NAG Library Routine Document

G02BCF

+− 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

G02BCF 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 omitting cases with missing values from only those calculations involving the variables for which the values are missing.

2 Specification

SUBROUTINE G02BCF (

N, M, X, LDX, MISS, XMISS, XBAR, STD, SSP, LDSSP, R, LDR, NCASES, CNT, LDCNT, IFAIL)

INTEGER	N, M, LDX, MISS(M), LDSSP, LDR, NCASES, LDCNT, IFAIL
REAL (KIND=nag_wp)	X(LDX,M), XMISS(M), XBAR(M), STD(M), SSP(LDSSP,M), R(LDR,M), CNT(LDCNT,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. 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 = 1, 2, \dots, m .

(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 = 1, 2, \dots, m .

(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 = 1, 2, \dots, m,

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 sums 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, = 1, 2, \dots, m,

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}

and

{\bar{x}}_{j (k)}

and

{\bar{x}}_{k (j)}

are as defined in (c) above

(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).

S_{j j (k)}

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 = 1, 2, \dots, m .

(The diagonal terms,

c_{j j}

, for

j = 1, 2, \dots, m

, 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 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 G02BCF is called.
Constraint: $LDX \geq N$ .
5: MISS(M) – INTEGER arrayInput: 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) arrayInput: 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: 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: 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 $k = 1, 2, \dots, m$ .
10: LDSSP – INTEGERInput: On entry: the first dimension of the array SSP as declared in the (sub)program from which G02BCF is called.

Constraint: $LDSSP \geq M$ .
11: 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 $k = 1, 2, \dots, m$ .
12: LDR – INTEGERInput: On entry: the first dimension of the array R as declared in the (sub)program from which G02BCF is called.
Constraint: $LDR \geq M$ .
13: NCASES – INTEGEROutput: 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).
14: CNT(LDCNT,M) – REAL (KIND=nag_wp) arrayOutput: 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 $j$ th and $k$ th variables, for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .
15: LDCNT – INTEGERInput: On entry: the first dimension of the array CNT as declared in the (sub)program from which G02BCF is called.

Constraint: $LDCNT \geq M$ .
16: 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, because for this routine the values of the output parameters may be useful even if $IFAIL \neq 0$ on exit, the recommended value is $- 1$ . 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).

Note: G02BCF may return useful information for one or more of the following detected errors or warnings.

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	$LDR < M$ ,
or	$LDCNT < M$ .

$IFAIL = 4$: After observations with missing values were omitted, fewer than two cases remained for at least one pair of variables. (The pairs of variables involved can be determined by examination of the contents of the array CNT.) All means, standard deviations, sums of squares and cross-products, and correlation coefficients based on two or more cases are returned by the routine even if $IFAIL = 4$ .

7 Accuracy

G02BCF 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. G02BCF 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 G02BCF 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

- 1.0

and

0.0

are declared for the first, second and third variables respectively. 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, omitting cases with missing values from only those calculations involving the variables for which the values are missing. The program therefore omits cases

4

and

5

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

3

and

4

for the first and third variables etc.

NAG Library Routine DocumentG02BCF

+− 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

NAG Library Routine Document

G02BCF