#LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL
#
#  ROC.pm - A Perl module implementing receiver-operator-characteristic (ROC)
#           curves with nonparametric confidence bounds
#
#     This program is free software; you may redistribute it and/or
#     modify it under the same terms as Perl itself.
#
#  This code implements a method for constructing nonparametric confidence
#  for ROC curves described in
#        R.A. Hilgers, Distribution-Free Confidence Bounds for ROC Curves,
#        Meth Inform Med 1991; 30:96-101
#  Additionally some auxilliary functions were ported (and corrected) from
#  Fortran (Applied Statistics, ACM).
#
#  Written in Perl by Hans A. Kestler.
#              Hans A. Kestler <hans.kestler@uni-ulm.de>
#                              <h.kestler@ieee.org>
#
#
#LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL

package Statistics::ROC;
require 5;
use Carp;
use strict;
use vars qw($VERSION @ISA @EXPORT @EXPORT_OK); require Exporter; @ISA = ('Exporter'); @EXPORT = qw( roc rank loggamma betain Betain xinbta Xinbta );$VERSION = '0.01';

# Algorithm 291, Logarithm of the gamma function.
#                in Collected Algorithms of the ACM, Vol II, 1980
# M.C. Pike and I.D. Hill with remark by M.R. Hoare
# see also Pike, M.C. and Hill, I.D. (1966). Algorithm 291. Logarithm of the
#          gamma function. Commun. Ass. Comput. Mach., 9,684.

sub loggamma($){ # This procedure evaluates the natural logarithm of gamma(x) for all # x>0, accurate to 10 decimal places. Stirlings formula is used for the # central polynomial part of the procedure my$x= shift; # default arg is @_
my ($f,$z);

if($x==0){return 743.746924740801} # this is: loggamma(9.9999999999E-324) if($x < 7)
{
for($z=$x,$f=1;$z<7;$z++) {$x=$z;$f*=$z; }$x++;
$f= -log($f); # log returns the natural logarithm
}
else{ $f=0;}$z=1/($x*$x);
return $f+($x-.5)*log($x)-$x+.918938533204673+
(((-.000595238095238*$z+.000793650793651)*$z-
.002777777777778)*$z+.083333333333333)/$x;
}

# Algorithm AS 63 with remark AS R19,
# Computes incomplete beta function ratio
#  K.L. Majumder and G.P. Bhattacharjee (1973). The Incomplete Beta Integral,
#  Appl. Statist.,22:409:411  and
#  G.W. Cran, K.J. Martin and G.E. Thomas (1977).Remark AS R19 and
#  Algorithm AS109, A Remark on Algorithms AS 63: The Incomplete Beta Integral
#  AS 64: Inverse of the Incomplete Beta Function Ratio,
#  Appl. Statist., 26:111-114.
#
# Remarks:
#    Complete beta function: B(p,q)=gamma(p)*gamma(q)/gamma(p+q)
#                       log(B(p,q))=ln(gamma(p))+ln(gamma(q))-ln(gamma(p+q))
#
#    Incomplete beta function ratio:
#                 I_x(p,q)=1/B(p,q) * \int_0^x t^{p-1}*(1-t)^{q-1} dt
#
#    --> log(B(p,q)) has to be supplied to calculate I_x(p,q)
#    log denotes the natural logarithm
#        $beta = log(B(p,q)) #$x    = x
#        $p = p #$q    = q
#    The subroutine returns I_x(p,q). If an error occurs a negative value
#    {-1,-2} is returned.

sub betain(){
# Computes incomplete beta function ratio for arguments
# $x between zero and one,$p and $q positive. # Log of complete beta function,$beta, is assumed to be known.

my ($x,$p, $q,$beta) = @_;
my ($xx,$psq, $cx,$pp, $qq,$index, $betain,$ns, $term,$ai, $rx,$temp);
my $ACU=1.0E-14; # accuracy # tests for admissibility of arguments if($p<=0 || $q<=0){ return -1;} if($x<0  || $x>1) { return -2;} # change tail if necessary and determine s$psq=$p+$q; $cx=1-$x;
if($p<$psq*$x){$xx=$cx;$cx=$x;$pp=$q;$qq=$p;$index=1;}
else{ $xx=$x; $pp=$p; $qq=$q; $index=0;}$term=1; $ai=1;$betain=1;
$ns=$qq+$cx*$psq;

