Engine: Statsample::Regression::Multiple::RubyEngine
Cases(listwise)=63(63)
R=0.978844
R^2=0.958137
R^2 Adj=0.956008
Std.Error R=3.092024
Equation=4.031667 + 0.018039cases + 0.244790vars + 0.001197c_v
| source | ss | df | ms | f | p |
|---|---|---|---|---|---|
| Regression | 12910.098 | 3 | 4303.366 | 450.114 | 0.000 |
| Error | 564.076 | 59 | 9.561 | ||
| Total | 13474.174 | 62 | 4312.927 |
| coeff | b | beta | se | t |
|---|---|---|---|---|
| Constant | 4.031667 | - | 0.752604 | 5.356953 |
| cases | 0.018039 | 0.381587 | 0.001961 | 9.200093 |
| vars | 0.244790 | 0.224390 | 0.036055 | 6.789335 |
| c_v | 0.001197 | 0.584174 | 0.000094 | 12.738410 |
Engine: Statsample::Regression::Multiple::RubyEngine
Cases(listwise)=63(63)
R=0.999637
R^2=0.999275
R^2 Adj=0.999238
Std.Error R=0.538365
Equation=-0.520303 + -0.000708cases + 1.234451vars + 0.000735c_v
| source | ss | df | ms | f | p |
|---|---|---|---|---|---|
| Regression | 23554.271 | 3 | 7851.424 | 27089.134 | 0.000 |
| Error | 17.100 | 59 | 0.290 | ||
| Total | 23571.372 | 62 | 7851.714 |
| coeff | b | beta | se | t |
|---|---|---|---|---|
| Constant | -0.520303 | - | 0.131039 | -3.970594 |
| cases | -0.000708 | -0.011324 | 0.000341 | -2.074007 |
| vars | 1.234451 | 0.855546 | 0.006278 | 196.641087 |
| c_v | 0.000735 | 0.271138 | 0.000016 | 44.912972 |
Levene's test ( Levene, 1960) is used to test if k samples have equal variances. Equal variances across samples is called homogeneity of variance. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples. The Levene test can be used to verify that assumption.# Use: # require 'statsample' # a=[1,2,3,4,5,6,7,8,100,10].to_scale # b=[30,40,50,60,70,80,90,100,110,120].to_scale # # levene=Statsample::Test::Levene.new([a,b]) # puts levene.summary # # Output: # Levene Test # F: 0.778121319848449 # p: 0.389344552595791 # # Reference: # * NIST/SEMATECH e-Handbook of Statistical Methods. Available on http://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm class Levene include Statsample::Test include Summarizable # Degrees of freedom 1 (k-1) attr_reader :d1 # Degrees of freedom 2 (n-k) attr_reader :d2 # Name of test attr_accessor :name # Input could be an array of vectors or a dataset def initialize(input, opts=Hash.new()) if input.is_a? Statsample::Dataset @vectors=input.vectors.values else @vectors=input end @name=_("Levene Test") opts.each{|k,v| self.send("#{k}=",v) if self.respond_to? k } compute end # Value of the test def f @w end def report_building(builder) # :nodoc: builder.text "%s : F(%d, %d) = %0.4f , p = %0.4f" % [@name, @d1, @d2, f, probability] end def compute n=@vectors.inject(0) {|ac,v| ac+v.n_valid} zi=@vectors.collect {|vector| mean=vector.mean vector.collect {|v| (v-mean).abs }.to_scale } total_mean=zi.inject([]) {|ac,vector| ac+vector.valid_data }.to_scale.mean k=@vectors.size sum_num=zi.inject(0) {|ac,vector| ac+(vector.size*(vector.mean-total_mean)**2) } sum_den=zi.inject(0) {|ac,vector| z_mean=vector.mean ac+vector.valid_data.inject(0) {|acp,zij| acp+(zij-z_mean)**2 } } @w=((n-k)*sum_num).quo((k-1)*sum_den) @d1=k-1 @d2=n-k end private :compute # Probability. # With H_0 = Sum(s2)=0, probability of getting a value of the test upper or equal to the obtained on the sample def probability p_using_cdf(Distribution::F.cdf(f, @d1, @d2), :right) end end end end ================================================ FILE: lib/statsample/test/t.rb ================================================ module Statsample module Test # A t-test is any statistical hypothesis test in which the test statistic follows a Student's t distribution, if the null hypothesis is supported class T class << self include Math # Test the null hypothesis that the population mean is equal to a specified value u, one uses the statistic. # Is the same formula used on t-test for paired sample. # * x: sample/differences mean # * u: population mean # * s: sample/differences standard deviation # * n: sample size def one_sample(x,u,s,n) (x-u)*Math::sqrt(n).quo(s) end # Test if means of two samples are different. # * x1: sample 1 mean # * x2: sample 2 mean # * s1: sample 1 standard deviation # * s2: sample 2 standard deviation # * n1: sample 1 size # * n2: sample 2 size # * equal_variance: true if equal_variance assumed # def two_sample_independent(x1, x2, s1, s2, n1, n2, equal_variance = false) num=x1-x2 if equal_variance sx1x2 = sqrt(((n1-1)*s1**2 + (n2-1)*s2**2).quo(n1+n2-2)) den = sx1x2*sqrt(1.quo(n1)+1.quo(n2)) else den=sqrt((s1**2).quo(n1) + (s2**2).quo(n2)) end num.quo(den) end # Degrees of freedom for equal variance on t test def df_equal_variance(n1,n2) n1+n2-2 end # Degrees of freedom for unequal variance # * s1: sample 1 standard deviation # * s2: sample 2 standard deviation # * n1: sample 1 size # * n2: sample 2 size # == Reference # * http://en.wikipedia.org/wiki/Welch-Satterthwaite_equation def df_not_equal_variance(s1,s2,n1,n2) s2_1=s1**2 s2_2=s2**2 num=(s2_1.quo(n1)+s2_2.quo(n2))**2 den=(s2_1.quo(n1)**2).quo(n1-1) + (s2_2.quo(n2)**2).quo(n2-1) num.quo(den) end end include Statsample::Test include Summarizable attr_reader :standard_error, :estimate, :df # Tails for p-value (:both, :left or :right). Default :both attr_accessor :tails # Name of F analysis attr_accessor :name attr_accessor :confidence_level attr_reader :t attr_accessor :estimate_name, :standard_error_name # Creates a generic t test. Use OneSample or TwoSamplesIndependent # classes for better summaries. # Parameters: # * estimate: estimate # * standard_error: standard error of estimate # * df: degrees of freedom def initialize(estimate, standard_error, df, opts=Hash.new) @estimate=estimate @standard_error=standard_error @df=df @t = @estimate / @standard_error.to_f opts_default={ :tails=>:both, :name=>_("T Test"), :estimate_name=>_("Estimate"), :standard_error_name=>_("Std.Err.of Estimate"), :confidence_level=>0.95} @opts = opts_default.merge(opts) @opts.keys.each {|k| send("#{k}=", @opts[k]) if respond_to? k } end alias :se :standard_error def to_f t end # probability def probability p_using_cdf(Distribution::T.cdf(t, df), tails) end def confidence_interval(cl=nil) cl||=confidence_level t_crit = t_critical(cl, df) [estimate - se*t_crit, estimate + se*t_crit] end alias :ci :confidence_interval def report_building(builder) #:nodoc: builder.section(:name=>@name) do |section| section.text _("%s: %0.4f | %s: %0.4f") % [@estimate_name, @estimate, @standard_error_name, se] report_building_t(section) end end def report_building_t(s) df_f=@df.is_a?(Integer) ? "%d" : "%0.4f" s.text _("t(%d) = %0.4f, p=%0.4f (%s tails)") % [df, t,probability, tails] s.text _("CI(%d%%): %0.4f - %0.4f") % [confidence_level*100, ci[0],ci[1]] end # One Sample t-test # == Usage # a=1000.times.map {rand(100)}.to_scale # t_1=Statsample::Test::T::OneSample.new(a, {:u=>50}) # t_1.summary # # === Output # # = One Sample T Test # Sample mean: 48.954 # Population mean:50 # Tails: both # t = -1.1573, p=0.2474, d.f=999 class OneSample include Math include Statsample::Test include Summarizable # Options attr_accessor :opts # Name of test attr_accessor :name # Population mean to contrast attr_accessor :u # Degress of freedom attr_reader :df # Tails for probability (:both, :left or :right) attr_accessor :tails # Create a One Sample T Test # Options: # * :u = Mean to compare. Default= 0 # * :name = Name of the analysis # * :tails = Tail for probability. Could be :both, :left, :right def initialize(vector, opts=Hash.new) @vector=vector default={:u=>0, :name=>"One Sample T Test", :tails=>:both} @opts=default.merge(opts) @name=@opts[:name] @u=@opts[:u] @tails=@opts[:tails] @confidence_level=@opts[:confidence_level] || 0.95 @df= @vector.n_valid-1 @t=nil end def t_object T.new(@vector.mean-u, @vector.se, @vector.n_valid-1, opts) end def t t_object.t end def probability t_object.probability end def standard_error t_object.standard_error end alias :se :standard_error def confidence_interval(cl=nil) t_object.confidence_interval(cl) end alias :ci :confidence_interval def report_building(b) # :nodoc: b.section(:name=>@name) {|s| s.text _("Sample mean: %0.4f | Sample sd: %0.4f | se : %0.4f") % [@vector.mean, @vector.sd, se] s.text _("Population mean: %0.4f") % u if u!