Apache Commons Math 3.6.1 · org.apache.commons.math3.stat.inference.TTest

Access	Constructor and Description
public	TTest()

Method Summary

Modifier and Type	Method and Description
private void	checkSampleData(final double[] Sample data. data) Check sample data.
private void	checkSampleData(final StatisticalSummary Statistical summary. stat) Check sample data.
private void	checkSignificanceLevel(final double significance level alpha) Check significance level.
protected double	Returns: approximate degrees of freedom df(double first sample variance v1, double second sample variance v2, double first sample n n1, double second sample n n2) Computes approximate degrees of freedom for 2-sample t-test.
public double	Returns: t statistic homoscedasticT(final double[] array of sample data values sample1, final double[] array of sample data values sample2) Computes a 2-sample t statistic, under the hypothesis of equal subpopulation variances.
public double	Returns: t statistic homoscedasticT(final StatisticalSummary StatisticalSummary describing data from the first sample sampleStats1, final StatisticalSummary StatisticalSummary describing data from the second sample 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.
protected double	Returns: t test statistic homoscedasticT(final double first sample mean m1, final double second sample mean m2, final double first sample variance v1, final double second sample variance v2, final double first sample n n1, final double second sample n n2) Computes t test statistic for 2-sample t-test under the hypothesis of equal subpopulation variances.
public double	Returns: p-value for t-test homoscedasticTTest(final double[] array of sample data values sample1, final double[] array of sample data values 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.
public boolean	Returns: true if the null hypothesis can be rejected with confidence 1 - alpha homoscedasticTTest(final double[] array of sample data values sample1, final double[] array of sample data values sample2, final double significance level of the test 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.
public double	Returns: p-value for t-test homoscedasticTTest(final StatisticalSummary StatisticalSummary describing data from the first sample sampleStats1, final StatisticalSummary StatisticalSummary describing data from the second sample 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.
protected double	Returns: p-value homoscedasticTTest(double first sample mean m1, double second sample mean m2, double first sample variance v1, double second sample variance v2, double first sample n n1, double second sample n n2) Computes p-value for 2-sided, 2-sample t-test, under the assumption of equal subpopulation variances.
public double	Returns: t statistic pairedT(final double[] array of sample data values sample1, final double[] array of sample data values sample2) Computes a paired, 2-sample t-statistic based on the data in the input arrays.
public double	Returns: p-value for t-test pairedTTest(final double[] array of sample data values sample1, final double[] array of sample data values 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.
public boolean	Returns: true if the null hypothesis can be rejected with confidence 1 - alpha pairedTTest(final double[] array of sample data values sample1, final double[] array of sample data values sample2, final double significance level of the test 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`.
public double	Returns: t statistic t(final double comparison constant mu, final double[] array of values observed) 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.
public double	Returns: t statistic t(final double comparison constant mu, final StatisticalSummary DescriptiveStatistics holding sample summary statitstics sampleStats) 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`.
public double	Returns: t statistic t(final double[] array of sample data values sample1, final double[] array of sample data values sample2) Computes a 2-sample t statistic, without the hypothesis of equal subpopulation variances.
public double	Returns: t statistic t(final StatisticalSummary StatisticalSummary describing data from the first sample sampleStats1, final StatisticalSummary StatisticalSummary describing data from the second sample 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.
protected double	Returns: t test statistic t(final double sample mean m, final double constant to test against mu, final double sample variance v, final double sample n n) Computes t test statistic for 1-sample t-test.
protected double	Returns: t test statistic t(final double first sample mean m1, final double second sample mean m2, final double first sample variance v1, final double second sample variance v2, final double first sample n n1, final double second sample n n2) Computes t test statistic for 2-sample t-test.
public double	Returns: p-value tTest(final double constant value to compare sample mean against mu, final double[] array of sample data values 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`.
public boolean	Returns: p-value tTest(final double constant value to compare sample mean against mu, final double[] array of sample data values sample, final double significance level of the test alpha) Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which `sample` is drawn equals `mu`.
public double	Returns: p-value tTest(final double constant value to compare sample mean against mu, final StatisticalSummary StatisticalSummary describing sample data 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`.
public boolean	Returns: p-value tTest(final double constant value to compare sample mean against mu, final StatisticalSummary StatisticalSummary describing sample data values sampleStats, final double significance level of the test 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`.
public double	Returns: p-value for t-test tTest(final double[] array of sample data values sample1, final double[] array of sample data values 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.
public boolean	Returns: true if the null hypothesis can be rejected with confidence 1 - alpha tTest(final double[] array of sample data values sample1, final double[] array of sample data values sample2, final double significance level of the test 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`.
public double	Returns: p-value for t-test tTest(final StatisticalSummary StatisticalSummary describing data from the first sample sampleStats1, final StatisticalSummary StatisticalSummary describing data from the second sample 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.
public boolean	Returns: true if the null hypothesis can be rejected with confidence 1 - alpha tTest(final StatisticalSummary StatisticalSummary describing sample data values sampleStats1, final StatisticalSummary StatisticalSummary describing sample data values sampleStats2, final double significance level of the test 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`.
protected double	Returns: p-value tTest(final double sample mean m, final double constant to test against mu, final double sample variance v, final double sample n n) Computes p-value for 2-sided, 1-sample t-test.
protected double	Returns: p-value tTest(final double first sample mean m1, final double second sample mean m2, final double first sample variance v1, final double second sample variance v2, final double first sample n n1, final double second sample n n2) Computes p-value for 2-sided, 2-sample t-test.

