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.MaxCountExceededException;
022import org.apache.commons.math3.exception.NoDataException;
023import org.apache.commons.math3.exception.NullArgumentException;
024import org.apache.commons.math3.stat.ranking.NaNStrategy;
025import org.apache.commons.math3.stat.ranking.NaturalRanking;
026import org.apache.commons.math3.stat.ranking.TiesStrategy;
027import org.apache.commons.math3.util.FastMath;
028
029/**
030 * An implementation of the Mann-Whitney U test (also called Wilcoxon rank-sum test).
031 *
032 * @version $Id: MannWhitneyUTest.java 1416643 2012-12-03 19:37:14Z tn $
033 */
034public class MannWhitneyUTest {
035
036    /** Ranking algorithm. */
037    private NaturalRanking naturalRanking;
038
039    /**
040     * Create a test instance using where NaN's are left in place and ties get
041     * the average of applicable ranks. Use this unless you are very sure of
042     * what you are doing.
043     */
044    public MannWhitneyUTest() {
045        naturalRanking = new NaturalRanking(NaNStrategy.FIXED,
046                TiesStrategy.AVERAGE);
047    }
048
049    /**
050     * Create a test instance using the given strategies for NaN's and ties.
051     * Only use this if you are sure of what you are doing.
052     *
053     * @param nanStrategy
054     *            specifies the strategy that should be used for Double.NaN's
055     * @param tiesStrategy
056     *            specifies the strategy that should be used for ties
057     */
058    public MannWhitneyUTest(final NaNStrategy nanStrategy,
059                            final TiesStrategy tiesStrategy) {
060        naturalRanking = new NaturalRanking(nanStrategy, tiesStrategy);
061    }
062
063    /**
064     * Ensures that the provided arrays fulfills the assumptions.
065     *
066     * @param x first sample
067     * @param y second sample
068     * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
069     * @throws NoDataException if {@code x} or {@code y} are zero-length.
070     */
071    private void ensureDataConformance(final double[] x, final double[] y)
072        throws NullArgumentException, NoDataException {
073
074        if (x == null ||
075            y == null) {
076            throw new NullArgumentException();
077        }
078        if (x.length == 0 ||
079            y.length == 0) {
080            throw new NoDataException();
081        }
082    }
083
084    /** Concatenate the samples into one array.
085     * @param x first sample
086     * @param y second sample
087     * @return concatenated array
088     */
089    private double[] concatenateSamples(final double[] x, final double[] y) {
090        final double[] z = new double[x.length + y.length];
091
092        System.arraycopy(x, 0, z, 0, x.length);
093        System.arraycopy(y, 0, z, x.length, y.length);
094
095        return z;
096    }
097
098    /**
099     * Computes the <a
100     * href="http://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U"> Mann-Whitney
101     * U statistic</a> comparing mean for two independent samples possibly of
102     * different length.
103     * <p>
104     * This statistic can be used to perform a Mann-Whitney U test evaluating
105     * the null hypothesis that the two independent samples has equal mean.
106     * </p>
107     * <p>
108     * Let X<sub>i</sub> denote the i'th individual of the first sample and
109     * Y<sub>j</sub> the j'th individual in the second sample. Note that the
110     * samples would often have different length.
111     * </p>
112     * <p>
113     * <strong>Preconditions</strong>:
114     * <ul>
115     * <li>All observations in the two samples are independent.</li>
116     * <li>The observations are at least ordinal (continuous are also ordinal).</li>
117     * </ul>
118     * </p>
119     *
120     * @param x the first sample
121     * @param y the second sample
122     * @return Mann-Whitney U statistic (maximum of U<sup>x</sup> and U<sup>y</sup>)
123     * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
124     * @throws NoDataException if {@code x} or {@code y} are zero-length.
