NAG Library Routine Document

G02BKF

+− 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

G02BKF computes means and standard deviations, sums of squares and cross-products about zero, and correlation-like coefficients for selected variables.

2 Specification

SUBROUTINE G02BKF (

N, M, X, LDX, NVARS, KVAR, XBAR, STD, SSPZ, LDSSPZ, RZ, LDRZ, IFAIL)

INTEGER	N, M, LDX, NVARS, KVAR(NVARS), LDSSPZ, LDRZ, IFAIL
REAL (KIND=nag_wp)	X(LDX,M), XBAR(NVARS), STD(NVARS), SSPZ(LDSSPZ,NVARS), RZ(LDRZ,NVARS)

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.

The quantities calculated are:

(a)

Means:

{\bar{x}}_{j} = \frac{\sum_{i = 1}^{n} x_{i j}}{n}, 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 about zero:

{\tilde{S}}_{j k} = \sum_{i = 1}^{n} x_{i j} x_{i k}, j, k = v_{1}, v_{2}, \dots, v_{p} .

(d)

Correlation-like coefficients:

{\tilde{R}}_{j k} = \frac{{\tilde{S}}_{j k}}{\sqrt{{\tilde{S}}_{j j} {\tilde{S}}_{k k}}}, j, k = v_{1}, v_{2}, \dots, v_{p} .

{\tilde{S}}_{j j}

{\tilde{S}}_{k k}

is zero,

{\tilde{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 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 G02BKF 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: SSPZ(LDSSPZ,NVARS) – REAL (KIND=nag_wp) arrayOutput: On exit: $SSPZ (j, k)$ is the cross-product about zero, ${\tilde{S}}_{j k}$ , for the variables specified in $KVAR (j)$ and $KVAR (k)$ , for $j = 1, 2, \dots, p$ and $k = 1, 2, \dots, p$ .
10: LDSSPZ – INTEGERInput: On entry: the first dimension of the array SSPZ as declared in the (sub)program from which G02BKF is called.

Constraint: $LDSSPZ \geq NVARS$ .
11: RZ(LDRZ,NVARS) – REAL (KIND=nag_wp) arrayOutput: On exit: $RZ (j, k)$ is the correlation-like coefficient, ${\tilde{R}}_{j k}$ , between the variables specified in $KVAR (j)$ and $KVAR (k)$ , for $j = 1, 2, \dots, p$ and $k = 1, 2, \dots, p$ .
12: LDRZ – INTEGERInput: On entry: the first dimension of the array RZ as declared in the (sub)program from which G02BKF is called.

Constraint: $LDRZ \geq NVARS$ .
13: 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,	$NVARS < 2$ ,
or	$NVARS > M$ .

$IFAIL = 3$

On entry,	$LDX < N$ ,
or	$LDSSPZ < NVARS$ ,
or	$LDRZ < NVARS$ .

$IFAIL = 4$

On entry,	$KVAR (j) < 1$ ,
or	$KVAR (j) > M$ for some $j = 1, 2, \dots, NVARS$ .

7 Accuracy

G02BKF 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 G02BKF depends on

n

and

p

The routine uses a two-pass algorithm.

9 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 about zero, and correlation-like coefficients for the fourth, first and second variables are then calculated and printed.

NAG Library Routine DocumentG02BKF