GTest.java
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package org.apache.commons.statistics.inference;
import org.apache.commons.numbers.core.Sum;
import org.apache.commons.statistics.descriptive.LongMean;
import org.apache.commons.statistics.distribution.ChiSquaredDistribution;
/**
* Implements G-test (Generalized Log-Likelihood Ratio Test) statistics.
*
* <p>This is known in statistical genetics as the McDonald-Kreitman test.
* The implementation handles both known and unknown distributions.
*
* <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.
*
* @see <a href="https://en.wikipedia.org/wiki/G-test">G-test (Wikipedia)</a>
* @since 1.1
*/
public final class GTest {
// Note:
// The g-test statistic is a summation of terms with positive and negative sign
// and thus the sum may exhibit cancellation. This class uses separate high precision
// sums of the positive and negative terms which are then combined.
// Total cancellation for a large number of terms will not impact
// p-values of interest around critical alpha values as the Chi^2
// distribution exhibits strong concentration around its mean (degrees of freedom, k).
// The summation only need maintain enough bits in the final sum to distinguish
// g values around critical alpha values where 0 << chisq.sf(g, k) << 0.5: g > k,
// with k = number of terms - 1.
/** Default instance. */
private static final GTest DEFAULT = new GTest(0);
/** Degrees of freedom adjustment. */
private final int degreesOfFreedomAdjustment;
/**
* @param degreesOfFreedomAdjustment Degrees of freedom adjustment.
*/
private GTest(int degreesOfFreedomAdjustment) {
this.degreesOfFreedomAdjustment = degreesOfFreedomAdjustment;
}
/**
* Return an instance using the default options.
*
* <ul>
* <li>{@linkplain #withDegreesOfFreedomAdjustment(int) Degrees of freedom adjustment = 0}
* </ul>
*
* @return default instance
*/
public static GTest withDefaults() {
return DEFAULT;
}
/**
* Return an instance with the configured degrees of freedom adjustment.
*
* <p>The default degrees of freedom for a sample of length {@code n} are
* {@code n - 1}. An intrinsic null hypothesis is one where you estimate one or
* more parameters from the data in order to get the numbers for your null
* hypothesis. For a distribution with {@code p} parameters where up to
* {@code p} parameters have been estimated from the data the degrees of freedom
* is in the range {@code [n - 1 - p, n - 1]}.
*
* @param v Value.
* @return an instance
* @throws IllegalArgumentException if the value is negative
*/
public GTest withDegreesOfFreedomAdjustment(int v) {
return new GTest(Arguments.checkNonNegative(v));
}
/**
* Computes the G-test goodness-of-fit statistic comparing the {@code observed} counts to
* a uniform expected value (each category is equally likely).
*
* <p>Note: This is a specialized version of a comparison of {@code observed}
* with an {@code expected} array of uniform values. The result is faster than
* calling {@link #statistic(double[], long[])} and the statistic is the same,
* with an allowance for accumulated floating-point error due to the optimized
* routine.
*
* @param observed Observed frequency counts.
* @return G-test statistic
* @throws IllegalArgumentException if the sample size is less than 2;
* {@code observed} has negative entries; or all the observations are zero.
* @see #test(long[])
*/
public double statistic(long[] observed) {
Arguments.checkValuesRequiredSize(observed.length, 2);
Arguments.checkNonNegative(observed);
final double e = LongMean.of(observed).getAsDouble();
if (e == 0) {
throw new InferenceException(InferenceException.NO_DATA);
}
// g = 2 * sum{o * ln(o/e)}
// = 2 * [ sum{o * ln(o)} - sum(o) * ln(e) ]
// The second form has more cancellation as the sums are larger.
// Separate sum for positive and negative terms.
final Sum sum = Sum.create();
final Sum sum2 = Sum.create();
for (final double o : observed) {
if (o > e) {
// Positive term
sum.add(o * Math.log(o / e));
} else if (o > 0) {
// Negative term
// Process non-zero counts to avoid 0 * -inf = NaN
sum2.add(o * Math.log(o / e));
}
}
return sum.add(sum2).getAsDouble() * 2;
}
/**
* Computes the G-test goodness-of-fit statistic comparing {@code observed} and {@code expected}
* frequency counts.
*
* <p><strong>Note:</strong>This implementation rescales the values
* if necessary to ensure that the sum of the expected and observed counts
* are equal.
*
* @param expected Expected frequency counts.
* @param observed Observed frequency counts.
* @return G-test statistic
* @throws IllegalArgumentException if the sample size is less than 2; the array
* sizes do not match; {@code expected} has entries that are not strictly
* positive; {@code observed} has negative entries; or all the observations are zero.
* @see #test(double[], long[])
*/
public double statistic(double[] expected, long[] observed) {
// g = 2 * sum{o * ln(o/e)}
// The sum of o and e must be the same.
final double ratio = StatisticUtils.computeRatio(expected, observed);
// High precision sum to reduce cancellation.
// Separate sum for positive and negative terms.
final Sum sum = Sum.create();
final Sum sum2 = Sum.create();
for (int i = 0; i < observed.length; i++) {
final long o = observed[i];
// Process non-zero counts to avoid 0 * -inf = NaN
if (o != 0) {
final double term = o * Math.log(o / (ratio * expected[i]));
if (term < 0) {
sum2.add(term);
} else {
sum.add(term);
}
}
}
return sum.add(sum2).getAsDouble() * 2;
}
/**
* Computes a G-test statistic associated with a G-test of
* independence based on the input {@code counts} array, viewed as a two-way
* table. The formula used to compute the test statistic is:
*
* <p>\[ G = 2 \cdot \sum_{ij}{O_{ij}} \cdot \left[ H(r) + H(c) - H(r,c) \right] \]
*
* <p>and \( H \) is the <a
* href="https://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:
*
* <p>\[ H(X) = - {\sum_{x \in \text{Supp}(X)} p(x) \ln p(x)} \]
*
* @param counts 2-way table.
