GTest.java

  1. /*
  2.  * Licensed to the Apache Software Foundation (ASF) under one or more
  3.  * contributor license agreements.  See the NOTICE file distributed with
  4.  * this work for additional information regarding copyright ownership.
  5.  * The ASF licenses this file to You under the Apache License, Version 2.0
  6.  * (the "License"); you may not use this file except in compliance with
  7.  * the License.  You may obtain a copy of the License at
  8.  *
  9.  *      http://www.apache.org/licenses/LICENSE-2.0
  10.  *
  11.  * Unless required by applicable law or agreed to in writing, software
  12.  * distributed under the License is distributed on an "AS IS" BASIS,
  13.  * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  14.  * See the License for the specific language governing permissions and
  15.  * limitations under the License.
  16.  */
  17. package org.apache.commons.math4.legacy.stat.inference;

  19. import org.apache.commons.statistics.distribution.ChiSquaredDistribution;
  20. import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
  21. import org.apache.commons.math4.legacy.exception.MaxCountExceededException;
  22. import org.apache.commons.math4.legacy.exception.NotPositiveException;
  23. import org.apache.commons.math4.legacy.exception.NotStrictlyPositiveException;
  24. import org.apache.commons.math4.legacy.exception.OutOfRangeException;
  25. import org.apache.commons.math4.legacy.exception.ZeroException;
  26. import org.apache.commons.math4.legacy.exception.util.LocalizedFormats;
  27. import org.apache.commons.math4.core.jdkmath.JdkMath;
  28. import org.apache.commons.math4.legacy.core.MathArrays;

  30. /**
  31.  * Implements <a href="http://en.wikipedia.org/wiki/G-test">G Test</a>
  32.  * statistics.
  33.  *
  34.  * <p>This is known in statistical genetics as the McDonald-Kreitman test.
  35.  * The implementation handles both known and unknown distributions.</p>
  36.  *
  37.  * <p>Two samples tests can be used when the distribution is unknown <i>a priori</i>
  38.  * but provided by one sample, or when the hypothesis under test is that the two
  39.  * samples come from the same underlying distribution.</p>
  40.  *
  41.  * @since 3.1
  42.  */
  43. public class GTest {

  45.     /**
  46.      * Computes the <a href="http://en.wikipedia.org/wiki/G-test">G statistic
  47.      * for Goodness of Fit</a> comparing {@code observed} and {@code expected}
  48.      * frequency counts.
  49.      *
  50.      * <p>This statistic can be used to perform a G test (Log-Likelihood Ratio
  51.      * Test) evaluating the null hypothesis that the observed counts follow the
  52.      * expected distribution.</p>
  53.      *
  54.      * <p><strong>Preconditions</strong>: <ul>
  55.      * <li>Expected counts must all be positive. </li>
  56.      * <li>Observed counts must all be &ge; 0. </li>
  57.      * <li>The observed and expected arrays must have the same length and their
  58.      * common length must be at least 2. </li></ul>
  59.      *
  60.      * <p>If any of the preconditions are not met, a
  61.      * {@code MathIllegalArgumentException} is thrown.</p>
  62.      *
  63.      * <p><strong>Note:</strong>This implementation rescales the
  64.      * {@code expected} array if necessary to ensure that the sum of the
  65.      * expected and observed counts are equal.</p>
  66.      *
  67.      * @param observed array of observed frequency counts
  68.      * @param expected array of expected frequency counts
  69.      * @return G-Test statistic
  70.      * @throws NotPositiveException if {@code observed} has negative entries
  71.      * @throws NotStrictlyPositiveException if {@code expected} has entries that
  72.      * are not strictly positive
  73.      * @throws DimensionMismatchException if the array lengths do not match or
  74.      * are less than 2.
  75.      */
  76.     public double g(final double[] expected, final long[] observed)
  77.             throws NotPositiveException, NotStrictlyPositiveException,
  78.             DimensionMismatchException {

  80.         if (expected.length < 2) {
  81.             throw new DimensionMismatchException(expected.length, 2);
  82.         }
  83.         if (expected.length != observed.length) {
  84.             throw new DimensionMismatchException(expected.length, observed.length);
  85.         }
  86.         MathArrays.checkPositive(expected);
  87.         MathArrays.checkNonNegative(observed);

