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# NAG Toolbox: nag_correg_coeffs_pearson_subset_miss_case (g02bh)

## Purpose

nag_correg_coeffs_pearson_subset_miss_case (g02bh) computes means and standard deviations, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for selected variables omitting completely any cases with a missing observation for any variable (either over all variables in the dataset or over only those variables in the selected subset).

## Syntax

[xbar, std, ssp, r, ncases, ifail] = g02bh(x, miss, xmiss, mistyp, kvar, 'n', n, 'm', m, 'nvars', nvars)
[xbar, std, ssp, r, ncases, ifail] = nag_correg_coeffs_pearson_subset_miss_case(x, miss, xmiss, mistyp, kvar, 'n', n, 'm', m, 'nvars', nvars)
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
 $xij, i=1,2,…,n n≥2,j=1,2,…,m m≥2,$
where ${x}_{ij}$ is the $i$th observation on the $j$th variable, together with the subset of these variables, ${v}_{1},{v}_{2},\dots ,{v}_{p}$, for which information is required.
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. The missing values can be utilized in two slightly different ways; you can indicate which scheme is required.
Firstly, let ${w}_{i}=0$ if observation $i$ contains a missing value for any of those variables in the set $1,2,\dots ,m$ for which missing values have been declared, i.e., if ${x}_{ij}={\mathit{xm}}_{j}$ for any $j$ ($j=1,2,\dots ,m$) for which an ${\mathit{xm}}_{j}$ has been assigned (see also Accuracy); and ${w}_{i}=1$ otherwise, for $\mathit{i}=1,2,\dots ,n$.
Secondly, let ${w}_{i}=0$ if observation $i$ contains a missing value for any of those variables in the selected subset ${v}_{1},{v}_{2},\dots ,{v}_{p}$ for which missing values have been declared, i.e., if ${x}_{ij}={\mathit{xm}}_{j}$ for any $j$ ($j={v}_{1},{v}_{2},\dots ,{v}_{p}$) 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:
 $x-j=∑i=1nwixij ∑i=1nwi , j=v1,v2,…,vp.$
(b) Standard deviations:
 $sj= ∑i= 1nwi xij-x-j 2 ∑i= 1nwi- 1 , j=v1,v2,…,vp.$
(c) Sums of squares and cross-products of deviations from means:
 $Sjk=∑i=1nwixij-x-jxik-x-k, j,k=v1,v2,…,vp.$
(d) Pearson product-moment correlation coefficients:
 $Rjk=SjkSjjSkk , j,k=v1,v2,…,vp.$
If ${S}_{jj}$ or ${S}_{kk}$ is zero, ${R}_{jk}$ 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).
4:     $\mathrm{mistyp}$int64int32nag_int scalar
Indicates the manner in which missing observations are to be treated.
${\mathbf{mistyp}}=1$
A case is excluded if it contains a missing value for any of the variables $1,2,\dots ,m$.
${\mathbf{mistyp}}=0$
A case is excluded if it contains a missing value for any of the $p\left(\le m\right)$ variables specified in the array kvar.
5:     $\mathrm{kvar}\left({\mathbf{nvars}}\right)$int64int32nag_int array
${\mathbf{kvar}}\left(\mathit{j}\right)$ must be set to the column number in x of the $\mathit{j}$th variable for which information is required, for $\mathit{j}=1,2,\dots ,p$.
Constraint: $1\le {\mathbf{kvar}}\left(\mathit{j}\right)\le {\mathbf{m}}$, for $\mathit{j}=1,2,\dots ,p$.

### 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: ${\mathbf{m}}\ge 2$.
3:     $\mathrm{nvars}$int64int32nag_int scalar
Default: the dimension of the array kvar.
$p$, the number of variables for which information is required.
Constraint: $2\le {\mathbf{nvars}}\le {\mathbf{m}}$.

