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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_coeffs_kspearman_overwrite (g02bn)

## Purpose

nag_correg_coeffs_kspearman_overwrite (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.

## Syntax

[x, rr, ifail] = g02bn(x, itype, 'n', n, 'm', m)
[x, rr, ifail] = nag_correg_coeffs_kspearman_overwrite(x, itype, 'n', n, 'm', m)
Note: the interface to this routine has changed since earlier releases of the toolbox:
 At Mark 22: n was made optional

## Description

The input data consists of $n$ observations for each of $m$ variables, given as an array
 $xij, i=1,2,…,n n≥2,j=1,2,…,mm≥2,$
where ${x}_{ij}$ is the $i$th observation of the $j$th variable.
The quantities calculated are:
(a) Ranks
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 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
 $h+1,h+2,…,h+k,$
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 actual observations ${x}_{ij}$ are replaced by the ranks ${y}_{ij}$.
(b) Nonparametric rank correlation coefficients
(i) Kendall's tau:
 $Rjk=∑h=1n∑i=1nsignyhj-yijsignyhk-yik nn-1-Tjnn-1-Tk , j,k=1,2,…,m,$
 where $\mathrm{sign}u=1$ if $u>0$, $\mathrm{sign}u=0$ if $u=0$, $\mathrm{sign}u=-1$ if $u<0$,
and ${T}_{j}=\sum {t}_{j}\left({t}_{j}-1\right)$, 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 $j$
(ii) Spearman's:
 $Rjk*=nn2-1-6∑i=1n yij-yik 2-12Tj*+Tk* nn2-1-Tj*nn2-1-Tk* , j,k=1,2,…,m,$
where ${T}_{j}^{*}=\sum {t}_{j}\left({t}_{j}^{2}-1\right)$, ${t}_{j}$ being the number of ties of a particular value of variable $j$, and the summation being over all tied values of variable $j$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{x}\left(\mathit{ldx},{\mathbf{m}}\right)$ – double array
ldx, the first dimension of the array, must satisfy the constraint $\mathit{ldx}\ge {\mathbf{n}}$.
${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must be set to ${x}_{\mathit{i}\mathit{j}}$, the value of the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
2:     $\mathrm{itype}$int64int32nag_int scalar
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 $1$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array x.
$n$, the number of observations or cases.
Constraint: ${\mathbf{n}}\ge 2$.
2:     $\mathrm{m}$int64int32nag_int scalar
Default: the second dimension of the array x.
$m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 2$.

### Output Parameters

1:     $\mathrm{x}\left(\mathit{ldx},{\mathbf{m}}\right)$ – double array
${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ contains the rank ${y}_{\mathit{i}\mathit{j}}$ of the observation ${x}_{\mathit{i}\mathit{j}}$, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
2:     $\mathrm{rr}\left(\mathit{ldrr},{\mathbf{m}}\right)$ – double array
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.)
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<2$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{m}}<2$.
${\mathbf{ifail}}=3$
 On entry, $\mathit{ldx}<{\mathbf{n}}$, or $\mathit{ldrr}<{\mathbf{m}}$.
${\mathbf{ifail}}=4$
 On entry, ${\mathbf{itype}}<-1$, or ${\mathbf{itype}}>1$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The method used is believed to be stable.

The time taken by nag_correg_coeffs_kspearman_overwrite (g02bn) depends on $n$ and $m$.

## Example

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.
```function g02bn_example

fprintf('g02bn example results\n\n');

x = [1.7,  1, 0.5;
2.8,  4, 3.0;
0.6,  6, 2.5;
1.8,  9, 6.0;
0.99, 4, 2.5;
1.4,  2, 5.5;
1.8,  9, 7.5;
2.5,  7, 0.0;
0.99, 5, 3.0];
[n,m] = size(x);
fprintf('Number of variables (columns) = %d\n', m);
fprintf('Number of cases     (rows)    = %d\n\n', n);
disp('Data matrix is:-');
disp(x);

itype = int64(0);
[x, rr, ifail] = g02bn(x, itype);

fprintf('\nMatrix of ranks:-\n');
disp(x);
fprintf('Matrix of rank correlation coefficients:\n');
fprintf('Upper triangle -- Spearman''s\n');
fprintf('Lower triangle -- Kendall''s tau\n\n');
disp(rr);

```
```g02bn example results

Number of variables (columns) = 3
Number of cases     (rows)    = 9

Data matrix is:-
1.7000    1.0000    0.5000
2.8000    4.0000    3.0000
0.6000    6.0000    2.5000
1.8000    9.0000    6.0000
0.9900    4.0000    2.5000
1.4000    2.0000    5.5000
1.8000    9.0000    7.5000
2.5000    7.0000         0
0.9900    5.0000    3.0000

Matrix of ranks:-
5.0000    1.0000    2.0000
9.0000    3.5000    5.5000
1.0000    6.0000    3.5000
6.5000    8.5000    8.0000
2.5000    3.5000    3.5000
4.0000    2.0000    7.0000
6.5000    8.5000    9.0000
8.0000    7.0000    1.0000
2.5000    5.0000    5.5000

Matrix of rank correlation coefficients:
Upper triangle -- Spearman's
Lower triangle -- Kendall's tau

1.0000    0.2246    0.1186
0.0294    1.0000    0.3814
0.1176    0.2353    1.0000

```