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* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
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* See the License for the specific language governing permissions and
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package org.apache.commons.math3.stat.inference;
import org.apache.commons.math3.distribution.ChiSquaredDistribution;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NotPositiveException;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.ZeroException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathArrays;
/**
* Implements <a href="http://en.wikipedia.org/wiki/G-test">G Test</a>
* statistics.
*
* <p>This is known in statistical genetics as the McDonald-Kreitman test.
* The implementation handles both known and unknown distributions.</p>
*
* <p>Two samples tests can be used when the distribution is unknown <i>a priori</i>
* but provided by one sample, or when the hypothesis under test is that the two
* samples come from the same underlying distribution.</p>
*
* @since 3.1
*/
public class GTest {
/**
* Computes the <a href="http://en.wikipedia.org/wiki/G-test">G statistic
* for Goodness of Fit</a> comparing {@code observed} and {@code expected}
* frequency counts.
*
* <p>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.</p>
*
* <p><strong>Preconditions</strong>: <ul>
* <li>Expected counts must all be positive. </li>
* <li>Observed counts must all be ≥ 0. </li>
* <li>The observed and expected arrays must have the same length and their
* common length must be at least 2. </li></ul></p>
*
* <p>If any of the preconditions are not met, a
* {@code MathIllegalArgumentException} is thrown.</p>
*
* <p><strong>Note:</strong>This implementation rescales the
* {@code expected} array if necessary to ensure that the sum of the
* expected and observed counts are equal.</p>
*
* @param observed array of observed frequency counts
* @param expected array of expected frequency counts
* @return G-Test statistic
* @throws NotPositiveException if {@code observed} has negative entries
* @throws NotStrictlyPositiveException if {@code expected} has entries that
* are not strictly positive
* @throws DimensionMismatchException if the array lengths do not match or
* are less than 2.
*/
public double g(final double[] expected, final long[] observed)
throws NotPositiveException, NotStrictlyPositiveException,
DimensionMismatchException {
if (expected.length < 2) {
throw new DimensionMismatchException(expected.length, 2);
}
if (expected.length != observed.length) {
throw new DimensionMismatchException(expected.length, observed.length);
}
MathArrays.checkPositive(expected);
MathArrays.checkNonNegative(observed);
double sumExpected = 0d;
double sumObserved = 0d;
for (int i = 0; i < observed.length; i++) {
sumExpected += expected[i];
sumObserved += observed[i];
}
double ratio = 1d;
boolean rescale = false;
if (FastMath.abs(sumExpected - sumObserved) > 10E-6) {
ratio = sumObserved / sumExpected;
rescale = true;
}
double sum = 0d;
for (int i = 0; i < observed.length; i++) {
final double dev = rescale ?
FastMath.log((double) observed[i] / (ratio * expected[i])) :
FastMath.log((double) observed[i] / expected[i]);
sum += ((double) observed[i]) * dev;
}
return 2d * sum;
}
/**
* Returns the <i>observed significance level</i>, or <a href=
* "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue"> p-value</a>,
* associated with a G-Test for goodness of fit</a> comparing the
* {@code observed} frequency counts to those in the {@code expected} array.
*
* <p>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.</p>
*
* <p>The probability returned is the tail probability beyond
* {@link #g(double[], long[]) g(expected, observed)}
* in the ChiSquare distribution with degrees of freedom one less than the
* common length of {@code expected} and {@code observed}.</p>
*
* <p> <strong>Preconditions</strong>: <ul>
* <li>Expected counts must all be positive. </li>
* <li>Observed counts must all be ≥ 0. </li>
* <li>The observed and expected arrays must have the
* same length and their common length must be at least 2.</li>
* </ul></p>
*
* <p>If any of the preconditions are not met, a
* {@code MathIllegalArgumentException} is thrown.</p>
*
* <p><strong>Note:</strong>This implementation rescales the
* {@code expected} array if necessary to ensure that the sum of the
* expected and observed counts are equal.</p>
*
* @param observed array of observed frequency counts
* @param expected array of expected frequency counts
* @return p-value
* @throws NotPositiveException if {@code observed} has negative entries
* @throws NotStrictlyPositiveException if {@code expected} has entries that
* are not strictly positive
* @throws DimensionMismatchException if the array lengths do not match or
* are less than 2.
* @throws MaxCountExceededException if an error occurs computing the
* p-value.
