/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* 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
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.stat.inference;
import org.apache.commons.math3.distribution.NormalDistribution;
import org.apache.commons.math3.exception.ConvergenceException;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NoDataException;
import org.apache.commons.math3.exception.NullArgumentException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.stat.ranking.NaNStrategy;
import org.apache.commons.math3.stat.ranking.NaturalRanking;
import org.apache.commons.math3.stat.ranking.TiesStrategy;
import org.apache.commons.math3.util.FastMath;
/**
* An implementation of the Wilcoxon signed-rank test.
*
*/
public class WilcoxonSignedRankTest {
/** Ranking algorithm. */
private NaturalRanking naturalRanking;
/**
* Create a test instance where NaN's are left in place and ties get
* the average of applicable ranks. Use this unless you are very sure
* of what you are doing.
*/
public WilcoxonSignedRankTest() {
naturalRanking = new NaturalRanking(NaNStrategy.FIXED,
TiesStrategy.AVERAGE);
}
/**
* Create a test instance using the given strategies for NaN's and ties.
* Only use this if you are sure of what you are doing.
*
* @param nanStrategy
* specifies the strategy that should be used for Double.NaN's
* @param tiesStrategy
* specifies the strategy that should be used for ties
*/
public WilcoxonSignedRankTest(final NaNStrategy nanStrategy,
final TiesStrategy tiesStrategy) {
naturalRanking = new NaturalRanking(nanStrategy, tiesStrategy);
}
/**
* Ensures that the provided arrays fulfills the assumptions.
*
* @param x first sample
* @param y second sample
* @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
* @throws NoDataException if {@code x} or {@code y} are zero-length.
* @throws DimensionMismatchException if {@code x} and {@code y} do not
* have the same length.
*/
private void ensureDataConformance(final double[] x, final double[] y)
throws NullArgumentException, NoDataException, DimensionMismatchException {
if (x == null ||
y == null) {
throw new NullArgumentException();
}
if (x.length == 0 ||
y.length == 0) {
throw new NoDataException();
}
if (y.length != x.length) {
throw new DimensionMismatchException(y.length, x.length);
}
}
/**
* Calculates y[i] - x[i] for all i
*
* @param x first sample
* @param y second sample
* @return z = y - x
*/
private double[] calculateDifferences(final double[] x, final double[] y) {
final double[] z = new double[x.length];
for (int i = 0; i < x.length; ++i) {
z[i] = y[i] - x[i];
}
return z;
}
/**
* Calculates |z[i]| for all i
*
* @param z sample
* @return |z|
* @throws NullArgumentException if {@code z} is {@code null}
* @throws NoDataException if {@code z} is zero-length.
*/
private double[] calculateAbsoluteDifferences(final double[] z)
throws NullArgumentException, NoDataException {
if (z == null) {
throw new NullArgumentException();
}
if (z.length == 0) {
throw new NoDataException();
}
final double[] zAbs = new double[z.length];
for (int i = 0; i < z.length; ++i) {
zAbs[i] = FastMath.abs(z[i]);
}
return zAbs;
}
/**
* Computes the <a
* href="http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test">
* Wilcoxon signed ranked statistic</a> comparing mean for two related
* samples or repeated measurements on a single sample.
* <p>
* This statistic can be used to perform a Wilcoxon signed ranked test
* evaluating the null hypothesis that the two related samples or repeated
* measurements on a single sample has equal mean.
* </p>
* <p>
* Let X<sub>i</sub> denote the i'th individual of the first sample and
* Y<sub>i</sub> the related i'th individual in the second sample. Let
* Z<sub>i</sub> = Y<sub>i</sub> - X<sub>i</sub>.
* </p>
* <p>
* <strong>Preconditions</strong>:
* <ul>
* <li>The differences Z<sub>i</sub> must be independent.</li>
* <li>Each Z<sub>i</sub> comes from a continuous population (they must be
* identical) and is symmetric about a common median.</li>
* <li>The values that X<sub>i</sub> and Y<sub>i</sub> represent are
* ordered, so the comparisons greater than, less than, and equal to are
* meaningful.</li>
* </ul>
* </p>
*
* @param x the first sample
* @param y the second sample
* @return wilcoxonSignedRank statistic (the larger of W+ and W-)
* @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
* @throws NoDataException if {@code x} or {@code y} are zero-length.
* @throws DimensionMismatchException if {@code x} and {@code y} do not
* have the same length.
