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.

2 Specification

Fortran Interface

Subroutine g02bcf (

n, m, x, ldx, miss, xmiss, xbar, std, ssp, ldssp, r, ldr, ncases, cnt, ldcnt, ifail)

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)

C Header Interface

#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.

3 Description

The input data consist of

n

observations for each of

m

variables, given as an array

[x_{i j}], i = 1, 2, \dots, n (n \geq 2), j = 1, 2, \dots, m (m \geq 2),

where

x_{i j}

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

{xm}_{j}

. Missing values need not be specified for all variables.

Let

w_{i j} = 0

if the

i

th observation for the

j

th variable is a missing value, i.e., if a missing value,

{xm}_{j}

, has been declared for the

j

th variable, and

x_{i j} = {xm}_{j}

(see also Section 7); and

w_{i j} = 1

otherwise, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

The quantities calculated are:

(a)Means:
${\bar{x}}_{j} = \frac{\sum_{i = 1}^{n} w_{i j} x_{i j}}{\sum_{i = 1}^{n} w_{i j}}, j = 1, 2, \dots, m .$
(b)Standard deviations:
$s_{j} = \sqrt{\frac{\sum_{i = 1}^{n} w_{i j} {(x_{i j} - {\bar{x}}_{j})}^{2}}{(\sum_{i = 1}^{n} w_{i j}) - 1}}, j = 1, 2, \dots, m .$
(c)Sums of squares and cross-products of deviations from means:
$S_{j k} = \sum_{i = 1}^{n} w_{i j} w_{i k} (x_{i j} - {\bar{x}}_{j (k)}) (x_{i k} - {\bar{x}}_{k (j)}), j, k = 1, 2, \dots, m,$
where
${\bar{x}}_{j (k)} = \frac{\sum_{i = 1}^{n} w_{i j} w_{i k} x_{i j}}{\sum_{i = 1}^{n} w_{i j} w_{i k}} and {\bar{x}}_{k (j)} = \frac{\sum_{i = 1}^{n} w_{i k} w_{i j} x_{i k}}{\sum_{i = 1}^{n} w_{i k} w_{i j}},$
(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:
$R_{j k} = \frac{S_{j k}}{\sqrt{S_{j j (k)} S_{k k (j)}}}, j, k, = 1, 2, \dots, m,$
where $S_{j j (k)} = \sum_{i = 1}^{n} w_{i j} w_{i k} {(x_{i j} - {\bar{x}}_{j (k)})}^{2}$ and $S_{k k (j)} = \sum_{i = 1}^{n} w_{i k} w_{i j} {(x_{i k} - {\bar{x}}_{k (j)})}^{2}$ and ${\bar{x}}_{j (k)}$ and ${\bar{x}}_{k (j)}$ 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 $S_{j j (k)}$ or $S_{k k (j)}$ is zero, $R_{j k}$ is set to zero.
(e)The number of cases used in the calculation of each of the correlation coefficients:
$c_{j k} = \sum_{i = 1}^{n} w_{i j} w_{i k}, j, k = 1, 2, \dots, m .$
(The diagonal terms, $c_{j j}$ , for $j = 1, 2, \dots, m$ , also give the number of cases used in the calculation of the means, ${\bar{x}}_{j}$ , and the standard deviations, $s_{j}$ .)

4 References

None.

5 Arguments

1: $n$ – Integer Input: On entry: $n$ , the number of observations or cases.

Constraint: $n \geq 2$ .
2: $m$ – Integer Input: On entry: $m$ , the number of variables.

Constraint: $m \geq 2$ .
3: $x (ldx, m)$ – Real (Kind=nag_wp) array Input: On entry: $x (i, j)$ must be set to $x_{i j}$ , the value of the $i$ th observation on the $j$ th variable, for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, m$ .
4: $ldx$ – Integer Input: On entry: the first dimension of the array x as declared in the (sub)program from which g02bcf is called.

Constraint: $ldx \geq n$ .
5: $miss (m)$ – Integer array Input: On entry: $miss (j)$ 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: $xmiss (m)$ – Real (Kind=nag_wp) array Input: On entry: $xmiss (j)$ 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: $xbar (m)$ – Real (Kind=nag_wp) array Output: On exit: the mean value, ${\bar{x}}_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
8: $std (m)$ – Real (Kind=nag_wp) array Output: On exit: the standard deviation, $s_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
9: $ssp (ldssp, m)$ – Real (Kind=nag_wp) array Output: On exit: $ssp (j, k)$ is the cross-product of deviations $S_{j k}$ , for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
10: $ldssp$ – Integer Input: On entry: the first dimension of the array ssp as declared in the (sub)program from which g02bcf is called.

Constraint: $ldssp \geq m$ .
11: $r (ldr, m)$ – Real (Kind=nag_wp) array Output: On exit: $r (j, k)$ is the product-moment correlation coefficient $R_{j k}$ between the $j$ th and $k$ th variables, for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
12: $ldr$ – Integer Input: On entry: the first dimension of the array r as declared in the (sub)program from which g02bcf is called.

Constraint: $ldr \geq m$ .
13: $ncases$ – 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: $cnt (ldcnt, m)$ – Real (Kind=nag_wp) array Output: On exit: $cnt (j, k)$ is the number of cases, $c_{j k}$ , actually used in the calculation of $S_{j k}$ , and $R_{j k}$ , the sum of cross-products and correlation coefficient for the $j$ th and $k$ th variables, for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
15: $ldcnt$ – Integer Input: On entry: the first dimension of the array cnt as declared in the (sub)program from which g02bcf is called.

Constraint: $ldcnt \geq m$ .
16: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if $ifail \neq 0$ on exit, the recommended value is $- 1$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 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.

$ifail = 1$: On entry, $n = 〈value〉$ .
Constraint: $n \geq 2$ .

$ifail = 2$: On entry, $m = 〈value〉$ .
Constraint: $m \geq 2$ .

$ifail = 3$: On entry, $ldcnt = 〈value〉$ and $m = 〈value〉$ .
Constraint: $ldcnt \geq m$ .

On entry, $ldr = 〈value〉$ and $m = 〈value〉$ .
Constraint: $ldr \geq m$ .

On entry, $ldssp = 〈value〉$ and $m = 〈value〉$ .
Constraint: $ldssp \geq m$ .

On entry, $ldx = 〈value〉$ and $n = 〈value〉$ .
Constraint: $ldx \geq n$ .

$ifail = 4$: 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 $ifail = 4$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

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

(1 \pm {0.1}^{(x02bef - 2)}) \times {x m}_{j}

, where

{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.

8 Parallelism 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.

9 Further Comments

The time taken by g02bcf depends on

n

and

m

, and the occurrence of missing values.

The routine uses a two-pass algorithm.

9.1 Internal Changes

Internal changes have been made to this routine as follows:

At Mark 27: The algorithm underlying this routine has been 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.

10 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.

Interfaces: FL CL AD

NAG FL Interface Introduction

G02 (Correg) Chapter Contents

G02 (Correg) Chapter Introduction

g02bc: FL CL AD

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

▸▿ Contents

1 Purpose