277 lines
8.5 KiB
Go
277 lines
8.5 KiB
Go
// Copyright 2015 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package stats
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import (
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"math"
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"sort"
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"github.com/aclements/go-moremath/mathx"
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)
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// A LocationHypothesis specifies the alternative hypothesis of a
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// location test such as a t-test or a Mann-Whitney U-test. The
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// default (zero) value is to test against the alternative hypothesis
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// that they differ.
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type LocationHypothesis int
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//go:generate stringer -type LocationHypothesis
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const (
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// LocationLess specifies the alternative hypothesis that the
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// location of the first sample is less than the second. This
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// is a one-tailed test.
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LocationLess LocationHypothesis = -1
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// LocationDiffers specifies the alternative hypothesis that
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// the locations of the two samples are not equal. This is a
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// two-tailed test.
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LocationDiffers LocationHypothesis = 0
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// LocationGreater specifies the alternative hypothesis that
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// the location of the first sample is greater than the
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// second. This is a one-tailed test.
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LocationGreater LocationHypothesis = 1
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)
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// A MannWhitneyUTestResult is the result of a Mann-Whitney U-test.
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type MannWhitneyUTestResult struct {
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// N1 and N2 are the sizes of the input samples.
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N1, N2 int
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// U is the value of the Mann-Whitney U statistic for this
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// test, generalized by counting ties as 0.5.
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//
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// Given the Cartesian product of the two samples, this is the
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// number of pairs in which the value from the first sample is
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// greater than the value of the second, plus 0.5 times the
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// number of pairs where the values from the two samples are
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// equal. Hence, U is always an integer multiple of 0.5 (it is
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// a whole integer if there are no ties) in the range [0, N1*N2].
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//
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// U statistics always come in pairs, depending on which
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// sample is "first". The mirror U for the other sample can be
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// calculated as N1*N2 - U.
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//
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// There are many equivalent statistics with slightly
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// different definitions. The Wilcoxon (1945) W statistic
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// (generalized for ties) is U + (N1(N1+1))/2. It is also
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// common to use 2U to eliminate the half steps and Smid
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// (1956) uses N1*N2 - 2U to additionally center the
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// distribution.
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U float64
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// AltHypothesis specifies the alternative hypothesis tested
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// by this test against the null hypothesis that there is no
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// difference in the locations of the samples.
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AltHypothesis LocationHypothesis
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// P is the p-value of the Mann-Whitney test for the given
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// null hypothesis.
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P float64
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}
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// MannWhitneyExactLimit gives the largest sample size for which the
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// exact U distribution will be used for the Mann-Whitney U-test.
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//
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// Using the exact distribution is necessary for small sample sizes
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// because the distribution is highly irregular. However, computing
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// the distribution for large sample sizes is both computationally
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// expensive and unnecessary because it quickly approaches a normal
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// approximation. Computing the distribution for two 50 value samples
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// takes a few milliseconds on a 2014 laptop.
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var MannWhitneyExactLimit = 50
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// MannWhitneyTiesExactLimit gives the largest sample size for which
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// the exact U distribution will be used for the Mann-Whitney U-test
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// in the presence of ties.
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//
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// Computing this distribution is more expensive than computing the
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// distribution without ties, so this is set lower. Computing this
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// distribution for two 25 value samples takes about ten milliseconds
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// on a 2014 laptop.
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var MannWhitneyTiesExactLimit = 25
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// MannWhitneyUTest performs a Mann-Whitney U-test [1,2] of the null
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// hypothesis that two samples come from the same population against
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// the alternative hypothesis that one sample tends to have larger or
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// smaller values than the other.
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//
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// This is similar to a t-test, but unlike the t-test, the
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// Mann-Whitney U-test is non-parametric (it does not assume a normal
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// distribution). It has very slightly lower efficiency than the
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// t-test on normal distributions.
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//
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// Computing the exact U distribution is expensive for large sample
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// sizes, so this uses a normal approximation for sample sizes larger
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// than MannWhitneyExactLimit if there are no ties or
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// MannWhitneyTiesExactLimit if there are ties. This normal
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// approximation uses both the tie correction and the continuity
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// correction.
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//
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// This can fail with ErrSampleSize if either sample is empty or
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// ErrSamplesEqual if all sample values are equal.
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//
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// This is also known as a Mann-Whitney-Wilcoxon test and is
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// equivalent to the Wilcoxon rank-sum test, though the Wilcoxon
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// rank-sum test differs in nomenclature.
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//
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// [1] Mann, Henry B.; Whitney, Donald R. (1947). "On a Test of
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// Whether one of Two Random Variables is Stochastically Larger than
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// the Other". Annals of Mathematical Statistics 18 (1): 50–60.
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//
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// [2] Klotz, J. H. (1966). "The Wilcoxon, Ties, and the Computer".
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// Journal of the American Statistical Association 61 (315): 772-787.
