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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_coeffs_pearson_miss_pair (g02bc)

## Purpose

nag_correg_coeffs_pearson_miss_pair (g02bc) computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment 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.

## Syntax

[xbar, std, ssp, r, ncases, cnt, ifail] = g02bc(x, miss, xmiss, 'n', n, 'm', m)
[xbar, std, ssp, r, ncases, cnt, ifail] = nag_correg_coeffs_pearson_miss_pair(x, miss, xmiss, '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 consist 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. 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 quantities calculated are:
(a) Means:
 $x-j=∑i=1nwijxij ∑i=1nwij , j=1,2,…,m.$
(b) Standard deviations:
 $sj=∑i= 1nwij xij-x-j 2 ∑i= 1nwij- 1 , j= 1,2,…,m.$
(c) Sums of squares and cross-products of deviations from means:
 $Sjk=∑i=1nwijwikxij-x-jkxik-x-kj, j,k=1,2,…,m,$
where
 $x-jk=∑i= 1nwijwikxij ∑i= 1nwijwik and x-kj=∑i= 1nwikwijxik ∑i= 1nwikwij ,$
(i.e., the means used in the calculation of the sums of squares and cross-products of deviations are based on the same set of observations as are the cross-products.)
(d) Pearson product-moment correlation coefficients:
 $Rjk=SjkSjjkSkkj , j,k,=1,2,…,m,$
where ${S}_{jj\left(k\right)}=\sum _{i=1}^{n}{w}_{ij}{w}_{ik}{\left({x}_{ij}-{\stackrel{-}{x}}_{j\left(k\right)}\right)}^{2}$ and ${S}_{kk\left(j\right)}=\sum _{i=1}^{n}{w}_{ik}{w}_{ij}{\left({x}_{ik}-{\stackrel{-}{x}}_{k\left(j\right)}\right)}^{2}$ and ${\stackrel{-}{x}}_{j\left(k\right)}$ and ${\stackrel{-}{x}}_{k\left(j\right)}$ are as defined in (c) above
(i.e., the sums of squares of deviations used in the denominator are based on the same set of observations as are used in the calculation of the numerator).
If ${S}_{jj\left(k\right)}$ or ${S}_{kk\left(j\right)}$ is zero, ${R}_{jk}$ is set to zero.
(e) The number of cases used in the calculation of each of the correlation coefficients:
 $cjk=∑i=1nwijwik, j,k=1,2,…,m.$
(The diagonal terms, ${c}_{\mathit{j}\mathit{j}}$, for $\mathit{j}=1,2,\dots ,m$, also give the number of cases used in the calculation of the means, ${\stackrel{-}{x}}_{\mathit{j}}$, and the standard deviations, ${s}_{\mathit{j}}$.)

None.

## 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).

### 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:     $\mathrm{xbar}\left({\mathbf{m}}\right)$ – double array
The mean value, ${\stackrel{-}{x}}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
2:     $\mathrm{std}\left({\mathbf{m}}\right)$ – double array
The standard deviation, ${s}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
3:     $\mathrm{ssp}\left(\mathit{ldssp},{\mathbf{m}}\right)$ – double array
${\mathbf{ssp}}\left(\mathit{j},\mathit{k}\right)$ is the cross-product of deviations ${S}_{\mathit{j}\mathit{k}}$, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
4:     $\mathrm{r}\left(\mathit{ldr},{\mathbf{m}}\right)$ – double array
${\mathbf{r}}\left(\mathit{j},\mathit{k}\right)$ is the product-moment correlation coefficient ${R}_{\mathit{j}\mathit{k}}$ between the $\mathit{j}$th and $\mathit{k}$th variables, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
5:     $\mathrm{ncases}$int64int32nag_int scalar
The minimum number of cases used in the calculation of any of the sums of squares and cross-products and correlation coefficients (when cases involving missing values have been eliminated).
6:     $\mathrm{cnt}\left(\mathit{ldcnt},{\mathbf{m}}\right)$ – double array
${\mathbf{cnt}}\left(\mathit{j},\mathit{k}\right)$ is the number of cases, ${c}_{\mathit{j}\mathit{k}}$, actually used in the calculation of ${S}_{\mathit{j}\mathit{k}}$, and ${R}_{\mathit{j}\mathit{k}}$, the sum of cross-products and correlation coefficient for the $\mathit{j}$th and $\mathit{k}$th variables, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
7:     $\mathrm{ifail}$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_pearson_miss_pair (g02bc) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}=1$
 On entry, ${\mathbf{n}}<2$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{m}}<2$.
${\mathbf{ifail}}=3$
 On entry, $\mathit{ldx}<{\mathbf{n}}$, or $\mathit{ldssp}<{\mathbf{m}}$, or $\mathit{ldr}<{\mathbf{m}}$, or $\mathit{ldcnt}<{\mathbf{m}}$.
W  ${\mathbf{ifail}}=4$
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 means, standard deviations, sums of squares and cross-products, and correlation coefficients based on two or more cases are returned by the function even if ${\mathbf{ifail}}={\mathbf{4}}$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

nag_correg_coeffs_pearson_miss_pair (g02bc) does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large $n$.
You are warned of the need to exercise extreme care in your selection of missing values. nag_correg_coeffs_pearson_miss_pair (g02bc) 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_pearson_miss_pair (g02bc) depends on $n$ and $m$, and the occurrence of missing values.
The function uses a two-pass algorithm.

## Example

This example reads in a set of data consisting of five observations on each of three variables. Missing values of $0.0$, $-1.0$ and $0.0$ are declared for the first, second and third variables respectively. The means, standard deviations, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for all three variables are then calculated and printed, omitting cases with missing values from only those calculations involving the variables for which the values are missing. The program therefore omits cases $4$ and $5$ in calculating the correlation between the first and second variables, and cases $3$ and $4$ for the first and third variables etc.
```function g02bc_example

fprintf('g02bc example results\n\n');

x = [ 2,  3, 3;
4,  6, 4;
9,  9, 0;
0, 12, 2;
12, -1, 5];
[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; -1; 0];

[xbar, std, ssp, r, ncases, count, ifail] = ...
g02bc(x, miss, xmiss);

fprintf('Variable   Mean     St. dev.\n');
fprintf('%5d%11.4f%11.4f\n',[[1:m]' xbar std]');
fprintf('\nSums of squares and cross-products of deviations\n');
disp(ssp)
fprintf('Correlation coefficients\n');
disp(r);
fprintf('Number of cases actually used  = %d\n', ncases);
fprintf('\nNumbers used for each pair are:\n');
disp(count);

```
```g02bc example results

Number of variables (columns) = 3
Number of cases     (rows)    = 5

Data matrix is:-
2     3     3
4     6     4
9     9     0
0    12     2
12    -1     5

Variable   Mean     St. dev.
1     6.7500     4.5735
2     7.5000     3.8730
3     3.5000     1.2910

Sums of squares and cross-products of deviations
62.7500   21.0000   10.0000
21.0000   45.0000   -6.0000
10.0000   -6.0000    5.0000

Correlation coefficients
1.0000    0.9707    0.9449
0.9707    1.0000   -0.6547
0.9449   -0.6547    1.0000

Number of cases actually used  = 3

Numbers used for each pair are:
4     3     3
3     4     3
3     3     4

```