
throws NullArgumentException, NoDataException, DimensionMismatchException,
NumberIsTooSmallException, OutOfRangeException, MaxCountExceededException {
checkSignificanceLevel(alpha);
return pairedTTest(sample1, sample2) < alpha;
}
/**
* Computes a
* 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 NullArgumentException if observed is null
* @throws NumberIsTooSmallException if the length of observed is < 2
*/
public double t(final double mu, final double[] observed)
throws NullArgumentException, NumberIsTooSmallException {
checkSampleData(observed);
// No try-catch or advertised exception because args have just been checked
return t(StatUtils.mean(observed), mu, StatUtils.variance(observed),
observed.length);
}
/**
* Computes a
* 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 NullArgumentException if sampleStats is null
* @throws NumberIsTooSmallException if the number of samples is < 2
*/
public double t(final double mu, final StatisticalSummary sampleStats)
throws NullArgumentException, NumberIsTooSmallException {
checkSampleData(sampleStats);
return t(sampleStats.getMean(), mu, sampleStats.getVariance(),
sampleStats.getN());
}
/**
* Computes a 2-sample t statistic, under the hypothesis of equal
* subpopulation variances. To compute a t-statistic without the
* equal variances hypothesis, use {@link #t(double[], double[])}.
*
* This statistic can be used to perform a (homoscedastic) two-sample
* t-test to compare sample means.
*
* The t-statistic 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 NullArgumentException if the arrays are null
* @throws NumberIsTooSmallException if the length of the arrays is < 2
*/
public double homoscedasticT(final double[] sample1, final double[] sample2)
throws NullArgumentException, NumberIsTooSmallException {
checkSampleData(sample1);
checkSampleData(sample2);
// No try-catch or advertised exception because args have just been checked
return homoscedasticT(StatUtils.mean(sample1), StatUtils.mean(sample2),
StatUtils.variance(sample1), StatUtils.variance(sample2),
sample1.length, sample2.length);
}
/**
* Computes a 2-sample t statistic, without the hypothesis of equal
* subpopulation variances. To compute a t-statistic assuming equal
* variances, use {@link #homoscedasticT(double[], double[])}.
*
* This statistic can be used to perform a two-sample t-test to compare
* sample means.
*
* The t-statistic 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 NullArgumentException if the arrays are null
* @throws NumberIsTooSmallException if the length of the arrays is < 2
*/
public double t(final double[] sample1, final double[] sample2)
throws NullArgumentException, NumberIsTooSmallException {
checkSampleData(sample1);
checkSampleData(sample2);
// No try-catch or advertised exception because args have just been checked
return t(StatUtils.mean(sample1), StatUtils.mean(sample2),
StatUtils.variance(sample1), StatUtils.variance(sample2),
sample1.length, sample2.length);
}
/**
* Computes a 2-sample t statistic , 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-statistic 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 NullArgumentException if the sample statistics are null
* @throws NumberIsTooSmallException if the number of samples is < 2
*/
public double t(final StatisticalSummary sampleStats1,
final StatisticalSummary sampleStats2)
throws NullArgumentException, NumberIsTooSmallException {
checkSampleData(sampleStats1);
checkSampleData(sampleStats2);
return t(sampleStats1.getMean(), sampleStats2.getMean(),
sampleStats1.getVariance(), sampleStats2.getVariance(),
sampleStats1.getN(), sampleStats2.getN());
}
/**
* Computes a 2-sample t statistic, comparing the means of the datasets
* described by two {@link StatisticalSummary} instances, under the
* assumption of equal subpopulation variances. To compute a t-statistic
* without the equal variances assumption, use
* {@link #t(StatisticalSummary, StatisticalSummary)}.
*
* This statistic can be used to perform a (homoscedastic) two-sample
* t-test to compare sample means.
