org.apache.commons.math3.stat.inference

## Class TTest

• public class TTest
extends Object
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 StatisticalSummary instances.

Uses commons-math TDistribution implementation to estimate exact p-values.

• ### Constructor Summary

Constructors
Constructor and Description
TTest()
• ### Method Summary

Methods
Modifier and Type Method and Description
protected double df(double v1, double v2, double n1, double n2)
Computes approximate degrees of freedom for 2-sample t-test.
double homoscedasticT(double[] sample1, double[] sample2)
Computes a 2-sample t statistic, under the hypothesis of equal subpopulation variances.
protected double homoscedasticT(double m1, double m2, double v1, double v2, double n1, double n2)
Computes t test statistic for 2-sample t-test under the hypothesis of equal subpopulation variances.
double homoscedasticT(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
Computes a 2-sample t statistic, comparing the means of the datasets described by two StatisticalSummary instances, under the assumption of equal subpopulation variances.
double homoscedasticTTest(double[] sample1, double[] sample2)
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.
boolean homoscedasticTTest(double[] sample1, double[] sample2, double 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.
protected double homoscedasticTTest(double m1, double m2, double v1, double v2, double n1, double n2)
Computes p-value for 2-sided, 2-sample t-test, under the assumption of equal subpopulation variances.
double homoscedasticTTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
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.
double pairedT(double[] sample1, double[] sample2)
Computes a paired, 2-sample t-statistic based on the data in the input arrays.
double pairedTTest(double[] sample1, double[] sample2)
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.
boolean pairedTTest(double[] sample1, double[] sample2, double alpha)
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.
double t(double[] sample1, double[] sample2)
Computes a 2-sample t statistic, without the hypothesis of equal subpopulation variances.
double t(double mu, double[] observed)
Computes a t statistic given observed values and a comparison constant.
protected double t(double m, double mu, double v, double n)
Computes t test statistic for 1-sample t-test.
protected double t(double m1, double m2, double v1, double v2, double n1, double n2)
Computes t test statistic for 2-sample t-test.
double t(double mu, StatisticalSummary sampleStats)
Computes a t statistic to use in comparing the mean of the dataset described by sampleStats to mu.
double t(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
Computes a 2-sample t statistic , comparing the means of the datasets described by two StatisticalSummary instances, without the assumption of equal subpopulation variances.
double tTest(double[] sample1, double[] sample2)
Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays.
boolean tTest(double[] sample1, double[] sample2, double 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.
double tTest(double mu, double[] sample)
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.
boolean tTest(double mu, double[] sample, double alpha)
Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which sample is drawn equals mu.
protected double tTest(double m, double mu, double v, double n)
Computes p-value for 2-sided, 1-sample t-test.
protected double tTest(double m1, double m2, double v1, double v2, double n1, double n2)
Computes p-value for 2-sided, 2-sample t-test.
double tTest(double mu, StatisticalSummary sampleStats)
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.
boolean tTest(double mu, StatisticalSummary sampleStats, double alpha)
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.
double tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
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.
boolean tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2, double alpha)
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.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### TTest

public TTest()
• ### Method Detail

• #### pairedT

public double pairedT(double[] sample1,
double[] sample2)
throws NullArgumentException,
NoDataException,
DimensionMismatchException,
NumberIsTooSmallException
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 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.

Parameters:
sample1 - array of sample data values
sample2 - array of sample data values
Returns:
t statistic
Throws:
NullArgumentException - if the arrays are null
NoDataException - if the arrays are empty
DimensionMismatchException - if the length of the arrays is not equal
NumberIsTooSmallException - if the length of the arrays is < 2
• #### pairedTTest

public double pairedTTest(double[] sample1,
double[] sample2)
throws NullArgumentException,
NoDataException,
DimensionMismatchException,
NumberIsTooSmallException,
MaxCountExceededException
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 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.

Parameters:
sample1 - array of sample data values
sample2 - array of sample data values
Returns:
p-value for t-test
Throws:
NullArgumentException - if the arrays are null
NoDataException - if the arrays are empty
DimensionMismatchException - if the length of the arrays is not equal
NumberIsTooSmallException - if the length of the arrays is < 2
MaxCountExceededException - if an error occurs computing the p-value
• #### pairedTTest

public boolean pairedTTest(double[] sample1,
double[] sample2,
double alpha)
throws NullArgumentException,
NoDataException,
DimensionMismatchException,
NumberIsTooSmallException,
OutOfRangeException,
MaxCountExceededException
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

Parameters:
sample1 - array of sample data values
sample2 - array of sample data values
alpha - significance level of the test
Returns:
true if the null hypothesis can be rejected with confidence 1 - alpha
Throws:
NullArgumentException - if the arrays are null
NoDataException - if the arrays are empty
DimensionMismatchException - if the length of the arrays is not equal
NumberIsTooSmallException - if the length of the arrays is < 2
OutOfRangeException - if alpha is not in the range (0, 0.5]
MaxCountExceededException - if an error occurs computing the p-value
• #### t

public double t(double mu,
double[] observed)
throws NullArgumentException,
NumberIsTooSmallException
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.

