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Statsample::Test::UMannWhitney

U Mann-Whitney test

Non-parametric test for assessing whether two independent samples of observations come from the same distribution.

Assumptions

Higher differences of distributions correspond to to lower values of U.

Constants

MAX_MN_EXACT

Max for m*n allowed for exact calculation of probability

Attributes

r1[R]

Sample 1 Rank sum

r2[R]

Sample 2 Rank sum

u1[R]

Sample 1 U (useful for demostration)

u2[R]

Sample 2 U (useful for demostration)

u[R]

U Value

t[R]

Value of compensation for ties (useful for demostration)

name[RW]

Name of test

Public Class Methods

distribution_permutations(n1,n2) click to toggle source

Generate distribution for permutations. Very expensive, but useful for demostrations

# File lib/statsample/test/umannwhitney.rb, line 78
def self.distribution_permutations(n1,n2)
  base=[0]*n1+[1]*n2
  po=Statsample::Permutation.new(base)
  
  total=n1*n2
  req={}
  po.each do |perm|
    r0,s0=0,0
    perm.each_index {|c_i|
      if perm[c_i]==0
        r0+=c_i+1
        s0+=1
      end
    }
    u1=r0-((s0*(s0+1)).quo(2))
    u2=total-u1
    temp_u= (u1 <= u2) ? u1 : u2
    req[perm]=temp_u
  end
  req
end
new(v1,v2, opts=Hash.new) click to toggle source

Create a new U Mann-Whitney test Params: Two Statsample::Vectors

# File lib/statsample/test/umannwhitney.rb, line 118
def initialize(v1,v2, opts=Hash.new)
  @v1=v1
  @v2=v2
  @n1=v1.valid_data.size
  @n2=v2.valid_data.size
  data=(v1.valid_data+v2.valid_data).to_scale
  groups=(([0]*@n1)+([1]*@n2)).to_vector
  ds={'g'=>groups, 'data'=>data}.to_dataset
  @t=nil
  @ties=data.data.size!=data.data.uniq.size        
  if(@ties)
    adjust_for_ties(ds['data'])
  end
  ds['ranked']=ds['data'].ranked(:scale)
  
  @n=ds.cases
    
  @r1=ds.filter{|r| r['g']==0}['ranked'].sum
  @r2=((ds.cases*(ds.cases+1)).quo(2))-r1
  @u1=r1-((@n1*(@n1+1)).quo(2))
  @u2=r2-((@n2*(@n2+1)).quo(2))
  @u=(u1<u2) ? u1 : u2
  opts_default={:name=>_("Mann-Whitney's U")}
  @opts=opts_default.merge(opts)
  opts_default.keys.each {|k|
    send("#{k}=", @opts[k])
  }
    
end
u_sampling_distribution_as62(n1,n2) click to toggle source

U sampling distribution, based on Dinneen & Blakesley (1973) algorithm. This is the algorithm used on SPSS.

Parameters:

  • n1: group 1 size

  • n2: group 2 size

Reference:

  • Dinneen, L., & Blakesley, B. (1973). Algorithm AS 62: A Generator for the Sampling Distribution of the Mann- Whitney U Statistic. Journal of the Royal Statistical Society, 22(2), 269-273

# File lib/statsample/test/umannwhitney.rb, line 31
def self.u_sampling_distribution_as62(n1,n2)

  freq=[]
  work=[]
  mn1=n1*n2+1
  max_u=n1*n2
  minmn=n1<n2 ? n1 : n2
  maxmn=n1>n2 ? n1 : n2
  n1=maxmn+1
  (1..n1).each{|i| freq[i]=1}
  n1+=1
  (n1..mn1).each{|i| freq[i]=0}
  work[1]=0
  xin=maxmn
  (2..minmn).each do |i|
    work[i]=0
    xin=xin+maxmn
    n1=xin+2
    l=1+xin.quo(2)
    k=i
    (1..l).each do |j|
      k=k+1
      n1=n1-1
      sum=freq[j]+work[j]
      freq[j]=sum
      work[k]=sum-freq[n1]
      freq[n1]=sum
    end
  end
  
  # Generate percentages for normal U
  dist=(1+max_u/2).to_i
  freq.shift
  total=freq.inject(0) {|a,v| a+v }
  (0...dist).collect {|i|
    if i!=max_u-i
      ues=freq[i]*2
    else
      ues=freq[i]
    end
    ues.quo(total)
  }
end

Public Instance Methods

probability_exact() click to toggle source

Exact probability of finding values of U lower or equal to sample on U distribution. Use with caution with m*n>100000. Uses u_sampling_distribution_as62

# File lib/statsample/test/umannwhitney.rb, line 162
def probability_exact
  dist=UMannWhitney.u_sampling_distribution_as62(@n1,@n2)
  sum=0
  (0..@u.to_i).each {|i|
    sum+=dist[i]
  }
  sum
end
probability_z() click to toggle source

Assuming H_0, the proportion of cdf with values of U lower than the sample, using normal approximation. Use with more than 30 cases per group.

# File lib/statsample/test/umannwhitney.rb, line 202
def probability_z
  (1-Distribution::Normal.cdf(z.abs()))*2
end
z() click to toggle source

Z value for U, with adjust for ties. For large samples, U is approximately normally distributed. In that case, you can use z to obtain probabily for U.

Reference:

# File lib/statsample/test/umannwhitney.rb, line 187
def z
  mu=(@n1*@n2).quo(2)
  if(!@ties)
    ou=Math::sqrt(((@n1*@n2)*(@n1+@n2+1)).quo(12))
  else
    n=@n1+@n2
    first=(@n1*@n2).quo(n*(n-1))
    second=((n**3-n).quo(12))-@t
    ou=Math::sqrt(first*second)
  end
  (@u-mu).quo(ou)
end

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