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 */ 035public class WilcoxonSignedRankTest { 036 037 /** Ranking algorithm. */ 038 private NaturalRanking naturalRanking; 039 040 /** 041 * Create a test instance where NaN's are left in place and ties get 042 * the average of applicable ranks. Use this unless you are very sure 043 * of what you are doing. 044 */ 045 public WilcoxonSignedRankTest() { 046 naturalRanking = new NaturalRanking(NaNStrategy.FIXED, 047 TiesStrategy.AVERAGE); 048 } 049 050 /** 051 * Create a test instance using the given strategies for NaN's and ties. 052 * Only use this if you are sure of what you are doing. 053 * 054 * @param nanStrategy 055 * specifies the strategy that should be used for Double.NaN's 056 * @param tiesStrategy 057 * specifies the strategy that should be used for ties 058 */ 059 public WilcoxonSignedRankTest(final NaNStrategy nanStrategy, 060 final TiesStrategy tiesStrategy) { 061 naturalRanking = new NaturalRanking(nanStrategy, tiesStrategy); 062 } 063 064 /** 065 * Ensures that the provided arrays fulfills the assumptions. 066 * 067 * @param x first sample 068 * @param y second sample 069 * @throws NullArgumentException if {@code x} or {@code y} are {@code null}. 070 * @throws NoDataException if {@code x} or {@code y} are zero-length. 071 * @throws DimensionMismatchException if {@code x} and {@code y} do not 072 * have the same length. 073 */ 074 private void ensureDataConformance(final double[] x, final double[] y) 075 throws NullArgumentException, NoDataException, DimensionMismatchException { 076 077 if (x == null || 078 y == null) { 079 throw new NullArgumentException(); 080 } 081 if (x.length == 0 || 082 y.length == 0) { 083 throw new NoDataException(); 084 } 085 if (y.length != x.length) { 086 throw new DimensionMismatchException(y.length, x.length); 087 } 088 } 089 090 /** 091 * Calculates y[i] - x[i] for all i 092 * 093 * @param x first sample 094 * @param y second sample 095 * @return z = y - x 096 */ 097 private double[] calculateDifferences(final double[] x, final double[] y) { 098 099 final double[] z = new double[x.length]; 100 101 for (int i = 0; i < x.length; ++i) { 102 z[i] = y[i] - x[i]; 103 } 104 105 return z; 106 } 107 108 /** 109 * Calculates |z[i]| for all i 110 * 111 * @param z sample 112 * @return |z| 113 * @throws NullArgumentException if {@code z} is {@code null} 114 * @throws NoDataException if {@code z} is zero-length. 115 */ 116 private double[] calculateAbsoluteDifferences(final double[] z) 117 throws NullArgumentException, NoDataException { 118 119 if (z == null) { 120 throw new NullArgumentException(); 121 } 122 123 if (z.length == 0) { 124 throw new NoDataException(); 125 } 126 127 final double[] zAbs = new double[z.length]; 128 129 for (int i = 0; i < z.length; ++i) { 130 zAbs[i] = FastMath.abs(z[i]); 131 } 132 133 return zAbs; 134 } 135 136 /** 137 * Computes the <a 138 * href="http://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test"> 139 * Wilcoxon signed ranked statistic</a> comparing mean for two related 140 * samples or repeated measurements on a single sample. 141 * <p> 142 * This statistic can be used to perform a Wilcoxon signed ranked test 143 * evaluating the null hypothesis that the two related samples or repeated 144 * measurements on a single sample has equal mean. 145 * </p> 146 * <p> 147 * Let X<sub>i</sub> denote the i'th individual of the first sample and 148 * Y<sub>i</sub> the related i'th individual in the second sample. Let 149 * Z<sub>i</sub> = Y<sub>i</sub> - X<sub>i</sub>. 