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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_coeffs_kspearman_miss_case (g02br)

## Purpose

nag_correg_coeffs_kspearman_miss_case (g02br) computes Kendall and/or Spearman nonparametric rank correlation coefficients for a set of data, omitting completely any cases with a missing observation for any variable; the data array is preserved, and the ranks of the observations are not available on exit from the function.

## Syntax

[rr, ncases, incase, ifail] = g02br(x, miss, xmiss, itype, 'n', n, 'm', m)
[rr, ncases, incase, ifail] = nag_correg_coeffs_kspearman_miss_case(x, miss, xmiss, itype, 'n', n, 'm', m)
Note: the interface to this routine has changed since earlier releases of the toolbox:
Mark 22: n has been made optional; miss, xmiss no longer output
.

## Description

The input data consists of n$n$ observations for each of m$m$ variables, given as an array
 [xij] ,   i = 1,2, … ,n (n ≥ 2) ,   j = 1,2, … ,m (m ≥ 2) , $[ xij ] , i=1,2,…,n (n≥2) , j=1,2,…,m (m≥2) ,$
where xij${x}_{ij}$ is the i$i$th observation on the j$j$th variable. In addition, each of the m$m$ 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$j$th variable is denoted by xmj${\mathit{xm}}_{j}$. Missing values need not be specified for all variables.
Let wi = 0${w}_{i}=0$ if observation i$i$ contains a missing value for any of those variables for which missing values have been declared, i.e., if xij = xmj${x}_{ij}={\mathit{xm}}_{j}$ for any j$j$ for which an xmj${\mathit{xm}}_{j}$ has been assigned (see also Section [Accuracy]); and wi = 1${w}_{i}=1$ otherwise, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
The observations are first ranked as follows.
For a given variable, j$j$ say, each of the observations xij${x}_{ij}$ for which wi = 1${w}_{i}=1$, (i = 1,2,,n$i=1,2,\dots ,n$) has associated with it an additional number, the ‘rank’ of the observation, which indicates the magnitude of that observation relative to the magnitudes of the other observations on that same variable for which wi = 1${w}_{i}=1$.
The smallest of these valid observations for variable j$j$ is assigned the rank 1$1$, the second smallest observation for variable j$j$ the rank 2$2$, the third smallest the rank 3$3$, and so on until the largest such observation is given the rank nc${n}_{c}$, where nc = i = 1nwi${n}_{c}=\sum _{i=1}^{n}{w}_{i}$.
If a number of cases all have the same value for the given variable, j$j$, then they are each given an ‘average’ rank, e.g., if in attempting to assign the rank h + 1$h+1$, k$k$ observations for which wi = 1${w}_{i}=1$ were found to have the same value, then instead of giving them the ranks
 h + 1, h + 2, … , h + k , $h+1, h+2, …, h+k ,$
all k$k$ observations would be assigned the rank
 (2h + k + 1)/2 $2h+k+1 2$
and the next value in ascending order would be assigned the rank
 h + k + 1 . $h+k+ 1 .$
The process is repeated for each of the m$m$ variables.
Let yij${y}_{ij}$ be the rank assigned to the observation xij${x}_{ij}$ when the j$j$th variable is being ranked. For those observations, i$i$, for which wi = 0${w}_{i}=0$, yij = 0${y}_{ij}=0$, for j = 1,2,,m$j=1,2,\dots ,m$.
The quantities calculated are:
(a) Kendall's tau rank correlation coefficients:
 Rjk = ( ∑ h = 1n ∑ i = 1n wh wi sign(yhj − yij) sign(yhk − yik) )/(sqrt([nc(nc − 1) − Tj][nc(nc − 1) − Tk])) ,   j,k = 1,2, … ,m , $Rjk = ∑ h=1 n ∑ i=1 n wh wi sign(yhj-yij) sign(yhk-yik) [nc(nc-1)-Tj][nc(nc-1)-Tk] , j,k=1,2,…,m ,$
 where nc = ∑ i = 1nwi${n}_{c}=\sum _{i=1}^{n}{w}_{i}$ and signu = 1$\mathrm{sign}u=1$ if u > 0$u>0$ signu = 0$\mathrm{sign}u=0$ if u = 0$u=0$ signu = − 1$\mathrm{sign}u=-1$ if u < 0$u<0$
and Tj = tj(tj1)${T}_{j}=\sum {t}_{j}\left({t}_{j}-1\right)$ where tj${t}_{j}$ is the number of ties of a particular value of variable j$j$, and the summation is over all tied values of variable j$j$.
(b) Spearman's rank correlation coefficients:
 Rjk * = (nc(nc2 − 1) − 6 ∑ i = 1nwi(yij − yik)2 − (1/2)(Tj * + Tk * ))/(sqrt([nc(nc2 − 1) − Tj * ][nc(nc2 − 1) − Tk * ])),  j,k = 1,2, … ,m, $Rjk*=nc(nc2-1)-6∑i=1nwi (yij-yik) 2-12(Tj*+Tk*) [nc(nc2-1)-Tj*][nc(nc2-1)-Tk*] , j,k=1,2,…,m,$
where nc = i = 1nwi${n}_{c}=\sum _{i=1}^{n}{w}_{i}$ and Tj * = tj(tj21)${T}_{j}^{*}=\sum {t}_{j}\left({t}_{j}^{2}-1\right)$ where tj${t}_{j}$ is the number of ties of a particular value of variable j$j$, and the summation is over all tied values of variable j$j$.