*/
public double gTest(final double[] expected, final long[] observed)
throws NotPositiveException, NotStrictlyPositiveException,
DimensionMismatchException, MaxCountExceededException {
// pass a null rng to avoid unneeded overhead as we will not sample from this distribution
final ChiSquaredDistribution distribution =
new ChiSquaredDistribution(null, expected.length - 1.0);
return 1.0 - distribution.cumulativeProbability(g(expected, observed));
}
/**
* Returns the intrinsic (Hardy-Weinberg proportions) p-Value, as described
* in p64-69 of McDonald, J.H. 2009. Handbook of Biological Statistics
* (2nd ed.). Sparky House Publishing, Baltimore, Maryland.
*
* <p> The probability returned is the tail probability beyond
* {@link #g(double[], long[]) g(expected, observed)}
* in the ChiSquare distribution with degrees of freedom two less than the
* common length of {@code expected} and {@code observed}.</p>
*
* @param observed array of observed frequency counts
* @param expected array of expected frequency counts
* @return p-value
* @throws NotPositiveException if {@code observed} has negative entries
* @throws NotStrictlyPositiveException {@code expected} has entries that are
* not strictly positive
* @throws DimensionMismatchException if the array lengths do not match or
* are less than 2.
* @throws MaxCountExceededException if an error occurs computing the
* p-value.
*/
public double gTestIntrinsic(final double[] expected, final long[] observed)
throws NotPositiveException, NotStrictlyPositiveException,
DimensionMismatchException, MaxCountExceededException {
// pass a null rng to avoid unneeded overhead as we will not sample from this distribution
final ChiSquaredDistribution distribution =
new ChiSquaredDistribution(null, expected.length - 2.0);
return 1.0 - distribution.cumulativeProbability(g(expected, observed));
}
/**
* 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 {@code alpha}. Returns true iff the null
* hypothesis can be rejected with {@code 100 * (1 - alpha)} percent confidence.
*
* <p><strong>Example:</strong><br> To test the hypothesis that
* {@code observed} follows {@code expected} at the 99% level,
* use </p><p>
* {@code gTest(expected, observed, 0.01)}</p>
*
* <p>Returns true iff {@link #gTest(double[], long[])
* gTestGoodnessOfFitPValue(expected, observed)} < alpha</p>
*
* <p><strong>Preconditions</strong>: <ul>
* <li>Expected counts must all be positive. </li>
* <li>Observed counts must all be ≥ 0. </li>
* <li>The observed and expected arrays must have the same length and their
* common length must be at least 2.
* <li> {@code 0 < alpha < 0.5} </li></ul></p>
*
* <p>If any of the preconditions are not met, a
* {@code MathIllegalArgumentException} is thrown.</p>
*
* <p><strong>Note:</strong>This implementation rescales the
* {@code expected} array if necessary to ensure that the sum of the
* expected and observed counts are equal.</p>
*
* @param observed array of observed frequency counts
* @param expected array of expected frequency counts
* @param alpha significance level of the test
* @return true iff null hypothesis can be rejected with confidence 1 -
* alpha
* @throws NotPositiveException if {@code observed} has negative entries
* @throws NotStrictlyPositiveException if {@code expected} has entries that
* are not strictly positive
* @throws DimensionMismatchException if the array lengths do not match or
* are less than 2.
* @throws MaxCountExceededException if an error occurs computing the
* p-value.
* @throws OutOfRangeException if alpha is not strictly greater than zero
* and less than or equal to 0.5
*/
public boolean gTest(final double[] expected, final long[] observed,
final double alpha)
throws NotPositiveException, NotStrictlyPositiveException,
DimensionMismatchException, OutOfRangeException, MaxCountExceededException {
if ((alpha <= 0) || (alpha > 0.5)) {
throw new OutOfRangeException(LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL,
alpha, 0, 0.5);
}
return gTest(expected, observed) < alpha;
}
/**
* Calculates the <a href=
* "http://en.wikipedia.org/wiki/Entropy_%28information_theory%29">Shannon
* entropy</a> for 2 Dimensional Matrix. The value returned is the entropy
* of the vector formed by concatenating the rows (or columns) of {@code k}
* to form a vector. See {@link #entropy(long[])}.