150 * </p> 151 * <p> 152 * <strong>Preconditions</strong>: 153 * <ul> 154 * <li>The differences Z<sub>i</sub> must be independent.</li> 155 * <li>Each Z<sub>i</sub> comes from a continuous population (they must be 156 * identical) and is symmetric about a common median.</li> 157 * <li>The values that X<sub>i</sub> and Y<sub>i</sub> represent are 158 * ordered, so the comparisons greater than, less than, and equal to are 159 * meaningful.</li> 160 * </ul> 161 * </p> 162 * 163 * @param x the first sample 164 * @param y the second sample 165 * @return wilcoxonSignedRank statistic (the larger of W+ and W-) 166 * @throws NullArgumentException if {@code x} or {@code y} are {@code null}. 167 * @throws NoDataException if {@code x} or {@code y} are zero-length. 168 * @throws DimensionMismatchException if {@code x} and {@code y} do not 169 * have the same length. 170 */ 171 public double wilcoxonSignedRank(final double[] x, final double[] y) 172 throws NullArgumentException, NoDataException, DimensionMismatchException { 173 174 ensureDataConformance(x, y); 175 176 // throws IllegalArgumentException if x and y are not correctly 177 // specified 178 final double[] z = calculateDifferences(x, y); 179 final double[] zAbs = calculateAbsoluteDifferences(z); 180 181 final double[] ranks = naturalRanking.rank(zAbs); 182 183 double Wplus = 0; 184 185 for (int i = 0; i < z.length; ++i) { 186 if (z[i] > 0) { 187 Wplus += ranks[i]; 188 } 189 } 190 191 final int N = x.length; 192 final double Wminus = (((double) (N * (N + 1))) / 2.0) - Wplus; 193 194 return FastMath.max(Wplus, Wminus); 195 } 196 197 /** 198 * Algorithm inspired by 199 * http://www.fon.hum.uva.nl/Service/Statistics/Signed_Rank_Algorihms.html#C 200 * by Rob van Son, Institute of Phonetic Sciences & IFOTT, 201 * University of Amsterdam 202 * 203 * @param Wmax largest Wilcoxon signed rank value 204 * @param N number of subjects (corresponding to x.length) 205 * @return two-sided exact p-value 206 */ 207 private double calculateExactPValue(final double Wmax, final int N) { 208 209 // Total number of outcomes (equal to 2^N but a lot faster) 210 final int m = 1 << N; 211 212 int largerRankSums = 0; 213 214 for (int i = 0; i < m; ++i) { 215 int rankSum = 0; 216 217 // Generate all possible rank sums 218 for (int j = 0; j < N; ++j) { 219 220 // (i >> j) & 1 extract i's j-th bit from the right 221 if (((i >> j) & 1) == 1) { 222 rankSum += j + 1; 223 } 224 } 225 226 if (rankSum >= Wmax) { 227 ++largerRankSums; 228 } 229 } 230 231 /* 232 * largerRankSums / m gives the one-sided p-value, so it's multiplied 233 * with 2 to get the two-sided p-value 234 */ 235 return 2 * ((double) largerRankSums) / ((double) m); 236 } 237 238 /** 239 * @param Wmin smallest Wilcoxon signed rank value 240 * @param N number of subjects (corresponding to x.length) 241 * @return two-sided asymptotic p-value 242 */ 243 private double calculateAsymptoticPValue(final double Wmin, final int N) { 244 245 final double ES = (double) (N * (N + 1)) / 4.0; 246 247 /* Same as (but saves computations): 248 * final double VarW = ((double) (N * (N + 1) * (2*N + 1))) / 24; 249 */ 250 final double VarS = ES * ((double) (2 * N + 1) / 6.0); 251 252 // - 0.5 is a continuity correction 253 final double z = (Wmin - ES - 0.5) / FastMath.sqrt(VarS); 254 255 // No try-catch or advertised exception because args are valid 256 // pass a null rng to avoid unneeded overhead as we will not sample from this distribution 257 final NormalDistribution standardNormal = new NormalDistribution(null, 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}