# NAG Library Routine Document

## 1Purpose

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.

## 2Specification

Fortran Interface
 Subroutine g02bqf ( n, m, x, ldx, rr, ldrr,
 Integer, Intent (In) :: n, m, ldx, itype, ldrr Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: kworka(n), kworkb(n) Real (Kind=nag_wp), Intent (In) :: x(ldx,m) Real (Kind=nag_wp), Intent (Inout) :: rr(ldrr,m) Real (Kind=nag_wp), Intent (Out) :: work1(n), work2(n)
C Header Interface
#include nagmk26.h
 void g02bqf_ (const Integer *n, const Integer *m, const double x[], const Integer *ldx, const Integer *itype, double rr[], const Integer *ldrr, Integer kworka[], Integer kworkb[], double work1[], double work2[], Integer *ifail)

## 3Description

The input data consists of $n$ observations for each of $m$ variables, given as an array
 $xij, i=1,2,…,nn≥2,j=1,2,…,mm≥2,$
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 $n-1$ 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 ${y}_{ij}$ 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:
 $Rjk=∑h=1n∑i=1nsignyhj-yijsignyhk-yik nn-1-Tjnn-1-Tk , j,k=1,2,…,m,$
 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:
 $Rjk*=nn2-1-6∑i=1n yij-yik 2-12Tj*+Tk* nn2-1-Tj*nn2-1-Tk* , j,k=1,2,…,m,$
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 $j$.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations or cases.
Constraint: ${\mathbf{n}}\ge 2$.
2:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 2$.
3:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{m}}\right)$ – 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:     $\mathbf{ldx}$ – 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:     $\mathbf{itype}$ – 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:     $\mathbf{rr}\left({\mathbf{ldrr}},{\mathbf{m}}\right)$ – 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:     $\mathbf{ldrr}$ – 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:     $\mathbf{kworka}\left({\mathbf{n}}\right)$ – Integer arrayWorkspace
9:     $\mathbf{kworkb}\left({\mathbf{n}}\right)$ – Integer arrayWorkspace
10:   $\mathbf{work1}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
11:   $\mathbf{work2}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
12:   $\mathbf{ifail}$ – 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).

## 6Error 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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 2$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ldrr}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldrr}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{itype}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{itype}}=-1$ or $1$.
${\mathbf{ifail}}=-99$
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.

## 7Accuracy

The method used is believed to be stable.

## 8Parallelism 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 implementation-specific information.

## 9Further Comments

The time taken by g02bqf depends on $n$ and $m$.

## 10Example

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.

### 10.1Program Text

Program Text (g02bqfe.f90)

### 10.2Program Data

Program Data (g02bqfe.d)

### 10.3Program Results

Program Results (g02bqfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017