g02bhf 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 completely any cases with a missing observation for any variable (either over all variables in the dataset or over only those variables in the selected subset).

2 Specification

Fortran Interface

Subroutine g02bhf (

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

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

C Header Interface

#include <nag.h>

void

g02bhf_ (const Integer *n, const Integer *m, const double x[], const Integer *ldx, Integer miss[], double xmiss[], const Integer *mistyp, const Integer *nvars, const Integer kvar[], double xbar[], double std[], double ssp[], const Integer *ldssp, double r[], const Integer *ldr, Integer *ncases, Integer *ifail)

The routine may be called by the names g02bhf or nagf_correg_coeffs_pearson_subset_miss_case.

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. The missing values can be utilized in two slightly different ways; you can indicate which scheme is required.

Firstly, let

w_{i} = 0

if observation

i

contains a missing value for any of those variables in the set

1, 2, \dots, m

for which missing values have been declared, i.e., if

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

for any

j

(

j = 1, 2, \dots, m

) for which an

{xm}_{j}

has been assigned (see also Section 7); and

w_{i} = 1

otherwise, for

i = 1, 2, \dots, n

Secondly, let

w_{i} = 0

if observation

i

contains a missing value for any of those variables in the selected subset

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

for which missing values have been declared, i.e., if

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

for any

j

(

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

) 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:

${\bar{x}}_{j} = \frac{\sum_{i = 1}^{n} w_{i} x_{i j}}{\sum_{i = 1}^{n} w_{i}}, j = v_{1}, v_{2}, \dots, v_{p} .$
(b)Standard deviations:

$s_{j} = \sqrt{\frac{\sum_{i = 1}^{n} w_{i} {(x_{i j} - {\bar{x}}_{j})}^{2}}{\sum_{i = 1}^{n} w_{i} - 1}}, j = v_{1}, v_{2}, \dots, v_{p} .$
(c)Sums of squares and cross-products of deviations from means:

$S_{j k} = \sum_{i = 1}^{n} w_{i} (x_{i j} - {\bar{x}}_{j}) (x_{i k} - {\bar{x}}_{k}), j, k = v_{1}, v_{2}, \dots, v_{p} .$
(d)Pearson product-moment correlation coefficients:

$R_{j k} = \frac{S_{j k}}{\sqrt{S_{j j} S_{k k}}}, j, k = v_{1}, v_{2}, \dots, v_{p} .$

If $S_{j j}$ or $S_{k k}$ is zero, $R_{j k}$ is set to zero.

4 References

None.

5 Arguments

1: $n$ – Integer Input

On entry:

n

, the number of observations or cases.

Constraint:

n \geq 2

2: $m$ – Integer Input

On entry:

m

, the number of variables.

Constraint:

m \geq 2

3: $x (ldx, m)$ – Real (Kind=nag_wp) array Input

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: $ldx$ – Integer Input

On entry: the first dimension of the array x as declared in the (sub)program from which g02bhf is called.

Constraint:

ldx \geq n

5: $miss (m)$ – Integer array Input/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) array Input/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: $mistyp$ – Integer Input

On entry: indicates the manner in which missing observations are to be treated.

$mistyp = 1$: A case is excluded if it contains a missing value for any of the variables $1, 2, \dots, m$ .
$mistyp = 0$: A case is excluded if it contains a missing value for any of the $p (\leq m)$ variables specified in the array kvar.

8: $nvars$ – Integer Input

On entry:

p

, the number of variables for which information is required.

Constraint:

2 \leq nvars \leq m

9: $kvar (nvars)$ – Integer array Input

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

10: $xbar (nvars)$ – Real (Kind=nag_wp) array Output

On exit: the mean value, of

{\bar{x}}_{j}

, of the variable specified in

kvar (j)

, for

j = 1, 2, \dots, p

11: $std (nvars)$ – Real (Kind=nag_wp) array Output

On exit: the standard deviation,

s_{j}

, of the variable specified in

kvar (j)

, for

j = 1, 2, \dots, p

12: $ssp (ldssp, nvars)$ – Real (Kind=nag_wp) array Output

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

13: $ldssp$ – Integer Input

On entry: the first dimension of the array ssp as declared in the (sub)program from which g02bhf is called.

Constraint:

ldssp \geq nvars

14: $r (ldr, nvars)$ – Real (Kind=nag_wp) array Output

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

15: $ldr$ – Integer Input

On entry: the first dimension of the array r as declared in the (sub)program from which g02bhf is called.

Constraint:

ldr \geq nvars

16: $ncases$ – Integer Output

On exit: the number of cases actually used in the calculations (when cases involving missing values have been eliminated).

17: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. 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, $n = ⟨ value ⟩$ .
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, $ldr = ⟨ value ⟩$ 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: $1 \leq kvar (i) \leq m$ .

$ifail = 5$: On entry, $mistyp = ⟨ value ⟩$ .
Constraint: $mistyp = 0$ or $1$ .

$ifail = 6$: After observations with missing values were omitted, no cases remained.

$ifail = 7$: After observations with missing values were omitted, only one case remained.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

g02bhf 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. g02bhf 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

g02bhf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

g02bhf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The time taken by g02bhf depends on

n

and

p

, and the occurrence of missing values.

The routine uses a two-pass algorithm.

9.1 Internal Changes

Internal changes have been made to this routine as follows:

At Mark 27: The algorithm underlying this routine has been altered to improve efficiency for large problem sizes on a multi-threaded system.

For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

10 Example

This example reads in a set of data consisting of five observations on each of four variables. Missing values of

0.0

are declared for the second and fourth variables; no missing values are specified for the first and third variables. 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 completely all cases containing missing values for these three selected variables; cases

3

and

4

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

g02bh: FL CL CPP AD

NAG FL Interfaceg02bhf (coeffs_​pearson_​subset_​miss_​case)

▸▿ 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

9.1 Internal Changes

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
g02bhf (coeffs_pearson_subset_miss_case)