WilcoxonSignedRankTest.java
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* http://www.apache.org/licenses/LICENSE-2.0
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package org.apache.commons.statistics.inference;
import java.util.Arrays;
import java.util.EnumSet;
import java.util.Objects;
import org.apache.commons.numbers.core.Sum;
import org.apache.commons.statistics.distribution.NormalDistribution;
import org.apache.commons.statistics.ranking.NaNStrategy;
import org.apache.commons.statistics.ranking.NaturalRanking;
import org.apache.commons.statistics.ranking.RankingAlgorithm;
import org.apache.commons.statistics.ranking.TiesStrategy;
/**
* Implements the Wilcoxon signed-rank test.
*
* @see <a href="https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test">Wilcoxon signed-rank test (Wikipedia)</a>
* @since 1.1
*/
public final class WilcoxonSignedRankTest {
/** Limit on sample size for the exact p-value computation. */
private static final int EXACT_LIMIT = 1023;
/** Limit on sample size for the exact p-value computation for the auto mode. */
private static final int AUTO_LIMIT = 50;
/** Ranking instance. */
private static final RankingAlgorithm RANKING = new NaturalRanking(NaNStrategy.FAILED, TiesStrategy.AVERAGE);
/** Default instance. */
private static final WilcoxonSignedRankTest DEFAULT = new WilcoxonSignedRankTest(
AlternativeHypothesis.TWO_SIDED, PValueMethod.AUTO, true, 0);
/** Alternative hypothesis. */
private final AlternativeHypothesis alternative;
/** Method to compute the p-value. */
private final PValueMethod pValueMethod;
/** Perform continuity correction. */
private final boolean continuityCorrection;
/** Expected location shift. */
private final double mu;
/**
* Result for the Wilcoxon signed-rank test.
*
* <p>This class is immutable.
*
* @since 1.1
*/
public static final class Result extends BaseSignificanceResult {
/** Flag indicating the data had tied values. */
private final boolean tiedValues;
/** Flag indicating the data had zero values. */
private final boolean zeroValues;
/**
* Create an instance.
*
* @param statistic Test statistic.
* @param tiedValues Flag indicating the data had tied values.
* @param zeroValues Flag indicating the data had zero values.
* @param p Result p-value.
*/
Result(double statistic, boolean tiedValues, boolean zeroValues, double p) {
super(statistic, p);
this.tiedValues = tiedValues;
this.zeroValues = zeroValues;
}
/**
* Return {@code true} if the data had tied values (with equal ranks).
*
* <p>Note: The exact computation cannot be used when there are tied values.
* The p-value uses the asymptotic approximation using a tie correction.
*
* @return {@code true} if there were tied values
*/
public boolean hasTiedValues() {
return tiedValues;
}
/**
* Return {@code true} if the data had zero values. This occurs when the differences between
* sample values matched the expected location shift: {@code z = x - y == mu}.
*
* <p>Note: The exact computation cannot be used when there are zero values.
* The p-value uses the asymptotic approximation.
*
* @return {@code true} if there were zero values
*/
public boolean hasZeroValues() {
return zeroValues;
}
}
/**
* @param alternative Alternative hypothesis.
* @param method P-value method.
* @param continuityCorrection true to perform continuity correction.
* @param mu Expected location shift.
*/
private WilcoxonSignedRankTest(AlternativeHypothesis alternative, PValueMethod method,
boolean continuityCorrection, double mu) {
this.alternative = alternative;
this.pValueMethod = method;
this.continuityCorrection = continuityCorrection;
this.mu = mu;
}
/**
* Return an instance using the default options.
*
* <ul>
* <li>{@link AlternativeHypothesis#TWO_SIDED}
* <li>{@link PValueMethod#AUTO}
* <li>{@link ContinuityCorrection#ENABLED}
* <li>{@linkplain #withMu(double) mu = 0}
* </ul>
*
* @return default instance
*/
public static WilcoxonSignedRankTest withDefaults() {
return DEFAULT;
}
/**
* Return an instance with the configured alternative hypothesis.
*
* @param v Value.
* @return an instance
*/
public WilcoxonSignedRankTest with(AlternativeHypothesis v) {
return new WilcoxonSignedRankTest(Objects.requireNonNull(v), pValueMethod, continuityCorrection, mu);
}
/**
* Return an instance with the configured p-value method.
*
* @param v Value.