## References

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

## Parameters

### Compulsory Input Parameters

1:     x(ldx,m) – double array
ldx, the first dimension of the array, must satisfy the constraint ldxn$\mathit{ldx}\ge {\mathbf{n}}$.
x(i,j)${\mathbf{x}}\left(i,j\right)$ must be set to xij${x}_{ij}$, the value of the i$i$th observation on the j$j$th variable, where i = 1,2,,n$i=1,2,\dots ,n$ and j = 1,2,,m.$j=1,2,\dots ,m\text{.}$
2:     miss(m) – int64int32nag_int array
m, the dimension of the array, must satisfy the constraint m2${\mathbf{m}}\ge 2$.
miss(j)${\mathbf{miss}}\left(j\right)$ must be set equal to 1$1$ if a missing value, xmj$x{m}_{j}$, is to be specified for the j$j$th variable in the array x, or set equal to 0$0$ otherwise. Values of miss must be given for all m$m$ variables in the array x.
3:     xmiss(m) – double array
m, the dimension of the array, must satisfy the constraint m2${\mathbf{m}}\ge 2$.
xmiss(j)${\mathbf{xmiss}}\left(j\right)$ must be set to the missing value, xmj$x{m}_{j}$, to be associated with the j$j$th variable in the array x, for those variables for which missing values are specified by means of the array miss (see Section [Accuracy]).
4:     itype – int64int32nag_int scalar
The type of correlation coefficients which are to be calculated.
itype = -1${\mathbf{itype}}=-1$
Only Kendall's tau coefficients are calculated.
itype = 0${\mathbf{itype}}=0$
Both Kendall's tau and Spearman's coefficients are calculated.
itype = 1${\mathbf{itype}}=1$
Only Spearman's coefficients are calculated.
Constraint: itype = -1${\mathbf{itype}}=-1$, 0$0$ or 1$1$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array x.
n$n$, the number of observations or cases.
Constraint: n2${\mathbf{n}}\ge 2$.
2:     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$m$, the number of variables.
Constraint: m2${\mathbf{m}}\ge 2$.