Parameters:
mu - comparison constant
observed - array of values
Returns:
t statistic
Throws:
NullArgumentException - if observed is null
NumberIsTooSmallException - if the length of observed is < 2
• #### t

public double t(double mu,
StatisticalSummary sampleStats)
throws NullArgumentException,
NumberIsTooSmallException
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.

Parameters:
mu - comparison constant
sampleStats - DescriptiveStatistics holding sample summary statitstics
Returns:
t statistic
Throws:
NullArgumentException - if sampleStats is null
NumberIsTooSmallException - if the number of samples is < 2
• #### homoscedasticT

public double homoscedasticT(double[] sample1,
double[] sample2)
throws NullArgumentException,
NumberIsTooSmallException
Computes a 2-sample t statistic, under the hypothesis of equal subpopulation variances. To compute a t-statistic without the equal variances hypothesis, use 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.

Parameters:
sample1 - array of sample data values
sample2 - array of sample data values
Returns:
t statistic
Throws:
NullArgumentException - if the arrays are null
NumberIsTooSmallException - if the length of the arrays is < 2
• #### t

public double t(double[] sample1,
double[] sample2)
throws NullArgumentException,
NumberIsTooSmallException
Computes a 2-sample t statistic, without the hypothesis of equal subpopulation variances. To compute a t-statistic assuming equal variances, use 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.

Parameters:
sample1 - array of sample data values
sample2 - array of sample data values
Returns:
t statistic
Throws:
NullArgumentException - if the arrays are null
NumberIsTooSmallException - if the length of the arrays is < 2
• #### t

public double t(StatisticalSummary sampleStats1,
StatisticalSummary sampleStats2)
throws NullArgumentException,
NumberIsTooSmallException
Computes a 2-sample t statistic , comparing the means of the datasets described by two StatisticalSummary instances, without the assumption of equal subpopulation variances. Use 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.

Parameters:
sampleStats1 - StatisticalSummary describing data from the first sample
sampleStats2 - StatisticalSummary describing data from the second sample
Returns:
t statistic
Throws:
NullArgumentException - if the sample statistics are null
NumberIsTooSmallException - if the number of samples is < 2
• #### homoscedasticT

public double homoscedasticT(StatisticalSummary sampleStats1,
StatisticalSummary sampleStats2)
throws NullArgumentException,
NumberIsTooSmallException
Computes a 2-sample t statistic, comparing the means of the datasets described by two StatisticalSummary instances, under the assumption of equal subpopulation variances. To compute a t-statistic without the equal variances assumption, use 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.

Parameters:
sampleStats1 - StatisticalSummary describing data from the first sample
sampleStats2 - StatisticalSummary describing data from the second sample
Returns:
t statistic
Throws:
NullArgumentException - if the sample statistics are null
NumberIsTooSmallException - if the number of samples is < 2
• #### tTest

public double tTest(double mu,
double[] sample)
throws NullArgumentException,
NumberIsTooSmallException,
MaxCountExceededException
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.

Parameters:
mu - constant value to compare sample mean against
sample - array of sample data values
Returns:
p-value
Throws:
NullArgumentException - if the sample array is null
NumberIsTooSmallException - if the length of the array is < 2
MaxCountExceededException - if an error occurs computing the p-value
• #### tTest

public boolean tTest(double mu,
double[] sample,
double alpha)
throws NullArgumentException,
NumberIsTooSmallException,
OutOfRangeException,
MaxCountExceededException
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:

1. To test the (2-sided) hypothesis sample mean = mu  at the 95% level, use
tTest(mu, sample, 0.05)
2. 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.

Parameters:
mu - constant value to compare sample mean against
sample - array of sample data values
alpha - significance level of the test
Returns:
p-value
Throws:
NullArgumentException - if the sample array is null
NumberIsTooSmallException - if the length of the array is < 2
OutOfRangeException - if alpha is not in the range (0, 0.5]
MaxCountExceededException - if an error computing the p-value
• #### tTest

public double tTest(double mu,
StatisticalSummary sampleStats)
throws NullArgumentException,
NumberIsTooSmallException,
MaxCountExceededException
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.

