g02bn Method
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g02bn computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data; the data array is overwritten with the ranks of the observations.

Syntax

C#
public static void g02bn(
	int n,
	int m,
	double[,] x,
	int itype,
	double[,] rr,
	out int ifail
)
Visual Basic (Declaration)
Public Shared Sub g02bn ( _
	n As Integer, _
	m As Integer, _
	x As Double(,), _
	itype As Integer, _
	rr As Double(,), _
	<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void g02bn(
	int n, 
	int m, 
	array<double,2>^ x, 
	int itype, 
	array<double,2>^ rr, 
	[OutAttribute] int% ifail
)
F#
static member g02bn : 
        n:int * 
        m:int * 
        x:float[,] * 
        itype:int * 
        rr:float[,] * 
        ifail:int byref -> unit

Parameters

n
Type: System..::.Int32
On entry: n, the number of observations or cases.
Constraint: n≥2.
m
Type: System..::.Int32
On entry: m, the number of variables.
Constraint: m≥2.
x
Type: array< System..::.Double ,2>[,](,)[,]
An array of size [ldx, m]
Note: ldx must satisfy the constraint: ldx≥n
On entry: x[i-1,j-1] must be set to xij, the value of the ith observation on the jth variable, for i=1,2,…,n and j=1,2,…,m.
On exit: x[i-1,j-1] contains the rank yij of the observation xij, for i=1,2,…,n and j=1,2,…,m.
itype
Type: System..::.Int32
On entry: the type of correlation coefficients which are to be calculated.
itype=-1
Only Kendall's tau coefficients are calculated.
itype=0
Both Kendall's tau and Spearman's coefficients are calculated.
itype=1
Only Spearman's coefficients are calculated.
Constraint: itype=-1, 0 or 1.
rr
Type: array< System..::.Double ,2>[,](,)[,]
An array of size [ldrr, m]
Note: ldrr must satisfy the constraint: ldrr≥m
On exit: the requested correlation coefficients.
If only Kendall's tau coefficients are requested (itype=-1), rr[j-1,k-1] contains Kendall's tau for the jth and kth variables.
If only Spearman's coefficients are requested (itype=1), rr[j-1,k-1] contains Spearman's rank correlation coefficient for the jth and kth variables.
If both Kendall's tau and Spearman's coefficients are requested (itype=0), the upper triangle of rr contains the Spearman coefficients and the lower triangle the Kendall coefficients. That is, for the jth and kth variables, where j is less than k, rr[j-1,k-1] contains the Spearman rank correlation coefficient, and rr[k-1,j-1] contains Kendall's tau, for j,k=1,2,…,m.
(Diagonal terms, rr[j-1,j-1], are unity for all three values of itype.)
ifail
Type: System..::.Int32 %
On exit: ifail=0 unless the method detects an error (see [Error Indicators and Warnings]).

Description

The input data consists of n observations for each of m variables, given as an array
xij,  i=1,2,…,n n≥2,j=1,2,…,mm≥2,
where xij is the ith observation of the jth variable.
The quantities calculated are:
(a) Ranks
For a given variable, j say, each of the n observations, x1j,x2j,…,xnj, has associated with it an additional number, the ‘rank’ of the observation, which indicates the magnitude of that observation relative to the magnitudes of the other n-1 observations on that same variable.
The smallest observation for variable j is assigned the rank 1, the second smallest observation for variable j the rank 2, the third smallest the rank 3, and so on until the largest observation for variable j is given the rank n.
If a number of cases all have the same value for the given variable, j, then they are each given an ‘average’ rank, e.g., if in attempting to assign the rank h+1, k observations were found to have the same value, then instead of giving them the ranks
h+1,h+2,…,h+k,
all k observations would be assigned the rank
2h+k+12
and the next value in ascending order would be assigned the rank
h+k+ 1.
The process is repeated for each of the m variables.
Let yij be the rank assigned to the observation xij when the jth variable is being ranked. The actual observations xij are replaced by the ranks yij.
(b) Nonparametric rank correlation coefficients
(i) Kendall's tau:
Rjk=∑h=1n∑i=1nsignyhj-yijsignyhk-yik nn-1-Tjnn-1-Tk ,  j,k=1,2,…,m,
where sign⁡u=1 if u>0,
sign⁡u=0 if u=0,
sign⁡u=-1 if u<0,
and Tj=∑tjtj-1, where tj is the number of ties of a particular value of variable j, and the summation is over all tied values of variable j.
(ii) Spearman's:
Rjk*=nn2-1-6∑i=1n yij-yik 2-12Tj*+Tk* nn2-1-Tj*nn2-1-Tk* ,  j,k=1,2,…,m,
where Tj*=∑tjtj2-1, tj being the number of ties of a particular value of variable j, and the summation being over all tied values of variable j.

References

Error Indicators and Warnings

Accuracy

Further Comments

The time taken by g02bn depends on n and m.

Example

See Also