* @return an instance
* @throws IllegalArgumentException if the value is not in the allowed options or is null
*/
public WilcoxonSignedRankTest with(PValueMethod v) {
return new WilcoxonSignedRankTest(alternative,
Arguments.checkOption(v, EnumSet.of(PValueMethod.AUTO, PValueMethod.EXACT, PValueMethod.ASYMPTOTIC)),
continuityCorrection, mu);
}
/**
* Return an instance with the configured continuity correction.
*
* <p>If {@code enabled}, adjust the Wilcoxon rank statistic by 0.5 towards the
* mean value when computing the z-statistic if a normal approximation is used
* to compute the p-value.
*
* @param v Value.
* @return an instance
*/
public WilcoxonSignedRankTest with(ContinuityCorrection v) {
return new WilcoxonSignedRankTest(alternative, pValueMethod,
Objects.requireNonNull(v) == ContinuityCorrection.ENABLED, mu);
}
/**
* Return an instance with the configured expected difference {@code mu}.
*
* @param v Value.
* @return an instance
* @throws IllegalArgumentException if the value is not finite
*/
public WilcoxonSignedRankTest withMu(double v) {
return new WilcoxonSignedRankTest(alternative, pValueMethod, continuityCorrection, Arguments.checkFinite(v));
}
/**
* Computes the Wilcoxon signed ranked statistic comparing the differences between
* sample values {@code z = x - y} to {@code mu}.
*
* <p>This method handles matching samples {@code z[i] == mu} (no difference)
* by including them in the ranking of samples but excludes them from the test statistic
* (<i>signed-rank zero procedure</i>).
*
* @param z Signed differences between sample values.
* @return Wilcoxon <i>positive-rank sum</i> statistic (W+)
* @throws IllegalArgumentException if {@code z} is zero-length; contains NaN values;
* or all differences are equal to the expected difference
* @see #withMu(double)
*/
public double statistic(double[] z) {
return computeStatistic(z, mu);
}
/**
* Computes the Wilcoxon signed ranked statistic comparing the differences between two related
* samples or repeated measurements on a single sample.
*
* <p>This method handles matching samples {@code x[i] - mu == y[i]} (no difference)
* by including them in the ranking of samples but excludes them from the test statistic
* (<i>signed-rank zero procedure</i>).
*
* <p>This method is functionally equivalent to creating an array of differences
* {@code z = x - y} and calling {@link #statistic(double[]) statistic(z)}; the
* implementation may use an optimised method to compute the differences and
* rank statistic if {@code mu != 0}.
*
* @param x First sample values.
* @param y Second sample values.
* @return Wilcoxon <i>positive-rank sum</i> statistic (W+)
* @throws IllegalArgumentException if {@code x} or {@code y} are zero-length; are not
* the same length; contain NaN values; or {@code x[i] == y[i]} for all data
* @see #withMu(double)
*/
public double statistic(double[] x, double[] y) {
checkSamples(x, y);
// Apply mu before creation of differences
final double[] z = calculateDifferences(mu, x, y);
return computeStatistic(z, 0);
}
/**
* Performs a Wilcoxon signed ranked statistic comparing the differences between
* sample values {@code z = x - y} to {@code mu}.
*
* <p>This method handles matching samples {@code z[i] == mu} (no difference)
* by including them in the ranking of samples but excludes them from the test statistic
* (<i>signed-rank zero procedure</i>).
*
* <p>The test is defined by the {@link AlternativeHypothesis}.
*
* <ul>
* <li>'two-sided': the distribution of the difference is not symmetric about {@code mu}.
* <li>'greater': the distribution of the difference is stochastically greater than a
* distribution symmetric about {@code mu}.
* <li>'less': the distribution of the difference is stochastically less than a distribution
* symmetric about {@code mu}.
* </ul>
*
* <p>If the p-value method is {@linkplain PValueMethod#AUTO auto} an exact p-value
* is computed if the samples contain less than 50 values; otherwise a normal
* approximation is used.
*
* <p>Computation of the exact p-value is only valid if there are no matching
* samples {@code z[i] == mu} and no tied ranks in the data; otherwise the
* p-value resorts to the asymptotic Cureton approximation using a tie
* correction and an optional continuity correction.
*
* <p><strong>Note: </strong>
* Computation of the exact p-value requires the
* sample size {@code <= 1023}. Exact computation requires tabulation of values
* not exceeding size {@code n(n+1)/2} and computes in Order(n*n/2). Maximum
* memory usage is approximately 4 MiB.
