GTest
org.apache.commons.math3.stat.inference

Class GTest



  • public class GTestextends Object
    Implements G Test statistics.

    This is known in statistical genetics as the McDonald-Kreitman test. The implementation handles both known and unknown distributions.

    Two samples tests can be used when the distribution is unknown a priori but provided by one sample, or when the hypothesis under test is that the two samples come from the same underlying distribution.

    • Constructor Summary

      Constructors 
      Constructor and Description
      GTest() 
    • Method Summary

      Methods 
      Modifier and TypeMethod and Description
      doubleg(double[] expected, long[] observed)
      Computes the G statistic for Goodness of Fit comparing observed and expected frequency counts.
      doublegDataSetsComparison(long[] observed1, long[] observed2)
      Computes a G (Log-Likelihood Ratio) two sample test statistic for independence comparing frequency counts in observed1 and observed2.
      doublegTest(double[] expected, long[] observed)
      Returns the observed significance level, or p-value, associated with a G-Test for goodness of fit comparing the observed frequency counts to those in the expected array.
      booleangTest(double[] expected, long[] observed, double alpha)
      Performs a G-Test (Log-Likelihood Ratio Test) for goodness of fit evaluating the null hypothesis that the observed counts conform to the frequency distribution described by the expected counts, with significance level alpha.
      doublegTestDataSetsComparison(long[] observed1, long[] observed2)
      Returns the observed significance level, or p-value, associated with a G-Value (Log-Likelihood Ratio) for two sample test comparing bin frequency counts in observed1 and observed2.
      booleangTestDataSetsComparison(long[] observed1, long[] observed2, double alpha)
      Performs a G-Test (Log-Likelihood Ratio Test) comparing two binned data sets.
      doublegTestIntrinsic(double[] expected, long[] observed)
      Returns the intrinsic (Hardy-Weinberg proportions) p-Value, as described in p64-69 of McDonald, J.H.
      doublerootLogLikelihoodRatio(long k11, long k12, long k21, long k22)
      Calculates the root log-likelihood ratio for 2 state Datasets.
    • Constructor Detail

      • GTest

        public GTest()
    • Method Detail

      • g

        public double g(double[] expected,       long[] observed)         throws NotPositiveException,                NotStrictlyPositiveException,                DimensionMismatchException
        Computes the G statistic for Goodness of Fit comparing observed and expected frequency counts.

        This statistic can be used to perform a G test (Log-Likelihood Ratio Test) evaluating the null hypothesis that the observed counts follow the expected distribution.

        Preconditions:

        • Expected counts must all be positive.
        • Observed counts must all be ≥ 0.
        • The observed and expected arrays must have the same length and their common length must be at least 2.

        If any of the preconditions are not met, a MathIllegalArgumentException is thrown.

        Note:This implementation rescales the expected array if necessary to ensure that the sum of the expected and observed counts are equal.

        Parameters:
        observed - array of observed frequency counts
        expected - array of expected frequency counts
        Returns:
        G-Test statistic
        Throws:
        NotPositiveException - if observed has negative entries
        NotStrictlyPositiveException - if expected has entries that are not strictly positive
        DimensionMismatchException - if the array lengths do not match or are less than 2.
      • gTest

        public double gTest(double[] expected,           long[] observed)             throws NotPositiveException,                    NotStrictlyPositiveException,                    DimensionMismatchException,                    MaxCountExceededException
        Returns the observed significance level, or p-value, associated with a G-Test for goodness of fit comparing the observed frequency counts to those in the expected array.

        The number returned is the smallest significance level at which one can reject the null hypothesis that the observed counts conform to the frequency distribution described by the expected counts.

        The probability returned is the tail probability beyond g(expected, observed) in the ChiSquare distribution with degrees of freedom one less than the common length of expected and observed.

        Preconditions:

        • Expected counts must all be positive.
        • Observed counts must all be ≥ 0.
        • The observed and expected arrays must have the same length and their common length must be at least 2.

        If any of the preconditions are not met, a MathIllegalArgumentException is thrown.

        Note:This implementation rescales the expected array if necessary to ensure that the sum of the expected and observed counts are equal.

