NAG Library Routine Document

G02BEF

+− 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

G02BEF computes means and standard deviations of variables, sums of squares and cross-products about zero, and correlation-like coefficients for a set of data omitting completely any cases with a missing observation for any variable.

2 Specification

SUBROUTINE G02BEF (

N, M, X, LDX, MISS, XMISS, XBAR, STD, SSPZ, LDSSPZ, RZ, LDRZ, NCASES, IFAIL)

INTEGER	N, M, LDX, MISS(M), LDSSPZ, LDRZ, NCASES, IFAIL
REAL (KIND=nag_wp)	X(LDX,M), XMISS(M), XBAR(M), STD(M), SSPZ(LDSSPZ,M), RZ(LDRZ,M)

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. 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} = 0

if observation

i

contains a missing value for any of those variables for which missing values have been declared, i.e., if

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

for any

j

for which an

{xm}_{j}

has been assigned (see also Section 7); and

w_{i} = 1

otherwise, for

i = 1, 2, \dots, n

The quantities calculated are:

(a)

Means:

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

(b)

Standard deviations:

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

(c)

Sums of squares and cross-products about zero:

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

(d)

Correlation-like coefficients:

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

{\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 G02BEF is called.
Constraint: $LDX \geq N$ .
5: MISS(M) – INTEGER arrayInput/Output: 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.

On exit: the array MISS is overwritten by the routine, and the information it contained on entry is lost.
6: XMISS(M) – REAL (KIND=nag_wp) arrayInput/Output: 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).

On exit: the array XMISS is overwritten by the routine, and the information it contained on entry is lost.
7: XBAR(M) – REAL (KIND=nag_wp) arrayOutput: On exit: the mean value, ${\bar{x}}_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
8: STD(M) – REAL (KIND=nag_wp) arrayOutput: On exit: the standard deviation, $s_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .
9: SSPZ(LDSSPZ,M) – REAL (KIND=nag_wp) arrayOutput: On exit: $SSPZ (j, k)$ is the cross-product about zero, ${\tilde{S}}_{j k}$ , for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
10: LDSSPZ – INTEGERInput: On entry: the first dimension of the array SSPZ as declared in the (sub)program from which G02BEF is called.

Constraint: $LDSSPZ \geq M$ .
11: RZ(LDRZ,M) – REAL (KIND=nag_wp) arrayOutput: On exit: $RZ (j, k)$ is the correlation-like coefficient, ${\tilde{R}}_{j k}$ , between the $j$ th and $k$ th variables, for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
12: LDRZ – INTEGERInput: On entry: the first dimension of the array RZ as declared in the (sub)program from which G02BEF is called.

Constraint: $LDRZ \geq M$ .
13: NCASES – INTEGEROutput: On exit: the number of cases actually used in the calculations (when cases involving missing values have been eliminated).
14: 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	$LDSSPZ < M$ ,
or	$LDRZ < M$ .

$IFAIL = 4$: After observations with missing values were omitted, no cases remained.

$IFAIL = 5$: After observations with missing values were omitted, only one case remained.

7 Accuracy

G02BEF 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. G02BEF 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 Further Comments

The time taken by G02BEF depends on

n

and

m

, and the occurrence of missing values.

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. Missing values of

0.0

are declared for the first and third variables; no missing value is specified for the second variable. The means, standard deviations, sums of squares and cross-products about zero, and correlation-like coefficients for all three variables are then calculated and printed, omitting completely all cases containing missing values; cases

3

and

4

are therefore eliminated, leaving only three cases in the calculations.

NAG Library Routine DocumentG02BEF