*
* @param z Differences between sample values.
* @return test result
* @throws IllegalArgumentException if {@code z} is zero-length; contains NaN values;
* or all differences are zero
* @see #withMu(double)
* @see #with(AlternativeHypothesis)
* @see #with(ContinuityCorrection)
*/
public Result test(double[] z) {
return computeTest(z, mu);
}
/**
* Performs a Wilcoxon signed ranked statistic comparing mean for two related
* samples or repeated measurements on a single sample.
*
* <p>This method handles matching samples {@code x[i] - mu == y[i]} (no difference)
* by including them in the ranking of samples but excludes them
* from the test statistic (<i>signed-rank zero procedure</i>).
*
* <p>This method is functionally equivalent to creating an array of differences
* {@code z = x - y} and calling {@link #test(double[]) test(z)}; the
* implementation may use an optimised method to compute the differences and
* rank statistic if {@code mu != 0}.
*
* @param x First sample values.
* @param y Second sample values.
* @return test result
* @throws IllegalArgumentException if {@code x} or {@code y} are zero-length; are not
* the same length; contain NaN values; or {@code x[i] - mu == y[i]} for all data
* @see #statistic(double[], double[])
* @see #test(double[])
*/
public Result test(double[] x, double[] y) {
checkSamples(x, y);
// Apply mu before creation of differences
final double[] z = calculateDifferences(mu, x, y);
return computeTest(z, 0);
}
/**
* Computes the Wilcoxon signed ranked statistic comparing the differences between
* sample values {@code z = x - y} to {@code mu}.
*
* @param z Signed differences between sample values.
* @param mu Expected difference.
* @return Wilcoxon <i>positive-rank sum</i> statistic (W+)
* @throws IllegalArgumentException if {@code z} is zero-length; contains NaN values;
* or all differences are equal to the expected difference
* @see #withMu(double)
*/
private static double computeStatistic(double[] z, double mu) {
Arguments.checkValuesRequiredSize(z.length, 1);
final double[] x = StatisticUtils.subtract(z, mu);
// Raises an error if all zeros
countZeros(x);
final double[] zAbs = calculateAbsoluteDifferences(x);
final double[] ranks = RANKING.apply(zAbs);
return calculateW(x, ranks);
}
/**
* Performs a Wilcoxon signed ranked statistic comparing the differences between
* sample values {@code z = x - y} to {@code mu}.
*
* @param z Differences between sample values.
* @param expectedMu Expected difference.
* @return test result
* @throws IllegalArgumentException if {@code z} is zero-length; contains NaN values;
* or all differences are zero
*/
private Result computeTest(double[] z, double expectedMu) {
// Computation as above. The ranks are required for tie correction.
Arguments.checkValuesRequiredSize(z.length, 1);
final double[] x = StatisticUtils.subtract(z, expectedMu);
// Raises an error if all zeros
final int zeros = countZeros(x);
final double[] zAbs = calculateAbsoluteDifferences(x);
final double[] ranks = RANKING.apply(zAbs);
final double wPlus = calculateW(x, ranks);
// Exact p has strict requirements for no zeros, no ties
final double c = calculateTieCorrection(ranks);
final boolean tiedValues = c != 0;
final int n = z.length;
// Exact p requires no ties and no zeros
final double p;
if (selectMethod(pValueMethod, n) == PValueMethod.EXACT && n <= EXACT_LIMIT && !tiedValues && zeros == 0) {
p = calculateExactPValue((int) wPlus, n, alternative);
} else {
p = calculateAsymptoticPValue(wPlus, n, zeros, c, alternative, continuityCorrection);
}
return new Result(wPlus, tiedValues, zeros != 0, p);
}
/**
* Ensures that the provided arrays fulfil the assumptions.
*
* @param x First sample.
* @param y Second sample.
* @throws IllegalArgumentException if {@code x} or {@code y} are zero-length; or do not
* have the same length
*/
private static void checkSamples(double[] x, double[] y) {
Arguments.checkValuesRequiredSize(x.length, 1);
Arguments.checkValuesRequiredSize(y.length, 1);
Arguments.checkValuesSizeMatch(x.length, y.length);
}
/**
* Calculates x[i] - mu - y[i] for all i.
*
* @param mu Expected difference.
* @param x First sample.
* @param y Second sample.