Parameters:
mu - constant value to compare sample mean against
sampleStats - StatisticalSummary describing sample data
Returns:
p-value
Throws:
NullArgumentException - if sampleStats is null
NumberIsTooSmallException - if the number of samples is < 2
MaxCountExceededException - if an error occurs computing the p-value
• #### tTest

public boolean tTest(double mu,
StatisticalSummary sampleStats,
double alpha)
throws NullArgumentException,
NumberIsTooSmallException,
OutOfRangeException,
MaxCountExceededException
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:

1. To test the (2-sided) hypothesis sample mean = mu  at the 95% level, use
tTest(mu, sampleStats, 0.05)
2. 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.

Parameters:
mu - constant value to compare sample mean against
sampleStats - StatisticalSummary describing sample data values
alpha - significance level of the test
Returns:
p-value
Throws:
NullArgumentException - if sampleStats is null
NumberIsTooSmallException - if the number of samples is < 2
OutOfRangeException - if alpha is not in the range (0, 0.5]
MaxCountExceededException - if an error occurs computing the p-value
• #### tTest

public double tTest(double[] sample1,
double[] sample2)
throws NullArgumentException,
NumberIsTooSmallException,
MaxCountExceededException
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 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 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.

Parameters:
sample1 - array of sample data values
sample2 - array of sample data values
Returns:
p-value for t-test
Throws:
NullArgumentException - if the arrays are null
NumberIsTooSmallException - if the length of the arrays is < 2
MaxCountExceededException - if an error occurs computing the p-value
• #### homoscedasticTTest

public double homoscedasticTTest(double[] sample1,
double[] sample2)
throws NullArgumentException,
NumberIsTooSmallException,
MaxCountExceededException
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 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 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.

Parameters:
sample1 - array of sample data values
sample2 - array of sample data values
Returns:
p-value for t-test
Throws:
NullArgumentException - if the arrays are null
NumberIsTooSmallException - if the length of the arrays is < 2
MaxCountExceededException - if an error occurs computing the p-value
• #### tTest

public boolean tTest(double[] sample1,
double[] sample2,
double alpha)
throws NullArgumentException,
NumberIsTooSmallException,
OutOfRangeException,
MaxCountExceededException
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 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 t(double[], double[]) for the formula used to compute the t-statistic. Degrees of freedom are approximated using the Welch-Satterthwaite approximation.

Examples:

1. To test the (2-sided) hypothesis mean 1 = mean 2  at the 95% level, use
tTest(sample1, sample2, 0.05).
2. 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

Parameters:
sample1 - array of sample data values
sample2 - array of sample data values
alpha - significance level of the test
Returns:
true if the null hypothesis can be rejected with confidence 1 - alpha
Throws:
NullArgumentException - if the arrays are null
NumberIsTooSmallException - if the length of the arrays is < 2
OutOfRangeException - if alpha is not in the range (0, 0.5]
MaxCountExceededException - if an error occurs computing the p-value
• #### homoscedasticTTest

public boolean homoscedasticTTest(double[] sample1,
double[] sample2,
double alpha)
throws NullArgumentException,
NumberIsTooSmallException,
OutOfRangeException,
MaxCountExceededException
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 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 tTest(double[], double[], double).

A pooled variance estimate is used to compute the t-statistic. See t(double[], double[]) for the formula. The sum of the sample sizes minus 2 is used as the degrees of freedom.

Examples:

1. To test the (2-sided) hypothesis mean 1 = mean 2  at the 95% level, use
tTest(sample1, sample2, 0.05).
2. 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

Parameters:
sample1 - array of sample data values
sample2 - array of sample data values
alpha - significance level of the test
Returns:
true if the null hypothesis can be rejected with confidence 1 - alpha
Throws:
NullArgumentException - if the arrays are null
NumberIsTooSmallException - if the length of the arrays is < 2
OutOfRangeException - if alpha is not in the range (0, 0.5]
MaxCountExceededException - if an error occurs computing the p-value
• #### tTest

public double tTest(StatisticalSummary sampleStats1,
StatisticalSummary sampleStats2)
throws NullArgumentException,
NumberIsTooSmallException,
MaxCountExceededException
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 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.