125     */
126    public double mannWhitneyU(final double[] x, final double[] y)
127        throws NullArgumentException, NoDataException {
128
129        ensureDataConformance(x, y);
130
131        final double[] z = concatenateSamples(x, y);
132        final double[] ranks = naturalRanking.rank(z);
133
134        double sumRankX = 0;
135
136        /*
137         * The ranks for x is in the first x.length entries in ranks because x
138         * is in the first x.length entries in z
139         */
140        for (int i = 0; i < x.length; ++i) {
141            sumRankX += ranks[i];
142        }
143
144        /*
145         * U1 = R1 - (n1 * (n1 + 1)) / 2 where R1 is sum of ranks for sample 1,
146         * e.g. x, n1 is the number of observations in sample 1.
147         */
148        final double U1 = sumRankX - (x.length * (x.length + 1)) / 2;
149
150        /*
151         * It can be shown that U1 + U2 = n1 * n2
152         */
153        final double U2 = x.length * y.length - U1;
154
155        return FastMath.max(U1, U2);
156    }
157
158    /**
159     * @param Umin smallest Mann-Whitney U value
160     * @param n1 number of subjects in first sample
161     * @param n2 number of subjects in second sample
162     * @return two-sided asymptotic p-value
163     * @throws ConvergenceException if the p-value can not be computed
164     * due to a convergence error
165     * @throws MaxCountExceededException if the maximum number of
166     * iterations is exceeded
167     */
168    private double calculateAsymptoticPValue(final double Umin,
169                                             final int n1,
170                                             final int n2)
171        throws ConvergenceException, MaxCountExceededException {
172
173        /* long multiplication to avoid overflow (double not used due to efficiency
174         * and to avoid precision loss)
175         */
176        final long n1n2prod = (long) n1 * n2;
177
178        // http://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U#Normal_approximation
179        final double EU = n1n2prod / 2.0;
180        final double VarU = n1n2prod * (n1 + n2 + 1) / 12.0;
181
182        final double z = (Umin - EU) / FastMath.sqrt(VarU);
183
184        // No try-catch or advertised exception because args are valid
185        final NormalDistribution standardNormal = new NormalDistribution(0, 1);
186
187        return 2 * standardNormal.cumulativeProbability(z);
188    }
189
190    /**
191     * Returns the asymptotic <i>observed significance level</i>, or <a href=
192     * "http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue">
193     * p-value</a>, associated with a <a
194     * href="http://en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U"> Mann-Whitney
195     * U statistic</a> comparing mean for two independent samples.
196     * <p>
197     * Let X<sub>i</sub> denote the i'th individual of the first sample and
198     * Y<sub>j</sub> the j'th individual in the second sample. Note that the
199     * samples would often have different length.
200     * </p>
201     * <p>
202     * <strong>Preconditions</strong>:
203     * <ul>
204     * <li>All observations in the two samples are independent.</li>
205     * <li>The observations are at least ordinal (continuous are also ordinal).</li>
206     * </ul>
207     * </p><p>
208     * Ties give rise to biased variance at the moment. See e.g. <a
209     * href="http://mlsc.lboro.ac.uk/resources/statistics/Mannwhitney.pdf"
210     * >http://mlsc.lboro.ac.uk/resources/statistics/Mannwhitney.pdf</a>.</p>
211     *
212     * @param x the first sample
213     * @param y the second sample
214     * @return asymptotic p-value
215     * @throws NullArgumentException if {@code x} or {@code y} are {@code null}.
216     * @throws NoDataException if {@code x} or {@code y} are zero-length.
217     * @throws ConvergenceException if the p-value can not be computed due to a
218     * convergence error
219     * @throws MaxCountExceededException if the maximum number of iterations
220     * is exceeded
221     */
222    public double mannWhitneyUTest(final double[] x, final double[] y)
223        throws NullArgumentException, NoDataException,
224        ConvergenceException, MaxCountExceededException {
225
226        ensureDataConformance(x, y);
227
228        final double Umax = mannWhitneyU(x, y);
229
230        /*
231         * It can be shown that U1 + U2 = n1 * n2
232         */
233        final double Umin = x.length * y.length - Umax;
234
235        return calculateAsymptoticPValue(Umin, x.length, y.length);
236    }
237
238}