NAG FL Interface
g02bcf (coeffs_​pearson_​miss_​pair)

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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

Fortran Interface
Subroutine g02bcf ( n, m, x, ldx, miss, xmiss, xbar, std, ssp, ldssp, r, ldr, ncases, cnt, ldcnt, ifail)
Integer, Intent (In) :: n, m, ldx, miss(m), 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,m), r(ldr,m), cnt(ldcnt,m)
Real (Kind=nag_wp), Intent (Out) :: xbar(m), std(m)
C Header Interface
#include <nag.h>
void  g02bcf_ (const Integer *n, const Integer *m, const double x[], const Integer *ldx, const Integer miss[], const double xmiss[], double xbar[], double std[], double ssp[], const Integer *ldssp, double r[], const Integer *ldr, Integer *ncases, double cnt[], const Integer *ldcnt, Integer *ifail)
The routine may be called by the names g02bcf or nagf_correg_coeffs_pearson_miss_pair.

3 Description

The input data consist of n observations for each of m variables, given as an array
[xij],  i=1,2,,n(n2),j=1,2,,m(m2),  
where xij is the ith observation on the jth 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 jth variable is denoted by xmj. Missing values need not be specified for all variables.
Let wij=0 if the ith observation for the jth variable is a missing value, i.e., if a missing value, xmj, has been declared for the jth variable, and xij=xmj (see also Section 7); and wij=1 otherwise, for i=1,2,,n and j=1,2,,m.
The quantities calculated are:
  1. (a)Means:
    x¯j=i=1nwijxij i=1nwij ,  j=1,2,,m.  
  2. (b)Standard deviations:
    sj=i= 1nwij (xij-x¯j) 2 (i= 1nwij)- 1 ,   j= 1,2,,m.  
  3. (c)Sums of squares and cross-products of deviations from means:
    Sjk=i=1nwijwik(xij-x¯j(k))(xik-x¯k(j)),  j,k=1,2,,m,  
    where
    x¯j(k)=i= 1nwijwikxij i= 1nwijwik   and   x¯k(j)=i= 1nwikwijxik i= 1nwikwij ,  
    (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.)
  4. (d)Pearson product-moment correlation coefficients:
    Rjk=SjkSjj(k)Skk(j) ,  j,k,=1,2,,m,  
    where Sjj(k)=i=1nwijwik(xij-x¯j(k))2 and Skk(j)=i=1nwikwij(xik-x¯k(j))2 and x¯j(k) and 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).
    If Sjj(k) or Skk(j) is zero, Rjk is set to zero.
  5. (e)The number of cases used in the calculation of each of the correlation coefficients:
    cjk=i=1nwijwik,  j,k=1,2,,m.  
    (The diagonal terms, cjj, for j=1,2,,m, also give the number of cases used in the calculation of the means, x¯j, and the standard deviations, sj.)

4 References

None.

5 Arguments

1: n Integer Input
On entry: n, the number of observations or cases.
Constraint: n2.
2: m Integer Input
On entry: m, the number of variables.
Constraint: m2.
3: x(ldx,m) Real (Kind=nag_wp) array Input
On entry: x(i,j) must be set to xij, the value of the ith observation on the jth variable, for i=1,2,,n and j=1,2,,m.
4: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02bcf is called.
Constraint: ldxn.
5: miss(m) Integer array Input
On entry: miss(j) must be set equal to 1 if a missing value, xmj, is to be specified for the jth 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) array Input
On entry: xmiss(j) must be set to the missing value, xmj, to be associated with the jth 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) array Output
On exit: the mean value, x¯j, of the jth variable, for j=1,2,,m.
8: std(m) Real (Kind=nag_wp) array Output
On exit: the standard deviation, sj, of the jth variable, for j=1,2,,m.
9: ssp(ldssp,m) Real (Kind=nag_wp) array Output
On exit: ssp(j,k) is the cross-product of deviations Sjk, for j=1,2,,m and k=1,2,,m.
10: ldssp Integer Input
On entry: the first dimension of the array ssp as declared in the (sub)program from which g02bcf is called.
Constraint: ldsspm.
11: r(ldr,m) Real (Kind=nag_wp) array Output
On exit: r(j,k) is the product-moment correlation coefficient Rjk between the jth and kth variables, for j=1,2,,m and k=1,2,,m.
12: ldr Integer Input
On entry: the first dimension of the array r as declared in the (sub)program from which g02bcf is called.
Constraint: ldrm.
13: ncases Integer Output
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) array Output
On exit: cnt(j,k) is the number of cases, cjk, actually used in the calculation of Sjk, and Rjk, the sum of cross-products and correlation coefficient for the jth and kth variables, for j=1,2,,m and k=1,2,,m.
15: ldcnt Integer Input
On entry: the first dimension of the array cnt as declared in the (sub)program from which g02bcf is called.
Constraint: ldcntm.
16: ifail Integer Input/Output
On entry: ifail must be set to 0, −1 or 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 or 1 is recommended. If message printing is undesirable, then the value 1 is recommended. Otherwise, the value −1 is recommended since useful values can be provided in some output arguments even when ifail0 on exit. 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 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases g02bcf may return useful information.
ifail=1
On entry, n=value.
Constraint: n2.
ifail=2
On entry, m=value.
Constraint: m2.
ifail=3
On entry, ldcnt=value and m=value.
Constraint: ldcntm.
On entry, ldr=value and m=value.
Constraint: ldrm.
On entry, ldssp=value and m=value.
Constraint: ldsspm.
On entry, ldx=value and n=value.
Constraint: ldxn.
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.
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

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±0.1(x02bef-2))×xmj, where xmj 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

g02bcf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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 g02bcf depends on n and m, 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:
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 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.

10.1 Program Text

Program Text (g02bcfe.f90)

10.2 Program Data

Program Data (g02bcfe.d)

10.3 Program Results

Program Results (g02bcfe.r)