g02bjf 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 cases with missing values from only those calculations involving the variables for which the values are missing.

2 Specification

Fortran Interface

Subroutine g02bjf (

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

Integer, Intent (In)	::	n, m, ldx, miss(m), nvars, kvar(nvars), 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,nvars), r(ldr,nvars), cnt(ldcnt,nvars)
Real (Kind=nag_wp), Intent (Out)	::	xbar(nvars), std(nvars)

C Header Interface

#include <nag.h>

void

g02bjf_ (const Integer *n, const Integer *m, const double x[], const Integer *ldx, const Integer miss[], const double xmiss[], const Integer *nvars, const Integer kvar[], 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 g02bjf or nagf_correg_coeffs_pearson_subset_miss_pair.

3 Description

The input data consists 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, 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

{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:

${\bar{x}}_{j} = \frac{\sum_{i = 1}^{n} w_{i j} x_{i j}}{\sum_{i = 1}^{n} w_{i j}}, j = v_{1}, v_{2}, \dots, v_{p} .$
(b)Standard deviations:

$s_{j} = \sqrt{\frac{\sum_{i = 1}^{n} w_{i j} {(x_{i j} - {\bar{x}}_{j})}^{2}}{\sum_{i = 1}^{n} w_{i j} - 1}}, j = v_{1}, v_{2}, \dots, v_{p} .$
(c)Sums of squares and cross-products of deviations from means:

$S_{j k} = \sum_{i = 1}^{n} w_{i j} w_{i k} (x_{i j} - {\bar{x}}_{j (k)}) (x_{i k} - {\bar{x}}_{k (j)}), j, k = v_{1}, v_{2}, \dots, v_{p},$

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 sum 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:

$R_{j k} = \frac{S_{j k}}{\sqrt{S_{j j (k)} S_{k k (j)}}}, j, k = v_{1}, v_{2}, \dots, v_{p},$

where

$S_{j j (k)} = \sum_{i = 1}^{n} w_{i j} w_{i k} {(x_{i j} - {\bar{x}}_{j (k)})}^{2} and S_{k k (j)} = \sum_{i = 1}^{n} w_{i k} w_{i j} {(x_{i k} - {\bar{x}}_{k (j)})}^{2},$

(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:

$c_{j k} = \sum_{i = 1}^{n} w_{i j} w_{i k}, j, k = v_{1}, v_{2}, \dots, v_{p} .$

(The diagonal terms, $c_{j j}$ , for $j = v_{1}, v_{2}, \dots, v_{p}$ , 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 g02bjf 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: $nvars$ – Integer Input: On entry: $p$ , the number of variables for which information is required.

Constraint: $2 \leq nvars \leq m$ .
8: $kvar (nvars)$ – Integer array Input: On entry: $kvar (j)$ must be set to the column number in x of the $j$ th variable for which information is required, for $j = 1, 2, \dots, p$ .

Constraint: $1 \leq kvar (j) \leq m$ , for $j = 1, 2, \dots, p$ .
9: $xbar (nvars)$ – Real (Kind=nag_wp) array Output: On exit: the mean value, ${\bar{x}}_{j}$ , of the variable specified in $kvar (j)$ , for $j = 1, 2, \dots, p$ .
10: $std (nvars)$ – Real (Kind=nag_wp) array Output: On exit: the standard deviation, $s_{j}$ , of the variable specified in $kvar (j)$ , for $j = 1, 2, \dots, p$ .
11: $ssp (ldssp, nvars)$ – Real (Kind=nag_wp) array Output: On exit: $ssp (j, k)$ is the cross-product of deviations, $S_{j k}$ , for the variables specified in $kvar (j)$ and $kvar (k)$ , for $j = 1, 2, \dots, p$ and $k = 1, 2, \dots, p$ .
12: $ldssp$ – Integer Input: On entry: the first dimension of the array ssp as declared in the (sub)program from which g02bjf is called.

Constraint: $ldssp \geq nvars$ .
13: $r (ldr, nvars)$ – Real (Kind=nag_wp) array Output: On exit: $r (j, k)$ is the product-moment correlation coefficient, $R_{j k}$ , between the variables specified in $kvar (j)$ and $kvar (k)$ , for $j = 1, 2, \dots, p$ and $k = 1, 2, \dots, p$ .
14: $ldr$ – Integer Input: On entry: the first dimension of the array r as declared in the (sub)program from which g02bjf is called.

Constraint: $ldr \geq nvars$ .
15: $ncases$ – 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).
16: $cnt (ldcnt, nvars)$ – 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 variables specified in $kvar (j)$ and $kvar (k)$ , for $j = 1, 2, \dots, p$ and $k = 1, 2, \dots, p$ .
17: $ldcnt$ – Integer Input: On entry: the first dimension of the array cnt as declared in the (sub)program from which g02bjf is called.

Constraint: $ldcnt \geq nvars$ .
18: $ifail$ – 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 $ifail \neq 0$ on exit. 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 g02bjf may return useful information.

$ifail = 1$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 2$ .

$ifail = 2$: On entry, $nvars = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $nvars \geq 2$ and $nvars \leq m$ .

$ifail = 3$: On entry, $ldcnt = ⟨ value ⟩$ and $nvars = ⟨ value ⟩$ .
Constraint: $ldcnt \geq nvars$ .

On entry, $ldr = ⟨ value ⟩$ and $nvars = ⟨ value ⟩$ .
Constraint: $ldr \geq nvars$ .

On entry, $ldssp = ⟨ value ⟩$ and $nvars = ⟨ value ⟩$ .
Constraint: $ldssp \geq nvars$ .

On entry, $ldx = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $ldx \geq n$ .

$ifail = 4$: On entry, $i = ⟨ value ⟩$ , $kvar (i) = ⟨ value ⟩$ and $m = ⟨ value ⟩$ .
Constraint: $1 \leq kvar (i) \leq m$ .

$ifail = 5$: After observations with missing values were omitted, fewer than two cases remained.

$ifail = - 99$: 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

g02bjf 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. g02bjf 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

Background information to multithreading can be found in the Multithreading documentation.

g02bjf is not threaded in any implementation.

9 Further Comments

The time taken by g02bjf depends on

n

and

p

, 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 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.

10 Example

This example reads in a set of data consisting of five observations on each of four variables. Missing values of

- 1.0

0.0

and

0.0

are declared for the first, second and fourth variables respectively; no missing value is specified for the third variable. 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 cases with missing values from only those calculations involving the variables for which the values are missing. The program, therefore, eliminates cases

4

and

5

in calculating the correlation between the fourth and first variables, and cases

3

and

4

for the fourth and second variables etc.

g02bj: FL CL CPP AD PY MB

NAG FL Interfaceg02bjf (coeffs_​pearson_​subset_​miss_​pair)

▸▿ Contents

1 Purpose