# NAG FL Interfaceg02bcf (coeffs_​pearson_​miss_​pair)

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## 1Purpose

g02bcf 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.

## 2Specification

Fortran Interface
 Subroutine g02bcf ( n, m, x, ldx, miss, xbar, std, ssp, r, ldr, cnt,
 Integer, Intent (In) :: n, m, ldx, miss(m), ldssp, ldr, ldcnt Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: ncases Real (Kind=nag_wp), Intent (In) :: x(ldx,m), xmiss(m) Real (Kind=nag_wp), Intent (Inout) :: ssp(ldssp,m), r(ldr,m), cnt(ldcnt,m) Real (Kind=nag_wp), Intent (Out) :: xbar(m), std(m)
#include <nag.h>
 void g02bcf_ (const Integer *n, const Integer *m, const double x[], const Integer *ldx, const Integer miss[], const double xmiss[], double xbar[], double std[], double ssp[], const Integer *ldssp, double r[], const Integer *ldr, Integer *ncases, double cnt[], const Integer *ldcnt, Integer *ifail)
The routine may be called by the names g02bcf or nagf_correg_coeffs_pearson_miss_pair.

## 3Description

The input data consist 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. 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 Section 7); 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:
1. (a)Means:
 $x¯j=∑i=1nwijxij ∑i=1nwij , j=1,2,…,m.$
2. (b)Standard deviations:
 $sj=∑i= 1nwij (xij-x¯j) 2 (∑i= 1nwij)- 1 , j= 1,2,…,m.$
3. (c)Sums of squares and cross-products of deviations from means:
 $Sjk=∑i=1nwijwik(xij-x¯j(k))(xik-x¯k(j)), j,k=1,2,…,m,$
where
 $x¯j(k)=∑i= 1nwijwikxij ∑i= 1nwijwik and x¯k(j)=∑i= 1nwikwijxik ∑i= 1nwikwij ,$
(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.)
4. (d)Pearson product-moment correlation coefficients:
 $Rjk=SjkSjj(k)Skk(j) , j,k,=1,2,…,m,$
where ${S}_{jj\left(k\right)}=\sum _{i=1}^{n}{w}_{ij}{w}_{ik}{\left({x}_{ij}-{\overline{x}}_{j\left(k\right)}\right)}^{2}$ and ${S}_{kk\left(j\right)}=\sum _{i=1}^{n}{w}_{ik}{w}_{ij}{\left({x}_{ik}-{\overline{x}}_{k\left(j\right)}\right)}^{2}$ and ${\overline{x}}_{j\left(k\right)}$ and ${\overline{x}}_{k\left(j\right)}$ 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.
5. (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, ${\overline{x}}_{\mathit{j}}$, and the standard deviations, ${s}_{\mathit{j}}$.)

None.

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of observations or cases.
Constraint: ${\mathbf{n}}\ge 2$.
2: $\mathbf{m}$Integer Input
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 2$.
3: $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ 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 $\mathit{j}=1,2,\dots ,m$.
4: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02bcf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
5: $\mathbf{miss}\left({\mathbf{m}}\right)$Integer array Input
On entry: ${\mathbf{miss}}\left(j\right)$ 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.
6: $\mathbf{xmiss}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{xmiss}}\left(j\right)$ 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 Section 7).
7: $\mathbf{xbar}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: the mean value, ${\overline{x}}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
8: $\mathbf{std}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: the standard deviation, ${s}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
9: $\mathbf{ssp}\left({\mathbf{ldssp}},{\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{ssp}}\left(\mathit{j},\mathit{k}\right)$ is the cross-product of deviations ${S}_{\mathit{j}\mathit{k}}$, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
10: $\mathbf{ldssp}$Integer Input
On entry: the first dimension of the array ssp as declared in the (sub)program from which g02bcf is called.
Constraint: ${\mathbf{ldssp}}\ge {\mathbf{m}}$.
11: $\mathbf{r}\left({\mathbf{ldr}},{\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{r}}\left(\mathit{j},\mathit{k}\right)$ 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 $\mathit{k}=1,2,\dots ,m$.
12: $\mathbf{ldr}$Integer Input
On entry: the first dimension of the array r as declared in the (sub)program from which g02bcf is called.
Constraint: ${\mathbf{ldr}}\ge {\mathbf{m}}$.
13: $\mathbf{ncases}$Integer Output
On exit: 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).
14: $\mathbf{cnt}\left({\mathbf{ldcnt}},{\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{cnt}}\left(\mathit{j},\mathit{k}\right)$ 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 $\mathit{k}=1,2,\dots ,m$.
15: $\mathbf{ldcnt}$Integer Input
On entry: the first dimension of the array cnt as declared in the (sub)program from which g02bcf is called.
Constraint: ${\mathbf{ldcnt}}\ge {\mathbf{m}}$.
16: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $-1$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases g02bcf may return useful information.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 2$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ldcnt}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldcnt}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldr}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldr}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldssp}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldssp}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=4$
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 routine even if ${\mathbf{ifail}}={\mathbf{4}}$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

g02bcf 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. g02bcf treats all values in the inclusive range $\left(1±{0.1}^{\left({\mathbf{x02bef}}-2\right)}\right)×{xm}_{j}$, 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.

## 8Parallelism and Performance

g02bcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by g02bcf depends on $n$ and $m$, and the occurrence of missing values.
The routine uses a two-pass algorithm.

### 9.1Internal Changes

Internal changes have been made to this routine as follows:
• At Mark 27: The algorithm underlying this routine was altered to improve efficiency for large problem sizes on a multi-threaded system.
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

## 10Example

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.

### 10.1Program Text

Program Text (g02bcfe.f90)

### 10.2Program Data

Program Data (g02bcfe.d)

### 10.3Program Results

Program Results (g02bcfe.r)