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.


public static void g02bn(
	int n,
	int m,
	double[,] x,
	int itype,
	double[,] rr,
	out int ifail
Visual Basic
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++
static void g02bn(
	int n, 
	int m, 
	array<double,2>^ x, 
	int itype, 
	array<double,2>^ rr, 
	[OutAttribute] int% ifail
static member g02bn : 
        n : int * 
        m : int * 
        x : float[,] * 
        itype : int * 
        rr : float[,] * 
        ifail : int byref -> unit 


Type: System..::..Int32
On entry: n, the number of observations or cases.
Constraint: n2.
Type: System..::..Int32
On entry: m, the number of variables.
Constraint: m2.
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, m]
Note: dim1 must satisfy the constraint: dim1n
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.
Type: System..::..Int32
On entry: the type of correlation coefficients which are to be calculated.
Only Kendall's tau coefficients are calculated.
Both Kendall's tau and Spearman's coefficients are calculated.
Only Spearman's coefficients are calculated.
Constraint: itype=-1, 0 or 1.
Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, m]
Note: dim1 must satisfy the constraint: dim1m
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=1,2,,m and k=1,2,,m.
(Diagonal terms, rr[j-1,j-1], are unity for all three values of itype.)
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).


The input data consists of n observations for each of m variables, given as an array
xij,  i=1,2,,nn2,j=1,2,,mm2,
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
all k observations would be assigned the rank
and the next value in ascending order would be assigned the rank
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=1ni=1nsignyhj-yijsignyhk-yiknn-1-Tjnn-1-Tk,  j,k=1,2,,m,
where signu=1 if u>0,
signu=0 if u=0,
signu=-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*=nn2-1-6i=1nyij-yik2-12Tj*+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.


Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill

Error Indicators and Warnings

Errors or warnings detected by the method:
Some error messages may refer to parameters that are dropped from this interface (LDX, LDRR) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
On entry,n<2.
On entry,m<2.
On entry,itype<-1,
An error occured, see message report.
Invalid Parameters value
Invalid dimension for array value
Negative dimension for array value
Invalid Parameters value


The method used is believed to be stable.

Parallelism and Performance


Further Comments

The time taken by g02bn depends on n and m.


This example reads in a set of data consisting of nine observations on each of three variables. The program then calculates and prints the rank of each observation, and both Kendall's tau and Spearman's rank correlation coefficients for all three variables.

Example program (C#): g02bne.cs

Example program data: g02bne.d

Example program results: g02bne.r

See Also