Package org.apache.commons.math.stat.inference

Interface TTest

  • All Known Implementing Classes:
    TTestImpl
    public interface TTest
    An interface for Student's t-tests.

    Tests can be:

    • One-sample or two-sample
    • One-sided or two-sided
    • Paired or unpaired (for two-sample tests)
    • Homoscedastic (equal variance assumption) or heteroscedastic (for two sample tests)
    • Fixed significance level (boolean-valued) or returning p-values.

    Test statistics are available for all tests. Methods including "Test" in in their names perform tests, all other methods return t-statistics. Among the "Test" methods, double-valued methods return p-values; boolean-valued methods perform fixed significance level tests. Significance levels are always specified as numbers between 0 and 0.5 (e.g. tests at the 95% level use alpha=0.05).

    Input to tests can be either double[] arrays or StatisticalSummary instances.

    Version:
    $Revision: 811786 $ $Date: 2009-09-06 11:36:08 +0200 (dim. 06 sept. 2009) $

    • Method Summary

      All Methods Instance Methods Abstract Methods 
      Modifier and Type Method Description
      double homoscedasticT​(double[] sample1, double[] sample2)
      Computes a 2-sample t statistic, under the hypothesis of equal subpopulation variances.
      double homoscedasticT​(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
      Computes a 2-sample t statistic, comparing the means of the datasets described by two StatisticalSummary instances, under the assumption of equal subpopulation variances.
      double homoscedasticTTest​(double[] sample1, double[] sample2)
      Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays, under the assumption that the two samples are drawn from subpopulations with equal variances.
      boolean homoscedasticTTest​(double[] sample1, double[] sample2, double alpha)
      Performs a two-sided t-test evaluating the null hypothesis that sample1 and sample2 are drawn from populations with the same mean, with significance level alpha, assuming that the subpopulation variances are equal.
      double homoscedasticTTest​(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
      Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the datasets described by two StatisticalSummary instances, under the hypothesis of equal subpopulation variances.
      double pairedT​(double[] sample1, double[] sample2)
      Computes a paired, 2-sample t-statistic based on the data in the input arrays.
      double pairedTTest​(double[] sample1, double[] sample2)
      Returns the observed significance level, or p-value, associated with a paired, two-sample, two-tailed t-test based on the data in the input arrays.
      boolean pairedTTest​(double[] sample1, double[] sample2, double alpha)
      Performs a paired t-test evaluating the null hypothesis that the mean of the paired differences between sample1 and sample2 is 0 in favor of the two-sided alternative that the mean paired difference is not equal to 0, with significance level alpha.
      double t​(double[] sample1, double[] sample2)
      Computes a 2-sample t statistic, without the hypothesis of equal subpopulation variances.
      double t​(double mu, double[] observed)
      Computes a t statistic given observed values and a comparison constant.
      double t​(double mu, StatisticalSummary sampleStats)
      Computes a t statistic to use in comparing the mean of the dataset described by sampleStats to mu.
      double t​(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
      Computes a 2-sample t statistic , comparing the means of the datasets described by two StatisticalSummary instances, without the assumption of equal subpopulation variances.
      double tTest​(double[] sample1, double[] sample2)
      Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays.
      boolean tTest​(double[] sample1, double[] sample2, double alpha)
      Performs a two-sided t-test evaluating the null hypothesis that sample1 and sample2 are drawn from populations with the same mean, with significance level alpha.
      double tTest​(double mu, double[] sample)
      Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the input array with the constant mu.
      boolean tTest​(double mu, double[] sample, double alpha)
      Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which sample is drawn equals mu.
      double tTest​(double mu, StatisticalSummary sampleStats)
      Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the dataset described by sampleStats with the constant mu.
      boolean tTest​(double mu, StatisticalSummary sampleStats, double alpha)
      Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which the dataset described by stats is drawn equals mu.
      double tTest​(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
      Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the datasets described by two StatisticalSummary instances.
      boolean tTest​(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2, double alpha)
      Performs a two-sided t-test evaluating the null hypothesis that sampleStats1 and sampleStats2 describe datasets drawn from populations with the same mean, with significance level alpha.

    • Method Detail

      • pairedT

        double pairedT​(double[] sample1,
               double[] sample2)
        throws java.lang.IllegalArgumentException,
               MathException
        Computes a paired, 2-sample t-statistic based on the data in the input arrays. The t-statistic returned is equivalent to what would be returned by computing the one-sample t-statistic t(double, double[]), with mu = 0 and the sample array consisting of the (signed) differences between corresponding entries in sample1 and sample2.