=0 t_object.report_building_t(s) } end end # Two Sample t-test. # # == Usage # a=1000.times.map {rand(100)}.to_scale # b=1000.times.map {rand(100)}.to_scale # t_2=Statsample::Test::T::TwoSamplesIndependent.new(a,b) # t_2.summary # === Output # = Two Sample T Test # Mean and standard deviation # +----------+---------+---------+------+ # | Variable | m | sd | n | # +----------+---------+---------+------+ # | 1 | 49.3310 | 29.3042 | 1000 | # | 2 | 47.8180 | 28.8640 | 1000 | # +----------+---------+---------+------+ # # == Levene Test # Levene Test # F: 0.3596 # p: 0.5488 # T statistics # +--------------------+--------+-----------+----------------+ # | Type | t | df | p (both tails) | # +--------------------+--------+-----------+----------------+ # | Equal variance | 1.1632 | 1998 | 0.2449 | # | Non equal variance | 1.1632 | 1997.5424 | 0.1362 | # +--------------------+--------+-----------+----------------+ class TwoSamplesIndependent include Math include Statsample::Test include DirtyMemoize include Summarizable # Options attr_accessor :opts # Name of test attr_accessor :name # Degress of freedom (equal variance) attr_reader :df_equal_variance # Degress of freedom (not equal variance) attr_reader :df_not_equal_variance # Value of t for equal_variance attr_reader :t_equal_variance # Value of t for non-equal_variance attr_reader :t_not_equal_variance # Probability(equal variance) attr_reader :probability_equal_variance # Probability(unequal variance) attr_reader :probability_not_equal_variance # Tails for probability (:both, :left or :right) attr_accessor :tails # Create the object dirty_writer :tails dirty_memoize :t_equal_variance, :t_not_equal_variance, :probability_equal_variance, :probability_not_equal_variance, :df_equal_variance, :df_not_equal_variance # Create a Two Independent T Test # Options: # * :name = Name of the analysis # * :tails = Tail for probability. Could be :both, :left, :right def initialize(v1, v2, opts=Hash.new) @v1=v1 @v2=v2 default={:u=>0, :name=>"Two Sample T Test", :tails=>:both} @opts=default.merge(opts) @name=@opts[:name] @tails=@opts[:tails] end # Set t and probability for given u def compute @t_equal_variance= T.two_sample_independent(@v1.mean, @v2.mean, @v1.sd, @v2.sd, @v1.n_valid, @v2.n_valid,true) @t_not_equal_variance= T.two_sample_independent(@v1.mean, @v2.mean, @v1.sd, @v2.sd, @v1.n_valid, @v2.n_valid, false) @df_equal_variance=T.df_equal_variance(@v1.n_valid, @v2.n_valid) @df_not_equal_variance=T.df_not_equal_variance(@v1.sd, @v2.sd, @v1.n_valid, @v2.n_valid) @probability_equal_variance = p_using_cdf(Distribution::T.cdf(@t_equal_variance, @df_equal_variance), tails) @probability_not_equal_variance = p_using_cdf(Distribution::T.cdf(@t_not_equal_variance, @df_not_equal_variance), tails) end # Cohen's d is a measure of effect size. Its defined as the difference between two means divided by a standard deviation for the data def d n1=@v1.n_valid n2=@v2.n_valid num=@v1.mean-@v2.mean den=Math::sqrt( ((n1-1)*@v1.sd+(n2-1)*@v2.sd).quo(n1+n2)) num.quo(den) end def report_building(b) # :nodoc: b.section(:name=>@name) {|g| g.table(:name=>_("Mean and standard deviation"), :header=>[_("Variable"), _("mean"), _("sd"),_("n")]) {|t| t.row([@v1.name,"%0.4f" % @v1.mean,"%0.4f" % @v1.sd,@v1.n_valid]) t.row([@v2.name,"%0.4f" % @v2.mean,"%0.4f" % @v2.sd, @v2.n_valid]) } g.parse_element(Statsample::Test.levene([@v1,@v2],:name=>_("Levene test for equality of variances"))) g.table(:name=>_("T statistics"),:header=>["Type","t","df", "p (#{tails} tails)"].map{|v| _(v)}) {|t| t.row([_("Equal variance"), "%0.4f" % t_equal_variance, df_equal_variance, "%0.4f" % probability_equal_variance]) t.row([_("Non equal variance"), "%0.4f" % t_not_equal_variance, "%0.4f" % df_not_equal_variance, "%0.4f" % probability_not_equal_variance]) } g.table(:name=>_("Effect size")) do |t| t.row ['x1-x2', "%0.4f" % (@v1.mean-@v2.mean)] t.row ['d', "%0.4f" % d] end } end end end end end ================================================ FILE: lib/statsample/test/umannwhitney.rb ================================================ module Statsample module Test # # = U Mann-Whitney test # # Non-parametric test for assessing whether two independent samples # of observations come from the same distribution. # # == Assumptions # # * The two samples under investigation in the test are independent of each other and the observations within each sample are independent. # * The observations are comparable (i.e., for any two observations, one can assess whether they are equal or, if not, which one is greater). # * The variances in the two groups are approximately equal. # # Higher differences of distributions correspond to # to lower values of U. # class UMannWhitney # Max for m*n allowed for exact calculation of probability MAX_MN_EXACT=10000 # 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 # def self.u_sampling_distribution_as62(n1,n2) freq=[] work=[] mn1=n1*n2+1 max_u=n1*n2 minmn=n1