class UMannWhitney
U Mann-Whitney test¶ ↑
Non-parametric test for assessing whether two independent samples of observations come from the same distribution.
Assumptions¶ ↑
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The two samples under investigation in the test are independent of each other and the observations within each sample are independent.
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The observations are comparable (i.e., for any two observations, one can assess whether they are equal or, if not, which one is greater).
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The variances in the two groups are approximately equal.
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
- VERSION
Attributes
Name of test
Sample 1 Rank sum
Sample 2 Rank sum
Value of compensation for ties (useful for demostration)
U Value
Sample 1 U (useful for demostration)
Sample 2 U (useful for demostration)
Public Class Methods
Create a new U Mann-Whitney test Params: Two Daru::Vectors
# File lib/u_mann_whitney.rb, line 99 def initialize(v1,v2, opts=Hash.new) @v1 = v1 @v2 = v2 v1_valid = v1.reject_values(*Daru::MISSING_VALUES).reset_index! v2_valid = v2.reject_values(*Daru::MISSING_VALUES).reset_index! @n1 = v1_valid.size @n2 = v2_valid.size data = Daru::Vector.new(v1_valid.to_a + v2_valid.to_a) groups = Daru::Vector.new(([0] * @n1) + ([1] * @n2)) ds = Daru::DataFrame.new({:g => groups, :data => data}) @t = nil @ties = data.to_a.size != data.to_a.uniq.size if @ties adjust_for_ties(ds[:data]) end ds[:ranked] = ds[:data].ranked @n = ds.nrows @r1 = ds.filter_rows { |r| r[:g] == 0}[:ranked].sum || 0 @r2 = ((ds.nrows * (ds.nrows + 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
# File lib/u_mann_whitney.rb, line 24 def self.u_mannwhitney(v1, v2) new(v1,v2) end
U sampling distribution, based on Dinneen & Blakesley (1973) algorithm. This is the algorithm used on SPSS.
Parameters:
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n1
: group 1 size -
n2
: group 2 size
Reference:¶ ↑
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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/u_mann_whitney.rb, line 37 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
Shim for gettext
# File lib/u_mann_whitney.rb, line 142 def _(t) t end
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/u_mann_whitney.rb, line 147 def probability_exact dist = UMannWhitney.u_sampling_distribution_as62(@n1,@n2) sum = 0 (0..@u.to_i).each {|i| sum+=dist[i] } sum end
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/u_mann_whitney.rb, line 187 def probability_z (1-Distribution::Normal.cdf(z.abs()))*2 end
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:¶ ↑
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SPSS Manual
# File lib/u_mann_whitney.rb, line 172 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