/// <summary>
/// Tests whether two samples comes from the
/// same distribution without assuming normality.
/// </summary>
///
/// <param name="sample1">The first sample.</param>
/// <param name="sample2">The second sample.</param>
/// <param name="alternate">The alternative hypothesis (research hypothesis) to test.</param>
/// <param name="exact">True to compute the exact distribution. May require a significant
/// amount of processing power for large samples (n > 12). If left at null, whether to
/// compute the exact or approximate distribution will depend on the number of samples.
/// Default is null.</param>
/// <param name="adjustForTies">Whether to account for ties when computing the
/// rank statistics or not. Default is true.</param>
///
public MannWhitneyWilcoxonTest(double[] sample1, double[] sample2,
TwoSampleHypothesis alternate = TwoSampleHypothesis.ValuesAreDifferent,
bool?exact = null, bool adjustForTies = true)
{
this.NumberOfSamples1 = sample1.Length;
this.NumberOfSamples2 = sample2.Length;
int n = NumberOfSamples1 + NumberOfSamples2;
// Concatenate both samples and rank them
double[] samples = sample1.Concatenate(sample2);
double[] rank = samples.Rank(adjustForTies: true);
// Split the rankings back and sum
Rank1 = rank.Get(0, NumberOfSamples1);
Rank2 = rank.Get(NumberOfSamples1, 0);
// Compute rank sum statistic
this.RankSum1 = Rank1.Sum();
this.RankSum2 = Rank2.Sum();
// It is not necessary to compute the sum for both ranks. The sum of ranks in the second
// sample can be determined from the first, since the sum of all the ranks equals n(n+1)/2
Accord.Diagnostics.Debug.Assert(RankSum2 == n * (n + 1) / 2 - RankSum1);
// The U statistic can be obtained from the sum of the ranks in the sample,
// minus the smallest value it can take (i.e. minus (n1 * (n1 + 1)) / 2.0),
// meaning there is an wasy way to convert from W to U:
// Compute Mann-Whitney's U statistic from the rank sum
// as in Zar, Jerrold H. Biostatistical Analysis, 1998:
this.Statistic1 = RankSum1 - (NumberOfSamples1 * (NumberOfSamples1 + 1)) / 2.0; // U1
this.Statistic2 = RankSum2 - (NumberOfSamples2 * (NumberOfSamples2 + 1)) / 2.0; // U2
// Again, it would not be necessary to compute U2 due the relation:
Accord.Diagnostics.Debug.Assert(Statistic1 + Statistic2 == NumberOfSamples1 * NumberOfSamples2);
Accord.Diagnostics.Debug.Assert(Statistic1 == MannWhitneyDistribution.MannWhitneyU(Rank1));
Accord.Diagnostics.Debug.Assert(Statistic2 == MannWhitneyDistribution.MannWhitneyU(Rank2));
// http://users.sussex.ac.uk/~grahamh/RM1web/WilcoxonHandoout2011.pdf
// https://onlinecourses.science.psu.edu/stat464/node/38
// http://www.real-statistics.com/non-parametric-tests/wilcoxon-rank-sum-test/
// http://personal.vu.nl/R.Heijungs/QM/201516/stat/bs/documents/Supplement%2016B%20-%20Wilcoxon%20Mann-Whitney%20Small%20Sample%20Test.pdf
// http://www.real-statistics.com/non-parametric-tests/wilcoxon-rank-sum-test/wilcoxon-rank-sum-exact-test/
// The smaller value of U1 and U2 is the one used when using significance tables
this.Statistic = (NumberOfSamples1 < NumberOfSamples2) ? Statistic1 : Statistic2;
this.Hypothesis = alternate;
this.Tail = (DistributionTail)alternate;
this.StatisticDistribution = new MannWhitneyDistribution(Rank1, Rank2, exact)
{
Correction = (Tail == DistributionTail.TwoTail) ? ContinuityCorrection.Midpoint : ContinuityCorrection.KeepInside
};
this.PValue = StatisticToPValue(Statistic);
this.OnSizeChanged();
}
/// <summary>
/// Tests whether two samples comes from the
/// same distribution without assuming normality.
/// </summary>
///
/// <param name="sample1">The first sample.</param>
/// <param name="sample2">The second sample.</param>
/// <param name="alternate">The alternative hypothesis (research hypothesis) to test.</param>
///
public MannWhitneyWilcoxonTest(double[] sample1, double[] sample2,
TwoSampleHypothesis alternate = TwoSampleHypothesis.ValuesAreDifferent)
{
int n1 = sample1.Length;
int n2 = sample2.Length;
int n = n1 + n2;
// Concatenate both samples and rank them
double[] samples = sample1.Concatenate(sample2);
double[] rank = Accord.Statistics.Tools.Rank(samples);
// Split the rankings back and sum
Rank1 = rank.Submatrix(0, n1 - 1);
Rank2 = rank.Submatrix(n1, n - 1);
double t1 = RankSum1 = Rank1.Sum();
double t2 = RankSum2 = Rank2.Sum();
// Estimated values for t under the null
double t1max = n1 * n2 + (n1 * (n1 + 1)) / 2.0;
double t2max = n1 * n2 + (n2 * (n2 + 1)) / 2.0;
// Diff in observed t and estimated t
double u1 = Statistic1 = t1max - t1;
double u2 = Statistic2 = t2max - t2;
double hypothesizedValue = (n1 * n2) / 2.0;
Compute(u1, rank, n1, n2, alternate);
}
public override string ToString()
{
return(string.Format("{0}{0}{0} {1} {2}", ThreeRank.ToDisplay(), Rank1.ToDisplay(), Rank2.ToDisplay()));
}
public override string ToString()
{
return(string.Format("{0}{0} {1} {2} {3}", PairRank.ToDisplay(), Rank1.ToDisplay(), Rank2.ToDisplay(), Rank3.ToDisplay()));
}