* @return z = x - y
*/
private static double[] calculateDifferences(double mu, double[] x, double[] y) {
final double[] z = new double[x.length];
for (int i = 0; i < x.length; ++i) {
z[i] = x[i] - mu - y[i];
}
return z;
}
/**
* Calculates |z[i]| for all i.
*
* @param z Sample.
* @return |z|
*/
private static double[] calculateAbsoluteDifferences(double[] z) {
final double[] zAbs = new double[z.length];
for (int i = 0; i < z.length; ++i) {
zAbs[i] = Math.abs(z[i]);
}
return zAbs;
}
/**
* Calculate the Wilcoxon <i>positive-rank sum</i> statistic.
*
* @param obs Observed signed value.
* @param ranks Ranks (including averages for ties).
* @return Wilcoxon <i>positive-rank sum</i> statistic (W+)
*/
private static double calculateW(final double[] obs, final double[] ranks) {
final Sum wPlus = Sum.create();
for (int i = 0; i < obs.length; ++i) {
// Must be positive (excludes zeros)
if (obs[i] > 0) {
wPlus.add(ranks[i]);
}
}
return wPlus.getAsDouble();
}
/**
* Count the number of zeros in the data.
*
* @param z Input data.
* @return the zero count
* @throws IllegalArgumentException if the data is all zeros
*/
private static int countZeros(final double[] z) {
int c = 0;
for (final double v : z) {
if (v == 0) {
c++;
}
}
if (c == z.length) {
throw new InferenceException("All signed differences are zero");
}
return c;
}
/**
* Calculate the tie correction.
* Destructively modifies ranks (by sorting).
* <pre>
* c = sum(t^3 - t)
* </pre>
* <p>where t is the size of each group of tied observations.
*
* @param ranks Ranks
* @return the tie correction
*/
static double calculateTieCorrection(double[] ranks) {
double c = 0;
int ties = 1;
Arrays.sort(ranks);
double last = Double.NaN;
for (final double rank : ranks) {
// Deliberate use of equals
if (last == rank) {
// Extend the tied group
ties++;
} else {
if (ties != 1) {
c += Math.pow(ties, 3) - ties;
ties = 1;
}
last = rank;
}
}
// Final ties count
c += Math.pow(ties, 3) - ties;
return c;
}
/**
* Select the method to compute the p-value.
*
* @param method P-value method.
* @param n Size of the data.
* @return p-value method.
*/
private static PValueMethod selectMethod(PValueMethod method, int n) {
return method == PValueMethod.AUTO && n <= AUTO_LIMIT ? PValueMethod.EXACT : method;
}
/**
* Compute the asymptotic p-value using the Cureton normal approximation. This
* corrects for zeros in the signed-rank zero procedure and/or ties corrected using
* the average method.
*
* @param wPlus Wilcoxon signed rank value (W+).
* @param n Number of subjects.
* @param z Count of number of zeros
* @param c Tie-correction
* @param alternative Alternative hypothesis.
* @param continuityCorrection true to use a continuity correction.
* @return two-sided asymptotic p-value
*/
private static double calculateAsymptoticPValue(double wPlus, int n, double z, double c,
AlternativeHypothesis alternative, boolean continuityCorrection) {
// E[W+] = n * (n + 1) / 4 - z * (z + 1) / 4
final double e = (n * (n + 1.0) - z * (z + 1.0)) * 0.25;
final double variance = ((n * (n + 1.0) * (2 * n + 1.0)) -
(z * (z + 1.0) * (2 * z + 1.0)) - c * 0.5) / 24;
double x = wPlus - e;
if (continuityCorrection) {
// +/- 0.5 is a continuity correction towards the expected.
if (alternative == AlternativeHypothesis.GREATER_THAN) {
x -= 0.5;
} else if (alternative == AlternativeHypothesis.LESS_THAN) {
x += 0.5;
} else {
// two-sided. Shift towards the expected of zero.
// Use of signum ignores x==0 (i.e. not copySign(0.5, z))
x -= Math.signum(x) * 0.5;
}
}
x /= Math.sqrt(variance);
final NormalDistribution standardNormal = NormalDistribution.of(0, 1);
if (alternative == AlternativeHypothesis.GREATER_THAN) {
return standardNormal.survivalProbability(x);
}
if (alternative == AlternativeHypothesis.LESS_THAN) {
return standardNormal.cumulativeProbability(x);
}
// two-sided
return 2 * standardNormal.survivalProbability(Math.abs(x));
}
/**
* Compute the exact p-value.