Inherited from java.lang.Object:: clone equals finalize getClass hashCode notify notifyAll toString wait wait wait

TTest	back to summary
public TTest()

Method Detail

checkSampleData back to summary

checkSampleData	back to summary
private void checkSampleData(final double[] data) throws NullArgumentException, NumberIsTooSmallException Check sample data. Parameters data:double[] Sample data. Exceptions NullArgumentException: if `data` is `null`. NumberIsTooSmallException: if there is not enough sample data.

private void checkSampleData(final double[] data) throws NullArgumentException, NumberIsTooSmallException

Check sample data.

Parameters

data:double[]: Sample data.

Exceptions

NullArgumentException:: if data is null.
NumberIsTooSmallException:: if there is not enough sample data.

checkSampleData back to summary

checkSampleData	back to summary
private void checkSampleData(final StatisticalSummary stat) throws NullArgumentException, NumberIsTooSmallException Check sample data. Parameters stat:StatisticalSummary Statistical summary. Exceptions NullArgumentException: if `data` is `null`. NumberIsTooSmallException: if there is not enough sample data.

private void checkSampleData(final StatisticalSummary stat) throws NullArgumentException, NumberIsTooSmallException

Check sample data.

Parameters

stat:StatisticalSummary: Statistical summary.

Exceptions

NullArgumentException:: if data is null.
NumberIsTooSmallException:: if there is not enough sample data.

checkSignificanceLevel	back to summary
private void checkSignificanceLevel(final double alpha) throws OutOfRangeException Check significance level. Parameters alpha:double significance level Exceptions OutOfRangeException: if the significance level is out of bounds.

checkSignificanceLevel

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private void checkSignificanceLevel(final double alpha) throws OutOfRangeException

Check significance level.

Parameters

alpha:double: significance level

Exceptions

OutOfRangeException:: if the significance level is out of bounds.

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

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protected double df(double v1, double v2, double n1, double n2)

Computes approximate degrees of freedom for 2-sample t-test.

Parameters

v1:double: first sample variance
v2:double: second sample variance
n1:double: first sample n
n2:double: second sample n

Returns:double

approximate degrees of freedom

homoscedasticT back to summary

homoscedasticT	back to summary
public double homoscedasticT(final double[] sample1, final 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:double[] array of sample data values sample2:double[] array of sample data values Returns:double t statistic Exceptions NullArgumentException: if the arrays are `null` NumberIsTooSmallException: if the length of the arrays is < 2

public double homoscedasticT(final double[] sample1, final 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:double[]: array of sample data values
sample2:double[]: array of sample data values

Returns:double

t statistic

Exceptions

NullArgumentException:: if the arrays are null
NumberIsTooSmallException:: if the length of the arrays is < 2

homoscedasticT back to summary

homoscedasticT	back to summary
public double homoscedasticT(final StatisticalSummary sampleStats1, final 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 StatisticalSummary describing data from the first sample sampleStats2:StatisticalSummary StatisticalSummary describing data from the second sample Returns:double t statistic Exceptions NullArgumentException: if the sample statistics are `null` NumberIsTooSmallException: if the number of samples is < 2

public double homoscedasticT(final StatisticalSummary sampleStats1, final 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))

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: StatisticalSummary describing data from the first sample
sampleStats2:StatisticalSummary: StatisticalSummary describing data from the second sample

Returns:double

t statistic

Exceptions

NullArgumentException:: if the sample statistics are null
NumberIsTooSmallException:: if the number of samples is < 2

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

homoscedasticT

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protected double homoscedasticT(final double m1, final double m2, final double v1, final double v2, final double n1, final double n2)

Computes t test statistic for 2-sample t-test under the hypothesis of equal subpopulation variances.

