001/*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements.  See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License.  You may obtain a copy of the License at
008 *
009 *      http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017package org.apache.commons.math3.stat.inference;
018
019import org.apache.commons.math3.distribution.NormalDistribution;
020import org.apache.commons.math3.exception.ConvergenceException;
021import org.apache.commons.math3.exception.DimensionMismatchException;
022import org.apache.commons.math3.exception.MaxCountExceededException;
023import org.apache.commons.math3.exception.NoDataException;
024import org.apache.commons.math3.exception.NullArgumentException;
025import org.apache.commons.math3.exception.NumberIsTooLargeException;
026import org.apache.commons.math3.stat.ranking.NaNStrategy;
027import org.apache.commons.math3.stat.ranking.NaturalRanking;
028import org.apache.commons.math3.stat.ranking.TiesStrategy;
029import org.apache.commons.math3.util.FastMath;
030
031/**
032 * An implementation of the Wilcoxon signed-rank test.
033 *
034 * @version $Id: WilcoxonSignedRankTest.java 1416643 2012-12-03 19:37:14Z tn $
035 */
036public class WilcoxonSignedRankTest {
037
038    /** Ranking algorithm. */
039    private NaturalRanking naturalRanking;
040
041    /**
042     * Create a test instance where NaN's are left in place and ties get
043     * the average of applicable ranks. Use this unless you are very sure
044     * of what you are doing.
045     */
046    public WilcoxonSignedRankTest() {
047        naturalRanking = new NaturalRanking(NaNStrategy.FIXED,
048                TiesStrategy.AVERAGE);
049    }
050
051    /**
052     * Create a test instance using the given strategies for NaN's and ties.
053     * Only use this if you are sure of what you are doing.
054     *
055     * @param nanStrategy
056     *            specifies the strategy that should be used for Double.NaN's
057     * @param tiesStrategy
058     *            specifies the strategy that should be used for ties
059     */
060    public WilcoxonSignedRankTest(final NaNStrategy nanStrategy,
061                                  final TiesStrategy tiesStrategy) {
062        naturalRanking = new NaturalRanking(nanStrategy, tiesStrategy);
063    }
064
065    /**
066     * Ensures that the provided arrays fulfills the assumptions.
067     *
068     * @param x first sample
069     * @param y second sample
070     * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
071     * @throws NoDataException if {@code x} or {@code y} are zero-length.
072     * @throws DimensionMismatchException if {@code x} and {@code y} do not
073     * have the same length.
074     */
075    private void ensureDataConformance(final double[] x, final double[] y)
076        throws NullArgumentException, NoDataException, DimensionMismatchException {
077
078        if (x == null ||
079            y == null) {
080                throw new NullArgumentException();
081        }
082        if (x.length == 0 ||
083            y.length == 0) {
084            throw new NoDataException();
085        }
086        if (y.length != x.length) {
087            throw new DimensionMismatchException(y.length, x.length);
088        }
089    }
090
091    /**
092     * Calculates y[i] - x[i] for all i
093     *
094     * @param x first sample
095     * @param y second sample
096     * @return z = y - x
097     */
098    private double[] calculateDifferences(final double[] x, final double[] y) {
099
100        final double[] z = new double[x.length];
101
102        for (int i = 0; i < x.length; ++i) {
103            z[i] = y[i] - x[i];
104        }
105
106        return z;
107    }
108
109    /**
110     * Calculates |z[i]| for all i
111     *
112     * @param z sample
113     * @return |z|
114     * @throws NullArgumentException if {@code z} is {@code null}
115     * @throws NoDataException if {@code z} is zero-length.
116     */
117    private double[] calculateAbsoluteDifferences(final double[] z)
118        throws NullArgumentException, NoDataException {
119
120        if (z == null) {
121            throw new NullArgumentException();
122        }
123
124        if (z.length == 0) {
125            throw new NoDataException();
126        }
127
128        final double[] zAbs = new double[z.length];
129
130        for (int i = 0; i < z.length; ++i) {
131            zAbs[i] = FastMath.abs(z[i]);
132        }
133
134        return zAbs;
135    }
136
137    /**
138     * Computes the <a
139     * href="http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test">
140     * Wilcoxon signed ranked statistic</a> comparing mean for two related
141     * samples or repeated measurements on a single sample.
142     * <p>
143     * This statistic can be used to perform a Wilcoxon signed ranked test
144     * evaluating the null hypothesis that the two related samples or repeated
145     * measurements on a single sample has equal mean.
146     * </p>
147     * <p>
148     * Let X<sub>i</sub> denote the i'th individual of the first sample and
149     * Y<sub>i</sub> the related i'th individual in the second sample. Let
150     * Z<sub>i</sub> = Y<sub>i</sub> - X<sub>i</sub>.
151     * </p>
152     * <p>
153     * <strong>Preconditions</strong>:
154     * <ul>
155     * <li>The differences Z<sub>i</sub> must be independent.</li>
156     * <li>Each Z<sub>i</sub> comes from a continuous population (they must be
157     * identical) and is symmetric about a common median.</li>
158     * <li>The values that X<sub>i</sub> and Y<sub>i</sub> represent are
159     * ordered, so the comparisons greater than, less than, and equal to are
160     * meaningful.</li>
161     * </ul>
162     * </p>
163     *
164     * @param x the first sample
165     * @param y the second sample
166     * @return wilcoxonSignedRank statistic (the larger of W+ and W-)
167     * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
168     * @throws NoDataException if {@code x} or {@code y} are zero-length.
169     * @throws DimensionMismatchException if {@code x} and {@code y} do not
170     * have the same length.