*
* The t-statistic 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 NullArgumentException if the sample statistics are null
* @throws NumberIsTooSmallException if the number of samples is < 2
*/
public double homoscedasticT(final StatisticalSummary sampleStats1,
final StatisticalSummary sampleStats2)
throws NullArgumentException, NumberIsTooSmallException {
checkSampleData(sampleStats1);
checkSampleData(sampleStats2);
return homoscedasticT(sampleStats1.getMean(), sampleStats2.getMean(),
sampleStats1.getVariance(), sampleStats2.getVariance(),
sampleStats1.getN(), sampleStats2.getN());
}
/**
* Returns the 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 NullArgumentException if the sample array is null
* @throws NumberIsTooSmallException if the length of the array is < 2
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public double tTest(final double mu, final double[] sample)
throws NullArgumentException, NumberIsTooSmallException,
MaxCountExceededException {
checkSampleData(sample);
// No try-catch or advertised exception because args have just been checked
return tTest(StatUtils.mean(sample), mu, StatUtils.variance(sample),
sample.length);
}
/**
* Performs a
* 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 NullArgumentException if the sample array is null
* @throws NumberIsTooSmallException if the length of the array is < 2
* @throws OutOfRangeException if alpha is not in the range (0, 0.5]
* @throws MaxCountExceededException if an error computing the p-value
*/
public boolean tTest(final double mu, final double[] sample, final double alpha)
throws NullArgumentException, NumberIsTooSmallException,
OutOfRangeException, MaxCountExceededException {
checkSignificanceLevel(alpha);
return tTest(mu, sample) < alpha;
}
/**
* Returns the 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 NullArgumentException if sampleStats is null
* @throws NumberIsTooSmallException if the number of samples is < 2
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public double tTest(final double mu, final StatisticalSummary sampleStats)
throws NullArgumentException, NumberIsTooSmallException,
MaxCountExceededException {
checkSampleData(sampleStats);
return tTest(sampleStats.getMean(), mu, sampleStats.getVariance(),
sampleStats.getN());
}
/**
* Performs a
* 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 NullArgumentException if sampleStats is null
* @throws NumberIsTooSmallException if the number of samples is < 2
* @throws OutOfRangeException if alpha is not in the range (0, 0.5]
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public boolean tTest(final double mu, final StatisticalSummary sampleStats,
final double alpha)
throws NullArgumentException, NumberIsTooSmallException,
OutOfRangeException, MaxCountExceededException {
checkSignificanceLevel(alpha);
return tTest(mu, sampleStats) < alpha;
}
/**
* Returns the 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 NullArgumentException if the arrays are null
* @throws NumberIsTooSmallException if the length of the arrays is < 2
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public double tTest(final double[] sample1, final double[] sample2)
throws NullArgumentException, NumberIsTooSmallException,
MaxCountExceededException {
checkSampleData(sample1);
checkSampleData(sample2);
// No try-catch or advertised exception because args have just been checked
return tTest(StatUtils.mean(sample1), StatUtils.mean(sample2),
StatUtils.variance(sample1), StatUtils.variance(sample2),
sample1.length, sample2.length);
}
/**
* Returns the 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 NullArgumentException if the arrays are null
* @throws NumberIsTooSmallException if the length of the arrays is < 2
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public double homoscedasticTTest(final double[] sample1, final double[] sample2)
throws NullArgumentException, NumberIsTooSmallException,
MaxCountExceededException {
checkSampleData(sample1);
checkSampleData(sample2);
// No try-catch or advertised exception because args have just been checked
return homoscedasticTTest(StatUtils.mean(sample1),
StatUtils.mean(sample2),
StatUtils.variance(sample1),
StatUtils.variance(sample2),
sample1.length, sample2.length);
}
/**
* Performs a
*
* 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 NullArgumentException if the arrays are null
* @throws NumberIsTooSmallException if the length of the arrays is < 2
* @throws OutOfRangeException if alpha is not in the range (0, 0.5]
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public boolean tTest(final double[] sample1, final double[] sample2,
final double alpha)
throws NullArgumentException, NumberIsTooSmallException,
OutOfRangeException, MaxCountExceededException {
checkSignificanceLevel(alpha);
return tTest(sample1, sample2) < alpha;
}
/**
* Performs a
*
* 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 NullArgumentException if the arrays are null
* @throws NumberIsTooSmallException if the length of the arrays is < 2
* @throws OutOfRangeException if alpha is not in the range (0, 0.5]
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public boolean homoscedasticTTest(final double[] sample1, final double[] sample2,
final double alpha)
throws NullArgumentException, NumberIsTooSmallException,
OutOfRangeException, MaxCountExceededException {
checkSignificanceLevel(alpha);
return homoscedasticTTest(sample1, sample2) < alpha;
}
/**
* Returns the 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 population variances are
* equal and it uses approximated degrees of freedom computed from the
* sample data to compute the p-value. To perform the test assuming
* equal variances, use
* {@link #homoscedasticTTest(StatisticalSummary, StatisticalSummary)}.