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func MannWhitneyUTest(x1, x2 []float64, alt LocationHypothesis) (*MannWhitneyUTestResult, error) {
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n1, n2 := len(x1), len(x2)
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if n1 == 0 || n2 == 0 {
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return nil, ErrSampleSize
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}
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// Compute the U statistic and tie vector T.
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x1 = append([]float64(nil), x1...)
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x2 = append([]float64(nil), x2...)
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sort.Float64s(x1)
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sort.Float64s(x2)
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merged, labels := labeledMerge(x1, x2)
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R1 := 0.0
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T, hasTies := []int{}, false
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for i := 0; i < len(merged); {
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rank1, nx1, v1 := i+1, 0, merged[i]
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// Consume samples that tie this sample (including itself).
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for ; i < len(merged) && merged[i] == v1; i++ {
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if labels[i] == 1 {
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nx1++
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}
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}
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// Assign all tied samples the average rank of the
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// samples, where merged[0] has rank 1.
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if nx1 != 0 {
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rank := float64(i+rank1) / 2
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R1 += rank * float64(nx1)
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}
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T = append(T, i-rank1+1)
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if i > rank1 {
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hasTies = true
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}
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}
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U1 := R1 - float64(n1*(n1+1))/2
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// Compute the smaller of U1 and U2
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U2 := float64(n1*n2) - U1
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Usmall := math.Min(U1, U2)
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var p float64
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if !hasTies && n1 <= MannWhitneyExactLimit && n2 <= MannWhitneyExactLimit ||
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hasTies && n1 <= MannWhitneyTiesExactLimit && n2 <= MannWhitneyTiesExactLimit {
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// Use exact U distribution. U1 will be an integer.
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if len(T) == 1 {
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// All values are equal. Test is meaningless.
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return nil, ErrSamplesEqual
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}
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dist := UDist{N1: n1, N2: n2, T: T}
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switch alt {
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case LocationDiffers:
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if U1 == U2 {
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// The distribution is symmetric about
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// Usmall. Since the distribution is
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// discrete, the CDF is discontinuous
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// and if simply double CDF(Usmall),
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// we'll double count the
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// (non-infinitesimal) probability
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// mass at Usmall. What we want is
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// just the integral of the whole CDF,
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// which is 1.
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p = 1
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} else {
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p = dist.CDF(Usmall) * 2
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}
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case LocationLess:
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p = dist.CDF(U1)
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case LocationGreater:
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p = 1 - dist.CDF(U1-1)
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}
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} else {
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// Use normal approximation (with tie and continuity
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// correction).
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t := tieCorrection(T)
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N := float64(n1 + n2)
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μ_U := float64(n1*n2) / 2
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σ_U := math.Sqrt(float64(n1*n2) * ((N + 1) - t/(N*(N-1))) / 12)
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if σ_U == 0 {
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return nil, ErrSamplesEqual
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}
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numer := U1 - μ_U
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// Perform continuity correction.
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switch alt {
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case LocationDiffers:
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numer -= mathx.Sign(numer) * 0.5
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case LocationLess:
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numer += 0.5
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case LocationGreater:
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numer -= 0.5
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}
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z := numer / σ_U
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switch alt {
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case LocationDiffers:
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p = 2 * math.Min(StdNormal.CDF(z), 1-StdNormal.CDF(z))
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case LocationLess:
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p = StdNormal.CDF(z)
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case LocationGreater:
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p = 1 - StdNormal.CDF(z)
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}
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}
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return &MannWhitneyUTestResult{N1: n1, N2: n2, U: U1,
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AltHypothesis: alt, P: p}, nil
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}
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// labeledMerge merges sorted lists x1 and x2 into sorted list merged.
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// labels[i] is 1 or 2 depending on whether merged[i] is a value from
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// x1 or x2, respectively.
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func labeledMerge(x1, x2 []float64) (merged []float64, labels []byte) {
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merged = make([]float64, len(x1)+len(x2))
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labels = make([]byte, len(x1)+len(x2))
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i, j, o := 0, 0, 0
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for i < len(x1) && j < len(x2) {
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if x1[i] < x2[j] {
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merged[o] = x1[i]
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labels[o] = 1
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i++
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} else {
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merged[o] = x2[j]
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labels[o] = 2
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j++
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}
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o++
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}
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for ; i < len(x1); i++ {
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merged[o] = x1[i]
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labels[o] = 1
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o++
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}
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for ; j < len(x2); j++ {
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merged[o] = x2[j]
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labels[o] = 2
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o++
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}
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return
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}
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// tieCorrection computes the tie correction factor Σ_j (t_j³ - t_j)
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// where t_j is the number of ties in the j'th rank.
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func tieCorrection(ties []int) float64 {
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t := 0
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for _, tie := range ties {
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t += tie*tie*tie - tie
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}
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return float64(t)
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}
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