Parameters

m1:double: first sample mean
m2:double: second sample mean
v1:double: first sample variance
v2:double: second sample variance
n1:double: first sample n
n2:double: second sample n

Returns:double

t test statistic

homoscedasticTTest back to summary

homoscedasticTTest	back to summary
public double homoscedasticTTest(final double[] sample1, final 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:double[] array of sample data values sample2:double[] array of sample data values Returns:double p-value for t-test Exceptions NullArgumentException: if the arrays are `null` NumberIsTooSmallException: if the length of the arrays is < 2 MaxCountExceededException: if an error occurs computing the p-value

public double homoscedasticTTest(final double[] sample1, final 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:double[]: array of sample data values
sample2:double[]: array of sample data values

Returns:double

p-value for t-test

Exceptions

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 back to summary

homoscedasticTTest	back to summary
public boolean homoscedasticTTest(final double[] sample1, final double[] sample2, final 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: 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` Parameters sample1:double[] array of sample data values sample2:double[] array of sample data values alpha:double significance level of the test Returns:boolean true if the null hypothesis can be rejected with confidence 1 - alpha Exceptions 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

public boolean homoscedasticTTest(final double[] sample1, final double[] sample2, final 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:

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

Parameters

sample1:double[]: array of sample data values
sample2:double[]: array of sample data values
alpha:double: significance level of the test

Returns:boolean

true if the null hypothesis can be rejected with confidence 1 - alpha

Exceptions

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 back to summary

homoscedasticTTest	back to summary
public double homoscedasticTTest(final StatisticalSummary sampleStats1, final 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 StatisticalSummary describing data from the first sample sampleStats2:StatisticalSummary StatisticalSummary describing data from the second sample Returns:double p-value for t-test Exceptions NullArgumentException: if the sample statistics are `null` NumberIsTooSmallException: if the number of samples is < 2 MaxCountExceededException: if an error occurs computing the p-value

public double homoscedasticTTest(final StatisticalSummary sampleStats1, final 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).

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: StatisticalSummary describing data from the first sample
sampleStats2:StatisticalSummary: StatisticalSummary describing data from the second sample

Returns:double

p-value for t-test

Exceptions

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

homoscedasticTTest

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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:double: first sample mean
m2:double: second sample mean
v1:double: first sample variance
v2:double: second sample variance
n1:double: first sample n
n2:double: second sample n

Returns:double

p-value

Exceptions

MaxCountExceededException:: if an error occurs computing the p-value
NotStrictlyPositiveException:: if the estimated degrees of freedom is not strictly positive

pairedT back to summary

pairedT	back to summary
public double pairedT(final double[] sample1, final 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:double[] array of sample data values sample2:double[] array of sample data values Returns:double t statistic Exceptions 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

public double pairedT(final double[] sample1, final 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:double[]: array of sample data values
sample2:double[]: array of sample data values

Returns:double

t statistic

Exceptions

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 back to summary

pairedTTest	back to summary
public double pairedTTest(final double[] sample1, final 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:double[] array of sample data values sample2:double[] array of sample data values Returns:double p-value for t-test Exceptions 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

public double pairedTTest(final double[] sample1, final 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:double[]: array of sample data values
sample2:double[]: array of sample data values

Returns:double

p-value for t-test

Exceptions

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 back to summary

pairedTTest	back to summary
public boolean pairedTTest(final double[] sample1, final double[] sample2, final 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:double[] array of sample data values sample2:double[] array of sample data values alpha:double significance level of the test Returns:boolean true if the null hypothesis can be rejected with confidence 1 - alpha Exceptions 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

public boolean pairedTTest(final double[] sample1, final double[] sample2, final 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:double[]: array of sample data values
sample2:double[]: array of sample data values
alpha:double: significance level of the test

Returns:boolean

true if the null hypothesis can be rejected with confidence 1 - alpha

Exceptions

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	back to summary
public double t(final double mu, final 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:double comparison constant observed:double[] array of values Returns:double t statistic Exceptions NullArgumentException: if `observed` is `null` NumberIsTooSmallException: if the length of `observed` is < 2

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public double t(final double mu, final 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:double: comparison constant
observed:double[]: array of values