*
* 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 NullArgumentException if the sample statistics are null
* @throws NumberIsTooSmallException if the number of samples is < 2
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public double tTest(final StatisticalSummary sampleStats1,
final StatisticalSummary sampleStats2)
throws NullArgumentException, NumberIsTooSmallException,
MaxCountExceededException {
checkSampleData(sampleStats1);
checkSampleData(sampleStats2);
return tTest(sampleStats1.getMean(), sampleStats2.getMean(),
sampleStats1.getVariance(), sampleStats2.getVariance(),
sampleStats1.getN(), sampleStats2.getN());
}
/**
* Returns the 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 NullArgumentException if the sample statistics are null
* @throws NumberIsTooSmallException if the number of samples is < 2
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public double homoscedasticTTest(final StatisticalSummary sampleStats1,
final StatisticalSummary sampleStats2)
throws NullArgumentException, NumberIsTooSmallException,
MaxCountExceededException {
checkSampleData(sampleStats1);
checkSampleData(sampleStats2);
return homoscedasticTTest(sampleStats1.getMean(),
sampleStats2.getMean(),
sampleStats1.getVariance(),
sampleStats2.getVariance(),
sampleStats1.getN(), sampleStats2.getN());
}
/**
* Performs a
*
* 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 NullArgumentException if the sample statistics are null
* @throws NumberIsTooSmallException if the number of samples is < 2
* @throws OutOfRangeException if alpha is not in the range (0, 0.5]
* @throws MaxCountExceededException if an error occurs computing the p-value
*/
public boolean tTest(final StatisticalSummary sampleStats1,
final StatisticalSummary sampleStats2,
final double alpha)
throws NullArgumentException, NumberIsTooSmallException,
OutOfRangeException, MaxCountExceededException {
checkSignificanceLevel(alpha);
return tTest(sampleStats1, sampleStats2) < alpha;
}
//----------------------------------------------- Protected methods
/**
* Computes approximate degrees of freedom for 2-sample t-test.
*
* @param v1 first sample variance
* @param v2 second sample variance
* @param n1 first sample n
* @param n2 second sample n
* @return approximate degrees of freedom
*/
protected double df(double v1, double v2, double n1, double n2) {
return (((v1 / n1) + (v2 / n2)) * ((v1 / n1) + (v2 / n2))) /
((v1 * v1) / (n1 * n1 * (n1 - 1d)) + (v2 * v2) /
(n2 * n2 * (n2 - 1d)));
}
/**
* Computes t test statistic for 1-sample t-test.
*
* @param m sample mean
* @param mu constant to test against
* @param v sample variance
* @param n sample n
* @return t test statistic
*/
protected double t(final double m, final double mu,
final double v, final double n) {
return (m - mu) / FastMath.sqrt(v / n);
}
/**
* Computes t test statistic for 2-sample t-test.
*
* 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(final double m1, final double m2,
final double v1, final double v2,
final double n1, final double n2) {
return (m1 - m2) / FastMath.sqrt((v1 / n1) + (v2 / n2));
}
/**
* Computes t test statistic for 2-sample t-test under the hypothesis
* of equal subpopulation variances.
*
* @param m1 first sample mean
* @param m2 second sample mean
* @param v1 first sample variance
* @param v2 second sample variance
* @param n1 first sample n
* @param n2 second sample n
* @return t test statistic
*/
protected double homoscedasticT(final double m1, final double m2,
final double v1, final double v2,
final double n1, final double n2) {
final double pooledVariance = ((n1 - 1) * v1 + (n2 -1) * v2 ) / (n1 + n2 - 2);
return (m1 - m2) / FastMath.sqrt(pooledVariance * (1d / n1 + 1d / n2));
}
/**
* Computes p-value for 2-sided, 1-sample t-test.
