NAG Library Routine Document

g02bjf computes means and standard deviations, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for selected variables omitting cases with missing values from only those calculations involving the variables for which the values are missing.

2

Specification

Fortran Interface

Subroutine g02bjf (

n, m, x, ldx, miss, xmiss, nvars, kvar, xbar, std, ssp, ldssp, r, ldr, ncases, cnt, ldcnt, ifail)

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

C Header Interface

#include <nagmk26.h>

void

g02bjf_ (const Integer *n, const Integer *m, const double x[], const Integer *ldx, const Integer miss[], const double xmiss[], const Integer *nvars, const Integer kvar[], double xbar[], double std[], double ssp[], const Integer *ldssp, double r[], const Integer *ldr, Integer *ncases, double cnt[], const Integer *ldcnt, Integer *ifail)

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.

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

(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 = v_{1}, v_{2}, \dots, v_{p} .

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

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

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},

(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 = v_{1}, v_{2}, \dots, v_{p} .

(The diagonal terms,

c_{j j}

, for

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

, 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

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 g02bjf 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: $nvars$ – IntegerInput: On entry: $p$ , the number of variables for which information is required.

Constraint: $2 \leq nvars \leq m$ .
8: $kvar (nvars)$ – 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 $j = 1, 2, \dots, p$ .
9: $xbar (nvars)$ – Real (Kind=nag_wp) arrayOutput: On exit: the mean value, ${\bar{x}}_{j}$ , of the variable specified in $kvar (j)$ , for $j = 1, 2, \dots, p$ .
10: $std (nvars)$ – Real (Kind=nag_wp) arrayOutput: On exit: the standard deviation, $s_{j}$ , of the variable specified in $kvar (j)$ , for $j = 1, 2, \dots, p$ .
11: $ssp (ldssp, nvars)$ – 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 $k = 1, 2, \dots, p$ .
12: $ldssp$ – IntegerInput: On entry: the first dimension of the array ssp as declared in the (sub)program from which g02bjf is called.

Constraint: $ldssp \geq nvars$ .
13: $r (ldr, nvars)$ – 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 $k = 1, 2, \dots, p$ .
14: $ldr$ – IntegerInput: On entry: the first dimension of the array r as declared in the (sub)program from which g02bjf is called.

Constraint: $ldr \geq nvars$ .
15: $ncases$ – 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).
16: $cnt (ldcnt, nvars)$ – 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 variables specified in $kvar (j)$ and $kvar (k)$ , for $j = 1, 2, \dots, p$ and $k = 1, 2, \dots, p$ .
17: $ldcnt$ – IntegerInput: On entry: the first dimension of the array cnt as declared in the (sub)program from which g02bjf is called.

Constraint: $ldcnt \geq nvars$ .
18: $ifail$ – 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, because for this routine the values of the output arguments 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: g02bjf 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, $n = 〈value〉$ .
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, $ldcnt = 〈value〉$ and $nvars = 〈value〉$ .
Constraint: $ldcnt \geq nvars$ .

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: $1 \leq kvar (i) \leq m$ .

$ifail = 5$: After observations with missing values were omitted, fewer than two cases remained.

$ifail = - 99$: 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

g02bjf 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. g02bjf 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

Parallelism and Performance

g02bjf is not threaded in any implementation.

9

Further Comments

The time taken by g02bjf depends on

n

and

p

, and the occurrence of missing values.

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. Missing values of

- 1.0

0.0

and

0.0

are declared for the first, second and fourth variables respectively; no missing value is specified for the third variable. 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, omitting cases with missing values from only those calculations involving the variables for which the values are missing. The program therefore eliminates cases

4

and

5

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

3

and

4

for the fourth and second variables etc.

NAG Library Routine Document

g02bjf (coeffs_pearson_subset_miss_pair)

▸▿ 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

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