        Preconditions:

        • The input arrays must have the same length and their common length must be at least 2.

        Parameters:
        sample1 - array of sample data values
        sample2 - array of sample data values
        Returns:
        t statistic
        Throws:
        java.lang.IllegalArgumentException - if the precondition is not met
        MathException - if the statistic can not be computed do to a convergence or other numerical error.

      • pairedTTest

        double pairedTTest​(double[] sample1,
                   double[] sample2)
            throws java.lang.IllegalArgumentException,
                   MathException
        Returns the observed significance level, or p-value, associated with a paired, two-sample, two-tailed t-test based on the data in the input arrays.

        The number returned is the smallest significance level at which one can reject the null hypothesis that the mean of the paired differences is 0 in favor of the two-sided alternative that the mean paired difference is not equal to 0. For a one-sided test, divide the returned value by 2.

        This test is equivalent to a one-sample t-test computed using tTest(double, double[]) with mu = 0 and the sample array consisting of the signed differences between corresponding elements of sample1 and sample2.

        Usage Note:
        The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed here

        Preconditions:

        • The input array lengths must be the same and their common length must be at least 2.

        Parameters:
        sample1 - array of sample data values
        sample2 - array of sample data values
        Returns:
        p-value for t-test
        Throws:
        java.lang.IllegalArgumentException - if the precondition is not met
        MathException - if an error occurs computing the p-value

      • pairedTTest

        boolean pairedTTest​(double[] sample1,
                    double[] sample2,
                    double alpha)
             throws java.lang.IllegalArgumentException,
                    MathException
        Performs a paired t-test evaluating the null hypothesis that the mean of the paired differences between sample1 and sample2 is 0 in favor of the two-sided alternative that the mean paired difference is not equal to 0, with significance level alpha.

        Returns true iff the null hypothesis can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2

        Usage Note:
        The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here

        Preconditions:

        • The input array lengths must be the same and their common length must be at least 2.
        • 0 < alpha < 0.5

        Parameters:
        sample1 - array of sample data values
        sample2 - array of sample data values
        alpha - significance level of the test
        Returns:
        true if the null hypothesis can be rejected with confidence 1 - alpha
        Throws:
        java.lang.IllegalArgumentException - if the preconditions are not met
        MathException - if an error occurs performing the test

      • t

        double t​(double mu,
         double[] observed)
  throws java.lang.IllegalArgumentException
        Computes a t statistic given observed values and a comparison constant.

        This statistic can be used to perform a one sample t-test for the mean.

        Preconditions:

        • The observed array length must be at least 2.

        Parameters:
        mu - comparison constant
        observed - array of values
        Returns:
        t statistic
        Throws:
        java.lang.IllegalArgumentException - if input array length is less than 2

      • t

        double t​(double mu,
         StatisticalSummary sampleStats)
  throws java.lang.IllegalArgumentException
        Computes a t statistic to use in comparing the mean of the dataset described by sampleStats to mu.

        This statistic can be used to perform a one sample t-test for the mean.

        Preconditions:

        • observed.getN() > = 2.

        Parameters:
        mu - comparison constant
        sampleStats - DescriptiveStatistics holding sample summary statitstics
        Returns:
        t statistic
        Throws:
        java.lang.IllegalArgumentException - if the precondition is not met

      • homoscedasticT

        double homoscedasticT​(double[] sample1,
                      double[] sample2)
               throws java.lang.IllegalArgumentException
        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-statisitc is

           t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))

        where n1 is the size of first sample; n2 is the size of second sample; m1 is the mean of first sample; m2 is the mean of second sample

      and var is the pooled variance estimate:

      var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))

      with var1 the variance of the first sample and var2 the variance of the second sample.

      Preconditions:

      • The observed array lengths must both be at least 2.

Parameters:
sample1 - array of sample data values
sample2 - array of sample data values
Returns:
t statistic
Throws:
java.lang.IllegalArgumentException - if the precondition is not met

  • t

    double t​(double[] sample1,
         double[] sample2)
  throws java.lang.IllegalArgumentException
    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-statisitc is

       t = (m1 - m2) / sqrt(var1/n1 + var2/n2)

    where n1 is the size of the first sample n2 is the size of the second sample; m1 is the mean of the first sample; m2 is the mean of the second sample; var1 is the variance of the first sample; var2 is the variance of the second sample;

    Preconditions:

    • The observed array lengths must both be at least 2.

    Parameters:
    sample1 - array of sample data values
    sample2 - array of sample data values
    Returns:
    t statistic
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met

  • t

    double t​(StatisticalSummary sampleStats1,
         StatisticalSummary sampleStats2)
  throws java.lang.IllegalArgumentException
    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-statisitc is

       t = (m1 - m2) / sqrt(var1/n1 + var2/n2)

    where n1 is the size of the first sample; n2 is the size of the second sample; m1 is the mean of the first sample; m2 is the mean of the second sample var1 is the variance of the first sample; var2 is the variance of the second sample

    Preconditions:

    • The datasets described by the two Univariates must each contain at least 2 observations.

    Parameters:
    sampleStats1 - StatisticalSummary describing data from the first sample
    sampleStats2 - StatisticalSummary describing data from the second sample
    Returns:
    t statistic
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met

  • homoscedasticT

    double homoscedasticT​(StatisticalSummary sampleStats1,
                      StatisticalSummary sampleStats2)
               throws java.lang.IllegalArgumentException
    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-statisitc returned is

       t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))

    where n1 is the size of first sample; n2 is the size of second sample; m1 is the mean of first sample; m2 is the mean of second sample and var is the pooled variance estimate:

    var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))

    with var1 the variance of the first sample and var2 the variance of the second sample.

    Preconditions:

    • The datasets described by the two Univariates must each contain at least 2 observations.

    Parameters:
    sampleStats1 - StatisticalSummary describing data from the first sample
    sampleStats2 - StatisticalSummary describing data from the second sample
    Returns:
    t statistic
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met

  • tTest

    double tTest​(double mu,
             double[] sample)
      throws java.lang.IllegalArgumentException,
             MathException
    Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the input array with the constant mu.

    The number returned is the smallest significance level at which one can reject the null hypothesis that the mean equals mu in favor of the two-sided alternative that the mean is different from mu. For a one-sided test, divide the returned value by 2.

    Usage Note:
    The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here

    Preconditions:

    • The observed array length must be at least 2.

    Parameters:
    mu - constant value to compare sample mean against
    sample - array of sample data values
    Returns:
    p-value
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met
    MathException - if an error occurs computing the p-value

  • tTest

    boolean tTest​(double mu,
              double[] sample,
              double alpha)
       throws java.lang.IllegalArgumentException,
              MathException
    Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which sample is drawn equals mu.

    Returns true iff the null hypothesis can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2

    Examples:

    1. To test the (2-sided) hypothesis sample mean = mu at the 95% level, use
      tTest(mu, sample, 0.05)
    2. To test the (one-sided) hypothesis sample mean < mu at the 99% level, first verify that the measured sample mean is less than mu and then use
      tTest(mu, sample, 0.02)

    Usage Note:
    The validity of the test depends on the assumptions of the one-sample parametric t-test procedure, as discussed here

    Preconditions:

    • The observed array length must be at least 2.

    Parameters:
    mu - constant value to compare sample mean against
    sample - array of sample data values
    alpha - significance level of the test
    Returns:
    p-value
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met
    MathException - if an error computing the p-value

  • tTest

    double tTest​(double mu,
             StatisticalSummary sampleStats)
      throws java.lang.IllegalArgumentException,
             MathException
    Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the dataset described by sampleStats with the constant mu.

    The number returned is the smallest significance level at which one can reject the null hypothesis that the mean equals mu in favor of the two-sided alternative that the mean is different from mu. For a one-sided test, divide the returned value by 2.

    Usage Note:
    The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here

    Preconditions:

    • The sample must contain at least 2 observations.

    Parameters:
    mu - constant value to compare sample mean against
    sampleStats - StatisticalSummary describing sample data
    Returns:
    p-value
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met
    MathException - if an error occurs computing the p-value

  • tTest

    boolean tTest​(double mu,
              StatisticalSummary sampleStats,
              double alpha)
       throws java.lang.IllegalArgumentException,
              MathException
    Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which the dataset described by stats is drawn equals mu.

    Returns true iff the null hypothesis can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2.

    Examples:

    1. To test the (2-sided) hypothesis sample mean = mu at the 95% level, use
      tTest(mu, sampleStats, 0.05)
    2. To test the (one-sided) hypothesis sample mean < mu at the 99% level, first verify that the measured sample mean is less than mu and then use
      tTest(mu, sampleStats, 0.02)

    Usage Note:
    The validity of the test depends on the assumptions of the one-sample parametric t-test procedure, as discussed here

    Preconditions:

    • The sample must include at least 2 observations.

    Parameters:
    mu - constant value to compare sample mean against
    sampleStats - StatisticalSummary describing sample data values
    alpha - significance level of the test
    Returns:
    p-value
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met
    MathException - if an error occurs computing the p-value

  • tTest

    double tTest​(double[] sample1,
             double[] sample2)
      throws java.lang.IllegalArgumentException,
             MathException
    Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays.

    The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the two-sided alternative that they are different. For a one-sided test, divide the returned value by 2.

    The test does not assume that the underlying popuation variances are equal and it uses approximated degrees of freedom computed from the sample data to compute the p-value. The t-statistic used is as defined in t(double[], double[]) and the Welch-Satterthwaite approximation to the degrees of freedom is used, as described here. To perform the test under the assumption of equal subpopulation variances, use homoscedasticTTest(double[], double[]).

    Usage Note:
    The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed here

    Preconditions:

    • The observed array lengths must both be at least 2.

    Parameters:
    sample1 - array of sample data values
    sample2 - array of sample data values
    Returns:
    p-value for t-test
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met
    MathException - if an error occurs computing the p-value

  • homoscedasticTTest

    double homoscedasticTTest​(double[] sample1,
                          double[] sample2)
                   throws java.lang.IllegalArgumentException,
                          MathException
    Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays, under the assumption that the two samples are drawn from subpopulations with equal variances. To perform the test without the equal variances assumption, use tTest(double[], double[]).

    The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the two-sided alternative that they are different. For a one-sided test, divide the returned value by 2.

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

    Usage Note:
    The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed here

    Preconditions:

    • The observed array lengths must both be at least 2.

    Parameters:
    sample1 - array of sample data values
    sample2 - array of sample data values
    Returns:
    p-value for t-test
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met
    MathException - if an error occurs computing the p-value

  • tTest

    boolean tTest​(double[] sample1,
              double[] sample2,
              double alpha)
       throws java.lang.IllegalArgumentException,
              MathException
    Performs a two-sided t-test evaluating the null hypothesis that sample1 and sample2 are drawn from populations with the same mean, with significance level alpha. This test does not assume that the subpopulation variances are equal. To perform the test assuming equal variances, use homoscedasticTTest(double[], double[], double).

    Returns true iff the null hypothesis that the means are equal can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2

    See t(double[], double[]) for the formula used to compute the t-statistic. Degrees of freedom are approximated using the Welch-Satterthwaite approximation.

    Examples:

    1. To test the (2-sided) hypothesis mean 1 = mean 2 at the 95% level, use
      tTest(sample1, sample2, 0.05).
    2. To test the (one-sided) hypothesis mean 1 < mean 2 , at the 99% level, first verify that the measured mean of sample 1 is less than the mean of sample 2 and then use
      tTest(sample1, sample2, 0.02)

    Usage Note:
    The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here

    Preconditions:

    • The observed array lengths must both be at least 2.
    • 0 < alpha < 0.5

    Parameters:
    sample1 - array of sample data values
    sample2 - array of sample data values
    alpha - significance level of the test
    Returns:
    true if the null hypothesis can be rejected with confidence 1 - alpha
    Throws:
    java.lang.IllegalArgumentException - if the preconditions are not met
    MathException - if an error occurs performing the test

  • homoscedasticTTest

    boolean homoscedasticTTest​(double[] sample1,
                           double[] sample2,
                           double alpha)
                    throws java.lang.IllegalArgumentException,
                           MathException
    Performs a two-sided t-test evaluating the null hypothesis that sample1 and sample2 are drawn from populations with the same mean, with significance level alpha, assuming that the subpopulation variances are equal. Use tTest(double[], double[], double) to perform the test without the assumption of equal variances.

    Returns true iff the null hypothesis that the means are equal can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2. To perform the test without the assumption of equal subpopulation variances, use tTest(double[], double[], double).

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

    Examples:

    1. To test the (2-sided) hypothesis mean 1 = mean 2 at the 95% level, use
      tTest(sample1, sample2, 0.05).
    2. To test the (one-sided) hypothesis mean 1 < mean 2, at the 99% level, first verify that the measured mean of sample 1 is less than the mean of sample 2 and then use
      tTest(sample1, sample2, 0.02)

    Usage Note:
    The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here

    Preconditions:

    • The observed array lengths must both be at least 2.
    • 0 < alpha < 0.5

    Parameters:
    sample1 - array of sample data values
    sample2 - array of sample data values
    alpha - significance level of the test
    Returns:
    true if the null hypothesis can be rejected with confidence 1 - alpha
    Throws:
    java.lang.IllegalArgumentException - if the preconditions are not met
    MathException - if an error occurs performing the test

  • tTest

    double tTest​(StatisticalSummary sampleStats1,
             StatisticalSummary sampleStats2)
      throws java.lang.IllegalArgumentException,
             MathException
    Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the datasets described by two StatisticalSummary instances.

    The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the two-sided alternative that they are different. For a one-sided test, divide the returned value by 2.

    The test does not assume that the underlying popuation variances are equal and it uses approximated degrees of freedom computed from the sample data to compute the p-value. To perform the test assuming equal variances, use homoscedasticTTest(StatisticalSummary, StatisticalSummary).

    Usage Note:
    The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed here

    Preconditions:

    • The datasets described by the two Univariates must each contain at least 2 observations.

    Parameters:
    sampleStats1 - StatisticalSummary describing data from the first sample
    sampleStats2 - StatisticalSummary describing data from the second sample
    Returns:
    p-value for t-test
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met
    MathException - if an error occurs computing the p-value

  • homoscedasticTTest

    double homoscedasticTTest​(StatisticalSummary sampleStats1,
                          StatisticalSummary sampleStats2)
                   throws java.lang.IllegalArgumentException,
                          MathException
    Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the datasets described by two StatisticalSummary instances, under the hypothesis of equal subpopulation variances. To perform a test without the equal variances assumption, use tTest(StatisticalSummary, StatisticalSummary).

    The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the two-sided alternative that they are different. For a one-sided test, divide the returned value by 2.

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

    Usage Note:
    The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed here

    Preconditions:

    • The datasets described by the two Univariates must each contain at least 2 observations.

    Parameters:
    sampleStats1 - StatisticalSummary describing data from the first sample
    sampleStats2 - StatisticalSummary describing data from the second sample
    Returns:
    p-value for t-test
    Throws:
    java.lang.IllegalArgumentException - if the precondition is not met
    MathException - if an error occurs computing the p-value

  • tTest

    boolean tTest​(StatisticalSummary sampleStats1,
              StatisticalSummary sampleStats2,
              double alpha)
       throws java.lang.IllegalArgumentException,
              MathException
    Performs a two-sided t-test evaluating the null hypothesis that sampleStats1 and sampleStats2 describe datasets drawn from populations with the same mean, with significance level alpha. This test does not assume that the subpopulation variances are equal. To perform the test under the equal variances assumption, use homoscedasticTTest(StatisticalSummary, StatisticalSummary).

    Returns true iff the null hypothesis that the means are equal can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2

    See t(double[], double[]) for the formula used to compute the t-statistic. Degrees of freedom are approximated using the Welch-Satterthwaite approximation.

    Examples:

    1. To test the (2-sided) hypothesis mean 1 = mean 2 at the 95%, use
      tTest(sampleStats1, sampleStats2, 0.05)
    2. To test the (one-sided) hypothesis mean 1 < mean 2 at the 99% level, first verify that the measured mean of sample 1 is less than the mean of sample 2 and then use
      tTest(sampleStats1, sampleStats2, 0.02)

    Usage Note:
    The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here

    Preconditions:

    • The datasets described by the two Univariates must each contain at least 2 observations.
    • 0 < alpha < 0.5

    Parameters:
    sampleStats1 - StatisticalSummary describing sample data values
    sampleStats2 - StatisticalSummary describing sample data values
    alpha - significance level of the test
    Returns:
    true if the null hypothesis can be rejected with confidence 1 - alpha
    Throws:
    java.lang.IllegalArgumentException - if the preconditions are not met
    MathException - if an error occurs performing the test