*
* @param m sample mean
* @param mu constant to test against
* @param v sample variance
* @param n sample n
* @return p-value
* @throws MaxCountExceededException if an error occurs computing the p-value
* @throws MathIllegalArgumentException if n is not greater than 1
*/
protected double tTest(final double m, final double mu,
final double v, final double n)
throws MaxCountExceededException, MathIllegalArgumentException {
final double t = FastMath.abs(t(m, mu, v, n));
// pass a null rng to avoid unneeded overhead as we will not sample from this distribution
final TDistribution distribution = new TDistribution(null, n - 1);
return 2.0 * distribution.cumulativeProbability(-t);
}
/**
* Computes p-value for 2-sided, 2-sample t-test.
*
* 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 MaxCountExceededException if an error occurs computing the p-value
* @throws NotStrictlyPositiveException if the estimated degrees of freedom is not
* strictly positive
*/
protected double tTest(final double m1, final double m2,
final double v1, final double v2,
final double n1, final double n2)
throws MaxCountExceededException, NotStrictlyPositiveException {
final double t = FastMath.abs(t(m1, m2, v1, v2, n1, n2));
final double degreesOfFreedom = df(v1, v2, n1, n2);
// pass a null rng to avoid unneeded overhead as we will not sample from this distribution
final TDistribution distribution = new TDistribution(null, degreesOfFreedom);
return 2.0 * distribution.cumulativeProbability(-t);
}
/**
* Computes p-value for 2-sided, 2-sample t-test, under the assumption
* of equal subpopulation variances.
*
* 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 MaxCountExceededException if an error occurs computing the p-value
* @throws NotStrictlyPositiveException if the estimated degrees of freedom is not
* strictly positive
*/
protected double homoscedasticTTest(double m1, double m2,
double v1, double v2,
double n1, double n2)
throws MaxCountExceededException, NotStrictlyPositiveException {
final double t = FastMath.abs(homoscedasticT(m1, m2, v1, v2, n1, n2));
final double degreesOfFreedom = n1 + n2 - 2;
// pass a null rng to avoid unneeded overhead as we will not sample from this distribution
final TDistribution distribution = new TDistribution(null, degreesOfFreedom);
return 2.0 * distribution.cumulativeProbability(-t);
}
/**
* Check significance level.
*
* @param alpha significance level
* @throws OutOfRangeException if the significance level is out of bounds.
*/
private void checkSignificanceLevel(final double alpha)
throws OutOfRangeException {
if (alpha <= 0 || alpha > 0.5) {
throw new OutOfRangeException(LocalizedFormats.SIGNIFICANCE_LEVEL,
alpha, 0.0, 0.5);
}
}
/**
* Check sample data.
*
* @param data Sample data.
* @throws NullArgumentException if {@code data} is {@code null}.
* @throws NumberIsTooSmallException if there is not enough sample data.
*/
private void checkSampleData(final double[] data)
throws NullArgumentException, NumberIsTooSmallException {
if (data == null) {
throw new NullArgumentException();
}
if (data.length < 2) {
throw new NumberIsTooSmallException(
LocalizedFormats.INSUFFICIENT_DATA_FOR_T_STATISTIC,
data.length, 2, true);
}
}
/**
* Check sample data.
*
* @param stat Statistical summary.
* @throws NullArgumentException if {@code data} is {@code null}.
* @throws NumberIsTooSmallException if there is not enough sample data.
*/
private void checkSampleData(final StatisticalSummary stat)
throws NullArgumentException, NumberIsTooSmallException {
if (stat == null) {
throw new NullArgumentException();
}
if (stat.getN() < 2) {
throw new NumberIsTooSmallException(
LocalizedFormats.INSUFFICIENT_DATA_FOR_T_STATISTIC,
stat.getN(), 2, true);
}
}
}