*/
public double wilcoxonSignedRank(final double[] x, final double[] y)
throws NullArgumentException, NoDataException, DimensionMismatchException {
ensureDataConformance(x, y);
// throws IllegalArgumentException if x and y are not correctly
// specified
final double[] z = calculateDifferences(x, y);
final double[] zAbs = calculateAbsoluteDifferences(z);
final double[] ranks = naturalRanking.rank(zAbs);
double Wplus = 0;
for (int i = 0; i < z.length; ++i) {
if (z[i] > 0) {
Wplus += ranks[i];
}
}
final int N = x.length;
final double Wminus = (((double) (N * (N + 1))) / 2.0) - Wplus;
return FastMath.max(Wplus, Wminus);
}
/**
* Algorithm inspired by
* http://www.fon.hum.uva.nl/Service/Statistics/Signed_Rank_Algorihms.html#C
* by Rob van Son, Institute of Phonetic Sciences & IFOTT,
* University of Amsterdam
*
* @param Wmax largest Wilcoxon signed rank value
* @param N number of subjects (corresponding to x.length)
* @return two-sided exact p-value
*/
private double calculateExactPValue(final double Wmax, final int N) {
// Total number of outcomes (equal to 2^N but a lot faster)
final int m = 1 << N;
int largerRankSums = 0;
for (int i = 0; i < m; ++i) {
int rankSum = 0;
// Generate all possible rank sums
for (int j = 0; j < N; ++j) {
// (i >> j) & 1 extract i's j-th bit from the right
if (((i >> j) & 1) == 1) {
rankSum += j + 1;
}
}
if (rankSum >= Wmax) {
++largerRankSums;
}
}
/*
* largerRankSums / m gives the one-sided p-value, so it's multiplied
* with 2 to get the two-sided p-value
*/
return 2 * ((double) largerRankSums) / ((double) m);
}
/**
* @param Wmin smallest Wilcoxon signed rank value
* @param N number of subjects (corresponding to x.length)
* @return two-sided asymptotic p-value
*/
private double calculateAsymptoticPValue(final double Wmin, final int N) {
final double ES = (double) (N * (N + 1)) / 4.0;
/* Same as (but saves computations):
* final double VarW = ((double) (N * (N + 1) * (2*N + 1))) / 24;
*/
final double VarS = ES * ((double) (2 * N + 1) / 6.0);
// - 0.5 is a continuity correction
final double z = (Wmin - ES - 0.5) / FastMath.sqrt(VarS);
// No try-catch or advertised exception because args are valid
// pass a null rng to avoid unneeded overhead as we will not sample from this distribution
final NormalDistribution standardNormal = new NormalDistribution(null, 0, 1);
return 2*standardNormal.cumulativeProbability(z);
}
/**
* 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 <a
* href="http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test">
* Wilcoxon signed ranked statistic</a> comparing mean for two related
* samples or repeated measurements on a single sample.
* <p>
* Let X<sub>i</sub> denote the i'th individual of the first sample and
* Y<sub>i</sub> the related i'th individual in the second sample. Let
* Z<sub>i</sub> = Y<sub>i</sub> - X<sub>i</sub>.
* </p>
* <p>
* <strong>Preconditions</strong>:
* <ul>
* <li>The differences Z<sub>i</sub> must be independent.</li>
* <li>Each Z<sub>i</sub> comes from a continuous population (they must be
* identical) and is symmetric about a common median.</li>
* <li>The values that X<sub>i</sub> and Y<sub>i</sub> represent are
* ordered, so the comparisons greater than, less than, and equal to are
* meaningful.</li>
* </ul>
* </p>
*
* @param x the first sample
* @param y the second sample
* @param exactPValue
* if the exact p-value is wanted (only works for x.length <= 30,
* if true and x.length > 30, this is ignored because
* calculations may take too long)
* @return p-value
* @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
* @throws NoDataException if {@code x} or {@code y} are zero-length.
* @throws DimensionMismatchException if {@code x} and {@code y} do not
* have the same length.
* @throws NumberIsTooLargeException if {@code exactPValue} is {@code true}
* and {@code x.length} > 30
* @throws ConvergenceException if the p-value can not be computed due to
* a convergence error
* @throws MaxCountExceededException if the maximum number of iterations
* is exceeded
*/
public double wilcoxonSignedRankTest(final double[] x, final double[] y,
final boolean exactPValue)
throws NullArgumentException, NoDataException, DimensionMismatchException,
NumberIsTooLargeException, ConvergenceException, MaxCountExceededException {
ensureDataConformance(x, y);
final int N = x.length;
final double Wmax = wilcoxonSignedRank(x, y);
if (exactPValue && N > 30) {
throw new NumberIsTooLargeException(N, 30, true);
}
if (exactPValue) {
return calculateExactPValue(Wmax, N);
} else {
final double Wmin = ( (double)(N*(N+1)) / 2.0 ) - Wmax;
return calculateAsymptoticPValue(Wmin, N);
}
}
}