Class: Statsample::Bivariate::Polychoric

Inherits:
Object
  • Object
show all
Includes:
DirtyMemoize, GetText
Defined in:
lib/statsample/bivariate/polychoric.rb

Overview

Polychoric correlation.

The polychoric correlation is a measure of bivariate association arising when both observed variates are ordered, categorical variables that result from polychotomizing the two undelying continuous variables (Drasgow, 2006)

According to Drasgow(2006), there are tree methods to estimate the polychoric correlation:

  1. Maximum Likehood Estimator

  2. Two-step estimator and

  3. Polychoric series estimate.

By default, two-step estimation are used. You can select the estimation method with method attribute. Joint estimate and polychoric series requires gsl library and rb-gsl.

Use

You should enter a Matrix with ordered data. For example:

      -------------------
      | y=0 | y=1 | y=2 | 
      -------------------
x = 0 |  1  |  10 | 20  |
      -------------------
x = 1 |  20 |  20 | 50  |
      -------------------

The code will be

matrix=Matrix[[1,10,20],[20,20,50]]
poly=Statsample::Bivariate::Polychoric.new(matrix, :method=>:joint)
puts poly.r

See extensive documentation on Uebersax(2002) and Drasgow(2006)

References

  • Uebersax, J.S. (2006). The tetrachoric and polychoric correlation coefficients. Statistical Methods for Rater Agreement web site. 2006. Available at: john-uebersax.com/stat/tetra.htm . Accessed February, 11, 2010

  • Drasgow F. (2006). Polychoric and polyserial correlations. In Kotz L, Johnson NL (Eds.), Encyclopedia of statistical sciences. Vol. 7 (pp. 69-74). New York: Wiley.

Constant Summary collapse

METHOD = 
:two_step
MAX_ITERATIONS = 
300
EPSILON = 
1e-6
MINIMIZER_TYPE_TWO_STEP = 
"brent"
MINIMIZER_TYPE_JOINT = 
"nmsimplex"

Instance Attribute Summary collapse

Instance Method Summary collapse

Constructor Details

#initialize(matrix, opts = Hash.new) ⇒ Polychoric

Params:

  • matrix: Contingence table

  • opts: Any attribute



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# File 'lib/statsample/bivariate/polychoric.rb', line 117

def initialize(matrix, opts=Hash.new)
  @matrix=matrix
  @n=matrix.column_size
  @m=matrix.row_size
  raise "row size <1" if @m<=1
  raise "column size <1" if @n<=1
  
  @method=METHOD
  @name="Polychoric correlation"
  @max_iterations=MAX_ITERATIONS
  @epsilon=EPSILON
  @minimizer_type_two_step=MINIMIZER_TYPE_TWO_STEP
  @minimizer_type_joint=MINIMIZER_TYPE_JOINT
  @debug=false
  @iteration=nil
  opts.each{|k,v|
    self.send("#{k}=",v) if self.respond_to? k
  }
  @r=nil
  compute_basic_parameters
end

Instance Attribute Details

#alpha ⇒ Object (readonly) Also known as: threshold_x

Returns the rows thresholds



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# File 'lib/statsample/bivariate/polychoric.rb', line 141

def alpha
  @alpha
end

#beta ⇒ Object (readonly) Also known as: threshold_y

Returns the columns thresholds



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# File 'lib/statsample/bivariate/polychoric.rb', line 143

def beta
  @beta
end

#debug ⇒ Object

Debug algorithm (See iterations, for example)



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# File 'lib/statsample/bivariate/polychoric.rb', line 75

def debug
  @debug
end

#epsilon ⇒ Object

Absolute error for iteration.



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# File 'lib/statsample/bivariate/polychoric.rb', line 94

def epsilon
  @epsilon
end

#iteration ⇒ Object (readonly)

Number of iterations



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# File 'lib/statsample/bivariate/polychoric.rb', line 97

def iteration
  @iteration
end

#log ⇒ Object (readonly)

Log of algorithm



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# File 'lib/statsample/bivariate/polychoric.rb', line 100

def log
  @log
end

#loglike_model ⇒ Object (readonly)

Returns the value of attribute loglike_model.