# use Soper's reduction formulae
$rx=$xx/$cx; do{ if($ns>=0){$temp=$qq-$ai; if($ns==0){$rx=$xx;}}
else{ $temp=$psq; $psq++;}$term *= $temp*$rx/($pp+$ai);
$betain+=$term;
$temp=abs($term); $ai++;$ns--;}
until($temp<=$ACU && $temp<=$ACU*$betain); # calculate result$betain *= exp($pp*log($xx)+($qq-1)*log($cx)-$beta)/$pp;
if($index){ return 1-$betain;}
else{ return $betain;} } sub Betain($$){ # Computes the incomplete beta function # by calling loggamma() and betain() my (x, p, q) = @_; if(x==1){return 1;} elsif(x==0){return 0;} else{ return betain(x, p, q,loggamma(p)+loggamma(q)-loggamma(p+q));} } sub max($$){ # computes the maximum of two numbers my ($a, $b) = @_; if($a>$b){ return$a;}
else{ return $b;} } # Algorithm AS 109, # Computes inverse of incomplete beta function ratio # G.W. Cran, K.J. Martin and G.E. Thomas (1977).Remark AS R19 and # Algorithm AS109, A Remark on Algorithms AS 63: The Incomplete Beta Integral # AS 64: Inverse of the Incomplete Beta Function Ratio, # Appl. Statist., 26:111-114. # # Remark AS R83 and the correction in vol40(1) of Appl. Statist.(1991), p.236 # have been incorporated in this version. # K.J. Berry, P.W. Mielke, Jr and G.W. Cran (1990) Algorithm AS R83, A Remark # on Algorithm AS 109: Inverse of the Incomplete Beta Function Ratio, # Appl. Statist., 39:309-310. # # Remarks: # # Complete beta function: B(p,q)=gamma(p)*gamma(q)/gamma(p+q) # log(B(p,q))=ln(gamma(p))+ln(gamma(q))-ln(gamma(p+q)) # # Incomplete beta function ratio: # alpha = I_x(p,q) = 1/B(p,q) * \int_0^x t^{p-1}*(1-t)^{q-1} dt # # --> log(B(p,q)) has to be supplied to calculate I_x(p,q) # log denotes the natural logarithm #$beta = log(B(p,q))
#        $alpha= I_x(p,q) #$p    = p
#        $q = q # The subroutine returns x. If an error occurs a negative value {-1,-2,-3} # is returned. sub xinbta(){ # Computes inverse of incomplete beta function ratio # for given positive values of the arguments$p and $q, #$alpha between zero and one.
# Log of complete beta function, $beta is assumed to be known. # # copyright by H.A. Kestler, 1998 my ($p, $q,$beta, $alpha) = @_; my ($a, $y,$pp, $qq,$index, $r,$h, $t,$w, $xinbta,$yprev, $prev,$s, $sq,$tx, $adj,$g);
my $SAE=-37; my$FPU=10**$SAE; my$ACU;

# test for admissibility of parameters
if($p<=0 ||$q<=0){ return -1;}
if($alpha<0 ||$alpha>1) { return -2;}
if($alpha==0 ||$alpha==1){ return $alpha;} # change tail if necessary if($alpha>.5){ $a=1-$alpha; $pp=$q; $qq=$p; $index=1;} else{$a=$alpha;$pp=$p;$qq=$q;$index=0;}