*
* <p>This computation requires that no zeros or ties are found in the data. The input
* value n is limited to 1023.
*
* @param w1 Wilcoxon signed rank value (W+, or W-).
* @param n Number of subjects.
* @param alternative Alternative hypothesis.
* @return exact p-value (two-sided, greater, or less using the options)
*/
private static double calculateExactPValue(int w1, int n, AlternativeHypothesis alternative) {
// T+ plus T- equals the sum of the ranks: n(n+1)/2
// Compute using the lower half.
// No overflow here if n <= 1023.
final int sum = n * (n + 1) / 2;
final int w2 = sum - w1;
// Return the correct side:
if (alternative == AlternativeHypothesis.GREATER_THAN) {
// sf(w1 - 1)
return sf(w1 - 1, w2 + 1, n);
}
if (alternative == AlternativeHypothesis.LESS_THAN) {
// cdf(w1)
return cdf(w1, w2, n);
}
// two-sided: 2 * sf(max(w1, w2) - 1) or 2 * cdf(min(w1, w2))
final double p = 2 * computeCdf(Math.min(w1, w2), n);
// Clip to range: [0, 1]
return Math.min(1, p);
}
/**
* Compute the cumulative density function of the Wilcoxon signed rank W+ statistic.
* The W- statistic is passed for convenience to exploit symmetry in the distribution.
*
* @param w1 Wilcoxon W+ statistic
* @param w2 Wilcoxon W- statistic
* @param n Number of subjects.
* @return {@code Pr(X <= k)}
*/
private static double cdf(int w1, int w2, int n) {
// Exploit symmetry. Note the distribution is discrete thus requiring (w2 - 1).
return w2 > w1 ?
computeCdf(w1, n) :
1 - computeCdf(w2 - 1, n);
}
/**
* Compute the survival function of the Wilcoxon signed rank W+ statistic.
* The W- statistic is passed for convenience to exploit symmetry in the distribution.
*
* @param w1 Wilcoxon W+ statistic
* @param w2 Wilcoxon W- statistic
* @param n Number of subjects.
* @return {@code Pr(X <= k)}
*/
private static double sf(int w1, int w2, int n) {
// Opposite of the CDF
return w2 > w1 ?
1 - computeCdf(w1, n) :
computeCdf(w2 - 1, n);
}
/**
* Compute the cumulative density function for the distribution of the Wilcoxon
* signed rank statistic. This is a discrete distribution and is only valid
* when no zeros or ties are found in the data.
*
* <p>This should be called with the lower of W+ or W- for computational efficiency.
* The input value n is limited to 1023.
*
* <p>Uses recursion to compute the density for {@code X <= t} and sums the values.
* See: https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test#Computing_the_null_distribution
*
* @param t Smallest Wilcoxon signed rank value (W+, or W-).
* @param n Number of subjects.
* @return {@code Pr(T <= t)}
*/
private static double computeCdf(int t, int n) {
// Currently limited to n=1023.
// Note:
// The limit for t is n(n+1)/2.
// The highest possible sum is bounded by the normalisation factor 2^n.
// n t sum support
// 31 [0, 496] < 2^31 int
// 63 [0, 2016] < 2^63 long
// 1023 [0, 523766] < 2^1023 double
if (t <= 0) {
// No recursion required
return t < 0 ? 0 : Math.scalb(1, -n);
}
// Define u_n(t) as the number of sign combinations for T = t
// Pr(T == t) = u_n(t) / 2^n
// Sum them to create the cumulative probability Pr(T <= t).
//
// Recursive formula:
// u_n(t) = u_{n-1}(t) + u_{n-1}(t-n)
// u_0(0) = 1
// u_0(t) = 0 : t != 0
// u_n(t) = 0 : t < 0 || t > n(n+1)/2
// Compute all u_n(t) up to t.
final double[] u = new double[t + 1];
// Initialize u_1(t) using base cases for recursion
u[0] = u[1] = 1;
// Each u_n(t) is created using the current correct values for u_{n-1}(t)
for (int nn = 2; nn < n + 1; nn++) {
// u[t] holds the correct value for u_{n-1}(t)
// u_n(t) = u_{n-1}(t) + u_{n-1}(t-n)
for (int tt = t; tt >= nn; tt--) {
u[tt] += u[tt - nn];
}
}
final double sum = Arrays.stream(u).sum();
// Finally divide by the number of possible sums: 2^n
return Math.scalb(sum, -n);
}
}