Statistics::ROC 0.04

Statistics::ROC is a Perl module with receiver-operator-characteristic (ROC) curves with nonparametric confidence bounds.

SYNOPSIS

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);

This program determines the ROC curve and its nonparametric confidence bounds for data categorized into two groups. A ROC curve shows the relationship of probability of false alarm (x-axis) to probability of detection (y-axis) for a certain test. Expressed 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 main function is roc(). The other exported functions are used by roc(), but might be useful for other nonparametric statistical procedures.

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 x=0 a value of 743.746924740801 will be returned: this is loggamma(9.9999999999E-324).

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.

Betain

Computes the incomplete beta function by calling loggamma() and betain().

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.

Xinbta

Computes the inverse of the incomplete beta function by calling loggamma() and xinbta().

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 ranks

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.

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";

Requirements:

· Perl



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Requirements:

· Perl