# calculate the initial approximation
$r=sqrt(-log($a*$a));$y=$r-(2.30753+.27061*$r)/(1+(.99229+.04481*$r)*$r);
if($pp>1 &&$qq > 1)
{
$r=($y*$y-3)/6;$s=1/($pp+$pp-1); $t=1/($qq+$qq-1);$h=2/($s+$t);
$w=$y*sqrt($h+$r)/$h-($t-$s)*($r+5/6-2/(3*$h));$xinbta=$pp/($pp+$qq*exp($w+$w)); } else {$r=$qq+$qq; $t=1/(9*$qq);
$t=$r*(1-$t+$y*sqrt($t))**3; if($t<=0){
$xinbta=1-exp((log((1-$a)*$qq)+$beta)/$qq); } else{$t=(4*$pp+$r-2)/$t; if($t<=1){ $xinbta=exp((log($a*$pp)+$beta)/$pp);} else{$xinbta=1-2/($t+1);} } } # solve for$x by a modified newton-raphson method
# using subroutine betain()
$r=1-$pp; $t=1-$qq; $yprev=0;$sq=1; $prev=1; if($xinbta<.0001){ $xinbta=.0001;} if($xinbta>.9999){ $xinbta=.9999;}$ACU=10**(max(-5/$pp**2-1/$a**.2-13,$SAE)); do{$y=betain($xinbta,$pp,$qq,$beta);
if($y==-1 ||$y==-2){ return -3;}  # betain returns an exception
$y=($y-$a)*exp($beta+$r*log($xinbta)+$t*log(1-$xinbta));
if($y*$y<=0){ $prev=max($sq,$FPU);}$g=1;
Label10: do{
do{ $adj=$g*$y;$sq=$adj*$adj; $g/=3;} while($sq>=$prev);$tx=$xinbta-$adj;}
until($tx>=0 &&$tx<=1);
if($prev<=$ACU || $y*$y<=$ACU){ goto Label12;} if($tx==0 || $tx==1){ goto Label10;}$xinbta=$tx;$yprev=$y;} until($adj==0);

Label12:
if($index){ return 1-$xinbta;}
else{ return $xinbta;} } sub Xinbta($$){ # Computes the inverse of the incomplete beta function # by calling loggamma() and xinbta() # # copyright by H.A. Kestler, 1998 my (p, q, alpha) = @_; if(alpha==1){return 1;} elsif(alpha==0){return 0;} else{ return xinbta(p, q,loggamma(p)+loggamma(q)-loggamma(p+q), alpha);} } sub rank(\@){ # Computes the ranks of the values specified as the second # argument (an array). Returns a vector of ranks # corresponding to the input vector. # Different types of ranking are possible ('high', 'low', 'mean'), # and are specified as first argument. # These differ in the way ties of the input vector, i.e. identical # values, are treated: # 'high' --> replace ranks of identical values with their # highest rank # 'low' --> replace ranks of identical values with their # lowest rank # 'mean' --> replace ranks of identical values with the mean # of their rank # # copyright by H.A. Kestler, 1998 my (type, r) = @_; # type: type of ranking 'high', 'low' or 'mean' # r: reference to array of values to be ranked my (@s, s, i, @e, @rk, rk_m); # calculate initial rank's @s=sort{$$r[$a]<=>$$r[b]} 0..#{r}; # sort idx num. by values of @r for(i=0,@rk=@s;i<@rk;i++){ rk[s[i]]=i+1;} # set rank's # treat ties for(i=1,@e=(); i<@s; i++){ if($$r[$s[$i]]==$$r[s[i-1]]){ # test if there are ties push @e,i-1;} # save index numbers of tied values (minus 1) elsif(@e){ # ties have occured and are now being treated if(type eq'mean'){ # calculate mean value of tied ranks rk_m=0; for(@e,e[-1]+1){ rk_m+=rk[s[_]];} rk_m/=@e+1; } for(@e,e[-1]+1){ if(type eq 'high'){ rk[s[_]]=rk[s[e[-1]+1]];} elsif(type eq 'low' ){ rk[s[_]]=rk[s[e[0]]];} elsif(type eq 'mean'){ rk[s[_]]=rk_m;} else{ croak "Wrong type of ranking (high|low|mean).\n";} } @e=(); # reinitialize @e } } return @rk; } sub locate(\@){ # Routine to find the index for table lookup which is below # the value to be interpolated. # Given a reference to an array xx and a value x a value j # is returned such that x is between xx[j] and xx[j+1]. # xx must be monotonic, either increasing or decreasing. # # This routine is adapted from "Numerical Recipes in C", # second edition, by Press, Teukolsky, Vetterling and Flannery, # Cambridge University Press, 1992. # It uses bisection to find the right place, which has a # comutational complexity of O(log_2(n)). # # copyright by H.A. Kestler, 1998 my (xx,x)=@_; my (jl,ju)=(0,#{xx}); # initialize lower and upper limits my (jm,ascend); ascend=$$xx[$ju] > $$xx[0]; # test if x is inside of the array if((x>$$xx[$ju] || $x<$$xx[jl]) && ascend) { croak "Value out of range for table lookup (1): x.\n";} if((x<$$xx[$ju] || $x>$$xx[jl]) && !ascend) { croak "Value out of range for table lookup (2): x.\n";} while((ju-jl)>1) { # If we are not yet done jm=int((ju+jl)/2); # compute a midpoint, if(x > xx->[jm] == ascend) { jl=jm;} # and replace either the lower limit else { ju=jm;} # or the upper limit, as appropriate. } return jl; } sub linlocate(\@$$){ # Routine to find the index for table lookup which is below # the value to be interpolated. # Given a reference to an array$xx and a value $x a value$j
# is returned such that $x is between$xx[$j] and$xx[$j+1]. #$xx must be monotonic, either increasing or decreasing.
#
# Starts searching linearly from an initial index value
# provided as the third argument.
# If no index value can be found a negative value is
# returned, i.e. -1.
#
# copyright by H.A. Kestler, 1998

