g02bp computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data omitting completely any cases with a missing observation for any variable; the data array is overwritten with the ranks of the observations.

Syntax

C#
public static void g02bp(
	int n,
	int m,
	double[,] x,
	int[] miss,
	double[] xmiss,
	int itype,
	double[,] rr,
	out int ncases,
	int[] incase,
	out int ifail
)
Visual Basic
Public Shared Sub g02bp ( _
	n As Integer, _
	m As Integer, _
	x As Double(,), _
	miss As Integer(), _
	xmiss As Double(), _
	itype As Integer, _
	rr As Double(,), _
	<OutAttribute> ByRef ncases As Integer, _
	incase As Integer(), _
	<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void g02bp(
	int n, 
	int m, 
	array<double,2>^ x, 
	array<int>^ miss, 
	array<double>^ xmiss, 
	int itype, 
	array<double,2>^ rr, 
	[OutAttribute] int% ncases, 
	array<int>^ incase, 
	[OutAttribute] int% ifail
)
F#
static member g02bp : 
        n : int * 
        m : int * 
        x : float[,] * 
        miss : int[] * 
        xmiss : float[] * 
        itype : int * 
        rr : float[,] * 
        ncases : int byref * 
        incase : int[] * 
        ifail : int byref -> unit 

Parameters

n
Type: System..::..Int32
On entry: n, the number of observations or cases.
Constraint: n2.
m
Type: System..::..Int32
On entry: m, the number of variables.
Constraint: m2.
x
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. (For those observations containing missing values, and therefore excluded from the calculation, yij=0, for j=1,2,,m.)
miss
Type: array<System..::..Int32>[]()[][]
An array of size [m]
On entry: miss[j-1] must be set to 1 if a missing value, xmj, is to be specified for the jth variable in the array x, or set equal to 0 otherwise. Values of miss must be given for all m variables in the array x.
On exit: the array miss is overwritten by the method, and the information it contained on entry is lost.
xmiss
Type: array<System..::..Double>[]()[][]
An array of size [m]
On entry: xmiss[j-1] must be set to the missing value, xmj, to be associated with the jth variable in the array x, for those variables for which missing values are specified by means of the array miss (see [Accuracy]).
On exit: the array xmiss is overwritten by the method, and the information it contained on entry is lost.
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 [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.)
ncases
Type: System..::..Int32%
On exit: the number of cases, nc, actually used in the calculations (when cases involving missing values have been eliminated).
incase
Type: array<System..::..Int32>[]()[][]
An array of size [n]
On exit: incase[i-1] holds the value 1 if the ith case was included in the calculations, and the value 0 if the ith case contained a missing value for at least one variable. That is, incase[i-1]=wi (see [Description]), for i=1,2,,n.
ifail
Type: System..::..Int32%
On exit: ifail=0 unless the method detects an error or a warning has been flagged (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 n2,j=1,2,,m m2,
where xij is the ith observation on the jth variable. In addition, each of the m variables may optionally have associated with it a value which is to be considered as representing a missing observation for that variable; the missing value for the jth variable is denoted by xmj. Missing values need not be specified for all variables.
Let wi=0 if observation i contains a missing value for any of those variables for which missing values have been declared; i.e., if xij=xmj for any j for which an xmj has been assigned (see also [Accuracy]); and wi=1 otherwise, for i=1,2,,n.
The quantities calculated are:
(a) Ranks
For a given variable, j say, each of the observations xij for which wi=1, for i=1,2,,n, 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 observations on that same variable for which wi=1.
The smallest of these valid observations 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 such observation is given the rank nc, where nc=i=1nwi.
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 for which wi=1 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. For those observations, i, for which wi=0, yij=0, for j=1,2,,m.
The actual observations xij are replaced by the ranks yij, for i=1,2,,n and j=1,2,,m.
(b) Nonparametric rank correlation coefficients
(i) Kendall's tau:
Rjk=h=1ni=1nwhwisignyhj-yijsignyhk-yikncnc-1-Tjncnc-1-Tk,  j,k=1,2,,m,
where nc=i=1nwi
and 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*=ncnc2-1-6i=1nwiyij-yik2-12Tj*+Tk*ncnc2-1-Tj*ncnc2-1-Tk*,  j,k=1,2,,m,
where nc=i=1nwi 
and Tj*=tjtj2-1 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.

References

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.
ifail=1
On entry,n<2.
ifail=2
On entry,m<2.
ifail=4
On entry,itype<-1,
oritype>1.
ifail=5
After observations with missing values were omitted, fewer than 2 cases remained.
ifail=-9000
An error occured, see message report.
ifail=-6000
Invalid Parameters value
ifail=-4000
Invalid dimension for array value
ifail=-8000
Negative dimension for array value
ifail=-6000
Invalid Parameters value

Accuracy

You are warned of the need to exercise extreme care in your selection of missing values. g02bp treats all values in the inclusive range 1±0.1x02be-2×xmj, where xmj is the missing value for variable j specified in xmiss.
You must therefore ensure that the missing value chosen for each variable is sufficiently different from all valid values for that variable so that none of the valid values fall within the range indicated above.

Parallelism and Performance

None.

Further Comments

The time taken by g02bp depends on n and m, and the occurrence of missing values.

Example

This example reads in a set of data consisting of nine observations on each of three variables. Missing values of 0.99 and 0.0 are declared for the first and third variables respectively; no missing value is specified for the second variable. 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, omitting completely all cases containing missing values; cases 5, 8 and 9 are therefore eliminated, leaving only six cases in the calculations.

Example program (C#): g02bpe.cs

Example program data: g02bpe.d

Example program results: g02bpe.r

See Also