  89.         double sumExpected = 0d;
  90.         double sumObserved = 0d;
  91.         for (int i = 0; i < observed.length; i++) {
  92.             sumExpected += expected[i];
  93.             sumObserved += observed[i];
  94.         }
  95.         double ratio = 1d;
  96.         boolean rescale = false;
  97.         if (JdkMath.abs(sumExpected - sumObserved) > 10E-6) {
  98.             ratio = sumObserved / sumExpected;
  99.             rescale = true;
  100.         }
  101.         double sum = 0d;
  102.         for (int i = 0; i < observed.length; i++) {
  103.             final double dev = rescale ?
  104.                     JdkMath.log((double) observed[i] / (ratio * expected[i])) :
  105.                         JdkMath.log((double) observed[i] / expected[i]);
  106.             sum += ((double) observed[i]) * dev;
  107.         }
  108.         return 2d * sum;
  109.     }

  111.     /**
  112.      * Returns the <i>observed significance level</i>, or <a href=
  113.      * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue"> p-value</a>,
  114.      * associated with a G-Test for goodness of fit comparing the
  115.      * {@code observed} frequency counts to those in the {@code expected} array.
  116.      *
  117.      * <p>The number returned is the smallest significance level at which one
  118.      * can reject the null hypothesis that the observed counts conform to the
  119.      * frequency distribution described by the expected counts.</p>
  120.      *
  121.      * <p>The probability returned is the tail probability beyond
  122.      * {@link #g(double[], long[]) g(expected, observed)}
  123.      * in the ChiSquare distribution with degrees of freedom one less than the
  124.      * common length of {@code expected} and {@code observed}.</p>
  125.      *
  126.      * <p> <strong>Preconditions</strong>: <ul>
  127.      * <li>Expected counts must all be positive. </li>
  128.      * <li>Observed counts must all be &ge; 0. </li>
  129.      * <li>The observed and expected arrays must have the
  130.      * same length and their common length must be at least 2.</li>
  131.      * </ul>
  132.      *
  133.      * <p>If any of the preconditions are not met, a
  134.      * {@code MathIllegalArgumentException} is thrown.</p>
  135.      *
  136.      * <p><strong>Note:</strong>This implementation rescales the
  137.      * {@code expected} array if necessary to ensure that the sum of the
  138.      *  expected and observed counts are equal.</p>
  139.      *
  140.      * @param observed array of observed frequency counts
  141.      * @param expected array of expected frequency counts
  142.      * @return p-value
  143.      * @throws NotPositiveException if {@code observed} has negative entries
  144.      * @throws NotStrictlyPositiveException if {@code expected} has entries that
  145.      * are not strictly positive
  146.      * @throws DimensionMismatchException if the array lengths do not match or
  147.      * are less than 2.
  148.      * @throws MaxCountExceededException if an error occurs computing the
  149.      * p-value.
  150.      */
  151.     public double gTest(final double[] expected, final long[] observed)
  152.             throws NotPositiveException, NotStrictlyPositiveException,
  153.             DimensionMismatchException, MaxCountExceededException {

  155.         // pass a null rng to avoid unneeded overhead as we will not sample from this distribution
  156.         final ChiSquaredDistribution distribution =
  157.                 ChiSquaredDistribution.of(expected.length - 1.0);
  158.         return distribution.survivalProbability(g(expected, observed));
  159.     }

  161.     /**
  162.      * Returns the intrinsic (Hardy-Weinberg proportions) p-Value, as described
  163.      * in p64-69 of McDonald, J.H. 2009. Handbook of Biological Statistics
  164.      * (2nd ed.). Sparky House Publishing, Baltimore, Maryland.
  165.      *
  166.      * <p> The probability returned is the tail probability beyond
  167.      * {@link #g(double[], long[]) g(expected, observed)}
  168.      * in the ChiSquare distribution with degrees of freedom two less than the
  169.      * common length of {@code expected} and {@code observed}.</p>
  170.      *
  171.      * @param observed array of observed frequency counts
  172.      * @param expected array of expected frequency counts
  173.      * @return p-value
  174.      * @throws NotPositiveException if {@code observed} has negative entries
  175.      * @throws NotStrictlyPositiveException {@code expected} has entries that are
  176.      * not strictly positive
  177.      * @throws DimensionMismatchException if the array lengths do not match or
  178.      * are less than 2.
  179.      * @throws MaxCountExceededException if an error occurs computing the
  180.      * p-value.
  181.      */
  182.     public double gTestIntrinsic(final double[] expected, final long[] observed)
  183.             throws NotPositiveException, NotStrictlyPositiveException,
  184.             DimensionMismatchException, MaxCountExceededException {