Parameters:
sampleStats1 - StatisticalSummary describing data from the first sample
sampleStats2 - StatisticalSummary describing data from the second sample
Returns:
p-value for t-test
Throws:
NullArgumentException - if the sample statistics are null
NumberIsTooSmallException - if the number of samples is < 2
MaxCountExceededException - if an error occurs computing the p-value
• #### homoscedasticTTest

public double homoscedasticTTest(StatisticalSummary sampleStats1,
StatisticalSummary sampleStats2)
throws NullArgumentException,
NumberIsTooSmallException,
MaxCountExceededException
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 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 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.

Parameters:
sampleStats1 - StatisticalSummary describing data from the first sample
sampleStats2 - StatisticalSummary describing data from the second sample
Returns:
p-value for t-test
Throws:
NullArgumentException - if the sample statistics are null
NumberIsTooSmallException - if the number of samples is < 2
MaxCountExceededException - if an error occurs computing the p-value
• #### tTest

public boolean tTest(StatisticalSummary sampleStats1,
StatisticalSummary sampleStats2,
double alpha)
throws NullArgumentException,
NumberIsTooSmallException,
OutOfRangeException,
MaxCountExceededException
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 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 t(double[], double[]) for the formula used to compute the t-statistic. Degrees of freedom are approximated using the Welch-Satterthwaite approximation.

Examples:

1. To test the (2-sided) hypothesis mean 1 = mean 2  at the 95%, use
tTest(sampleStats1, sampleStats2, 0.05)
2. 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

Parameters:
sampleStats1 - StatisticalSummary describing sample data values
sampleStats2 - StatisticalSummary describing sample data values
alpha - significance level of the test
Returns:
true if the null hypothesis can be rejected with confidence 1 - alpha
Throws:
NullArgumentException - if the sample statistics are null
NumberIsTooSmallException - if the number of samples is < 2
OutOfRangeException - if alpha is not in the range (0, 0.5]
MaxCountExceededException - if an error occurs computing the p-value
• #### df

protected double df(double v1,
double v2,
double n1,
double n2)
Computes approximate degrees of freedom for 2-sample t-test.
Parameters:
v1 - first sample variance
v2 - second sample variance
n1 - first sample n
n2 - second sample n
Returns:
approximate degrees of freedom
• #### t

protected double t(double m,
double mu,
double v,
double n)
Computes t test statistic for 1-sample t-test.
Parameters:
m - sample mean
mu - constant to test against
v - sample variance
n - sample n
Returns:
t test statistic
• #### t

protected double t(double m1,
double m2,
double v1,
double v2,
double n1,
double n2)
Computes t test statistic for 2-sample t-test.

Does not assume that subpopulation variances are equal.

Parameters:
m1 - first sample mean
m2 - second sample mean
v1 - first sample variance
v2 - second sample variance
n1 - first sample n
n2 - second sample n
Returns:
t test statistic
• #### homoscedasticT

protected double homoscedasticT(double m1,
double m2,
double v1,
double v2,
double n1,
double n2)
Computes t test statistic for 2-sample t-test under the hypothesis of equal subpopulation variances.
Parameters:
m1 - first sample mean
m2 - second sample mean
v1 - first sample variance
v2 - second sample variance
n1 - first sample n
n2 - second sample n
Returns:
t test statistic
• #### tTest

protected double tTest(double m,
double mu,
double v,
double n)
throws MaxCountExceededException,
MathIllegalArgumentException
Computes p-value for 2-sided, 1-sample t-test.
Parameters:
m - sample mean
mu - constant to test against
v - sample variance
n - sample n
Returns:
p-value
Throws:
MaxCountExceededException - if an error occurs computing the p-value
MathIllegalArgumentException - if n is not greater than 1
• #### tTest

protected double tTest(double m1,
double m2,
double v1,
double v2,
double n1,
double n2)
throws MaxCountExceededException,
NotStrictlyPositiveException
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.

Parameters:
m1 - first sample mean
m2 - second sample mean
v1 - first sample variance
v2 - second sample variance
n1 - first sample n
n2 - second sample n
Returns:
p-value
Throws:
MaxCountExceededException - if an error occurs computing the p-value
NotStrictlyPositiveException - if the estimated degrees of freedom is not strictly positive
• #### homoscedasticTTest

protected double homoscedasticTTest(double m1,
double m2,
double v1,
double v2,
double n1,
double n2)
throws MaxCountExceededException,
NotStrictlyPositiveException
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.

Parameters:
m1 - first sample mean
m2 - second sample mean
v1 - first sample variance
v2 - second sample variance
n1 - first sample n
n2 - second sample n
Returns:
p-value
Throws:
MaxCountExceededException - if an error occurs computing the p-value
NotStrictlyPositiveException - if the estimated degrees of freedom is not strictly positive