NAG Library Routine Document
G02BEF
1 Purpose
G02BEF computes means and standard deviations of variables, sums of squares and crossproducts about zero, and correlationlike coefficients for a set of data omitting completely any cases with a missing observation for any variable.
2 Specification
SUBROUTINE G02BEF ( 
N, M, X, LDX, MISS, XMISS, XBAR, STD, SSPZ, LDSSPZ, RZ, LDRZ, NCASES, IFAIL) 
INTEGER 
N, M, LDX, MISS(M), LDSSPZ, LDRZ, NCASES, IFAIL 
REAL (KIND=nag_wp) 
X(LDX,M), XMISS(M), XBAR(M), STD(M), SSPZ(LDSSPZ,M), RZ(LDRZ,M) 

3 Description
The input data consists of
$n$ observations for each of
$m$ variables, given as an array
where
${x}_{ij}$ 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
${\mathit{xm}}_{j}$. Missing values need not be specified for all variables.
Let
${w}_{i}=0$ if observation
$i$ contains a missing value for any of those variables for which missing values have been declared, i.e., if
${x}_{ij}={\mathit{xm}}_{j}$ for any
$j$ for which an
${\mathit{xm}}_{j}$ has been assigned (see also
Section 7); and
${w}_{i}=1$ otherwise, for
$\mathit{i}=1,2,\dots ,n$.
The quantities calculated are:
(a) 
Means:

(b) 
Standard deviations:

(c) 
Sums of squares and crossproducts about zero:

(d) 
Correlationlike coefficients:
If ${\stackrel{~}{S}}_{jj}$ or ${\stackrel{~}{S}}_{kk}$ is zero, ${\stackrel{~}{R}}_{jk}$ is set to zero. 
4 References
None.
5 Arguments
 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the number of observations or cases.
Constraint:
${\mathbf{N}}\ge 2$.
 2: $\mathrm{M}$ – INTEGERInput

On entry: $m$, the number of variables.
Constraint:
${\mathbf{M}}\ge 2$.
 3: $\mathrm{X}\left({\mathbf{LDX}},{\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: ${\mathbf{X}}\left(\mathit{i},\mathit{j}\right)$ must be set to ${x}_{\mathit{i}\mathit{j}}$, the value of the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
 4: $\mathrm{LDX}$ – INTEGERInput

On entry: the first dimension of the array
X as declared in the (sub)program from which G02BEF is called.
Constraint:
${\mathbf{LDX}}\ge {\mathbf{N}}$.
 5: $\mathrm{MISS}\left({\mathbf{M}}\right)$ – INTEGER arrayInput/Output

On entry:
${\mathbf{MISS}}\left(j\right)$ 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: $\mathrm{XMISS}\left({\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry:
${\mathbf{XMISS}}\left(j\right)$ 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: $\mathrm{XBAR}\left({\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the mean value,
${\stackrel{}{x}}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
 8: $\mathrm{STD}\left({\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the standard deviation,
${s}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
 9: $\mathrm{SSPZ}\left({\mathbf{LDSSPZ}},{\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: ${\mathbf{SSPZ}}\left(\mathit{j},\mathit{k}\right)$ is the crossproduct about zero, ${\stackrel{~}{S}}_{\mathit{j}\mathit{k}}$, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
 10: $\mathrm{LDSSPZ}$ – INTEGERInput

On entry: the first dimension of the array
SSPZ as declared in the (sub)program from which G02BEF is called.
Constraint:
${\mathbf{LDSSPZ}}\ge {\mathbf{M}}$.
 11: $\mathrm{RZ}\left({\mathbf{LDRZ}},{\mathbf{M}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: ${\mathbf{RZ}}\left(\mathit{j},\mathit{k}\right)$ is the correlationlike coefficient, ${\stackrel{~}{R}}_{\mathit{j}\mathit{k}}$, between the $\mathit{j}$th and $\mathit{k}$th variables, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
 12: $\mathrm{LDRZ}$ – INTEGERInput

On entry: the first dimension of the array
RZ as declared in the (sub)program from which G02BEF is called.
Constraint:
${\mathbf{LDRZ}}\ge {\mathbf{M}}$.
 13: $\mathrm{NCASES}$ – INTEGEROutput

On exit: the number of cases actually used in the calculations (when cases involving missing values have been eliminated).
 14: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$

On entry,  ${\mathbf{N}}<2$. 
 ${\mathbf{IFAIL}}=2$

On entry,  ${\mathbf{M}}<2$. 
 ${\mathbf{IFAIL}}=3$

On entry,  ${\mathbf{LDX}}<{\mathbf{N}}$, 
or  ${\mathbf{LDSSPZ}}<{\mathbf{M}}$, 
or  ${\mathbf{LDRZ}}<{\mathbf{M}}$. 
 ${\mathbf{IFAIL}}=4$

After observations with missing values were omitted, no cases remained.
 ${\mathbf{IFAIL}}=5$

After observations with missing values were omitted, only one case remained.
 ${\mathbf{IFAIL}}=99$
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.
 ${\mathbf{IFAIL}}=399$
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.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7 Accuracy
G02BEF 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. G02BEF treats all values in the inclusive range
$\left(1\pm {0.1}^{\left({\mathbf{X02BEF}}2\right)}\right)\times {xm}_{j}$, where
${\mathit{xm}}_{j}$ 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
G02BEF is not threaded in any implementation.
The time taken by G02BEF depends on $n$ and $m$, and the occurrence of missing values.
The routine uses a twopass algorithm.
10 Example
This example reads in a set of data consisting of five observations on each of three variables. Missing values of $0.0$ are declared for the first and third variables; no missing value is specified for the second variable. The means, standard deviations, sums of squares and crossproducts about zero, and correlationlike coefficients for all three variables are then calculated and printed, omitting completely all cases containing missing values; cases $3$ and $4$ are therefore eliminated, leaving only three cases in the calculations.
10.1 Program Text
Program Text (g02befe.f90)
10.2 Program Data
Program Data (g02befe.d)
10.3 Program Results
Program Results (g02befe.r)