my ($xx,$x,$index)=@_; my ($jl,$ju)=(0,$#{$xx}); # initialize lower and upper limits my$ascend;

$ascend=$$xx[ju] >$$xx[0]; # test if$x is inside of the array
if(($x>$$xx[ju] || x<$$xx[$jl]) && $ascend) { croak "Value out of range for table lookup.\n";} if(($x<$$xx[ju] || x>$$xx[$jl]) && !$ascend)
{ croak "Value out of range for table lookup.\n";}

# step through the table sequentially
if($ascend &&$xx->[$index]<$x){     # ascending
while($x>$xx->[$index] and$index<=$ju) {$index++;}}
elsif(!$ascend &&$xx->[$index]>$x){ # descending
while($x<$xx->[$index] and$index<=$ju) {$index++;}}
else{ return -1;}    # starting index is too high

return $index-1; } sub interp(\@\@\@){ # Interpolates (table lookup) piecewise linearly an # array (third argument). Returns # The table is represented by the first two arguments, i.e. @xx and @yy. # Assumes the @xx values to be monotonically increasing. # # copyright by H.A. Kestler, 1998 use vars ('@xx', '@yy', '@x'); local (*xx, *yy, *x)=@_; my ($i, $index, @y); # make checks if(@xx != @yy) {croak "Sizes of xx and yy arrays are not equal.\n";} for($i=0; $i<@x;$i++)
{
$index=locate(@xx,$x[$i]);$y[$i]=($yy[$index+1]-$yy[$index])/($xx[$index+1]-$xx[$index])* ($x[$i]-$xx[$index]) +$yy[$index]; } return @y; } sub roc($$\@){ # ROC (receiver operator characteristic) curves with confidence bounds # # Determines the ROC curve and its nonparametric confidence bounds. # The ROC curve shows the relationship of "probability of false # alarm" (x-axis) to "probability of detection" (y-axis) for a # certain test. # Or in medical terms: the "probability of a positive test, given no # disease" to the "probability of a positive test, given disease". # The ROC curve may be used to determine an "optimal" cutoff # point for the test. # # The routine takes three arguments: # (1) type of model: 'decrease' or 'increase', this states the assumption # that a higher ('increase') value of the data tends to be an # indicator of a positive test result or for the model 'decrease' # a lower value. # (2) two-sided confidence interval (usually 0.95 is chosen). # (3) the data stored as a list-of-lists: # each entry in this list consits of an "value / true group" pair, # i.e. value / disease present. Group values are from {0,1}. # 0 stands for disease (or signal) not present (prior knowledge) and # 1 for disease (or signal) present (prior knowledge). # Example: @s=([2, 0], [12.5, 1], [3, 0], [10, 1], [9.5, 0], [9, 1]); # Notice the small overlap of the groups. The # optimal cutoff point to separate the two groups would be between # 9 and 9.5 if the criterion of optimality is to maximize the # probability of detection and simultaneously minimize the # probability of false alarm. # # Returns a list-of-lists with the three curves: # @ROC=([@lower_b], [@roc], [@upper_b]) each of the curves is # again a list-of-lists with each entry consisting of one (x,y) pair. # The routine impelements the method described in: # R.A. Hilgers, Distribution-Free Confidence Bounds for ROC Curves, # Meth Inform Med 1991; 30:96-101 # # copyright by H.A. Kestler, 1998 my$model_type = shift;   # assign
my $conf = shift; use vars '@val_grp'; local (*val_grp)=@_; my ($cu, $cl,$elem, $n1,$n0, $i,$j);
my @grp1=();my @grp0=();
my (@f_l_1,@f_m_1,@f_h_1,@f_l_0,@f_m_0,@f_h_0,@mat,@xx,@yy,@x,@y,@index);
my (@lower_b ,@roc ,@upper_b, @ROC);