### Input Parameters Omitted from the MATLAB Interface

ldx ldrr kworka kworkb kworkc work1 work2

### Output Parameters

1:     rr(ldrr,m) – double array
ldrrm$\mathit{ldrr}\ge {\mathbf{m}}$.
The requested correlation coefficients.
If only Kendall's tau coefficients are requested (itype = 1${\mathbf{itype}}=-1$), rr(j,k)${\mathbf{rr}}\left(j,k\right)$ contains Kendall's tau for the j$j$th and k$k$th variables.
If only Spearman's coefficients are requested (itype = 1${\mathbf{itype}}=1$), rr(j,k)${\mathbf{rr}}\left(j,k\right)$ contains Spearman's rank correlation coefficient for the j$j$th and k$k$th variables.
If both Kendall's tau and Spearman's coefficients are requested (itype = 0${\mathbf{itype}}=0$), the upper triangle of rr contains the Spearman coefficients and the lower triangle the Kendall coefficients. That is, for the j$\mathit{j}$th and k$\mathit{k}$th variables, where j$\mathit{j}$ is less than k$\mathit{k}$, rr(j,k)${\mathbf{rr}}\left(\mathit{j},\mathit{k}\right)$ contains the Spearman rank correlation coefficient, and rr(k,j)${\mathbf{rr}}\left(\mathit{k},\mathit{j}\right)$ contains Kendall's tau, for j = 1,2,,m$\mathit{j}=1,2,\dots ,m$ and k = 1,2,,m$\mathit{k}=1,2,\dots ,m$.
(Diagonal terms, rr(j,j)${\mathbf{rr}}\left(j,j\right)$, are unity for all three values of itype.)
2:     ncases – int64int32nag_int scalar
The number of cases, nc${n}_{\mathrm{c}}$, actually used in the calculations (when cases involving missing values have been eliminated).
3:     incase(n) – int64int32nag_int array
incase(i)${\mathbf{incase}}\left(\mathit{i}\right)$ holds the value 1$1$ if the i$\mathit{i}$th case was included in the calculations, and the value 0$0$ if the i$\mathit{i}$th case contained a missing value for at least one variable. That is, incase(i) = wi${\mathbf{incase}}\left(\mathit{i}\right)={w}_{\mathit{i}}$ (see Section [Description]), for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
4:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, n < 2${\mathbf{n}}<2$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, m < 2${\mathbf{m}}<2$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, ldx < n$\mathit{ldx}<{\mathbf{n}}$, or ldrr < m$\mathit{ldrr}<{\mathbf{m}}$.
ifail = 4${\mathbf{ifail}}=4$
 On entry, itype < − 1${\mathbf{itype}}<-1$, or itype > 1${\mathbf{itype}}>1$.
ifail = 5${\mathbf{ifail}}=5$
After observations with missing values were omitted, fewer than 2$2$ cases remained.

## Accuracy

You are warned of the need to exercise extreme care in your selection of missing values. nag_correg_coeffs_kspearman_miss_case (g02br) treats all values in the inclusive range (1 ± 0.1(x02be2)) × xmj$\left(1±{0.1}^{\left(\mathbf{x02be}-2\right)}\right)×{xm}_{j}$, where xmj${\mathit{xm}}_{j}$ is the missing value for variable j$j$ 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_case (g02br) depends on n$n$ and m$m$, and the occurrence of missing values.

## Example

```function nag_correg_coeffs_kspearman_miss_case_example
x = [1.7, 1, 0.5;
2.8, 4, 3;
0.6, 6, 2.5;
1.8, 9, 6;
0.99, 4, 2.5;
1.4, 2, 5.5;
1.8, 9, 7.5;
2.5, 7, 0;
0.99, 5, 3];
miss = [int64(1);0;1];
xmiss = [0.99;
0;
0];
itype = int64(0);
[rr, ncases, incase, ifail] = nag_correg_coeffs_kspearman_miss_case(x, miss, xmiss, itype)
```
```

rr =

1.0000    0.2941    0.4058
0.1429    1.0000    0.7537
0.2760    0.5521    1.0000

ncases =

6

incase =

1
1
1
1
0
1
1
0
0

ifail =

0

```
```function g02br_example
x = [1.7, 1, 0.5;
2.8, 4, 3;
0.6, 6, 2.5;
1.8, 9, 6;
0.99, 4, 2.5;
1.4, 2, 5.5;
1.8, 9, 7.5;
2.5, 7, 0;
0.99, 5, 3];
miss = [int64(1);0;1];
xmiss = [0.99;
0;
0];
itype = int64(0);
[rr, ncases, incase, ifail] = g02br(x, miss, xmiss, itype)
```
```

rr =

1.0000    0.2941    0.4058
0.1429    1.0000    0.7537
0.2760    0.5521    1.0000

ncases =

6

incase =

1
1
1
1
0
1
1
0
0

ifail =

0

```