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# File 'lib/statsample/bivariate/polychoric.rb', line 103

def loglike_model
  @loglike_model
end

#max_iterations ⇒ Object

Max number of iterations used on iterative methods. Default to MAX_ITERATIONS



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# File 'lib/statsample/bivariate/polychoric.rb', line 73

def max_iterations
  @max_iterations
end

#method ⇒ Object

Method of calculation of polychoric series.

:two_step

two-step ML, based on code by Gegenfurtner(1992).

:polychoric_series

polychoric series estimate, using algorithm AS87 by Martinson and Hamdan (1975).

:joint

one-step ML, based on R package ‘polycor’ by J.Fox.



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# File 'lib/statsample/bivariate/polychoric.rb', line 92

def method
  @method
end

#minimizer_type_joint ⇒ Object

Minimizer type for joint estimate. Default “nmsimplex” See rb-gsl.rubyforge.org/min.html for reference.



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# File 'lib/statsample/bivariate/polychoric.rb', line 82

def minimizer_type_joint
  @minimizer_type_joint
end

#minimizer_type_two_step ⇒ Object

Minimizer type for two step. Default “brent” See rb-gsl.rubyforge.org/min.html for reference.



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# File 'lib/statsample/bivariate/polychoric.rb', line 78

def minimizer_type_two_step
  @minimizer_type_two_step
end

#name ⇒ Object

Name of the analysis



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# File 'lib/statsample/bivariate/polychoric.rb', line 71

def name
  @name
end

#r ⇒ Object (readonly)

Returns the polychoric correlation



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# File 'lib/statsample/bivariate/polychoric.rb', line 139

def r
  @r
end

Instance Method Details

#chi_square ⇒ Object 



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# File 'lib/statsample/bivariate/polychoric.rb', line 179

def chi_square
  if @loglike_model.nil?
    compute
  end
  -2*(@loglike_model-loglike_data)
end

#chi_square_df ⇒ Object 



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# File 'lib/statsample/bivariate/polychoric.rb', line 185

def chi_square_df
  (@nr*@nc)-@nc-@nr
end

#compute ⇒ Object

Start the computation of polychoric correlation based on attribute method



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# File 'lib/statsample/bivariate/polychoric.rb', line 154

def compute
  if @method==:two_step
    compute_two_step_mle_drasgow
  elsif @method==:joint
    compute_one_step_mle
  elsif @method==:polychoric_series
    compute_polychoric_series
  else
    raise "Not implemented"
  end
end

#compute_basic_parameters ⇒ Object 



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# File 'lib/statsample/bivariate/polychoric.rb', line 269

def compute_basic_parameters
  @nr=@matrix.row_size
  @nc=@matrix.column_size
  @sumr=[0]*@matrix.row_size
  @sumrac=[0]*@matrix.row_size
  @sumc=[0]*@matrix.column_size
  @sumcac=[0]*@matrix.column_size
  @alpha=[0]*(@nr-1)
  @beta=[0]*(@nc-1)
  @total=0
  @nr.times do |i|
    @nc.times do |j|
      @sumr[i]+=@matrix[i,j]
      @sumc[j]+=@matrix[i,j]
      @total+=@matrix[i,j]
    end
  end
  ac=0
  (@nr-1).times do |i|
    @sumrac[i]=@sumr[i]+ac
    @alpha[i]=Distribution::Normal.p_value(@sumrac[i] / @total.to_f)
    ac=@sumrac[i]
  end
  ac=0
  (@nc-1).times do |i|
    @sumcac[i]=@sumc[i]+ac
    @beta[i]=Distribution::Normal.p_value(@sumcac[i] / @total.to_f)
    ac=@sumcac[i]
  end
end

#compute_one_step_mle ⇒ Object

Compute Polychoric correlation with joint estimate. Rho and thresholds are estimated at same time. Code based on R package “polycor”, by J.Fox.



