NAG Library Routine Document
G02BQF
1 Purpose
G02BQF computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data; the data array is preserved, and the ranks of the observations are not available on exit from the routine.
2 Specification
SUBROUTINE G02BQF ( 
N, M, X, LDX, ITYPE, RR, LDRR, KWORKA, KWORKB, WORK1, WORK2, IFAIL) 
INTEGER 
N, M, LDX, ITYPE, LDRR, KWORKA(N), KWORKB(N), IFAIL 
REAL (KIND=nag_wp) 
X(LDX,M), RR(LDRR,M), WORK1(N), WORK2(N) 

3 Description
The input data consists of
$n$ observations for each of
$m$ variables, given as an array
where
${x}_{ij}$ is the
$i$th observation on the
$j$th variable.
The observations are first ranked, as follows.
For a given variable, $j$ say, each of the $n$ observations, ${x}_{1j},{x}_{2j},\dots ,{x}_{nj}$, 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 $n1$ 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 ${y}_{ij}$ be the rank assigned to the observation ${x}_{ij}$ when the $j$th variable is being ranked.
The quantities calculated are:
(a) 
Kendall's tau rank correlation coefficients:
and 
$\mathrm{sign}u=1$ if $u>0$ 

$\mathrm{sign}u=0$ if $u=0$ 

$\mathrm{sign}u=1$ if $u<0$ 
and ${T}_{j}=\sum {t}_{j}\left({t}_{j}1\right)$, ${t}_{j}$ being the number of ties of a particular value of variable $j$, and the summation being over all tied values of variable $j$. 
(b) 
Spearman's rank correlation coefficients:
where ${T}_{j}^{*}=\sum {t}_{j}\left({t}_{j}^{2}1\right)$ where ${t}_{j}$ is the number of ties of a particular value of variable $j$, and the summation is over all tied values of variable $j$. 
4 References
Siegel S (1956) Nonparametric Statistics for the Behavioral Sciences McGraw–Hill
5 Arguments
 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the number of observations or cases.
Constraint:
${\mathbf{N}}\ge 2$.
 2: $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of variables.
Constraint:
${\mathbf{M}}\ge 2$.
 3: $\mathrm{X}\left({\mathbf{LDX}},{\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: ${\mathbf{X}}\left(\mathit{i},\mathit{j}\right)$ must be set to data value ${x}_{\mathit{i}\mathit{j}}$, the value of the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
 4: $\mathrm{LDX}$ – INTEGERInput

On entry: the first dimension of the array
X as declared in the (sub)program from which G02BQF is called.
Constraint:
${\mathbf{LDX}}\ge {\mathbf{N}}$.
 5: $\mathrm{ITYPE}$ – INTEGERInput

On entry: the type of correlation coefficients which are to be calculated.
 ${\mathbf{ITYPE}}=1$
 Only Kendall's tau coefficients are calculated.
 ${\mathbf{ITYPE}}=0$
 Both Kendall's tau and Spearman's coefficients are calculated.
 ${\mathbf{ITYPE}}=1$
 Only Spearman's coefficients are calculated.
Constraint:
${\mathbf{ITYPE}}=1$, $0$ or $1$.
 6: $\mathrm{RR}\left({\mathbf{LDRR}},{\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the requested correlation coefficients.
If only Kendall's tau coefficients are requested (${\mathbf{ITYPE}}=1$), ${\mathbf{RR}}\left(j,k\right)$ contains Kendall's tau for the $j$th and $k$th variables.
If only Spearman's coefficients are requested (${\mathbf{ITYPE}}=1$), ${\mathbf{RR}}\left(j,k\right)$ contains Spearman's rank correlation coefficient for the $j$th and $k$th variables.
If both Kendall's tau and Spearman's coefficients are requested (
${\mathbf{ITYPE}}=0$), the upper triangle of
RR contains the Spearman coefficients and the lower triangle the Kendall coefficients. That is, for the
$\mathit{j}$th and
$\mathit{k}$th variables, where
$\mathit{j}$ is less than
$\mathit{k}$,
${\mathbf{RR}}\left(\mathit{j},\mathit{k}\right)$ contains the Spearman rank correlation coefficient, and
${\mathbf{RR}}\left(\mathit{k},\mathit{j}\right)$ contains Kendall's tau, for
$\mathit{j}=1,2,\dots ,m$ and
$\mathit{k}=1,2,\dots ,m$.
(Diagonal terms,
${\mathbf{RR}}\left(j,j\right)$, are unity for all three values of
ITYPE.)
 7: $\mathrm{LDRR}$ – INTEGERInput

On entry: the first dimension of the array
RR as declared in the (sub)program from which G02BQF is called.
Constraint:
${\mathbf{LDRR}}\ge {\mathbf{M}}$.
 8: $\mathrm{KWORKA}\left({\mathbf{N}}\right)$ – INTEGER arrayWorkspace
 9: $\mathrm{KWORKB}\left({\mathbf{N}}\right)$ – INTEGER arrayWorkspace
 10: $\mathrm{WORK1}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayWorkspace
 11: $\mathrm{WORK2}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayWorkspace

 12: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry,  ${\mathbf{N}}<2$. 
 ${\mathbf{IFAIL}}=2$

On entry,  ${\mathbf{M}}<2$. 
 ${\mathbf{IFAIL}}=3$

On entry,  ${\mathbf{LDX}}<{\mathbf{N}}$, 
or  ${\mathbf{LDRR}}<{\mathbf{M}}$. 
 ${\mathbf{IFAIL}}=4$

On entry,  ${\mathbf{ITYPE}}<1$, 
or  ${\mathbf{ITYPE}}>1$. 
 ${\mathbf{IFAIL}}=99$
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.
 ${\mathbf{IFAIL}}=399$
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.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7 Accuracy
The method used is believed to be stable.
8 Parallelism and Performance
G02BQF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time taken by G02BQF depends on $n$ and $m$.
10 Example
This example reads in a set of data consisting of nine observations on each of three variables. The program then calculates and prints both Kendall's tau and Spearman's rank correlation coefficients for all three variables.
10.1 Program Text
Program Text (g02bqfe.f90)
10.2 Program Data
Program Data (g02bqfe.d)
10.3 Program Results
Program Results (g02bqfe.r)