Returns:double

t statistic

Exceptions

NullArgumentException:: if observed is null
NumberIsTooSmallException:: if the length of observed is < 2

t back to summary

t	back to summary
public double t(final double mu, final 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:double comparison constant sampleStats:StatisticalSummary DescriptiveStatistics holding sample summary statitstics Returns:double t statistic Exceptions NullArgumentException: if `sampleStats` is `null` NumberIsTooSmallException: if the number of samples is < 2

public double t(final double mu, final 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

t back to summary

t	back to summary
public double t(final double[] sample1, final 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:double[] array of sample data values sample2:double[] array of sample data values Returns:double t statistic Exceptions NullArgumentException: if the arrays are `null` NumberIsTooSmallException: if the length of the arrays is < 2

public double t(final double[] sample1, final 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:double[]: array of sample data values
sample2:double[]: array of sample data values

Returns:double

t statistic

Exceptions

NullArgumentException:: if the arrays are null
NumberIsTooSmallException:: if the length of the arrays is < 2

t back to summary

t	back to summary
public double t(final StatisticalSummary sampleStats1, final 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 StatisticalSummary describing data from the first sample sampleStats2:StatisticalSummary StatisticalSummary describing data from the second sample Returns:double t statistic Exceptions NullArgumentException: if the sample statistics are `null` NumberIsTooSmallException: if the number of samples is < 2

public double t(final StatisticalSummary sampleStats1, final 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: StatisticalSummary describing data from the first sample
sampleStats2:StatisticalSummary: StatisticalSummary describing data from the second sample

Returns:double

t statistic

Exceptions

NullArgumentException:: if the sample statistics are null
NumberIsTooSmallException:: if the number of samples is < 2

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

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protected double t(final double m, final double mu, final double v, final double n)

Computes t test statistic for 1-sample t-test.

Parameters

m:double: sample mean
mu:double: constant to test against
v:double: sample variance
n:double: sample n

Returns:double

t test statistic

t	back to summary
protected double t(final double m1, final double m2, final double v1, final double v2, final double n1, final double n2) Computes t test statistic for 2-sample t-test. Does not assume that subpopulation variances are equal. Parameters m1:double first sample mean m2:double second sample mean v1:double first sample variance v2:double second sample variance n1:double first sample n n2:double second sample n Returns:double t test statistic

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protected double t(final double m1, final double m2, final double v1, final double v2, final double n1, final double n2)

Computes t test statistic for 2-sample t-test.

Does not assume that subpopulation variances are equal.

Parameters

m1:double: first sample mean
m2:double: second sample mean
v1:double: first sample variance
v2:double: second sample variance
n1:double: first sample n
n2:double: second sample n

Returns:double

t test statistic

tTest back to summary

tTest	back to summary
public double tTest(final double mu, final 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:double constant value to compare sample mean against sample:double[] array of sample data values Returns:double p-value Exceptions 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

public double tTest(final double mu, final 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:double: constant value to compare sample mean against
sample:double[]: array of sample data values

Returns:double

p-value

Exceptions

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 back to summary

tTest	back to summary
public boolean tTest(final double mu, final double[] sample, final 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: 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. Parameters mu:double constant value to compare sample mean against sample:double[] array of sample data values alpha:double significance level of the test Returns:boolean p-value Exceptions 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

public boolean tTest(final double mu, final double[] sample, final 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:

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.

Parameters

mu:double: constant value to compare sample mean against
sample:double[]: array of sample data values
alpha:double: significance level of the test

Returns:boolean

p-value

Exceptions

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 back to summary

tTest	back to summary
public double tTest(final double mu, final 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:double constant value to compare sample mean against sampleStats:StatisticalSummary StatisticalSummary describing sample data Returns:double p-value Exceptions NullArgumentException: if `sampleStats` is `null` NumberIsTooSmallException: if the number of samples is < 2 MaxCountExceededException: if an error occurs computing the p-value

public double tTest(final double mu, final 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.

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:double: constant value to compare sample mean against
sampleStats:StatisticalSummary: StatisticalSummary describing sample data

Returns:double

p-value

Exceptions

NullArgumentException:: if sampleStats is null
NumberIsTooSmallException:: if the number of samples is < 2
MaxCountExceededException:: if an error occurs computing the p-value

tTest back to summary

tTest	back to summary
public boolean tTest(final double mu, final StatisticalSummary sampleStats, final 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: 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. Parameters mu:double constant value to compare sample mean against sampleStats:StatisticalSummary StatisticalSummary describing sample data values alpha:double significance level of the test Returns:boolean p-value Exceptions 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

public boolean tTest(final double mu, final StatisticalSummary sampleStats, final 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:

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.

