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NAG Toolbox

# NAG Toolbox: nag_correg_coeffs_zero_miss_case (g02be)

## Purpose

nag_correg_coeffs_zero_miss_case (g02be) computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data omitting completely any cases with a missing observation for any variable.

## Syntax

[xbar, std, sspz, rz, ncases, ifail] = g02be(x, miss, xmiss, 'n', n, 'm', m)
[xbar, std, sspz, rz, ncases, ifail] = nag_correg_coeffs_zero_miss_case(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; miss and xmiss are no longer output parameters

## 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 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 ${w}_{i}=0$ if observation $i$ contains a missing value for any of those variables for which missing values have been declared, i.e., if ${x}_{ij}={\mathit{xm}}_{j}$ for any $j$ for which an ${\mathit{xm}}_{j}$ has been assigned (see also Accuracy); and ${w}_{i}=1$ otherwise, for $\mathit{i}=1,2,\dots ,n$.
The quantities calculated are:
(a) Means:
 $x-j=∑i=1nwixij ∑i=1nwi , j=1,2,…,m.$
(b) Standard deviations:
 $sj= ∑i= 1nwi xij-x-j 2 ∑i= 1nwi- 1 , j= 1,2,…,m.$
(c) Sums of squares and cross-products about zero:
 $S~jk=∑i=1nwixijxik, j,k=1,2,…,m.$
(d) Correlation-like coefficients:
 $R~jk=S~jkS~jj S~kk , j,k= 1,2,…,m.$
If ${\stackrel{~}{S}}_{jj}$ or ${\stackrel{~}{S}}_{kk}$ is zero, ${\stackrel{~}{R}}_{jk}$ is set to zero.

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:     $\mathrm{sspz}\left(\mathit{ldsspz},{\mathbf{m}}\right)$ – double array
${\mathbf{sspz}}\left(\mathit{j},\mathit{k}\right)$ is the cross-product about zero, ${\stackrel{~}{S}}_{\mathit{j}\mathit{k}}$, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
4:     $\mathrm{rz}\left(\mathit{ldrz},{\mathbf{m}}\right)$ – double array
${\mathbf{rz}}\left(\mathit{j},\mathit{k}\right)$ is the correlation-like coefficient, ${\stackrel{~}{R}}_{\mathit{j}\mathit{k}}$, 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 number of cases actually used in the calculations (when cases involving missing values have been eliminated).
6:     $\mathrm{ifail}$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 $\mathit{ldsspz}<{\mathbf{m}}$, or $\mathit{ldrz}<{\mathbf{m}}$.
${\mathbf{ifail}}=4$
After observations with missing values were omitted, no cases remained.
${\mathbf{ifail}}=5$
After observations with missing values were omitted, only one case remained.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

nag_correg_coeffs_zero_miss_case (g02be) 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_zero_miss_case (g02be) 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_zero_miss_case (g02be) 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 $0.0$ are declared for the first and third variables; no missing value is specified for the second variable. The means, standard deviations, sums of squares and cross-products about zero, and correlation-like coefficients for all three variables are then calculated and printed, omitting completely all cases containing missing values; cases $3$ and $4$ are therefore eliminated, leaving only three cases in the calculations.
```function g02be_example

fprintf('g02be 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);  0;  1];
xmiss = [         0;  0;  0];
[xbar, std, sspz, rz, ncases, ifail] = ...
g02be(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 about zero\n');
disp(sspz)
fprintf('Correlation-like coefficients\n');
disp(rz);
fprintf('Number of cases actually used  = %d\n', ncases);

```
```g02be 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     6.0000     5.2915
2     2.6667     3.5119
3     4.0000     1.0000

Sums of squares and cross-products about zero
164    18    82
18    46    28
82    28    50

Correlation-like coefficients
1.0000    0.2072    0.9055
0.2072    1.0000    0.5838
0.9055    0.5838    1.0000

Number of cases actually used  = 3
```