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# File 'lib/statsample/bivariate/polychoric.rb', line 385

def compute_one_step_mle
  # Get initial values with two-step aproach
  compute_two_step_mle_drasgow
  # Start iteration with past values
  rho=@r
  cut_alpha=@alpha
  cut_beta=@beta
  parameters=[rho]+cut_alpha+cut_beta
  minimization = Proc.new { |v, params|
   rho=v[0]
   alpha=v[1,@nr-1]
   beta=v[@nr,@nc-1]
   loglike(alpha,beta,rho)
  }
  np=@nc-1+@nr
  my_func = GSL::MultiMin::Function.alloc(minimization, np)
  my_func.set_params(parameters)      # parameters
  
  x = GSL::Vector.alloc(parameters.dup)
  
  ss = GSL::Vector.alloc(np)
  ss.set_all(1.0)
  
  minimizer = GSL::MultiMin::FMinimizer.alloc(minimizer_type_joint,np)
  minimizer.set(my_func, x, ss)
  
  iter = 0
  message=""
  begin
    iter += 1
    status = minimizer.iterate()
    status = minimizer.test_size(@epsilon)
    if status == GSL::SUCCESS
      message="Joint MLE converged to minimum at\n"
    end
    x = minimizer.x
    message+= sprintf("%5d iterations", iter)+"\n";
    for i in 0...np do
      message+=sprintf("%10.3e ", x[i])
    end
    message+=sprintf("f() = %7.3f size = %.3f\n", minimizer.fval, minimizer.size)+"\n";
  end while status == GSL::CONTINUE and iter < @max_iterations
  @iteration=iter
  @log+=message
  puts message if @debug
  @r=minimizer.x[0]
  @alpha=minimizer.x[1,@nr-1].to_a
  @beta=minimizer.x[@nr,@nc-1].to_a
  @loglike_model= -minimizer.minimum
end

#compute_polychoric_series ⇒ Object

Compute polychoric correlation using polychoric series. Algorithm: AS87, by Martinson and Hamdam(1975).

Warning: According to Drasgow(2006), this computation diverges greatly of joint and two-step methods.



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# File 'lib/statsample/bivariate/polychoric.rb', line 483

def compute_polychoric_series 
  @nn=@n-1
  @mm=@m-1
  @nn7=7*@nn
  @mm7=7*@mm
  @mn=@n*@m
  @cont=[nil]
  @n.times {|j|
    @m.times {|i|
      @cont.push(@matrix[i,j])
    }
  }

  pcorl=0
  cont=@cont
  xmean=0.0
  sum=0.0
  row=[]
  colmn=[]
  (1..@m).each do |i|
    row[i]=0.0
    l=i
    (1..@n).each do |j|
      row[i]=row[i]+cont[l]
      l+=@m
    end
    raise "Should not be empty rows" if(row[i]==0.0)
    xmean=xmean+row[i]*i.to_f
    sum+=row[i]
  end
  xmean=xmean/sum.to_f
  ymean=0.0
  (1..@n).each do |j|
    colmn[j]=0.0
    l=(j-1)*@m
    (1..@m).each do |i|
      l=l+1
      colmn[j]=colmn[j]+cont[l] #12
    end
    raise "Should not be empty cols" if colmn[j]==0
    ymean=ymean+colmn[j]*j.to_f
  end
  ymean=ymean/sum.to_f
  covxy=0.0
  (1..@m).each do |i|
    l=i
    (1..@n).each do |j|
      conxy=covxy+cont[l]*(i.to_f-xmean)*(j.to_f-ymean)
      l=l+@m
    end
  end
  
  chisq=0.0
  (1..@m).each do |i|
    l=i
    (1..@n).each do |j|
      chisq=chisq+((cont[l]**2).quo(row[i]*colmn[j]))
      l=l+@m
    end
  end
  
  phisq=chisq-1.0-(@mm*@nn).to_f / sum.to_f
  phisq=0 if(phisq<0.0) 
  # Compute cumulative sum of columns and rows
  sumc=[]
  sumr=[]
  sumc[1]=colmn[1]
  sumr[1]=row[1]
  cum=0
  (1..@nn).each do |i| # goto 17 r20
    cum=cum+colmn[i]
    sumc[i]=cum
  end
  cum=0
  (1..@mm).each do |i| 
    cum=cum+row[i]
    sumr[i]=cum
  end
  alpha=[]
  beta=[]
  # Compute points of polytomy
  (1..@mm).each do |i| #do 21
    alpha[i]=Distribution::Normal.p_value(sumr[i] / sum.to_f)
  end # 21
  (1..@nn).each do |i| #do 22
    beta[i]=Distribution::Normal.p_value(sumc[i] / sum.to_f)
  end # 21
  @alpha=alpha[1,alpha.size] 
  @beta=beta[1,beta.size]
  @sumr=row[1,row.size]
  @sumc=colmn[1,colmn.size]
  @total=sum
  