        Parameters:
        observed - array of observed frequency counts
        expected - array of expected frequency counts
        Returns:
        p-value
        Throws:
        NotPositiveException - if observed has negative entries
        NotStrictlyPositiveException - if expected has entries that are not strictly positive
        DimensionMismatchException - if the array lengths do not match or are less than 2.
        MaxCountExceededException - if an error occurs computing the p-value.
      • gTest

        public boolean gTest(double[] expected,            long[] observed,            double alpha)              throws NotPositiveException,                     NotStrictlyPositiveException,                     DimensionMismatchException,                     OutOfRangeException,                     MaxCountExceededException
        Performs a G-Test (Log-Likelihood Ratio Test) for goodness of fit evaluating the null hypothesis that the observed counts conform to the frequency distribution described by the expected counts, with significance level alpha. Returns true iff the null hypothesis can be rejected with 100 * (1 - alpha) percent confidence.

        Example:
        To test the hypothesis that observed follows expected at the 99% level, use

        gTest(expected, observed, 0.01)

        Returns true iff gTestGoodnessOfFitPValue(expected, observed) < alpha

        Preconditions:

        • Expected counts must all be positive.
        • Observed counts must all be ≥ 0.
        • The observed and expected arrays must have the same length and their common length must be at least 2.
        • 0 < alpha < 0.5

        If any of the preconditions are not met, a MathIllegalArgumentException is thrown.

        Note:This implementation rescales the expected array if necessary to ensure that the sum of the expected and observed counts are equal.

        Parameters:
        observed - array of observed frequency counts
        expected - array of expected frequency counts
        alpha - significance level of the test
        Returns:
        true iff null hypothesis can be rejected with confidence 1 - alpha
        Throws:
        NotPositiveException - if observed has negative entries
        NotStrictlyPositiveException - if expected has entries that are not strictly positive
        DimensionMismatchException - if the array lengths do not match or are less than 2.
        MaxCountExceededException - if an error occurs computing the p-value.
        OutOfRangeException - if alpha is not strictly greater than zero and less than or equal to 0.5
      • gDataSetsComparison

        public double gDataSetsComparison(long[] observed1,                         long[] observed2)                           throws DimensionMismatchException,                                  NotPositiveException,                                  ZeroException

        Computes a G (Log-Likelihood Ratio) two sample test statistic for independence comparing frequency counts in observed1 and observed2. The sums of frequency counts in the two samples are not required to be the same. The formula used to compute the test statistic is

        2 * totalSum * [H(rowSums) + H(colSums) - H(k)]

        where H is the Shannon Entropy of the random variable formed by viewing the elements of the argument array as incidence counts;
        k is a matrix with rows [observed1, observed2];
        rowSums, colSums are the row/col sums of k;
        and totalSum is the overall sum of all entries in k.

        This statistic can be used to perform a G test evaluating the null hypothesis that both observed counts are independent

        Preconditions:

        • Observed counts must be non-negative.
        • Observed counts for a specific bin must not both be zero.
        • Observed counts for a specific sample must not all be 0.
        • The arrays observed1 and observed2 must have the same length and their common length must be at least 2.

        If any of the preconditions are not met, a MathIllegalArgumentException is thrown.

        Parameters:
        observed1 - array of observed frequency counts of the first data set
        observed2 - array of observed frequency counts of the second data set
        Returns:
        G-Test statistic
        Throws:
        DimensionMismatchException - the the lengths of the arrays do not match or their common length is less than 2
        NotPositiveException - if any entry in observed1 or observed2 is negative
        ZeroException - if either all counts of observed1 or observed2 are zero, or if the count at the same index is zero for both arrays.
      • rootLogLikelihoodRatio

        public double rootLogLikelihoodRatio(long k11,                            long k12,                            long k21,                            long k22)
        Calculates the root log-likelihood ratio for 2 state Datasets. See gDataSetsComparison(long[], long[] ).