* @return G-test statistic
* @throws IllegalArgumentException if the number of rows or columns is less
* than 2; the array is non-rectangular; the array has negative entries; or the
* sum of a row or column is zero.
* @see ChiSquareTest#test(long[][])
*/
public double statistic(long[][] counts) {
Arguments.checkCategoriesRequiredSize(counts.length, 2);
Arguments.checkValuesRequiredSize(counts[0].length, 2);
Arguments.checkRectangular(counts);
Arguments.checkNonNegative(counts);
final int ni = counts.length;
final int nj = counts[0].length;
// Compute row, column and total sums
final double[] sumi = new double[ni];
final double[] sumj = new double[nj];
double n = 0;
// We can sum data on the first pass. See below for computation details.
final Sum sum = Sum.create();
for (int i = 0; i < ni; i++) {
for (int j = 0; j < nj; j++) {
final long c = counts[i][j];
sumi[i] += c;
sumj[j] += c;
if (c > 1) {
sum.add(c * Math.log(c));
}
}
checkNonZero(sumi[i], "Row", i);
n += sumi[i];
}
for (int j = 0; j < nj; j++) {
checkNonZero(sumj[j], "Column", j);
}
// This computes a modified form of the Shannon entropy H without requiring
// normalisation of observations to probabilities and without negation,
// i.e. we compute n * [ H(r) + H(c) - H(r,c) ] as [ H'(r,c) - H'(r) - H'(c) ].
// H = -sum (p * log(p))
// H' = n * sum (p * log(p))
// = n * sum (o/n * log(o/n))
// = n * [ sum(o/n * log(o)) - sum(o/n * log(n)) ]
// = sum(o * log(o)) - n log(n)
// After 3 modified entropy sums H'(r,c) - H'(r) - H'(c) compensation is (-1 + 2) * n log(n)
sum.addProduct(n, Math.log(n));
// Negative terms
final Sum sum2 = Sum.create();
// All these counts are above zero so no check for zeros
for (final double c : sumi) {
sum2.add(c * -Math.log(c));
}
for (final double c : sumj) {
sum2.add(c * -Math.log(c));
}
return sum.add(sum2).getAsDouble() * 2;
}
/**
* Perform a G-test for goodness-of-fit evaluating the null hypothesis that the {@code observed}
* counts conform to a uniform distribution (each category is equally likely).
*
* @param observed Observed frequency counts.
* @return test result
* @throws IllegalArgumentException if the sample size is less than 2;
* {@code observed} has negative entries; or all the observations are zero
* @see #statistic(long[])
*/
public SignificanceResult test(long[] observed) {
final int df = observed.length - 1;
final double g = statistic(observed);
final double p = computeP(g, df);
return new BaseSignificanceResult(g, p);
}
/**
* Perform a G-test for goodness-of-fit evaluating the null hypothesis that the {@code observed}
* counts conform to the {@code expected} counts.
*
* <p>The test can be configured to apply an adjustment to the degrees of freedom
* if the observed data has been used to create the expected counts.
*
* @param expected Expected frequency counts.
* @param observed Observed frequency counts.
* @return test result
* @throws IllegalArgumentException if the sample size is less than 2; the array
* sizes do not match; {@code expected} has entries that are not strictly
* positive; {@code observed} has negative entries; all the observations are zero; or
* the adjusted degrees of freedom are not strictly positive
* @see #withDegreesOfFreedomAdjustment(int)
* @see #statistic(double[], long[])
*/
public SignificanceResult test(double[] expected, long[] observed) {
final int df = StatisticUtils.computeDegreesOfFreedom(observed.length, degreesOfFreedomAdjustment);
final double g = statistic(expected, observed);
final double p = computeP(g, df);
return new BaseSignificanceResult(g, p);
}
/**
* Perform a G-test of independence based on the input
* {@code counts} array, viewed as a two-way table.
*
* @param counts 2-way table.
* @return test result
* @throws IllegalArgumentException if the number of rows or columns is less
* than 2; the array is non-rectangular; the array has negative entries; or the
* sum of a row or column is zero.
* @see #statistic(long[][])
*/
public SignificanceResult test(long[][] counts) {
final double g = statistic(counts);
final double df = (counts.length - 1.0) * (counts[0].length - 1.0);
final double p = computeP(g, df);
return new BaseSignificanceResult(g, p);
}
/**
* Compute the G-test p-value.
*
* @param g G-test statistic.
* @param degreesOfFreedom Degrees of freedom.
* @return p-value
*/
private static double computeP(double g, double degreesOfFreedom) {
return ChiSquaredDistribution.of(degreesOfFreedom).survivalProbability(g);
}
/**
* Check the array value is non-zero.
*
* @param value Value
* @param name Name of the array
* @param index Index in the array
* @throws IllegalArgumentException if the value is zero
*/
private static void checkNonZero(double value, String name, int index) {
if (value == 0) {
throw new InferenceException(InferenceException.ZERO_AT, name, index);
}
}
}