  # Compute Fourier coefficients a and b. Verified
  h=hermit(alpha,@mm)
  hh=hermit(beta,@nn)
  a=[]
  b=[]
  if @m!=2 # goto 24
    mmm=@m-2
    (1..mmm).each do |i| #do 23
      a1=sum.quo(row[i+1] * sumr[i] * sumr[i+1])
      a2=sumr[i]   * xnorm(alpha[i+1])
      a3=sumr[i+1] * xnorm(alpha[i])
      l=i
      (1..7).each do |j| #do 23
        a[l]=Math::sqrt(a1.quo(j))*(h[l+1] * a2 - h[l] * a3)
        l=l+@mm
      end
    end #23
  end
  # 24
  
  
  if @n!=2 # goto 26
    nnn=@n-2
    (1..nnn).each do |i| #do 25
      a1=sum.quo(colmn[i+1] * sumc[i] * sumc[i+1])
      a2=sumc[i] * xnorm(beta[i+1])
      a3=sumc[i+1] * xnorm(beta[i])
      l=i
      (1..7).each do |j| #do 25
        b[l]=Math::sqrt(a1.quo(j))*(a2 * hh[l+1] - a3*hh[l])
        l=l+@nn
      end # 25
    end # 25
  end
  #26 r20
  l = @mm
  a1 = -sum * xnorm(alpha[@mm])
  a2 = row[@m] * sumr[@mm] 
  (1..7).each do |j| # do 27
    a[l]=a1 * h[l].quo(Math::sqrt(j*a2))
    l=l+@mm
  end # 27
  
  l = @nn
  a1 = -sum * xnorm(beta[@nn])
  a2 = colmn[@n] * sumc[@nn]

  (1..7).each do |j| # do 28
    b[l]=a1 * hh[l].quo(Math::sqrt(j*a2))
    l = l + @nn
  end # 28
  rcof=[]
  # compute coefficients rcof of polynomial of order 8
  rcof[1]=-phisq
  (2..9).each do |i| # do 30
    rcof[i]=0.0
  end #30 
  m1=@mm
  (1..@mm).each do |i| # do 31
    m1=m1+1
    m2=m1+@mm
    m3=m2+@mm
    m4=m3+@mm
    m5=m4+@mm
    m6=m5+@mm
    n1=@nn
    (1..@nn).each do |j| # do 31
      n1=n1+1
      n2=n1+@nn
      n3=n2+@nn
      n4=n3+@nn
      n5=n4+@nn
      n6=n5+@nn
      
      rcof[3] = rcof[3] + a[i]**2 * b[j]**2
      
      rcof[4] = rcof[4] + 2.0 * a[i] * a[m1] * b[j] * b[n1]
      
      rcof[5] = rcof[5] + a[m1]**2 * b[n1]**2 +
        2.0 * a[i] * a[m2] * b[j] * b[n2]
      
      rcof[6] = rcof[6] + 2.0 * (a[i] * a[m3] * b[j] *
        b[n3] + a[m1] * a[m2] * b[n1] * b[n2])
      
      rcof[7] = rcof[7] + a[m2]**2 * b[n2]**2 +
        2.0 * (a[i] * a[m4] * b[j] * b[n4] + a[m1] * a[m3] *
          b[n1] * b[n3])
      
      rcof[8] = rcof[8] + 2.0 * (a[i] * a[m5] * b[j] * b[n5] +
        a[m1] * a[m4] * b[n1] * b[n4] + a[m2] *  a[m3] * b[n2] * b[n3])
      
      rcof[9] = rcof[9] + a[m3]**2 * b[n3]**2 +
        2.0 * (a[i] * a[m6] * b[j] * b[n6] + a[m1] * a[m5] * b[n1] *
        b[n5] + (a[m2] * a[m4] * b[n2] * b[n4]))
    end # 31
  end # 31