# make checks
if($conf>=1 ||$conf<=0){ croak
"The nominal 2-sided confidence limit must be a number of [0,1].\n";}
if($model_type ne 'increase' &&$model_type ne 'decrease'){ croak
"Wrong model type specified!\n";}

$cu=(sqrt($conf)+1)/2;   # calculate the one-sided upper
$cl=1-$cu;               # and lower confidence limits

# extract values
for($i=0;$i<@val_grp;$i++){ if($val_grp[$i][1]==1) { push @grp1,$val_grp[$i][0];} else { push @grp0,$val_grp[$i][0];} } # compute ranks and values of inverse incomplete beta function @f_l_1=rank('low' ,@grp1); @f_m_1=rank('mean',@grp1); @f_h_1=rank('high',@grp1); @f_l_0=rank('low' ,@grp0); @f_m_0=rank('mean',@grp0); @f_h_0=rank('high',@grp0);$n1=@grp1; $n0=@grp0; # number of elements in both arrays for$elem (@f_l_1){ $elem=Xinbta($elem,$n1+1-$elem,$cl);} for$elem (@f_m_1){ $elem=Xinbta($elem,$n1+1-$elem,0.5);}
for $elem (@f_h_1){$elem=Xinbta($elem,$n1+1-$elem,$cu);}
for $elem (@f_l_0){$elem=Xinbta($elem,$n0+1-$elem,$cl);}
for $elem (@f_m_0){$elem=Xinbta($elem,$n0+1-$elem,0.5);} for$elem (@f_h_0){ $elem=Xinbta($elem,$n0+1-$elem,$cu);} # merge and sort @mat=(); for($i=0;$i<$n1;$i++){ push @mat, [($grp1[$i], -1, -1, -1,$f_l_1[$i],$f_m_1[$i],$f_h_1[$i])];} for($i=0;$i<$n0;$i++){ push @mat, [($grp0[$i],$f_l_0[$i],$f_m_0[$i],$f_h_0[$i], -1, -1, -1)];} # sort numerically according to value in first column @mat=@mat[sort{$mat[$a][0] <=>$mat[$b][0]} 0..$#mat];

# for practical purposes augment @mat and fill missing data (-1)
# at the beginning and end of the matrix
unshift @mat,[-1, 0, 0, Xinbta(1,$n0,$cu), 0, 0, Xinbta(1,$n1,$cu)];
push    @mat,[-1, Xinbta($n0,1,$cl), 1, 1, Xinbta($n1,1,$cl), 1, 1];
for($i=1;$mat[$i][1]==-1;$i++){
$mat[$i][1]=0; $mat[$i][2]=0; $mat[$i][3]=$mat[0][3];} for($i=1;$mat[$i][4]==-1; $i++){$mat[$i][4]=0;$mat[$i][5]=0;$mat[$i][6]=$mat[0][6];}
for($i=$#mat-1;$mat[$i][1]==-1; $i--){$mat[$i][1]=$mat[$#mat][1];$mat[$i][2]=1;$mat[$i][3]=1;} for($i=$#mat-1;$mat[$i][4]==-1;$i--){
$mat[$i][4]=$mat[$#mat][4]; $mat[$i][5]=1; $mat[$i][6]=1;}

