Class TTest
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 Hipparchus TDistribution
implementation to estimate exact p-values.
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Constructor Summary
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Method Summary
Modifier and TypeMethodDescriptionprotected 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 twoStatisticalSummary
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 thatsample1
andsample2
are drawn from populations with the same mean, with significance levelalpha
, 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 betweensample1
andsample2
is 0 in favor of the two-sided alternative that the mean paired difference is not equal to 0, with significance levelalpha
.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) double
t
(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) Computes a 2-sample t statistic, comparing the means of the datasets described by twoStatisticalSummary
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 thatsample1
andsample2
are drawn from populations with the same mean, with significance levelalpha
.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 constantmu
.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 whichsample
is drawn equalsmu
.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 bysampleStats
with the constantmu
.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 bystats
is drawn equalsmu
.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 thatsampleStats1
andsampleStats2
describe datasets drawn from populations with the same mean, with significance levelalpha
.
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Constructor Details
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TTest
public TTest()Empty constructor.This constructor is not strictly necessary, but it prevents spurious javadoc warnings with JDK 18 and later.
- Since:
- 3.0
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Method Details
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pairedT
public double pairedT(double[] sample1, double[] sample2) throws MathIllegalArgumentException, NullArgumentException 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-statistict(double, double[])
, withmu = 0
and the sample array consisting of the (signed) differences between corresponding entries insample1
andsample2.
* Preconditions:
- The input arrays must have the same length and their common length must be at least 2.
- Parameters:
sample1
- array of sample data valuessample2
- array of sample data values- Returns:
- t statistic
- Throws:
NullArgumentException
- if the arrays arenull
MathIllegalArgumentException
- if the arrays are emptyMathIllegalArgumentException
- if the length of the arrays is not equalMathIllegalArgumentException
- if the length of the arrays is < 2
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pairedTTest
public double pairedTTest(double[] sample1, double[] sample2) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException 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[])
withmu = 0
and the sample array consisting of the signed differences between corresponding elements ofsample1
andsample2.
Usage Note:
The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed herePreconditions:
- The input array lengths must be the same and their common length must be at least 2.
- Parameters:
sample1
- array of sample data valuessample2
- array of sample data values- Returns:
- p-value for t-test
- Throws:
NullArgumentException
- if the arrays arenull
MathIllegalArgumentException
- if the arrays are emptyMathIllegalArgumentException
- if the length of the arrays is not equalMathIllegalArgumentException
- if the length of the arrays is < 2MathIllegalStateException
- if an error occurs computing the p-value
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pairedTTest
public boolean pairedTTest(double[] sample1, double[] sample2, double alpha) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException Performs a paired t-test evaluating the null hypothesis that the mean of the paired differences betweensample1
andsample2
is 0 in favor of the two-sided alternative that the mean paired difference is not equal to 0, with significance levelalpha
.Returns
true
iff the null hypothesis can be rejected with confidence1 - alpha
. To perform a 1-sided test, usealpha * 2
Usage Note:
The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed herePreconditions:
- 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 valuessample2
- array of sample data valuesalpha
- significance level of the test- Returns:
- true if the null hypothesis can be rejected with confidence 1 - alpha
- Throws:
NullArgumentException
- if the arrays arenull
MathIllegalArgumentException
- if the arrays are emptyMathIllegalArgumentException
- if the length of the arrays is not equalMathIllegalArgumentException
- if the length of the arrays is < 2MathIllegalArgumentException
- ifalpha
is not in the range (0, 0.5]MathIllegalStateException
- if an error occurs computing the p-value
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t
public double t(double mu, double[] observed) throws MathIllegalArgumentException, NullArgumentException 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 constantobserved
- array of values- Returns:
- t statistic
- Throws:
NullArgumentException
- ifobserved
isnull
MathIllegalArgumentException
- if the length ofobserved
is < 2
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t
public double t(double mu, StatisticalSummary sampleStats) throws MathIllegalArgumentException, NullArgumentException Computes a t statistic to use in comparing the mean of the dataset described bysampleStats
tomu
.This statistic can be used to perform a one sample t-test for the mean.
Preconditions:
observed.getN() ≥ 2
.