  186.         // pass a null rng to avoid unneeded overhead as we will not sample from this distribution
  187.         final ChiSquaredDistribution distribution =
  188.                 ChiSquaredDistribution.of(expected.length - 2.0);
  189.         return distribution.survivalProbability(g(expected, observed));
  190.     }

  192.     /**
  193.      * Performs a G-Test (Log-Likelihood Ratio Test) for goodness of fit
  194.      * evaluating the null hypothesis that the observed counts conform to the
  195.      * frequency distribution described by the expected counts, with
  196.      * significance level {@code alpha}. Returns true iff the null
  197.      * hypothesis can be rejected with {@code 100 * (1 - alpha)} percent confidence.
  198.      *
  199.      * <p><strong>Example:</strong><br> To test the hypothesis that
  200.      * {@code observed} follows {@code expected} at the 99% level,
  201.      * use </p><p>
  202.      * {@code gTest(expected, observed, 0.01)}</p>
  203.      *
  204.      * <p>Returns true iff {@link #gTest(double[], long[])
  205.      *  gTestGoodnessOfFitPValue(expected, observed)} &lt; alpha</p>
  206.      *
  207.      * <p><strong>Preconditions</strong>: <ul>
  208.      * <li>Expected counts must all be positive. </li>
  209.      * <li>Observed counts must all be &ge; 0. </li>
  210.      * <li>The observed and expected arrays must have the same length and their
  211.      * common length must be at least 2.
  212.      * <li> {@code 0 < alpha < 0.5} </li></ul>
  213.      *
  214.      * <p>If any of the preconditions are not met, a
  215.      * {@code MathIllegalArgumentException} is thrown.</p>
  216.      *
  217.      * <p><strong>Note:</strong>This implementation rescales the
  218.      * {@code expected} array if necessary to ensure that the sum of the
  219.      * expected and observed counts are equal.</p>
  220.      *
  221.      * @param observed array of observed frequency counts
  222.      * @param expected array of expected frequency counts
  223.      * @param alpha significance level of the test
  224.      * @return true iff null hypothesis can be rejected with confidence 1 -
  225.      * alpha
  226.      * @throws NotPositiveException if {@code observed} has negative entries
  227.      * @throws NotStrictlyPositiveException if {@code expected} has entries that
  228.      * are not strictly positive
  229.      * @throws DimensionMismatchException if the array lengths do not match or
  230.      * are less than 2.
  231.      * @throws MaxCountExceededException if an error occurs computing the
  232.      * p-value.
  233.      * @throws OutOfRangeException if alpha is not strictly greater than zero
  234.      * and less than or equal to 0.5
  235.      */
  236.     public boolean gTest(final double[] expected, final long[] observed,
  237.             final double alpha)
  238.             throws NotPositiveException, NotStrictlyPositiveException,
  239.             DimensionMismatchException, OutOfRangeException, MaxCountExceededException {

  241.         if (alpha <= 0 || alpha > 0.5) {
  242.             throw new OutOfRangeException(LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL,
  243.                     alpha, 0, 0.5);
  244.         }
  245.         return gTest(expected, observed) < alpha;
  246.     }

  248.     /**
  249.      * Calculates the <a href=
  250.      * "http://en.wikipedia.org/wiki/Entropy_%28information_theory%29">Shannon
  251.      * entropy</a> for 2 Dimensional Matrix.  The value returned is the entropy
  252.      * of the vector formed by concatenating the rows (or columns) of {@code k}
  253.      * to form a vector. See {@link #entropy(long[])}.
  254.      *
  255.      * @param k 2 Dimensional Matrix of long values (for ex. the counts of a
  256.      * trials)
  257.      * @return Shannon Entropy of the given Matrix
  258.      *
  259.      */
  260.     private double entropy(final long[][] k) {
  261.         double h = 0d;
  262.         double sumK = 0d;
  263.         for (int i = 0; i < k.length; i++) {
  264.             for (int j = 0; j < k[i].length; j++) {
  265.                 sumK += (double) k[i][j];
  266.             }
  267.         }
  268.         for (int i = 0; i < k.length; i++) {
  269.             for (int j = 0; j < k[i].length; j++) {
  270.                 if (k[i][j] != 0) {
  271.                     final double pIJ = (double) k[i][j] / sumK;
  272.                     h += pIJ * JdkMath.log(pIJ);
  273.                 }
  274.             }
  275.         }
  276.         return -h;
  277.     }