  rcof=rcof[1,rcof.size]
  poly = GSL::Poly.alloc(rcof)
  roots=poly.solve
  rootr=[nil]
  rooti=[nil]
  roots.each {|c|
    rootr.push(c.real)
    rooti.push(c.im)
  }
  @rootr=rootr
  @rooti=rooti
  
  norts=0
  (1..7).each do |i| # do 43
    
    next if rooti[i]!=0.0 
    if (covxy>=0.0)
      next if(rootr[i]<0.0 or rootr[i]>1.0)
      pcorl=rootr[i]
      norts=norts+1
    else
      if (rootr[i]>=-1.0 and rootr[i]<0.0)
        pcorl=rootr[i]
        norts=norts+1              
      end
    end
  end # 43
  raise "Error" if norts==0
  @r=pcorl
  
  @loglike_model=-loglike(@alpha, @beta, @r)
  
end

#compute_two_step_mle_drasgow ⇒ Object

Computation of polychoric correlation usign two-step ML estimation.

Two-step ML estimation “first estimates the thresholds from the one-way marginal frequencies, then estimates rho, conditional on these thresholds, via maximum likelihood” (Uebersax, 2006).

The algorithm is based on code by Gegenfurtner(1992).

References:

  • Gegenfurtner, K. (1992). PRAXIS: Brent’s algorithm for function minimization. Behavior Research Methods, Instruments & Computers, 24(4), 560-564. Available on www.allpsych.uni-giessen.de/karl/pdf/03.praxis.pdf

  • Uebersax, J.S. (2006). The tetrachoric and polychoric correlation coefficients. Statistical Methods for Rater Agreement web site. 2006. Available at: john-uebersax.com/stat/tetra.htm . Accessed February, 11, 2010



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# File 'lib/statsample/bivariate/polychoric.rb', line 311

def compute_two_step_mle_drasgow
  if Statsample.has_gsl?
    compute_two_step_mle_drasgow_gsl
  else
    compute_two_step_mle_drasgow_ruby
  end
end

#compute_two_step_mle_drasgow_gsl ⇒ Object

:nodoc:



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# File 'lib/statsample/bivariate/polychoric.rb', line 339

def compute_two_step_mle_drasgow_gsl #:nodoc:
  
  fn1=GSL::Function.alloc {|rho| 
    loglike(@alpha,@beta, rho)
  }
@iteration = 0
max_iter = @max_iterations
m = 0             # initial guess
m_expected = 0
a=-0.9999
b=+0.9999
gmf = GSL::Min::FMinimizer.alloc(@minimizer_type_two_step)
gmf.set(fn1, m, a, b)
header=sprintf("Two step minimization using %s method\n", gmf.name)
header+=sprintf("%5s [%9s, %9s] %9s %10s %9s\n", "iter", "lower", "upper", "min",
   "err", "err(est)")
  
header+=sprintf("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n", @iteration, a, b, m, m - m_expected, b - a)
@log=header
puts header if @debug
begin
  @iteration += 1
  status = gmf.iterate
  status = gmf.test_interval(@epsilon, 0.0)
  
  if status == GSL::SUCCESS
    @log+="converged:"
    puts "converged:" if @debug
  end
  a = gmf.x_lower
  b = gmf.x_upper
  m = gmf.x_minimum
  message=sprintf("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
    @iteration, a, b, m, m - m_expected, b - a);
  @log+=message
  puts message if @debug
end while status == GSL::CONTINUE and @iteration < @max_iterations
@r=gmf.x_minimum
@loglike_model=-gmf.f_minimum
end

#compute_two_step_mle_drasgow_ruby ⇒ Object

Depends on minimization algorithm.



