NAG Library Routine Document

g02bgf computes means and standard deviations, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for selected variables.

2

Specification

Fortran Interface

Subroutine g02bgf (

n, m, x, ldx, nvars, kvar, xbar, std, ssp, ldssp, r, ldr, ifail)

Integer, Intent (In)	::	n, m, ldx, nvars, kvar(nvars), ldssp, ldr
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (In)	::	x(ldx,m)
Real (Kind=nag_wp), Intent (Inout)	::	ssp(ldssp,nvars), r(ldr,nvars)
Real (Kind=nag_wp), Intent (Out)	::	xbar(nvars), std(nvars)

C Header Interface

#include <nagmk26.h>

void	g02bgf_ (const Integer n, const Integer m, const double x[], const Integer ldx, const Integer nvars, const Integer kvar[], double xbar[], double std[], double ssp[], const Integer ldssp, double r[], const Integer ldr, Integer *ifail)

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, together with the subset of these variables,

v_{1}, v_{2}, \dots, v_{p}

, for which information is required.

The quantities calculated are:

(a)

Means:

{\bar{x}}_{j} = \frac{1}{n} \sum_{i = 1}^{n} x_{i j}, j = v_{1}, v_{2}, \dots, v_{p} .

(b)

Standard deviations:

s_{j} = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} {(x_{i j} - {\bar{x}}_{j})}^{2}}, j = v_{1}, v_{2}, \dots, v_{p} .

(c)

Sums of squares and cross-products of deviations from zero:

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

(d)

Pearson product-moment correlation coefficients:

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

S_{j j}

S_{k k}

is zero,

R_{j k}

is set to zero.

4

References

None.

5

Arguments

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

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

Constraint: $m \geq 2$ .
3: $x (ldx, m)$ – Real (Kind=nag_wp) arrayInput: 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: $ldx$ – IntegerInput: On entry: the first dimension of the array x as declared in the (sub)program from which g02bgf is called.

Constraint: $ldx \geq n$ .
5: $nvars$ – IntegerInput: On entry: $p$ , the number of variables for which information is required.

Constraint: $2 \leq nvars \leq m$ .
6: $kvar (nvars)$ – Integer arrayInput: 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 $j = 1, 2, \dots, p$ .
7: $xbar (nvars)$ – Real (Kind=nag_wp) arrayOutput: On exit: the mean value, ${\bar{x}}_{j}$ , of the variable specified in $kvar (j)$ , for $j = 1, 2, \dots, p$ .
8: $std (nvars)$ – Real (Kind=nag_wp) arrayOutput: On exit: the standard deviation, $s_{j}$ , of the variable specified in $kvar (j)$ , for $j = 1, 2, \dots, p$ .
9: $ssp (ldssp, nvars)$ – Real (Kind=nag_wp) arrayOutput: 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 $k = 1, 2, \dots, p$ .
10: $ldssp$ – IntegerInput: On entry: the first dimension of the array ssp as declared in the (sub)program from which g02bgf is called.

Constraint: $ldssp \geq nvars$ .
11: $r (ldr, nvars)$ – Real (Kind=nag_wp) arrayOutput: 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 $k = 1, 2, \dots, p$ .
12: $ldr$ – IntegerInput: On entry: the first dimension of the array r as declared in the (sub)program from which g02bgf is called.

Constraint: $ldr \geq nvars$ .
13: $ifail$ – IntegerInput/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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, if you are not familiar with this argument, the recommended value is $0$ . 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:

$ifail = 1$: 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, $ldr = 〈value〉$ 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: $1 \leq kvar (i) \leq m$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

g02bgf does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large

n

8

Parallelism and Performance

g02bgf is not threaded in any implementation.

9

Further Comments

The time taken by g02bgf depends on

n

and

p

The routine uses a two pass algorithm.

10

Example

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

NAG Library Routine Document

g02bgf (coeffs_pearson_subset)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose