Package org.hipparchus.stat.inference

Class TTest

  • public class TTest
extends Object
    An implementation for Student's t-tests.

    Tests can be:

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

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

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

    Uses Hipparchus TDistribution implementation to estimate exact p-values.

    • Constructor Summary

      Constructors 
      Constructor Description
      TTest()  

    • Method Summary

      All Methods Instance Methods Concrete Methods 
      Modifier and Type Method Description
      protected double df​(double v1, double v2, double n1, double n2)
      Computes approximate degrees of freedom for 2-sample t-test.
      double homoscedasticT​(double[] sample1, double[] sample2)
      Computes a 2-sample t statistic, under the hypothesis of equal subpopulation variances.
      protected double homoscedasticT​(double m1, double m2, double v1, double v2, double n1, double n2)
      Computes t test statistic for 2-sample t-test under the hypothesis of equal subpopulation variances.
      double homoscedasticT​(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
      Computes a 2-sample t statistic, comparing the means of the datasets described by two StatisticalSummary instances, under the assumption of equal subpopulation variances.
      double homoscedasticTTest​(double[] sample1, double[] sample2)
      Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays, under the assumption that the two samples are drawn from subpopulations with equal variances.
      boolean homoscedasticTTest​(double[] sample1, double[] sample2, double alpha)
      Performs a two-sided t-test evaluating the null hypothesis that sample1 and sample2 are drawn from populations with the same mean, with significance level alpha, assuming that the subpopulation variances are equal.
      protected double homoscedasticTTest​(double m1, double m2, double v1, double v2, double n1, double n2)
      Computes p-value for 2-sided, 2-sample t-test, under the assumption of equal subpopulation variances.
      double homoscedasticTTest​(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
      Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the datasets described by two StatisticalSummary instances, under the hypothesis of equal subpopulation variances.
      double pairedT​(double[] sample1, double[] sample2)
      Computes a paired, 2-sample t-statistic based on the data in the input arrays.
      double pairedTTest​(double[] sample1, double[] sample2)
      Returns the observed significance level, or p-value, associated with a paired, two-sample, two-tailed t-test based on the data in the input arrays.
      boolean pairedTTest​(double[] sample1, double[] sample2, double alpha)
      Performs a paired t-test evaluating the null hypothesis that the mean of the paired differences between sample1 and sample2 is 0 in favor of the two-sided alternative that the mean paired difference is not equal to 0, with significance level alpha.
      double t​(double[] sample1, double[] sample2)
      Computes a 2-sample t statistic, without the hypothesis of equal subpopulation variances.
      double t​(double mu, double[] observed)
      Computes a t statistic given observed values and a comparison constant.
      protected double t​(double m, double mu, double v, double n)
      Computes t test statistic for 1-sample t-test.
      protected double t​(double m1, double m2, double v1, double v2, double n1, double n2)
      Computes t test statistic for 2-sample t-test.
      double t​(double mu, StatisticalSummary sampleStats)
      Computes a t statistic to use in comparing the mean of the dataset described by sampleStats to mu.
      double t​(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
      Computes a 2-sample t statistic , comparing the means of the datasets described by two StatisticalSummary instances, without the assumption of equal subpopulation variances.
      double tTest​(double[] sample1, double[] sample2)
      Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays.
      boolean tTest​(double[] sample1, double[] sample2, double alpha)
      Performs a two-sided t-test evaluating the null hypothesis that sample1 and sample2 are drawn from populations with the same mean, with significance level alpha.
      double tTest​(double mu, double[] sample)
      Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the input array with the constant mu.
      boolean tTest​(double mu, double[] sample, double alpha)
      Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which sample is drawn equals mu.
      protected double tTest​(double m, double mu, double v, double n)
      Computes p-value for 2-sided, 1-sample t-test.
      protected double tTest​(double m1, double m2, double v1, double v2, double n1, double n2)
      Computes p-value for 2-sided, 2-sample t-test.
      double tTest​(double mu, StatisticalSummary sampleStats)
      Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the dataset described by sampleStats with the constant mu.
      boolean tTest​(double mu, StatisticalSummary sampleStats, double alpha)
      Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which the dataset described by stats is drawn equals mu.
      double tTest​(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)
      Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the datasets described by two StatisticalSummary instances.
      boolean tTest​(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2, double alpha)
      Performs a two-sided t-test evaluating the null hypothesis that sampleStats1 and sampleStats2 describe datasets drawn from populations with the same mean, with significance level alpha.

    • Constructor Detail

      • TTest

        public TTest()

    • Method Detail

      • pairedT

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

        Preconditions:

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

        Parameters:
        sample1 - array of sample data values
        sample2 - array of sample data values
        Returns:
        t statistic
        Throws:
        NullArgumentException - if the arrays are null
        MathIllegalArgumentException - if the arrays are empty
        MathIllegalArgumentException - if the length of the arrays is not equal
        MathIllegalArgumentException - if the length of the arrays is < 2

      • pairedTTest

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

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

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

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

        Preconditions:

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

        Parameters:
        sample1 - array of sample data values
        sample2 - array of sample data values
        Returns:
        p-value for t-test
        Throws:
        NullArgumentException - if the arrays are null
        MathIllegalArgumentException - if the arrays are empty
        MathIllegalArgumentException - if the length of the arrays is not equal
        MathIllegalArgumentException - if the length of the arrays is < 2
        MathIllegalStateException - if an error occurs computing the p-value

      • pairedTTest

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

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

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

        Preconditions:

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

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

      • homoscedasticT

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

        This statistic can be used to perform a (homoscedastic) two-sample t-test to compare sample means.

        The t-statistic is

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

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

      and var is the pooled variance estimate:

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

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

      Preconditions:

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

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