*
* @param k 2 Dimensional Matrix of long values (for ex. the counts of a
* trials)
* @return Shannon Entropy of the given Matrix
*
*/
private double entropy(final long[][] k) {
double h = 0d;
double sum_k = 0d;
for (int i = 0; i < k.length; i++) {
for (int j = 0; j < k[i].length; j++) {
sum_k += (double) k[i][j];
}
}
for (int i = 0; i < k.length; i++) {
for (int j = 0; j < k[i].length; j++) {
if (k[i][j] != 0) {
final double p_ij = (double) k[i][j] / sum_k;
h += p_ij * FastMath.log(p_ij);
}
}
}
return -h;
}
/**
* Calculates the <a href="http://en.wikipedia.org/wiki/Entropy_%28information_theory%29">
* Shannon entropy</a> for a vector. The values of {@code k} are taken to be
* incidence counts of the values of a random variable. What is returned is <br/>
* ∑p<sub>i</sub>log(p<sub>i</sub><br/>
* where p<sub>i</sub> = k[i] / (sum of elements in k)
*
* @param k Vector (for ex. Row Sums of a trials)
* @return Shannon Entropy of the given Vector
*
*/
private double entropy(final long[] k) {
double h = 0d;
double sum_k = 0d;
for (int i = 0; i < k.length; i++) {
sum_k += (double) k[i];
}
for (int i = 0; i < k.length; i++) {
if (k[i] != 0) {
final double p_i = (double) k[i] / sum_k;
h += p_i * FastMath.log(p_i);
}
}
return -h;
}
/**
* <p>Computes a G (Log-Likelihood Ratio) two sample test statistic for
* independence comparing frequency counts in
* {@code observed1} and {@code 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 </p>
*
* <p>{@code 2 * totalSum * [H(rowSums) + H(colSums) - H(k)]}</p>
*
* <p> where {@code H} is the
* <a href="http://en.wikipedia.org/wiki/Entropy_%28information_theory%29">
* Shannon Entropy</a> of the random variable formed by viewing the elements
* of the argument array as incidence counts; <br/>
* {@code k} is a matrix with rows {@code [observed1, observed2]}; <br/>
* {@code rowSums, colSums} are the row/col sums of {@code k}; <br>
* and {@code totalSum} is the overall sum of all entries in {@code k}.</p>
*
* <p>This statistic can be used to perform a G test evaluating the null
* hypothesis that both observed counts are independent </p>
*
* <p> <strong>Preconditions</strong>: <ul>
* <li>Observed counts must be non-negative. </li>
* <li>Observed counts for a specific bin must not both be zero. </li>
* <li>Observed counts for a specific sample must not all be 0. </li>
* <li>The arrays {@code observed1} and {@code observed2} must have
* the same length and their common length must be at least 2. </li></ul></p>
*
* <p>If any of the preconditions are not met, a
* {@code MathIllegalArgumentException} is thrown.</p>
*
* @param observed1 array of observed frequency counts of the first data set
* @param observed2 array of observed frequency counts of the second data
* set
* @return G-Test statistic
* @throws DimensionMismatchException the the lengths of the arrays do not
* match or their common length is less than 2
* @throws NotPositiveException if any entry in {@code observed1} or
* {@code observed2} is negative
* @throws ZeroException if either all counts of
* {@code observed1} or {@code observed2} are zero, or if the count
* at the same index is zero for both arrays.
*/
public double gDataSetsComparison(final long[] observed1, final long[] observed2)
throws DimensionMismatchException, NotPositiveException, ZeroException {
// Make sure lengths are same
if (observed1.length < 2) {
throw new DimensionMismatchException(observed1.length, 2);
}
if (observed1.length != observed2.length) {
throw new DimensionMismatchException(observed1.length, observed2.length);
}
// Ensure non-negative counts
MathArrays.checkNonNegative(observed1);
MathArrays.checkNonNegative(observed2);
// Compute and compare count sums
long countSum1 = 0;
long countSum2 = 0;
// Compute and compare count sums
final long[] collSums = new long[observed1.length];
final long[][] k = new long[2][observed1.length];
for (int i = 0; i < observed1.length; i++) {
if (observed1[i] == 0 && observed2[i] == 0) {
throw new ZeroException(LocalizedFormats.OBSERVED_COUNTS_BOTTH_ZERO_FOR_ENTRY, i);
} else {
countSum1 += observed1[i];
countSum2 += observed2[i];
collSums[i] = observed1[i] + observed2[i];
k[0][i] = observed1[i];
k[1][i] = observed2[i];
}
}
// Ensure neither sample is uniformly 0
if (countSum1 == 0 || countSum2 == 0) {
throw new ZeroException();
}
final long[] rowSums = {countSum1, countSum2};
final double sum = (double) countSum1 + (double) countSum2;
return 2 * sum * (entropy(rowSums) + entropy(collSums) - entropy(k));
}
/**
* Calculates the root log-likelihood ratio for 2 state Datasets. See
* {@link #gDataSetsComparison(long[], long[] )}.
