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NAG Toolbox

# NAG Toolbox: nag_correg_coeffs_pearson_miss_case (g02bb)

## Purpose

nag_correg_coeffs_pearson_miss_case (g02bb) 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 completely any cases with a missing observation for any variable.

## Syntax

[xbar, std, ssp, r, ncases, ifail] = g02bb(x, miss, xmiss, 'n', n, 'm', m)
[xbar, std, ssp, r, ncases, ifail] = nag_correg_coeffs_pearson_miss_case(x, miss, xmiss, '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 consist 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$x{m}_{j}$. 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${x}_{ij}=x{m}_{j}$ for any j$j$ for which an xmj$x{m}_{j}$ 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 quantities calculated are:
(a) Means:
 xj = ( ∑ i = 1nwixij)/( ∑ i = 1nwi),  j = 1,2, … ,m. $x-j=∑i=1nwixij ∑i=1nwi , j=1,2,…,m.$
(b) Standard deviations:
 sj = sqrt( ( ∑ i = 1nwi(xij − xj)2)/( ∑ i = 1nwi − 1)),   j = 1,2, … ,m. $sj= ∑i= 1nwi (xij-x-j) 2 ∑i= 1nwi- 1 , j= 1,2,…,m.$
(c) Sums of squares and cross-products of deviations from means:
 n Sjk = ∑ wi(xij − xj)(xik − xk),  j,k = 1,2, … ,m. i = 1
$Sjk=∑i=1nwi(xij-x-j)(xik-x-k), j,k=1,2,…,m.$
(d) Pearson product-moment correlation coefficients:
 Rjk = (Sjk)/(sqrt(SjjSkk)),   j,k = 1,2, … ,m. $Rjk=SjkSjjSkk , j,k= 1,2,…,m.$
If Sjj${S}_{jj}$ or Skk${S}_{kk}$ is zero, Rjk${R}_{jk}$ is set to zero.

None.

## 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)${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must be set to xij${x}_{\mathit{i}\mathit{j}}$, the value of the i$\mathit{i}$th observation on the j$\mathit{j}$th variable, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$ and j = 1,2,,m$\mathit{j}=1,2,\dots ,m$.
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]).

### 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${\mathbf{m}}\ge 2$.

ldx ldssp ldr

### Output Parameters

1:     xbar(m) – double array
The mean value, xj${\stackrel{-}{x}}_{\mathit{j}}$, of the j$\mathit{j}$th variable, for j = 1,2,,m$\mathit{j}=1,2,\dots ,m$.
2:     std(m) – double array
The standard deviation, sj${s}_{\mathit{j}}$, of the j$\mathit{j}$th variable, for j = 1,2,,m$\mathit{j}=1,2,\dots ,m$.
3:     ssp(ldssp,m) – double array
ldsspm$\mathit{ldssp}\ge {\mathbf{m}}$.
ssp(j,k)${\mathbf{ssp}}\left(\mathit{j},\mathit{k}\right)$ is the cross-product of deviations Sjk${S}_{\mathit{j}\mathit{k}}$, for j = 1,2,,m$\mathit{j}=1,2,\dots ,m$ and k = 1,2,,m$\mathit{k}=1,2,\dots ,m$.
4:     r(ldr,m) – double array
ldrm$\mathit{ldr}\ge {\mathbf{m}}$.
r(j,k)${\mathbf{r}}\left(\mathit{j},\mathit{k}\right)$ is the product-moment correlation coefficient Rjk${R}_{\mathit{j}\mathit{k}}$ between the j$\mathit{j}$th and k$\mathit{k}$th variables, for j = 1,2,,m$\mathit{j}=1,2,\dots ,m$ and k = 1,2,,m$\mathit{k}=1,2,\dots ,m$.
5:     ncases – int64int32nag_int scalar
The number of cases actually used in the calculations (when cases involving missing values have been eliminated).
6:     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 ldssp < m$\mathit{ldssp}<{\mathbf{m}}$, or ldr < m$\mathit{ldr}<{\mathbf{m}}$.
ifail = 4${\mathbf{ifail}}=4$
After observations with missing values were omitted, no cases remained.
ifail = 5${\mathbf{ifail}}=5$
After observations with missing values were omitted, only one case remained.

## Accuracy

nag_correg_coeffs_pearson_miss_case (g02bb) does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large n$n$.
You are warned of the need to exercise extreme care in your selection of missing values. nag_correg_coeffs_pearson_miss_case (g02bb) 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_pearson_miss_case (g02bb) depends on n$n$ and m$m$, and the occurrence of missing values.
The function uses a two-pass algorithm.

## Example

```function nag_correg_coeffs_pearson_miss_case_example
x = [2, 3, 3;
4, 6, 4;
9, 9, 0;
0, 12, 2;
12, -1, 5];
miss = [int64(1);0;1];
xmiss = [0;
0;
0];
[xbar, std, ssp, r, ncases, ifail] = nag_correg_coeffs_pearson_miss_case(x, miss, xmiss)
```
```

xbar =

6.0000
2.6667
4.0000

std =

5.2915
3.5119
1.0000

ssp =

56.0000  -30.0000   10.0000
-30.0000   24.6667   -4.0000
10.0000   -4.0000    2.0000

r =

1.0000   -0.8072    0.9449
-0.8072    1.0000   -0.5695
0.9449   -0.5695    1.0000

ncases =

3

ifail =

0

```
```function g02bb_example
x = [2, 3, 3;
4, 6, 4;
9, 9, 0;
0, 12, 2;
12, -1, 5];
miss = [int64(1);0;1];
xmiss = [0;
0;
0];
[xbar, std, ssp, r, ncases, ifail] = g02bb(x, miss, xmiss)
```
```

xbar =

6.0000
2.6667
4.0000

std =

5.2915
3.5119
1.0000

ssp =

56.0000  -30.0000   10.0000
-30.0000   24.6667   -4.0000
10.0000   -4.0000    2.0000

r =

1.0000   -0.8072    0.9449
-0.8072    1.0000   -0.5695
0.9449   -0.5695    1.0000

ncases =

3

ifail =

0

```