G02BSF (PDF version)
G02 Chapter Contents
G02 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

G02BSF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

G02BSF computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data omitting cases with missing values from only those calculations involving the variables for which the values are missing; the data array is preserved, and the ranks of the observations are not available on exit from the routine.

2  Specification

SUBROUTINE G02BSF ( N, M, X, LDX, MISS, XMISS, ITYPE, RR, LDRR, NCASES, CNT, LDCNT, KWORKA, KWORKB, KWORKC, KWORKD, WORK1, WORK2, IFAIL)
INTEGER  N, M, LDX, MISS(M), ITYPE, LDRR, NCASES, LDCNT, KWORKA(N), KWORKB(N), KWORKC(N), KWORKD(N), IFAIL
REAL (KIND=nag_wp)  X(LDX,M), XMISS(M), RR(LDRR,M), CNT(LDCNT,M), WORK1(N), WORK2(N)

3  Description

The input data consists of n observations for each of m variables, given as an array
xij ,   i=1,2,,n   n2 ​ and ​ 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 wij=0 if the ith observation for the jth variable is a missing value, i.e., if a missing value, xmj, has been declared for the jth variable, and xij=xmj (see also Section 7); and wij=1 otherwise, for i=1,2,,n and j=1,2,,m.
The observations are first ranked, a pair of variables at a time as follows:
For a given pair of variables, j and l say, each of the observations xij for which the product wijwil=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 magnitude of the other observations on variable j for which wijwil=1.
The smallest of these valid observations for variable j is assigned to rank 1, the second smallest valid observation for variable j the rank 2, the third smallest rank 3, and so on until the largest such observation is given the rank njl, where
njl=i=1nwijwil.  
If a number of cases all have the same value for the 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 wijwil=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 variable l is then ranked in a similar way. The process is then repeated for all pairs of variables j and l, for j=1,2,,m and l=j,,m. Let yijl be the rank assigned to the observation xij when the jth and lth variables are being ranked, and yilj be the rank assigned to the observation xil during the same process, for i=1,2,,n, j=1,2,,m and l=j,,m.
The quantities calculated are:
(a) Kendall's tau rank correlation coefficients:
Rjk=h=1ni=1nwhjwhkwijwiksignyhjk-yijksignyhkj-yikj njknjk-1-Tjknjknjk-1-Tkj ,  j,k=1,2,,m,  
where njk=i=1nwijwik
and signu=1 if u>0
signu=0 if u=0
signu=-1 if u<0
and Tjk=tjtj-1 where tj is the number of ties of a particular value of variable j when the jth and kth variables are being ranked, and the summation is over all tied values of variable j.
(b) Spearman's rank correlation coefficients:
Rjk*= njknjk2-1-6i=1nwijwikyijk-yikj2-12Tjk*+Tkj* njknjk2-1-Tjk* njknjk2-1-Tkj* ,   j,k=1,2,,m,  
where njk=i=1nwijwik 
and Tjk*=tjtj2-1, where tj is the number of ties of a particular value of variable j when the jth and kth variables are being ranked, and the summation is over all tied values of variable j.

4  References

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

5  Arguments

1:     N – INTEGERInput
On entry: n, the number of observations or cases.
Constraint: N2.
2:     M – INTEGERInput
On entry: m, the number of variables.
Constraint: M2.
3:     XLDXM – REAL (KIND=nag_wp) arrayInput
On entry: Xij 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.
4:     LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which G02BSF is called.
Constraint: LDXN.
5:     MISSM – INTEGER arrayInput
On entry: MISSj must be set equal 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.
6:     XMISSM – REAL (KIND=nag_wp) arrayInput
On entry: XMISSj 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 Section 7).
7:     ITYPE – INTEGERInput
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.
8:     RRLDRRM – REAL (KIND=nag_wp) arrayOutput
On exit: the requested correlation coefficients.
If only Kendall's tau coefficients are requested (ITYPE=-1), RRjk contains Kendall's tau for the jth and kth variables.
If only Spearman's coefficients are requested (ITYPE=1), RRjk 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, RRjk contains the Spearman rank correlation coefficient, and RRkj contains Kendall's tau, for j=1,2,,m and k=1,2,,m.
(Diagonal terms, RRjj, are unity for all three values of ITYPE.)
9:     LDRR – INTEGERInput
On entry: the first dimension of the array RR as declared in the (sub)program from which G02BSF is called.
Constraint: LDRRM.
10:   NCASES – INTEGEROutput
On exit: the minimum number of cases used in the calculation of any of the correlation coefficients (when cases involving missing values have been eliminated).
11:   CNTLDCNTM – REAL (KIND=nag_wp) arrayOutput
On exit: the number of cases, njk, actually used in the calculation of the rank correlation coefficient for the jth and kth variables, for j=1,2,,m and k=1,2,,m.
12:   LDCNT – INTEGERInput
On entry: the first dimension of the array CNT as declared in the (sub)program from which G02BSF is called.
Constraint: LDCNTM.
13:   KWORKAN – INTEGER arrayWorkspace
14:   KWORKBN – INTEGER arrayWorkspace
15:   KWORKCN – INTEGER arrayWorkspace
16:   KWORKDN – INTEGER arrayWorkspace
17:   WORK1N – REAL (KIND=nag_wp) arrayWorkspace
18:   WORK2N – REAL (KIND=nag_wp) arrayWorkspace
19:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if IFAIL0 on exit, the recommended value is -1. When the value -1​ or ​1 is used it is essential to test the value of IFAIL on exit.
On exit: IFAIL=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

If on entry IFAIL=0 or -1, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Note: G02BSF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
IFAIL=1
On entry,N<2.
IFAIL=2
On entry,M<2.
IFAIL=3
On entry,LDX<N,
orLDRR<M,
orLDCNT<M.
IFAIL=4
On entry,ITYPE<-1,
orITYPE>1.
IFAIL=5
After observations with missing values were omitted, fewer than two cases remained for at least one pair of variables. (The pairs of variables involved can be determined by examination of the contents of the array CNT.) All correlation coefficients based on two or more cases are returned by the routine even if IFAIL=5.
IFAIL=-99
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
IFAIL=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
IFAIL=-999
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7  Accuracy

You are warned of the need to exercise extreme care in your selection of missing values. G02BSF treats all values in the inclusive range 1±0.1X02BEF-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.

8  Parallelism and Performance

G02BSF is not threaded in any implementation.

9  Further Comments

The time taken by G02BSF depends on n and m, and the occurrence of missing values.

10  Example

This example reads in a set of data consisting of nine observations on each of three variables. Missing values of 0.99, 9.0 and 0.0 are declared for the first, second and third variables respectively. The program then calculates and prints both Kendall's tau and Spearman's rank correlation coefficients for all three variables, omitting cases with missing values from only those calculations involving the variables for which the values are missing. The program therefore eliminates cases 4, 5, 7 and 9 in calculating and correlation between the first and second variables, cases 5, 8 and 9 for the first and third variables, and cases 4, 7 and 8 for the second and third variables.

10.1  Program Text

Program Text (g02bsfe.f90)

10.2  Program Data

Program Data (g02bsfe.d)

10.3  Program Results

Program Results (g02bsfe.r)


G02BSF (PDF version)
G02 Chapter Contents
G02 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2016