- Parameters:
mu
- comparison constantsampleStats
- DescriptiveStatistics holding sample summary statitstics- Returns:
- t statistic
- Throws:
NullArgumentException
- ifsampleStats
isnull
MathIllegalArgumentException
- if the number of samples is < 2
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homoscedasticT
public double homoscedasticT(double[] sample1, double[] sample2) throws MathIllegalArgumentException, NullArgumentException Computes a 2-sample t statistic, under the hypothesis of equal subpopulation variances. To compute a t-statistic without the equal variances hypothesis, uset(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 andvar
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 andvar2
the variance of the second sample.Preconditions:
- The observed array lengths must both be at least 2.
- Parameters:
sample1
- array of sample data valuessample2
- array of sample data values- Returns:
- t statistic
- Throws:
NullArgumentException
- if the arrays arenull
MathIllegalArgumentException
- if the length of the arrays is < 2
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t
public double t(double[] sample1, double[] sample2) throws MathIllegalArgumentException, NullArgumentException Computes a 2-sample t statistic, without the hypothesis of equal subpopulation variances. To compute a t-statistic assuming equal variances, usehomoscedasticT(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 samplen2
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 valuessample2
- array of sample data values- Returns:
- t statistic
- Throws:
NullArgumentException
- if the arrays arenull
MathIllegalArgumentException
- if the length of the arrays is < 2
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t
public double t(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws MathIllegalArgumentException, NullArgumentException Computes a 2-sample t statistic, comparing the means of the datasets described by twoStatisticalSummary
instances, without the assumption of equal subpopulation variances. UsehomoscedasticT(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 samplevar1
is the variance of the first sample;var2
is the variance of the second samplePreconditions:
- The datasets described by the two Univariates must each contain at least 2 observations.
- Parameters:
sampleStats1
- StatisticalSummary describing data from the first samplesampleStats2
- StatisticalSummary describing data from the second sample- Returns:
- t statistic
- Throws:
NullArgumentException
- if the sample statistics arenull
MathIllegalArgumentException
- if the number of samples is < 2
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homoscedasticT
public double homoscedasticT(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws MathIllegalArgumentException, NullArgumentException Computes a 2-sample t statistic, comparing the means of the datasets described by twoStatisticalSummary
instances, under the assumption of equal subpopulation variances. To compute a t-statistic without the equal variances assumption, uset(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 andvar
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 andvar2
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 samplesampleStats2
- StatisticalSummary describing data from the second sample- Returns:
- t statistic
- Throws:
NullArgumentException
- if the sample statistics arenull
MathIllegalArgumentException
- if the number of samples is < 2
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tTest
public double tTest(double mu, double[] sample) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException 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 constantmu
.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 frommu
. 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 herePreconditions:
- The observed array length must be at least 2.
- Parameters:
mu
- constant value to compare sample mean againstsample
- array of sample data values- Returns:
- p-value
- Throws:
NullArgumentException
- if the sample array isnull
MathIllegalArgumentException
- if the length of the array is < 2MathIllegalStateException
- if an error occurs computing the p-value
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tTest
public boolean tTest(double mu, double[] sample, double alpha) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from whichsample
is drawn equalsmu
.Returns
true
iff the null hypothesis can be rejected with confidence1 - alpha
. To perform a 1-sided test, usealpha * 2
* Examples:
- To test the (2-sided) hypothesis
sample mean = mu
at the 95% level, usetTest(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 thanmu
and then usetTest(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 herePreconditions:
- The observed array length must be at least 2.
- Parameters:
mu
- constant value to compare sample mean againstsample
- array of sample data valuesalpha
- significance level of the test- Returns:
- p-value
- Throws:
NullArgumentException
- if the sample array isnull
MathIllegalArgumentException
- if the length of the array is < 2MathIllegalArgumentException
- ifalpha
is not in the range (0, 0.5]MathIllegalStateException
- if an error computing the p-value
- To test the (2-sided) hypothesis
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tTest
public double tTest(double mu, StatisticalSummary sampleStats) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the dataset described bysampleStats
with the constantmu
.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 frommu
. 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 herePreconditions:
- The sample must contain at least 2 observations.
- Parameters:
mu
- constant value to compare sample mean againstsampleStats
- StatisticalSummary describing sample data- Returns:
- p-value
- Throws:
NullArgumentException
- ifsampleStats
isnull
MathIllegalArgumentException
- if the number of samples is < 2MathIllegalStateException
- if an error occurs computing the p-value
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tTest
public boolean tTest(double mu, StatisticalSummary sampleStats, double alpha) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which the dataset described bystats
is drawn equalsmu
.Returns
true
iff the null hypothesis can be rejected with confidence1 - alpha
. To perform a 1-sided test, usealpha * 2.