171     */
172    public double wilcoxonSignedRank(final double[] x, final double[] y)
173        throws NullArgumentException, NoDataException, DimensionMismatchException {
174
175        ensureDataConformance(x, y);
176
177        // throws IllegalArgumentException if x and y are not correctly
178        // specified
179        final double[] z = calculateDifferences(x, y);
180        final double[] zAbs = calculateAbsoluteDifferences(z);
181
182        final double[] ranks = naturalRanking.rank(zAbs);
183
184        double Wplus = 0;
185
186        for (int i = 0; i < z.length; ++i) {
187            if (z[i] > 0) {
188                Wplus += ranks[i];
189            }
190        }
191
192        final int N = x.length;
193        final double Wminus = (((double) (N * (N + 1))) / 2.0) - Wplus;
194
195        return FastMath.max(Wplus, Wminus);
196    }
197
198    /**
199     * Algorithm inspired by
200     * http://www.fon.hum.uva.nl/Service/Statistics/Signed_Rank_Algorihms.html#C
201     * by Rob van Son, Institute of Phonetic Sciences & IFOTT,
202     * University of Amsterdam
203     *
204     * @param Wmax largest Wilcoxon signed rank value
205     * @param N number of subjects (corresponding to x.length)
206     * @return two-sided exact p-value
207     */
208    private double calculateExactPValue(final double Wmax, final int N) {
209
210        // Total number of outcomes (equal to 2^N but a lot faster)
211        final int m = 1 << N;
212
213        int largerRankSums = 0;
214
215        for (int i = 0; i < m; ++i) {
216            int rankSum = 0;
217
218            // Generate all possible rank sums
219            for (int j = 0; j < N; ++j) {
220
221                // (i >> j) & 1 extract i's j-th bit from the right
222                if (((i >> j) & 1) == 1) {
223                    rankSum += j + 1;
224                }
225            }
226
227            if (rankSum >= Wmax) {
228                ++largerRankSums;
229            }
230        }
231
232        /*
233         * largerRankSums / m gives the one-sided p-value, so it's multiplied
234         * with 2 to get the two-sided p-value
235         */
236        return 2 * ((double) largerRankSums) / ((double) m);
237    }
238
239    /**
240     * @param Wmin smallest Wilcoxon signed rank value
241     * @param N number of subjects (corresponding to x.length)
242     * @return two-sided asymptotic p-value
243     */
244    private double calculateAsymptoticPValue(final double Wmin, final int N) {
245
246        final double ES = (double) (N * (N + 1)) / 4.0;
247
248        /* Same as (but saves computations):
249         * final double VarW = ((double) (N * (N + 1) * (2*N + 1))) / 24;
250         */
251        final double VarS = ES * ((double) (2 * N + 1) / 6.0);
252
253        // - 0.5 is a continuity correction
254        final double z = (Wmin - ES - 0.5) / FastMath.sqrt(VarS);
255
256        // No try-catch or advertised exception because args are valid
257        final NormalDistribution standardNormal = new NormalDistribution(0, 1);
258
259        return 2*standardNormal.cumulativeProbability(z);
260    }
261
262    /**
263     * Returns the <i>observed significance level</i>, or <a href=
264     * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue">
265     * p-value</a>, associated with a <a
266     * href="http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test">
267     * Wilcoxon signed ranked statistic</a> comparing mean for two related
268     * samples or repeated measurements on a single sample.
269     * <p>
270     * Let X<sub>i</sub> denote the i'th individual of the first sample and
271     * Y<sub>i</sub> the related i'th individual in the second sample. Let
272     * Z<sub>i</sub> = Y<sub>i</sub> - X<sub>i</sub>.
273     * </p>
274     * <p>
275     * <strong>Preconditions</strong>:
276     * <ul>
277     * <li>The differences Z<sub>i</sub> must be independent.</li>
278     * <li>Each Z<sub>i</sub> comes from a continuous population (they must be
279     * identical) and is symmetric about a common median.</li>
280     * <li>The values that X<sub>i</sub> and Y<sub>i</sub> represent are
281     * ordered, so the comparisons greater than, less than, and equal to are
282     * meaningful.</li>
283     * </ul>
284     * </p>
285     *
286     * @param x the first sample
287     * @param y the second sample
288     * @param exactPValue
289     *            if the exact p-value is wanted (only works for x.length <= 30,
290     *            if true and x.length > 30, this is ignored because
291     *            calculations may take too long)
292     * @return p-value
293     * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
294     * @throws NoDataException if {@code x} or {@code y} are zero-length.
295     * @throws DimensionMismatchException if {@code x} and {@code y} do not
296     * have the same length.
297     * @throws NumberIsTooLargeException if {@code exactPValue} is {@code true}
298     * and {@code x.length} > 30
299     * @throws ConvergenceException if the p-value can not be computed due to
300     * a convergence error
301     * @throws MaxCountExceededException if the maximum number of iterations
302     * is exceeded
303     */
304    public double wilcoxonSignedRankTest(final double[] x, final double[] y,
305                                         final boolean exactPValue)
306        throws NullArgumentException, NoDataException, DimensionMismatchException,
307        NumberIsTooLargeException, ConvergenceException, MaxCountExceededException {
308
309        ensureDataConformance(x, y);
310
311        final int N = x.length;
312        final double Wmax = wilcoxonSignedRank(x, y);
313
314        if (exactPValue && N > 30) {
315            throw new NumberIsTooLargeException(N, 30, true);
316        }
317
318        if (exactPValue) {
319            return calculateExactPValue(Wmax, N);
320        } else {
321            final double Wmin = ( (double)(N*(N+1)) / 2.0 ) - Wmax;
322            return calculateAsymptoticPValue(Wmin, N);
323        }
324    }
325}