  279.     /**
  280.      * Calculates the <a href="http://en.wikipedia.org/wiki/Entropy_%28information_theory%29">
  281.      * Shannon entropy</a> for a vector.  The values of {@code k} are taken to be
  282.      * incidence counts of the values of a random variable. What is returned is <br>
  283.      * &sum;p<sub>i</sub>log(p<sub>i</sub><br>
  284.      * where p<sub>i</sub> = k[i] / (sum of elements in k)
  285.      *
  286.      * @param k Vector (for ex. Row Sums of a trials)
  287.      * @return Shannon Entropy of the given Vector
  288.      *
  289.      */
  290.     private double entropy(final long[] k) {
  291.         double h = 0d;
  292.         double sumK = 0d;
  293.         for (int i = 0; i < k.length; i++) {
  294.             sumK += (double) k[i];
  295.         }
  296.         for (int i = 0; i < k.length; i++) {
  297.             if (k[i] != 0) {
  298.                 final double pI = (double) k[i] / sumK;
  299.                 h += pI * JdkMath.log(pI);
  300.             }
  301.         }
  302.         return -h;
  303.     }

  305.     /**
  306.      * <p>Computes a G (Log-Likelihood Ratio) two sample test statistic for
  307.      * independence comparing frequency counts in
  308.      * {@code observed1} and {@code observed2}. The sums of frequency
  309.      * counts in the two samples are not required to be the same. The formula
  310.      * used to compute the test statistic is </p>
  311.      *
  312.      * <p>{@code 2 * totalSum * [H(rowSums) + H(colSums) - H(k)]}</p>
  313.      *
  314.      * <p> where {@code H} is the
  315.      * <a href="http://en.wikipedia.org/wiki/Entropy_%28information_theory%29">
  316.      * Shannon Entropy</a> of the random variable formed by viewing the elements
  317.      * of the argument array as incidence counts; <br>
  318.      * {@code k} is a matrix with rows {@code [observed1, observed2]}; <br>
  319.      * {@code rowSums, colSums} are the row/col sums of {@code k}; <br>
  320.      * and {@code totalSum} is the overall sum of all entries in {@code k}.</p>
  321.      *
  322.      * <p>This statistic can be used to perform a G test evaluating the null
  323.      * hypothesis that both observed counts are independent </p>
  324.      *
  325.      * <p> <strong>Preconditions</strong>: <ul>
  326.      * <li>Observed counts must be non-negative. </li>
  327.      * <li>Observed counts for a specific bin must not both be zero. </li>
  328.      * <li>Observed counts for a specific sample must not all be  0. </li>
  329.      * <li>The arrays {@code observed1} and {@code observed2} must have
  330.      * the same length and their common length must be at least 2. </li></ul>
  331.      *
  332.      * <p>If any of the preconditions are not met, a
  333.      * {@code MathIllegalArgumentException} is thrown.</p>
  334.      *
  335.      * @param observed1 array of observed frequency counts of the first data set
  336.      * @param observed2 array of observed frequency counts of the second data
  337.      * set
  338.      * @return G-Test statistic
  339.      * @throws DimensionMismatchException the lengths of the arrays do not
  340.      * match or their common length is less than 2
  341.      * @throws NotPositiveException if any entry in {@code observed1} or
  342.      * {@code observed2} is negative
  343.      * @throws ZeroException if either all counts of
  344.      * {@code observed1} or {@code observed2} are zero, or if the count
  345.      * at the same index is zero for both arrays.
  346.      */
  347.     public double gDataSetsComparison(final long[] observed1, final long[] observed2)
  348.             throws DimensionMismatchException, NotPositiveException, ZeroException {

  350.         // Make sure lengths are same
  351.         if (observed1.length < 2) {
  352.             throw new DimensionMismatchException(observed1.length, 2);
  353.         }
  354.         if (observed1.length != observed2.length) {
  355.             throw new DimensionMismatchException(observed1.length, observed2.length);
  356.         }

  358.         // Ensure non-negative counts
  359.         MathArrays.checkNonNegative(observed1);
  360.         MathArrays.checkNonNegative(observed2);

  362.         // Compute and compare count sums
  363.         long countSum1 = 0;
  364.         long countSum2 = 0;