* Examples:
- To test the (2-sided) hypothesis
sample mean = mu
at the 95% level, usetTest(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 thanmu
and then usetTest(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 herePreconditions:
- The sample must include at least 2 observations.
- Parameters:
mu
- constant value to compare sample mean againstsampleStats
- StatisticalSummary describing sample data valuesalpha
- significance level of the test- Returns:
- p-value
- Throws:
NullArgumentException
- ifsampleStats
isnull
MathIllegalArgumentException
- if the number of samples is < 2MathIllegalArgumentException
- ifalpha
is not in the range (0, 0.5]MathIllegalStateException
- if an error occurs computing the p-value
- To test the (2-sided) hypothesis
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tTest
public double tTest(double[] sample1, double[] sample2) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException 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, usehomoscedasticTTest(double[], double[])
.Usage Note:
The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed herePreconditions:
- The observed array lengths must both be at least 2.
- Parameters:
sample1
- array of sample data valuessample2
- array of sample data values- Returns:
- p-value for t-test
- Throws:
NullArgumentException
- if the arrays arenull
MathIllegalArgumentException
- if the length of the arrays is < 2MathIllegalStateException
- if an error occurs computing the p-value
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homoscedasticTTest
public double homoscedasticTTest(double[] sample1, double[] sample2) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException 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, usetTest(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 herePreconditions:
- The observed array lengths must both be at least 2.
- Parameters:
sample1
- array of sample data valuessample2
- array of sample data values- Returns:
- p-value for t-test
- Throws:
NullArgumentException
- if the arrays arenull
MathIllegalArgumentException
- if the length of the arrays is < 2MathIllegalStateException
- if an error occurs computing the p-value
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tTest
public boolean tTest(double[] sample1, double[] sample2, double alpha) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException Performs a two-sided t-test evaluating the null hypothesis thatsample1
andsample2
are drawn from populations with the same mean, with significance levelalpha
. This test does not assume that the subpopulation variances are equal. To perform the test assuming equal variances, usehomoscedasticTTest(double[], double[], double)
.Returns
true
iff the null hypothesis that the means are equal can be rejected with confidence1 - alpha
. To perform a 1-sided test, usealpha * 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:
- To test the (2-sided) hypothesis
mean 1 = mean 2
at the 95% level, usetTest(sample1, sample2, 0.05).
- To test the (one-sided) hypothesis
mean 1 < mean 2
, at the 99% level, first verify that the measured mean ofsample 1
is less than the mean ofsample 2
and then usetTest(sample1, sample2, 0.02)
Usage Note:
The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed herePreconditions:
- The observed array lengths must both be at least 2.
-
0 < alpha < 0.5
- Parameters:
sample1
- array of sample data valuessample2
- array of sample data valuesalpha
- significance level of the test- Returns:
- true if the null hypothesis can be rejected with confidence 1 - alpha
- Throws:
NullArgumentException
- if the arrays arenull
MathIllegalArgumentException
- if the length of the arrays is < 2MathIllegalArgumentException
- ifalpha
is not in the range (0, 0.5]MathIllegalStateException
- if an error occurs computing the p-value
- To test the (2-sided) hypothesis
-
homoscedasticTTest
public boolean homoscedasticTTest(double[] sample1, double[] sample2, double alpha) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException Performs a two-sided t-test evaluating the null hypothesis thatsample1
andsample2
are drawn from populations with the same mean, with significance levelalpha
, assuming that the subpopulation variances are equal. UsetTest(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 confidence1 - alpha
. To perform a 1-sided test, usealpha * 2.
To perform the test without the assumption of equal subpopulation variances, usetTest(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:
- To test the (2-sided) hypothesis
mean 1 = mean 2
at the 95% level, usetTest(sample1, sample2, 0.05).
- To test the (one-sided) hypothesis
mean 1 < mean 2,
at the 99% level, first verify that the measured mean ofsample 1
is less than the mean ofsample 2
and then usetTest(sample1, sample2, 0.02)
Usage Note:
The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed herePreconditions:
- The observed array lengths must both be at least 2.
-
0 < alpha < 0.5
- Parameters:
sample1
- array of sample data valuessample2
- array of sample data valuesalpha
- significance level of the test- Returns:
- true if the null hypothesis can be rejected with confidence 1 - alpha
- Throws:
NullArgumentException
- if the arrays arenull
MathIllegalArgumentException
- if the length of the arrays is < 2MathIllegalArgumentException
- ifalpha
is not in the range (0, 0.5]MathIllegalStateException
- if an error occurs computing the p-value
- To test the (2-sided) hypothesis
-
tTest
public double tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException 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 herePreconditions:
- The datasets described by the two Univariates must each contain at least 2 observations.