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# File 'lib/statsample/bivariate/polychoric.rb', line 321

def compute_two_step_mle_drasgow_ruby #:nodoc:
  
  f=proc {|rho| 
    loglike(@alpha,@beta, rho)
  }
  @log="Minimizing using GSL Brent method\n"
  min=Minimization::Brent.new(-0.9999,0.9999,f)
  min.epsilon=@epsilon
  min.expected=0
  min.iterate
  @log+=min.log
  @r=min.x_minimum
  @loglike_model=-min.f_minimum
  puts @log if @debug
  
end

#expected ⇒ Object

:nodoc:



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# File 'lib/statsample/bivariate/polychoric.rb', line 452

def expected # :nodoc:
  rt=[]
  ct=[]
  t=0
  @matrix.row_size.times {|i|
    @matrix.column_size.times {|j|
      rt[i]=0 if rt[i].nil?
      ct[j]=0 if ct[j].nil?
      rt[i]+=@matrix[i,j]
      ct[j]+=@matrix[i,j]
      t+=@matrix[i,j]
    }
  }
  m=[]
  @matrix.row_size.times {|i|
    row=[]
    @matrix.column_size.times {|j|
      row[j]=(rt[i]*ct[j]).quo(t)
    }
    m.push(row)
  }
  
  Matrix.rows(m)
end

#hermit(s, k) ⇒ Object

Computes vector h(mm7) of orthogonal hermite…



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# File 'lib/statsample/bivariate/polychoric.rb', line 707

def hermit(s,k) # :nodoc:
  h=[]
  (1..k).each do |i| # do 14
    l=i
    ll=i+k
    lll=ll+k
    h[i]=1.0
    h[ll]=s[i]
    v=1.0
    (2..6).each do |j| #do 14
      w=Math::sqrt(j)
      h[lll]=(s[i]*h[ll] - v*h[l]).quo(w)
      v=w
      l=l+k
      ll=ll+k
      lll=lll+k
    end
  end
  h
end

#loglike(alpha, beta, rho) ⇒ Object 



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# File 'lib/statsample/bivariate/polychoric.rb', line 234

def loglike(alpha,beta,rho)
  if rho.abs>0.9999
    rho= (rho>0) ? 0.9999 : -0.9999
  end
  
  loglike=0
  pd=@nr.times.collect{ [0]*@nc}
  pc=@nr.times.collect{ [0]*@nc}
  @nr.times { |i|
    @nc.times { |j|
      #puts "i:#{i} | j:#{j}"
      if i==@nr-1 and j==@nc-1
        pd[i][j]=1.0
      else
        a=(i==@nr-1) ? 100: alpha[i]
        b=(j==@nc-1) ? 100: beta[j]
        #puts "a:#{a} b:#{b}"
        pd[i][j]=Distribution::NormalBivariate.cdf(a, b, rho)
      end
      pc[i][j] = pd[i][j]
      pd[i][j] = pd[i][j] - pc[i-1][j] if i>0
      pd[i][j] = pd[i][j] - pc[i][j-1] if j>0
      pd[i][j] = pd[i][j] + pc[i-1][j-1] if (i>0 and j>0)
      res= pd[i][j]
       #puts "i:#{i} | j:#{j} | ac: #{sprintf("%0.4f", pc[i][j])} | pd: #{sprintf("%0.4f", pd[i][j])} | res:#{sprintf("%0.4f", res)}"
    if (res==0)
     #    puts "Correccion"
      res=1e-16
    end
      loglike+= @matrix[i,j]  * Math::log( res )
    }
  }
  @pd=pd
  -loglike
end

#loglike_data ⇒ Object 



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# File 'lib/statsample/bivariate/polychoric.rb', line 166

def loglike_data
  loglike=0
  @nr.times do |i|
    @nc.times do |j|
      res=@matrix[i,j].quo(@total)
      if (res==0)
        res=1e-16
      end
    loglike+= @matrix[i,j]  * Math::log(res )
    end
  end
  loglike
end

#loglike_fd_rho(alpha, beta, rho) ⇒ Object 



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# File 'lib/statsample/bivariate/polychoric.rb', line 189

def loglike_fd_rho(alpha,beta,rho)
  if rho.abs>0.9999
    rho= (rho>0) ? 0.9999 : -0.9999
  end
   #puts "rho: #{rho}"
  