# replace missing data (-1) with a piecewise linear interpolation
for($j=1;$j<7;$j++) # iterate thru columns { for($i=1,@xx=(),@yy=(),@x=(),@index=();$i<$#mat;$i++){ push @xx,$mat[$i][0] if$mat[$i][$j] !=-1;
push @yy, $mat[$i][$j] if$mat[$i][$j] !=-1;
push @x,  $mat[$i][0]  if $mat[$i][$j] ==-1; push @index,$i        if $mat[$i][$j] ==-1; } @y=interp(@xx,@yy,@x); for($i=0;$i<@index;$i++){ $mat[$index[$i]][$j]=$y[$i];}
}

# calculate (x,y) pairs of ROC curve and its limit curves
# (lower, ROC, upper) according to specified model
for($i=0,@lower_b=(),@roc=(),@upper_b=(),;$i<@mat;$i++){ if($model_type eq 'decrease'){
push @lower_b, [($mat[$i][3], $mat[$i][4])];
push @roc,     [($mat[$i][2], $mat[$i][5])];
push @upper_b, [($mat[$i][1], $mat[$i][6])];
}
else{
push @lower_b, [(1-$mat[$i][3], 1-$mat[$i][4])];
push @roc,     [(1-$mat[$i][2], 1-$mat[$i][5])];
push @upper_b, [(1-$mat[$i][1], 1-$mat[$i][6])];
}
}

@ROC=([@lower_b], [@roc], [@upper_b]);
return  @ROC;
}

# Autoload methods go after =cut, and are processed by the autosplit program.

1;
__END__

Statistics::ROC - receiver-operator-characteristic (ROC) curves with nonparametric confidence bounds

use Statistics::ROC;

my ($y) = loggamma($x);
my ($y) = betain($x, $p,$q, $beta); my ($y)    = Betain($x,$p, $q); my ($y)    = xinbta($p,$q, $beta,$alpha);
my ($y) = Xinbta($p, $q,$alpha);
my (@rk)   = rank($type, \@r); my (@ROC) = roc($model_type,$conf,\@val_grp); =head1 DESCRIPTION This program determines the ROC curve and its nonparametric confidence bounds for data categorized into two groups. A ROC curve shows the relationship of B<probability of false alarm> (x-axis) to B<probability of detection> (y-axis) for a certain test. Expressed in medical terms: the B<probability of a positive test, given no disease> to the B<probability of a positive test, given disease>. The ROC curve may be used to determine an I<optimal> cutoff point for the test. The main function is B<roc()>. The other exported functions are used by B<roc()>, but might be useful for other nonparametric statistical procedures. =over 4 =item B<loggamma> This procedure evaluates the natural logarithm of gamma(x) for all x>0, accurate to 10 decimal places. Stirlings formula is used for the central polynomial part of the procedure. For C<x=0> a value of 743.746924740801 will be returned: this is loggamma(9.9999999999E-324). =item B<betain> Computes incomplete beta function ratio Remarks: Complete beta function: B(p,q)=gamma(p)*gamma(q)/gamma(p+q) log(B(p,q))=ln(gamma(p))+ln(gamma(q))-ln(gamma(p+q)) Incomplete beta function ratio: I_x(p,q)=1/B(p,q) * \int_0^x t^{p-1}*(1-t)^{q-1} dt --> log(B(p,q)) has to be supplied to calculate I_x(p,q) log denotes the natural logarithm$beta = log(B(p,q))
$x = x$p    = p
$q = q The subroutine returns I_x(p,q). If an error occurs a negative value {-1,-2} is returned. =item B<Betain> Computes the incomplete beta function by calling B<loggamma()> and B<betain()>. =item B<xinbta> Computes inverse of incomplete beta function ratio Remarks: Complete beta function: B(p,q)=gamma(p)*gamma(q)/gamma(p+q) log(B(p,q))=ln(gamma(p))+ln(gamma(q))-ln(gamma(p+q)) Incomplete beta function ratio: alpha = I_x(p,q) = 1/B(p,q) * \int_0^x t^{p-1}*(1-t)^{q-1} dt --> log(B(p,q)) has to be supplied to calculate I_x(p,q) log denotes the natural logarithm$beta = log(B(p,q))
$alpha= I_x(p,q)$p    = p
$q = q The subroutine returns x. If an error occurs a negative value {-1,-2,-3} is returned. =item B<Xinbta> Computes the inverse of the incomplete beta function by calling B<loggamma()> and B<xinbta()>. =item B<rank> Computes the ranks of the values specified as the second argument (an array). Returns a vector of ranks corresponding to the input vector. Different types of ranking are possible ('high', 'low', 'mean'), and are specified as first argument. These differ in the way ties of the input vector, i.e. identical values, are treated: =over 10 =item * B<high>: replace ranks of identical values with their highest rank =item * B<low>: replace ranks of identical values with their lowest rank =item * B<mean>: replace ranks of identical values with the mean of their ranks =back =item B<roc> Determines the ROC curve and its nonparametric confidence bounds. The ROC curve shows the relationship of "probability of false alarm" (x-axis) to "probability of detection" (y-axis) for a certain test. Or in medical terms: the "probability of a positive test, given no disease" to the "probability of a positive test, given disease". The ROC curve may be used to determine an "optimal" cutoff point for the test. The routine takes three arguments: (1) type of model: 'decrease' or 'increase', this states the assumption that a higher ('increase') value of the data tends to be an indicator of a positive test result or for the model 'decrease' a lower value. (2) two-sided confidence interval (usually 0.95 is chosen). (3) the data stored as a list-of-lists: each entry in this list consits of an "value / true group" pair, i.e. value / disease present. Group values are from {0,1}. 0 stands for disease (or signal) not present (prior knowledge) and 1 for disease (or signal) present (prior knowledge). Example: @s=([2, 0], [12.5, 1], [3, 0], [10, 1], [9.5, 0], [9, 1]); Notice the small overlap of the groups. The optimal cutoff point to separate the two groups would be between 9 and 9.5 if the criterion of optimality is to maximize the probability of detection and simultaneously minimize the probability of false alarm. Returns a list-of-lists with the three curves: @ROC=([@lower_b], [@roc], [@upper_b]) each of the curves is again a list-of-lists with each entry consisting of one (x,y) pair. =back =over 4 =head2 Examples$,=" ";
print loggamma(10), "\n";
print Xinbta(3,4,Betain(.6,3,4)),"\n";

