NAG Library Routine Document

G02BAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

G02BAF 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.

2 Specification

SUBROUTINE G02BAF (

N, M, X, LDX, XBAR, STD, SSP, LDSSP, R, LDR, IFAIL)

INTEGER	N, M, LDX, LDSSP, LDR, IFAIL
REAL (KIND=nag_wp)	X(LDX,M), XBAR(M), STD(M), SSP(LDSSP,M), R(LDR,M)

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.

The quantities calculated are:

(a)

Means:

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

(b)

Standard deviations:

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

(c)

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

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

(d)

Pearson product-moment correlation coefficients:

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

S_{j j}

S_{k k}

is zero,

R_{j k}

is set to zero.

4 References

None.

5 Parameters

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 $i$ 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 G02BAF is called.
Constraint: $LDX \geq N$ .
5: XBAR(M) – REAL (KIND=nag_wp) arrayOutput: On exit: the mean value, ${\bar{x}}_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
6: STD(M) – REAL (KIND=nag_wp) arrayOutput: On exit: the standard deviation, $s_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
7: SSP(LDSSP,M) – REAL (KIND=nag_wp) arrayOutput: On exit: $SSP (j, k)$ is the cross-product of deviations $S_{j k}$ , for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
8: LDSSP – INTEGERInput: On entry: the first dimension of the array SSP as declared in the (sub)program from which G02BAF is called.

Constraint: $LDSSP \geq M$ .
9: R(LDR,M) – REAL (KIND=nag_wp) arrayOutput: 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 $k = 1, 2, \dots, m$ .
10: LDR – INTEGERInput: On entry: the first dimension of the array R as declared in the (sub)program from which G02BAF is called.
Constraint: $LDR \geq M$ .
11: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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 parameter, 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 < 2

$IFAIL = 2$

On entry,

M < 2

$IFAIL = 3$

On entry,	$LDX < N$ ,
or	$LDSSP < M$ ,
or	$LDR < M$ .

7 Accuracy

G02BAF 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 Further Comments

The time taken by G02BAF depends on

n

and

m

The routine uses a two-pass algorithm.

9 Example

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

NAG Library Routine DocumentG02BAF