  loglike=0
  pd=@nr.times.collect{ [0]*@nc}
  pc=@nr.times.collect{ [0]*@nc}
  @nr.times { |i|
    @nc.times { |j|
      if i==@nr-1 and j==@nc-1
        pd[i][j]=1.0
        a=100
        b=100
      else
        a=(i==@nr-1) ? 100: alpha[i]
        b=(j==@nc-1) ? 100: beta[j]
        pd[i][j]=Distribution::NormalBivariate.cdf(a, b, rho)
      end
      pc[i][j] = pd[i][j]
      pd[i][j] = pd[i][j] - pc[i-1][j] if i>0
      pd[i][j] = pd[i][j] - pc[i][j-1] if j>0
      pd[i][j] = pd[i][j] + pc[i-1][j-1] if (i>0 and j>0)
      
      pij= pd[i][j]+EPSILON
      if i==0
        alpha_m1=-10
      else
        alpha_m1=alpha[i-1]
      end
      
      if j==0
        beta_m1=-10
      else
        beta_m1=beta[j-1]
      end
      
      loglike+= (@matrix[i,j].quo(pij))*(Distribution::NormalBivariate.pdf(a,b,rho) - Distribution::NormalBivariate.pdf(alpha_m1, b,rho) - Distribution::NormalBivariate.pdf(a, beta_m1,rho) + Distribution::NormalBivariate.pdf(alpha_m1, beta_m1,rho) )
      
    }
  }
  #puts "derivative: #{loglike}"
  -loglike
end

#matrix_for_rho(rho) ⇒ Object

:nodoc:



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# File 'lib/statsample/bivariate/polychoric.rb', line 436

def matrix_for_rho(rho) # :nodoc:
  pd=@nr.times.collect{ [0]*@nc}
  pc=@nr.times.collect{ [0]*@nc}
  @nr.times { |i|
      @nc.times { |j|
        pd[i][j]=Distribution::NormalBivariate.cdf(@alpha[i], @beta[j], rho)
        pc[i][j] = pd[i][j]
        pd[i][j] = pd[i][j] - pc[i-1][j] if i>0
        pd[i][j] = pd[i][j] - pc[i][j-1] if j>0
        pd[i][j] = pd[i][j] + pc[i-1][j-1] if (i>0 and j>0)
        res= pd[i][j]
      }
   }
   Matrix.rows(pc)
end

#new_with_vectors(v1, v2) ⇒ Object 



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# File 'lib/statsample/bivariate/polychoric.rb', line 110

def new_with_vectors(v1,v2)
  Polychoric.new(Crosstab.new(v1,v2).to_matrix)
end

#report_building(generator) ⇒ Object

:nodoc:



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# File 'lib/statsample/bivariate/polychoric.rb', line 736

def report_building(generator) # :nodoc: 
  compute if dirty?
  section=ReportBuilder::Section.new(:name=>@name)
  t=ReportBuilder::Table.new(:name=>_("Contingence Table"), :header=>[""]+(@n.times.collect {|i| "Y=#{i}"})+["Total"])
  @m.times do |i|
    t.row(["X = #{i}"]+(@n.times.collect {|j| @matrix[i,j]}) + [@sumr[i]])
  end
  t.hr
  t.row(["T"]+(@n.times.collect {|j| @sumc[j]})+[@total])
  section.add(t)
  section.add(sprintf("r: %0.4f",r))
  t=ReportBuilder::Table.new(:name=>_("Thresholds"), :header=>["","Value"])
  threshold_x.each_with_index {|val,i|
    t.row(["Threshold X #{i}", sprintf("%0.4f", val)])
  }
  threshold_y.each_with_index {|val,i|
    t.row(["Threshold Y #{i}", sprintf("%0.4f", val)])
  }
  section.add(t)
  section.add(_("Test of bivariate normality: X2 = %0.3f, df = %d, p= %0.5f" % [ chi_square, chi_square_df, 1-Distribution::ChiSquare.cdf(chi_square, chi_square_df)])) 
  generator.parse_element(section)
end

#summary ⇒ Object 



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# File 'lib/statsample/bivariate/polychoric.rb', line 731

def summary
  rp=ReportBuilder.new(:no_title=>true).add(self).to_text
end

#xnorm(t) ⇒ Object

:nodoc:



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# File 'lib/statsample/bivariate/polychoric.rb', line 727

def xnorm(t) # :nodoc:
  Math::exp(-0.5 * t **2) * (1.0/Math::sqrt(2*Math::PI))
end