        Given two events A and B, let k11 be the number of times both events occur, k12 the incidence of B without A, k21 the count of A without B, and k22 the number of times neither A nor B occurs. What is returned by this method is

        (sgn) sqrt(gValueDataSetsComparison({k11, k12}, {k21, k22})

        where sgn is -1 if k11 / (k11 + k12) < k21 / (k21 + k22));
        1 otherwise.

        Signed root LLR has two advantages over the basic LLR: a) it is positive where k11 is bigger than expected, negative where it is lower b) if there is no difference it is asymptotically normally distributed. This allows one to talk about "number of standard deviations" which is a more common frame of reference than the chi^2 distribution.

        Parameters:
        k11 - number of times the two events occurred together (AB)
        k12 - number of times the second event occurred WITHOUT the first event (notA,B)
        k21 - number of times the first event occurred WITHOUT the second event (A, notB)
        k22 - number of times something else occurred (i.e. was neither of these events (notA, notB)
        Returns:
        root log-likelihood ratio
      • gTestDataSetsComparison

        public double gTestDataSetsComparison(long[] observed1,                             long[] observed2)                               throws DimensionMismatchException,                                      NotPositiveException,                                      ZeroException,                                      MaxCountExceededException

        Returns the observed significance level, or p-value, associated with a G-Value (Log-Likelihood Ratio) for two sample test comparing bin frequency counts in observed1 and observed2.

        The number returned is the smallest significance level at which one can reject the null hypothesis that the observed counts conform to the same distribution.

        See gTest(double[], long[]) for details on how the p-value is computed. The degrees of of freedom used to perform the test is one less than the common length of the input observed count arrays.

        Preconditions:

        • Observed counts must be non-negative.
        • Observed counts for a specific bin must not both be zero.
        • Observed counts for a specific sample must not all be 0.
        • The arrays observed1 and observed2 must have the same length and their common length must be at least 2.

        If any of the preconditions are not met, a MathIllegalArgumentException is thrown.

        Parameters:
        observed1 - array of observed frequency counts of the first data set
        observed2 - array of observed frequency counts of the second data set
        Returns:
        p-value
        Throws:
        DimensionMismatchException - the the length of the arrays does not match or their common length is less than 2
        NotPositiveException - if any of the entries in observed1 or observed2 are negative
        ZeroException - if either all counts of observed1 or observed2 are zero, or if the count at some index is zero for both arrays
        MaxCountExceededException - if an error occurs computing the p-value.
      • gTestDataSetsComparison

        public boolean gTestDataSetsComparison(long[] observed1,                              long[] observed2,                              double alpha)                                throws DimensionMismatchException,                                       NotPositiveException,                                       ZeroException,                                       OutOfRangeException,                                       MaxCountExceededException

        Performs a G-Test (Log-Likelihood Ratio Test) comparing two binned data sets. The test evaluates the null hypothesis that the two lists of observed counts conform to the same frequency distribution, with significance level alpha. Returns true iff the null hypothesis can be rejected with 100 * (1 - alpha) percent confidence.

        See gDataSetsComparison(long[], long[]) for details on the formula used to compute the G (LLR) statistic used in the test and gTest(double[], long[]) for information on how the observed significance level is computed. The degrees of of freedom used to perform the test is one less than the common length of the input observed count arrays.

        Preconditions:
        • Observed counts must be non-negative.
        • Observed counts for a specific bin must not both be zero.
        • Observed counts for a specific sample must not all be 0.
        • The arrays observed1 and observed2 must have the same length and their common length must be at least 2.
        • 0 < alpha < 0.5

        If any of the preconditions are not met, a MathIllegalArgumentException is thrown.

        Parameters:
        observed1 - array of observed frequency counts of the first data set
        observed2 - array of observed frequency counts of the second data set
        alpha - significance level of the test
        Returns:
        true iff null hypothesis can be rejected with confidence 1 - alpha
        Throws:
        DimensionMismatchException - the the length of the arrays does not match
        NotPositiveException - if any of the entries in observed1 or observed2 are negative
        ZeroException - if either all counts of observed1 or observed2 are zero, or if the count at some index is zero for both arrays
        OutOfRangeException - if alpha is not in the range (0, 0.5]
        MaxCountExceededException - if an error occurs performing the test

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