### Output Parameters

1:     $\mathrm{xbar}\left({\mathbf{nvars}}\right)$ – double array
The mean value, of ${\stackrel{-}{x}}_{\mathit{j}}$, of the variable specified in ${\mathbf{kvar}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,p$.
2:     $\mathrm{std}\left({\mathbf{nvars}}\right)$ – double array
The standard deviation, ${s}_{\mathit{j}}$, of the variable specified in ${\mathbf{kvar}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,p$.
3:     $\mathrm{ssp}\left(\mathit{ldssp},{\mathbf{nvars}}\right)$ – double array
${\mathbf{ssp}}\left(\mathit{j},\mathit{k}\right)$ is the cross-product of deviations, ${S}_{\mathit{j}\mathit{k}}$, for the variables specified in ${\mathbf{kvar}}\left(\mathit{j}\right)$ and ${\mathbf{kvar}}\left(\mathit{k}\right)$, for $\mathit{j}=1,2,\dots ,p$ and $\mathit{k}=1,2,\dots ,p$.
4:     $\mathrm{r}\left(\mathit{ldr},{\mathbf{nvars}}\right)$ – double array
${\mathbf{r}}\left(\mathit{j},\mathit{k}\right)$ is the product-moment correlation coefficient, ${R}_{\mathit{j}\mathit{k}}$, between the variables specified in ${\mathbf{kvar}}\left(\mathit{j}\right)$ and ${\mathbf{kvar}}\left(\mathit{k}\right)$, for $\mathit{j}=1,2,\dots ,p$ and $\mathit{k}=1,2,\dots ,p$.
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, ${\mathbf{nvars}}<2$, or ${\mathbf{nvars}}>{\mathbf{m}}$.
${\mathbf{ifail}}=3$
 On entry, $\mathit{ldx}<{\mathbf{n}}$, or $\mathit{ldssp}<{\mathbf{nvars}}$, or $\mathit{ldr}<{\mathbf{nvars}}$.
${\mathbf{ifail}}=4$
 On entry, ${\mathbf{kvar}}\left(j\right)<1$, or ${\mathbf{kvar}}\left(j\right)>{\mathbf{m}}$ for some $j=1,2,\dots ,{\mathbf{nvars}}$.
${\mathbf{ifail}}=5$
 On entry, ${\mathbf{mistyp}}\ne 1$ or $0$
${\mathbf{ifail}}=6$
After observations with missing values were omitted, no cases remained.
${\mathbf{ifail}}=7$
After observations with missing values were omitted, only one case remained.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
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_subset_miss_case (g02bh) 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_subset_miss_case (g02bh) 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_subset_miss_case (g02bh) depends on $n$ and $p$, 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 four variables. Missing values of $0.0$ are declared for the second and fourth variables; no missing values are specified for the first and third variables. The means, standard deviations, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for the fourth, first and second variables are then calculated and printed, omitting completely all cases containing missing values for these three selected variables; cases $3$ and $4$ are therefore eliminated, leaving only three cases in the calculations.
function g02bh_example

fprintf('g02bh example results\n\n');

x = [ 3,  3,  1,  2;
6,  4, -1,  4;
9,  0,  5,  9;
12,  2,  0,  0;
-1,  5,  4, 12];
[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(0); 1; 0; 1];
xmiss  = [        0;  0; 0; 0];
mistyp =  int64(0);
kvar   = [int64(4); 1; 2];
nvar   = size(kvar,1);

[xbar, std, ssp, r, ncases, ifail] = ...
g02bh( ...
x, miss, xmiss, mistyp, kvar);

fprintf('Variable   Mean     St. dev.\n');
fprintf('%5d%11.4f%11.4f\n',[double(kvar) xbar(1:nvar) std(1:nvar)]');
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);

g02bh example results

Number of variables (columns) = 4
Number of cases     (rows)    = 5

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

Variable   Mean     St. dev.
4     6.0000     5.2915
1     2.6667     3.5119
2     4.0000     1.0000

Sums of squares and cross-products of deviations
56.0000  -30.0000   10.0000
-30.0000   24.6667   -4.0000
10.0000   -4.0000    2.0000

Correlation coefficients
1.0000   -0.8072    0.9449
-0.8072    1.0000   -0.5695
0.9449   -0.5695    1.0000

Number of cases actually used  = 3

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