NAG Library Routine Document
G02BFF
1 Purpose
G02BFF 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 cases with missing values from only those calculations involving the variables for which the values are missing.
2 Specification
SUBROUTINE G02BFF ( |
N, M, X, LDX, MISS, XMISS, XBAR, STD, SSPZ, LDSSPZ, RZ, LDRZ, NCASES, CNT, LDCNT, IFAIL) |
INTEGER |
N, M, LDX, MISS(M), LDSSPZ, LDRZ, NCASES, LDCNT, IFAIL |
REAL (KIND=nag_wp) |
X(LDX,M), XMISS(M), XBAR(M), STD(M), SSPZ(LDSSPZ,M), RZ(LDRZ,M), CNT(LDCNT,M) |
|
3 Description
The input data consists of
n observations for each of
m variables, given as an array
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:
(a) |
Means:
|
(b) |
Standard deviations:
|
(c) |
Sums of squares and cross-products about zero:
|
(d) |
Correlation-like coefficients:
where S~jjk=∑i=1nwijwikxij2 and S~kkj=∑i=1nwikwijxik2 (i.e., the sums of squares about zero are based on the same set of observations as are used in the calculation of the numerator).
If S~jjk or S~kkj is zero, R~jk is set to zero. |
(e) |
The number of cases used in the calculation of each of the correlation-like coefficients:
(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 Parameters
- 1: N – INTEGERInput
On entry: n, the number of observations or cases.
Constraint:
N≥2.
- 2: M – INTEGERInput
On entry: m, the number of variables.
Constraint:
M≥2.
- 3: X(LDX,M) – REAL (KIND=nag_wp) arrayInput
On entry: Xij 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 – INTEGERInput
On entry: the first dimension of the array
X as declared in the (sub)program from which G02BFF is called.
Constraint:
LDX≥N.
- 5: MISS(M) – INTEGER arrayInput
On entry:
MISSj 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) arrayInput
On entry:
XMISSj 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) arrayOutput
On exit: the mean value,
x-j, of the jth variable, for j=1,2,…,m.
- 8: STD(M) – REAL (KIND=nag_wp) arrayOutput
On exit: the standard deviation,
sj, of the jth variable, for j=1,2,…,m.
- 9: SSPZ(LDSSPZ,M) – REAL (KIND=nag_wp) arrayOutput
On exit: SSPZjk is the cross-product about zero, S~jk, for j=1,2,…,m and k=1,2,…,m.
- 10: LDSSPZ – INTEGERInput
On entry: the first dimension of the array
SSPZ as declared in the (sub)program from which G02BFF is called.
Constraint:
LDSSPZ≥M.
- 11: RZ(LDRZ,M) – REAL (KIND=nag_wp) arrayOutput
On exit: RZjk is the correlation-like coefficient, R~jk, between the jth and kth variables, for j=1,2,…,m and k=1,2,…,m.
- 12: LDRZ – INTEGERInput
On entry: the first dimension of the array
RZ as declared in the (sub)program from which G02BFF is called.
Constraint:
LDRZ≥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-like coefficients (when cases involving missing values have been eliminated).
- 14: CNT(LDCNT,M) – REAL (KIND=nag_wp) arrayOutput
On exit: CNTjk is the number of cases, cjk, actually used in the calculation of S~jk, and R~jk, the sum of cross-products and correlation-like coefficient for the jth and kth variables, for j=1,2,…,m and k=1,2,…,m.
- 15: LDCNT – INTEGERInput
On entry: must specify the first dimension of the array
CNT as declared in the (sub)program from which G02BFF is called.
Constraint:
LDCNT≥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≠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 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Note: G02BFF may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
- IFAIL=1
-
- IFAIL=2
-
- IFAIL=3
On entry, | LDX<N, |
or | LDSSPZ<M, |
or | LDRZ<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-like coefficients based on two or more cases are returned by the routine even if
IFAIL=4.
7 Accuracy
G02BFF 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. G02BFF treats all values in the inclusive range
1±0.1X02BEF-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 Further Comments
The time taken by G02BFF 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 about zero, and correlation-like 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.
9.1 Program Text
Program Text (g02bffe.f90)
9.2 Program Data
Program Data (g02bffe.d)
9.3 Program Results
Program Results (g02bffe.r)