@e=(0.7, 0.7, 0.9, 0.6, 1.0, 1.1, 1,.7,.6);
print rank('low',@e),"\n";
print rank('high',@e),"\n";
print rank('mean',@e),"\n";

@var_grp=([1.5,0],[1.4,0],[1.4,0],[1.3,0],[1.2,0],[1,0],[0.8,0],
[1.1,1],[1,1],[1,1],[0.9,1],[0.7,1],[0.7,1],[0.6,1]);
@curves=roc('decrease',0.95,@var_grp);
print "$curves[0][2][0]$curves[0][2][1] \n";

Hans A. Kestler,  I<hans.kestler@uni-ulm.de>     B<or>
I<h.kestler@ieee.org>

Perl/Tk userinterface for drawing ROC curves (requires installed Tk and X11 on MacOS X).

R.A. Hilgers, Distribution-Free Confidence Bounds for ROC Curves (1991),
I<Meth Inform Med>, 30:96-101

Algorithm 291, Logarithm of the gamma function.
I<Collected Algorithms of the ACM>, Vol II, 1980

I<Numerical Recipes in C>, second edition, by Press, Teukolsky, Vetterling and Flannery,
Cambridge University Press, 1992.

G.W. Cran, K.J. Martin and G.E. Thomas (1977).Remark AS R19 and
Algorithm AS109, A Remark on Algorithms AS 63: The Incomplete Beta Integral
AS 64: Inverse of the Incomplete Beta Function Ratio,
I<Appl Statist>, 26:111-114.

K.J. Berry, P.W. Mielke, Jr and G.W. Cran (1990) Algorithm AS R83, A Remark
on Algorithm AS 109: Inverse of the Incomplete Beta Function Ratio,
I<Appl Statist>, 39:309-310.

=cut