  366.         // Compute and compare count sums
  367.         final long[] collSums = new long[observed1.length];
  368.         final long[][] k = new long[2][observed1.length];

  370.         for (int i = 0; i < observed1.length; i++) {
  371.             if (observed1[i] == 0 && observed2[i] == 0) {
  372.                 throw new ZeroException(LocalizedFormats.OBSERVED_COUNTS_BOTTH_ZERO_FOR_ENTRY, i);
  373.             } else {
  374.                 countSum1 += observed1[i];
  375.                 countSum2 += observed2[i];
  376.                 collSums[i] = observed1[i] + observed2[i];
  377.                 k[0][i] = observed1[i];
  378.                 k[1][i] = observed2[i];
  379.             }
  380.         }
  381.         // Ensure neither sample is uniformly 0
  382.         if (countSum1 == 0 || countSum2 == 0) {
  383.             throw new ZeroException();
  384.         }
  385.         final long[] rowSums = {countSum1, countSum2};
  386.         final double sum = (double) countSum1 + (double) countSum2;
  387.         return 2 * sum * (entropy(rowSums) + entropy(collSums) - entropy(k));
  388.     }

  390.     /**
  391.      * Calculates the root log-likelihood ratio for 2 state Datasets. See
  392.      * {@link #gDataSetsComparison(long[], long[] )}.
  393.      *
  394.      * <p>Given two events A and B, let k11 be the number of times both events
  395.      * occur, k12 the incidence of B without A, k21 the count of A without B,
  396.      * and k22 the number of times neither A nor B occurs.  What is returned
  397.      * by this method is </p>
  398.      *
  399.      * <p>{@code (sgn) sqrt(gValueDataSetsComparison({k11, k12}, {k21, k22})}</p>
  400.      *
  401.      * <p>where {@code sgn} is -1 if {@code k11 / (k11 + k12) < k21 / (k21 + k22))};<br>
  402.      * 1 otherwise.</p>
  403.      *
  404.      * <p>Signed root LLR has two advantages over the basic LLR: a) it is positive
  405.      * where k11 is bigger than expected, negative where it is lower b) if there is
  406.      * no difference it is asymptotically normally distributed. This allows one
  407.      * to talk about "number of standard deviations" which is a more common frame
  408.      * of reference than the chi^2 distribution.</p>
  409.      *
  410.      * @param k11 number of times the two events occurred together (AB)
  411.      * @param k12 number of times the second event occurred WITHOUT the
  412.      * first event (notA,B)
  413.      * @param k21 number of times the first event occurred WITHOUT the
  414.      * second event (A, notB)
  415.      * @param k22 number of times something else occurred (i.e. was neither
  416.      * of these events (notA, notB)
  417.      * @return root log-likelihood ratio
  418.      *
  419.      */
  420.     public double rootLogLikelihoodRatio(final long k11, long k12,
  421.             final long k21, final long k22) {
  422.         final double llr = gDataSetsComparison(
  423.                 new long[]{k11, k12}, new long[]{k21, k22});
  424.         double sqrt = JdkMath.sqrt(llr);
  425.         if ((double) k11 / (k11 + k12) < (double) k21 / (k21 + k22)) {
  426.             sqrt = -sqrt;
  427.         }
  428.         return sqrt;
  429.     }