*
* <p>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 </p>
*
* <p>{@code (sgn) sqrt(gValueDataSetsComparison({k11, k12}, {k21, k22})}</p>
*
* <p>where {@code sgn} is -1 if {@code k11 / (k11 + k12) < k21 / (k21 + k22))};<br/>
* 1 otherwise.</p>
*
* <p>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.</p>
*
* @param k11 number of times the two events occurred together (AB)
* @param k12 number of times the second event occurred WITHOUT the
* first event (notA,B)
* @param k21 number of times the first event occurred WITHOUT the
* second event (A, notB)
* @param k22 number of times something else occurred (i.e. was neither
* of these events (notA, notB)
* @return root log-likelihood ratio
*
*/
public double rootLogLikelihoodRatio(final long k11, long k12,
final long k21, final long k22) {
final double llr = gDataSetsComparison(
new long[]{k11, k12}, new long[]{k21, k22});
double sqrt = FastMath.sqrt(llr);
if ((double) k11 / (k11 + k12) < (double) k21 / (k21 + k22)) {
sqrt = -sqrt;
}
return sqrt;
}
/**
* <p>Returns the <i>observed significance level</i>, or <a href=
* "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue">
* p-value</a>, associated with a G-Value (Log-Likelihood Ratio) for two
* sample test comparing bin frequency counts in {@code observed1} and
* {@code observed2}.</p>
*
* <p>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. </p>
*
* <p>See {@link #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.</p>
*
* <p><strong>Preconditions</strong>:
* <ul> <li>Observed counts must be non-negative. </li>
* <li>Observed counts for a specific bin must not both be zero. </li>
* <li>Observed counts for a specific sample must not all be 0. </li>
* <li>The arrays {@code observed1} and {@code observed2} must
* have the same length and their common length must be at least 2. </li>
* </ul><p>
* <p> If any of the preconditions are not met, a
* {@code MathIllegalArgumentException} is thrown.</p>
*
* @param observed1 array of observed frequency counts of the first data set
* @param observed2 array of observed frequency counts of the second data
* set
* @return p-value
* @throws DimensionMismatchException the the length of the arrays does not
* match or their common length is less than 2
* @throws NotPositiveException if any of the entries in {@code observed1} or
* {@code observed2} are negative
* @throws ZeroException if either all counts of {@code observed1} or
* {@code observed2} are zero, or if the count at some index is
* zero for both arrays
* @throws MaxCountExceededException if an error occurs computing the
* p-value.
*/
public double gTestDataSetsComparison(final long[] observed1,
final long[] observed2)
throws DimensionMismatchException, NotPositiveException, ZeroException,
MaxCountExceededException {
// pass a null rng to avoid unneeded overhead as we will not sample from this distribution
final ChiSquaredDistribution distribution =
new ChiSquaredDistribution(null, (double) observed1.length - 1);
return 1 - distribution.cumulativeProbability(
gDataSetsComparison(observed1, observed2));
}
/**
* <p>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 {@code alpha}. Returns true iff the null
* hypothesis can be rejected with 100 * (1 - alpha) percent confidence.
* </p>
* <p>See {@link #gDataSetsComparison(long[], long[])} for details
* on the formula used to compute the G (LLR) statistic used in the test and
* {@link #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. </p>
*
* <strong>Preconditions</strong>: <ul>
* <li>Observed counts must be non-negative. </li>
* <li>Observed counts for a specific bin must not both be zero. </li>
* <li>Observed counts for a specific sample must not all be 0. </li>
* <li>The arrays {@code observed1} and {@code observed2} must
* have the same length and their common length must be at least 2. </li>
* <li>{@code 0 < alpha < 0.5} </li></ul></p>
*
* <p>If any of the preconditions are not met, a
* {@code MathIllegalArgumentException} is thrown.</p>
*
* @param observed1 array of observed frequency counts of the first data set
* @param observed2 array of observed frequency counts of the second data
* set
* @param alpha significance level of the test
* @return true iff null hypothesis can be rejected with confidence 1 -
* alpha
* @throws DimensionMismatchException the the length of the arrays does not
* match
* @throws NotPositiveException if any of the entries in {@code observed1} or
* {@code observed2} are negative
* @throws ZeroException if either all counts of {@code observed1} or
* {@code observed2} are zero, or if the count at some index is
* zero for both arrays
* @throws OutOfRangeException if {@code alpha} is not in the range
* (0, 0.5]
* @throws MaxCountExceededException if an error occurs performing the test
*/
public boolean gTestDataSetsComparison(
final long[] observed1,
final long[] observed2,
final double alpha)
throws DimensionMismatchException, NotPositiveException,
ZeroException, OutOfRangeException, MaxCountExceededException {
if (alpha <= 0 || alpha > 0.5) {
throw new OutOfRangeException(
LocalizedFormats.OUT_OF_BOUND_SIGNIFICANCE_LEVEL, alpha, 0, 0.5);
}
return gTestDataSetsComparison(observed1, observed2) < alpha;
}
}