Parameters

mu:double: constant value to compare sample mean against
sampleStats:StatisticalSummary: StatisticalSummary describing sample data values
alpha:double: significance level of the test

Returns:boolean

p-value

Exceptions

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 back to summary

tTest	back to summary
public double tTest(final double[] sample1, final 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:double[] array of sample data values sample2:double[] array of sample data values Returns:double p-value for t-test Exceptions NullArgumentException: if the arrays are `null` NumberIsTooSmallException: if the length of the arrays is < 2 MaxCountExceededException: if an error occurs computing the p-value

public double tTest(final double[] sample1, final 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 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:double[]: array of sample data values
sample2:double[]: array of sample data values

Returns:double

p-value for t-test

Exceptions

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 back to summary

tTest	back to summary
public boolean tTest(final double[] sample1, final double[] sample2, final 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: 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` Parameters sample1:double[] array of sample data values sample2:double[] array of sample data values alpha:double significance level of the test Returns:boolean true if the null hypothesis can be rejected with confidence 1 - alpha Exceptions 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

public boolean tTest(final double[] sample1, final double[] sample2, final 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:

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

Parameters

sample1:double[]: array of sample data values
sample2:double[]: array of sample data values
alpha:double: significance level of the test

Returns:boolean

true if the null hypothesis can be rejected with confidence 1 - alpha

Exceptions

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 back to summary

tTest	back to summary
public double tTest(final StatisticalSummary sampleStats1, final 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 StatisticalSummary describing data from the first sample sampleStats2:StatisticalSummary StatisticalSummary describing data from the second sample Returns:double p-value for t-test Exceptions NullArgumentException: if the sample statistics are `null` NumberIsTooSmallException: if the number of samples is < 2 MaxCountExceededException: if an error occurs computing the p-value

public double tTest(final StatisticalSummary sampleStats1, final 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 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: StatisticalSummary describing data from the first sample
sampleStats2:StatisticalSummary: StatisticalSummary describing data from the second sample

Returns:double

p-value for t-test

Exceptions

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 back to summary

tTest	back to summary
public boolean tTest(final StatisticalSummary sampleStats1, final StatisticalSummary sampleStats2, final 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: 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` Parameters sampleStats1:StatisticalSummary StatisticalSummary describing sample data values sampleStats2:StatisticalSummary StatisticalSummary describing sample data values alpha:double significance level of the test Returns:boolean true if the null hypothesis can be rejected with confidence 1 - alpha Exceptions 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

public boolean tTest(final StatisticalSummary sampleStats1, final StatisticalSummary sampleStats2, final 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:

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

Parameters

sampleStats1:StatisticalSummary: StatisticalSummary describing sample data values
sampleStats2:StatisticalSummary: StatisticalSummary describing sample data values
alpha:double: significance level of the test

Returns:boolean

true if the null hypothesis can be rejected with confidence 1 - alpha

Exceptions

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

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

tTest

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protected double tTest(final double m, final double mu, final double v, final double n) throws MaxCountExceededException, MathIllegalArgumentException

Computes p-value for 2-sided, 1-sample t-test.

Parameters

m:double: sample mean
mu:double: constant to test against
v:double: sample variance
n:double: sample n

Returns:double

p-value

Exceptions

MaxCountExceededException:: if an error occurs computing the p-value
MathIllegalArgumentException:: if n is not greater than 1

tTest	back to summary
protected double tTest(final double m1, final double m2, final double v1, final double v2, final double n1, final 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:double first sample mean m2:double second sample mean v1:double first sample variance v2:double second sample variance n1:double first sample n n2:double second sample n Returns:double p-value Exceptions MaxCountExceededException: if an error occurs computing the p-value NotStrictlyPositiveException: if the estimated degrees of freedom is not strictly positive

tTest

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protected double tTest(final double m1, final double m2, final double v1, final double v2, final double n1, final 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:double: first sample mean
m2:double: second sample mean
v1:double: first sample variance
v2:double: second sample variance
n1:double: first sample n
n2:double: second sample n

Returns:double

p-value

Exceptions

MaxCountExceededException:: if an error occurs computing the p-value
NotStrictlyPositiveException:: if the estimated degrees of freedom is not strictly positive

public Class TTest

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