  431.     /**
  432.      * <p>Returns the <i>observed significance level</i>, or <a href=
  433.      * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue">
  434.      * p-value</a>, associated with a G-Value (Log-Likelihood Ratio) for two
  435.      * sample test comparing bin frequency counts in {@code observed1} and
  436.      * {@code observed2}.</p>
  437.      *
  438.      * <p>The number returned is the smallest significance level at which one
  439.      * can reject the null hypothesis that the observed counts conform to the
  440.      * same distribution. </p>
  441.      *
  442.      * <p>See {@link #gTest(double[], long[])} for details
  443.      * on how the p-value is computed.  The degrees of of freedom used to
  444.      * perform the test is one less than the common length of the input observed
  445.      * count arrays.</p>
  446.      *
  447.      * <p><strong>Preconditions</strong>:
  448.      * <ul> <li>Observed counts must be non-negative. </li>
  449.      * <li>Observed counts for a specific bin must not both be zero. </li>
  450.      * <li>Observed counts for a specific sample must not all be 0. </li>
  451.      * <li>The arrays {@code observed1} and {@code observed2} must
  452.      * have the same length and their common length must be at least 2. </li>
  453.      * </ul>
  454.      * <p> If any of the preconditions are not met, a
  455.      * {@code MathIllegalArgumentException} is thrown.</p>
  456.      *
  457.      * @param observed1 array of observed frequency counts of the first data set
  458.      * @param observed2 array of observed frequency counts of the second data
  459.      * set
  460.      * @return p-value
  461.      * @throws DimensionMismatchException the length of the arrays does not
  462.      * match or their common length is less than 2
  463.      * @throws NotPositiveException if any of the entries in {@code observed1} or
  464.      * {@code observed2} are negative
  465.      * @throws ZeroException if either all counts of {@code observed1} or
  466.      * {@code observed2} are zero, or if the count at some index is
  467.      * zero for both arrays
  468.      * @throws MaxCountExceededException if an error occurs computing the
  469.      * p-value.
  470.      */
  471.     public double gTestDataSetsComparison(final long[] observed1,
  472.             final long[] observed2)
  473.             throws DimensionMismatchException, NotPositiveException, ZeroException,
  474.             MaxCountExceededException {

  476.         // pass a null rng to avoid unneeded overhead as we will not sample from this distribution
  477.         final ChiSquaredDistribution distribution =
  478.                 ChiSquaredDistribution.of((double) observed1.length - 1);
  479.         return distribution.survivalProbability(
  480.                 gDataSetsComparison(observed1, observed2));
  481.     }

  483.     /**
  484.      * <p>Performs a G-Test (Log-Likelihood Ratio Test) comparing two binned
  485.      * data sets. The test evaluates the null hypothesis that the two lists
  486.      * of observed counts conform to the same frequency distribution, with
  487.      * significance level {@code alpha}. Returns true iff the null
  488.      * hypothesis can be rejected  with 100 * (1 - alpha) percent confidence.
  489.      * </p>
  490.      * <p>See {@link #gDataSetsComparison(long[], long[])} for details
  491.      * on the formula used to compute the G (LLR) statistic used in the test and
  492.      * {@link #gTest(double[], long[])} for information on how
  493.      * the observed significance level is computed. The degrees of of freedom used
  494.      * to perform the test is one less than the common length of the input observed
  495.      * count arrays. </p>
  496.      *
  497.      * <strong>Preconditions</strong>: <ul>
  498.      * <li>Observed counts must be non-negative. </li>
  499.      * <li>Observed counts for a specific bin must not both be zero. </li>
  500.      * <li>Observed counts for a specific sample must not all be 0. </li>
  501.      * <li>The arrays {@code observed1} and {@code observed2} must
  502.      * have the same length and their common length must be at least 2. </li>
  503.      * <li>{@code 0 < alpha < 0.5} </li></ul>
  504.      *
  505.      * <p>If any of the preconditions are not met, a
  506.      * {@code MathIllegalArgumentException} is thrown.</p>
  507.      *
  508.      * @param observed1 array of observed frequency counts of the first data set
  509.      * @param observed2 array of observed frequency counts of the second data
  510.      * set
  511.      * @param alpha significance level of the test
  512.      * @return true iff null hypothesis can be rejected with confidence 1 -
  513.      * alpha
  514.      * @throws DimensionMismatchException the length of the arrays does not
  515.      * match
  516.      * @throws NotPositiveException if any of the entries in {@code observed1} or
  517.      * {@code observed2} are negative
  518.      * @throws ZeroException if either all counts of {@code observed1} or
  519.      * {@code observed2} are zero, or if the count at some index is
  520.      * zero for both arrays
  521.      * @throws OutOfRangeException if {@code alpha} is not in the range
  522.      * (0, 0.5]
  523.      * @throws MaxCountExceededException if an error occurs performing the test
  524.      */
  525.     public boolean gTestDataSetsComparison(
  526.             final long[] observed1,
  527.             final long[] observed2,
  528.             final double alpha)
  529.             throws DimensionMismatchException, NotPositiveException,
  530.             ZeroException, OutOfRangeException, MaxCountExceededException {

  532.         if (alpha <= 0 || alpha > 0.5) {
  533.             throw new OutOfRangeException(
  534.                     LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL, alpha, 0, 0.5);
  535.         }
  536.         return gTestDataSetsComparison(observed1, observed2) < alpha;
  537.     }
  538. }
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