- Parameters:
sampleStats1
- StatisticalSummary describing data from the first samplesampleStats2
- StatisticalSummary describing data from the second sample- Returns:
- p-value for t-test
- Throws:
NullArgumentException
- if the sample statistics arenull
MathIllegalArgumentException
- if the number of samples is < 2MathIllegalStateException
- if an error occurs computing the p-value
-
homoscedasticTTest
public double homoscedasticTTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException 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, usetTest(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 herePreconditions:
- The datasets described by the two Univariates must each contain at least 2 observations.
- Parameters:
sampleStats1
- StatisticalSummary describing data from the first samplesampleStats2
- StatisticalSummary describing data from the second sample- Returns:
- p-value for t-test
- Throws:
NullArgumentException
- if the sample statistics arenull
MathIllegalArgumentException
- if the number of samples is < 2MathIllegalStateException
- if an error occurs computing the p-value
-
tTest
public boolean tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2, double alpha) throws MathIllegalArgumentException, NullArgumentException, MathIllegalStateException Performs a two-sided t-test evaluating the null hypothesis thatsampleStats1
andsampleStats2
describe datasets drawn from populations with the same mean, with significance levelalpha
. This test does not assume that the subpopulation variances are equal. To perform the test under the equal variances assumption, usehomoscedasticTTest(StatisticalSummary, StatisticalSummary)
.Returns
true
iff the null hypothesis that the means are equal can be rejected with confidence1 - alpha
. To perform a 1-sided test, usealpha * 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:
- To test the (2-sided) hypothesis
mean 1 = mean 2
at the 95%, usetTest(sampleStats1, sampleStats2, 0.05)
- To test the (one-sided) hypothesis
mean 1 < mean 2
at the 99% level, first verify that the measured mean ofsample 1
is less than the mean ofsample 2
and then usetTest(sampleStats1, sampleStats2, 0.02)
Usage Note:
The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed herePreconditions:
- The datasets described by the two Univariates must each contain at least 2 observations.
-
0 < alpha < 0.5
- Parameters:
sampleStats1
- StatisticalSummary describing sample data valuessampleStats2
- StatisticalSummary describing sample data valuesalpha
- significance level of the test- Returns:
- true if the null hypothesis can be rejected with confidence 1 - alpha
- Throws:
NullArgumentException
- if the sample statistics arenull
MathIllegalArgumentException
- if the number of samples is < 2MathIllegalArgumentException
- ifalpha
is not in the range (0, 0.5]MathIllegalStateException
- if an error occurs computing the p-value
- To test the (2-sided) hypothesis
-
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 variancev2
- second sample variancen1
- first sample nn2
- 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 meanmu
- constant to test againstv
- sample variancen
- 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 meanm2
- second sample meanv1
- first sample variancev2
- second sample variancen1
- first sample nn2
- 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 meanm2
- second sample meanv1
- first sample variancev2
- second sample variancen1
- first sample nn2
- second sample n- Returns:
- t test statistic
-
tTest
protected double tTest(double m, double mu, double v, double n) throws MathIllegalArgumentException, MathIllegalStateException Computes p-value for 2-sided, 1-sample t-test.- Parameters:
m
- sample meanmu
- constant to test againstv
- sample variancen
- sample n- Returns:
- p-value
- Throws:
MathIllegalStateException
- if an error occurs computing the p-valueMathIllegalArgumentException
- if n is not greater than 1
-
tTest
protected double tTest(double m1, double m2, double v1, double v2, double n1, double n2) throws MathIllegalArgumentException, MathIllegalStateException 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 meanm2
- second sample meanv1
- first sample variancev2
- second sample variancen1
- first sample nn2
- second sample n- Returns:
- p-value
- Throws:
MathIllegalStateException
- if an error occurs computing the p-valueMathIllegalArgumentException
- 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 MathIllegalArgumentException, MathIllegalStateException 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 meanm2
- second sample meanv1
- first sample variancev2
- second sample variancen1
- first sample nn2
- second sample n- Returns:
- p-value
- Throws:
MathIllegalStateException
- if an error occurs computing the p-valueMathIllegalArgumentException
- if the estimated degrees of freedom is not strictly positive
-