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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_coeffs_kspearman_miss_pair (g02bs)

## Purpose

nag_correg_coeffs_kspearman_miss_pair (g02bs) 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 function.

## Syntax

[rr, ncases, cnt, ifail] = g02bs(x, miss, xmiss, itype, 'n', n, 'm', m)
[rr, ncases, cnt, ifail] = nag_correg_coeffs_kspearman_miss_pair(x, miss, xmiss, 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
where ${x}_{ij}$ is the $i$th observation on the $j$th 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 $j$th variable is denoted by ${\mathit{xm}}_{j}$. Missing values need not be specified for all variables.
Let ${w}_{\mathit{i}\mathit{j}}=0$ if the $\mathit{i}$th observation for the $\mathit{j}$th variable is a missing value, i.e., if a missing value, ${\mathit{xm}}_{\mathit{j}}$, has been declared for the $\mathit{j}$th variable, and ${x}_{\mathit{i}\mathit{j}}={\mathit{xm}}_{\mathit{j}}$ (see also Accuracy); and ${w}_{\mathit{i}\mathit{j}}=1$ otherwise, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
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 ${x}_{\mathit{i}j}$ for which the product ${w}_{\mathit{i}j}{w}_{\mathit{i}l}=1$, for $\mathit{i}=1,2,\dots ,n$, 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 ${w}_{ij}{w}_{il}=1$.
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 ${n}_{jl}$, 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 ${w}_{ij}{w}_{il}=1$ 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 $\mathit{l}$ is then ranked in a similar way. The process is then repeated for all pairs of variables $\mathit{j}$ and $\mathit{l}$, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{l}=\mathit{j},\dots ,m$. Let ${y}_{\mathit{i}\mathit{j}\left(\mathit{l}\right)}$ be the rank assigned to the observation ${x}_{\mathit{i}\mathit{j}}$ when the $\mathit{j}$th and $\mathit{l}$th variables are being ranked, and ${y}_{\mathit{i}\mathit{l}\left(\mathit{j}\right)}$ be the rank assigned to the observation ${x}_{\mathit{i}\mathit{l}}$ during the same process, for $\mathit{i}=1,2,\dots ,n$, $\mathit{j}=1,2,\dots ,m$ and $\mathit{l}=j,\dots ,m$.
The quantities calculated are:
(a) Kendall's tau rank correlation coefficients:
 $Rjk=∑h=1n∑i=1nwhjwhkwijwiksignyhjk-yijksignyhkj-yikj njknjk-1-Tjknjknjk-1-Tkj , j,k=1,2,…,m,$
 where ${n}_{jk}=\sum _{i=1}^{n}{w}_{ij}{w}_{ik}$ and $\mathrm{sign}u=1$ if $u>0$ $\mathrm{sign}u=0$ if $u=0$ $\mathrm{sign}u=-1$ if $u<0$
and ${T}_{j\left(k\right)}=\sum {t}_{j}\left({t}_{j}-1\right)$ where ${t}_{j}$ is the number of ties of a particular value of variable $j$ when the $j$th and $k$th variables are being ranked, and the summation is over all tied values of variable $j$
(b) Spearman's rank correlation coefficients:
 $Rjk*= njknjk2-1-6∑i=1nwijwikyijk-yikj2-12Tjk*+Tkj* njknjk2-1-Tjk* njknjk2-1-Tkj* , j,k=1,2,…,m,$
where ${n}_{jk}=\sum _{i=1}^{n}{w}_{ij}{w}_{ik}$
and ${T}_{j\left(k\right)}^{*}=\sum {t}_{j}\left({t}_{j}^{2}-1\right)$, where ${t}_{j}$ is the number of ties of a particular value of variable $j$ when the $j$th and $k$th variables are being ranked, and the summation is 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:     $\mathrm{miss}\left({\mathbf{m}}\right)$int64int32nag_int array
${\mathbf{miss}}\left(j\right)$ must be set equal to $1$ if a missing value, $x{m}_{j}$, is to be specified for the $j$th 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.
3:     $\mathrm{xmiss}\left({\mathbf{m}}\right)$ – double array
${\mathbf{xmiss}}\left(j\right)$ must be set to the missing value, $x{m}_{j}$, to be associated with the $j$th variable in the array x, for those variables for which missing values are specified by means of the array miss (see Accuracy).
4:     $\mathrm{itype}$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 dimension of the arrays miss, xmiss and the second dimension of the array x. (An error is raised if these dimensions are not equal.)
$m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 2$.

### Output Parameters

1:     $\mathrm{rr}\left(\mathit{ldrr},{\mathbf{m}}\right)$ – 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.)
2:     $\mathrm{ncases}$int64int32nag_int scalar
The minimum number of cases used in the calculation of any of the correlation coefficients (when cases involving missing values have been eliminated).
3:     $\mathrm{cnt}\left(\mathit{ldcnt},{\mathbf{m}}\right)$ – double array
The number of cases, ${n}_{\mathit{j}\mathit{k}}$, actually used in the calculation of the rank correlation coefficient for the $\mathit{j}$th and $\mathit{k}$th variables, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
4:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Note: nag_correg_coeffs_kspearman_miss_pair (g02bs) may return useful information for one or more of the following detected errors or 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 $\mathit{ldrr}<{\mathbf{m}}$, or $\mathit{ldcnt}<{\mathbf{m}}$.
${\mathbf{ifail}}=4$
 On entry, ${\mathbf{itype}}<-1$, or ${\mathbf{itype}}>1$.
${\mathbf{ifail}}=5$
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 function even if ${\mathbf{ifail}}={\mathbf{5}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

You are warned of the need to exercise extreme care in your selection of missing values. nag_correg_coeffs_kspearman_miss_pair (g02bs) treats all values in the inclusive range $\left(1±{0.1}^{\left({\mathbf{x02be}}-2\right)}\right)×{xm}_{j}$, where ${\mathit{xm}}_{j}$ 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.

The time taken by nag_correg_coeffs_kspearman_miss_pair (g02bs) 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 $0.99$, $9.0$ 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.
```function g02bs_example

fprintf('g02bs 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);

miss  = [int64(1); 1; 1];
xmiss = [0.99;       9; 0];
itype = int64(0);

[rr, ncases, count, ifail] = ...
g02bs( ...
x, miss, xmiss, itype);

fprintf('Matrix of rank correlation coefficients:\n');
fprintf('Upper triangle -- Spearman''s\n');
fprintf('Lower triangle -- Kendall''s tau\n\n');
disp(rr);
fprintf('Number of cases used for any pair of variables = %d\n', ncases);
fprintf('Numbers used for each pair are:\n');
disp(count);

```
```g02bs 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 rank correlation coefficients:
Upper triangle -- Spearman's
Lower triangle -- Kendall's tau

1.0000    0.1000    0.4058
0    1.0000    0.0896
0.2760         0    1.0000

Number of cases used for any pair of variables = 5
Numbers used for each pair are:
7     5     6
5     7     6
6     6     8

```