# NAG Library Routine Document

## 1Purpose

g02bmf computes means and standard deviations, sums of squares and cross-products about zero, and correlation-like coefficients for selected variables omitting cases with missing values from only those calculations involving the variables for which the values are missing.

## 2Specification

Fortran Interface
 Subroutine g02bmf ( n, m, x, ldx, miss, kvar, xbar, std, sspz, rz, ldrz, cnt,
 Integer, Intent (In) :: n, m, ldx, miss(m), nvars, kvar(nvars), ldsspz, ldrz, 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) :: sspz(ldsspz,nvars), rz(ldrz,nvars), cnt(ldcnt,nvars) Real (Kind=nag_wp), Intent (Out) :: xbar(nvars), std(nvars)
#include nagmk26.h
 void g02bmf_ (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 sspz[], const Integer *ldsspz, double rz[], const Integer *ldrz, Integer *ncases, double cnt[], const Integer *ldcnt, Integer *ifail)

## 3Description

The input data consists of $n$ observations for each of $m$ variables, given as an array
 $xij, i=1,2,…,n n≥2,j=1,2,…,m m≥2,$
where ${x}_{ij}$ 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 ${\mathit{xm}}_{j}$. Missing values need not be specified for all variables.
Let ${w}_{\mathit{i}\mathit{j}}=0$, if the $\mathit{i}$th observation for the $\mathit{j}$th variable is a missing value, i.e., if a missing value, ${\mathit{xm}}_{\mathit{j}}$, has been declared for the $\mathit{j}$th variable, and ${x}_{\mathit{i}\mathit{j}}={\mathit{xm}}_{\mathit{j}}$ (see also Section 7); and ${w}_{\mathit{i}\mathit{j}}=1$ otherwise, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
The quantities calculated are:
(a) Means:
 $x-j=∑i=1nwijxij ∑i=1nwij , j=v1,v2,…,vp.$
(b) Standard deviations:
 $sj=∑i= 1nwij xij-x-j 2 ∑i= 1nwij- 1 , j=v1,v2,…,vp.$
(c) Sums of squares and cross-products about zero:
 $S~jk=∑i=1nwijwikxijxik, j,k=v1,v2,…,vp.$
(d) Correlation-like coefficients:
 $R~jk=S~jkS~jjk S~kkj , j,k=v1,v2,…,vp ,$
where ${\stackrel{~}{S}}_{jj\left(k\right)}=\sum _{i=1}^{n}{w}_{ij}{w}_{ik}{x}_{ij}^{2}$ and ${\stackrel{~}{S}}_{kk\left(j\right)}=\sum _{i=1}^{n}{w}_{ik}{w}_{ij}{x}_{ik}^{2}$
(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 ${\stackrel{~}{S}}_{jj\left(k\right)}$ or ${\stackrel{~}{S}}_{kk\left(j\right)}$ is zero, ${\stackrel{~}{R}}_{jk}$ is set to zero.
(e) The number of cases used in the calculation of each of the correlation-like coefficients:
 $cjk=∑i=1nwijwik, j,k=v1,v2,…,vp.$
(The diagonal terms, ${c}_{\mathit{j}\mathit{j}}$, for $\mathit{j}=1,2,\dots ,n$, also give the number of cases used in the calculation of the means ${\stackrel{-}{x}}_{\mathit{j}}$ and the standard deviations ${s}_{\mathit{j}}$.)
None.

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations or cases.
Constraint: ${\mathbf{n}}\ge 2$.
2:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 2$.
3:     $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{m}}\right)$ – 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:     $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which g02bmf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
5:     $\mathbf{miss}\left({\mathbf{m}}\right)$ – Integer arrayInput
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.
6:     $\mathbf{xmiss}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput
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).
7:     $\mathbf{nvars}$ – IntegerInput
On entry: $p$, the number of variables for which information is required.
Constraint: $2\le {\mathbf{nvars}}\le {\mathbf{m}}$.
8:     $\mathbf{kvar}\left({\mathbf{nvars}}\right)$ – Integer arrayInput
On entry: ${\mathbf{kvar}}\left(\mathit{j}\right)$ must be set to the column number in x of the $\mathit{j}$th variable for which information is required, for $\mathit{j}=1,2,\dots ,p$.
Constraint: $1\le {\mathbf{kvar}}\left(\mathit{j}\right)\le {\mathbf{m}}$, for $\mathit{j}=1,2,\dots ,p$.
9:     $\mathbf{xbar}\left({\mathbf{nvars}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the mean value, ${\stackrel{-}{x}}_{\mathit{j}}$, of the variable specified in ${\mathbf{kvar}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,p$.
10:   $\mathbf{std}\left({\mathbf{nvars}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the standard deviation, ${s}_{\mathit{j}}$, of the variable specified in ${\mathbf{kvar}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,p$.
11:   $\mathbf{sspz}\left({\mathbf{ldsspz}},{\mathbf{nvars}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{sspz}}\left(\mathit{j},\mathit{k}\right)$ is the cross-product about zero, ${\stackrel{~}{S}}_{\mathit{j}\mathit{k}}$, for the variables specified in ${\mathbf{kvar}}\left(\mathit{j}\right)$ and ${\mathbf{kvar}}\left(\mathit{k}\right)$, for $\mathit{j}=1,2,\dots ,p$ and $\mathit{k}=1,2,\dots ,p$.
12:   $\mathbf{ldsspz}$ – IntegerInput
On entry: the first dimension of the array sspz as declared in the (sub)program from which g02bmf is called.
Constraint: ${\mathbf{ldsspz}}\ge {\mathbf{nvars}}$.
13:   $\mathbf{rz}\left({\mathbf{ldrz}},{\mathbf{nvars}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{rz}}\left(\mathit{j},\mathit{k}\right)$ is the correlation-like coefficient, ${\stackrel{~}{R}}_{\mathit{j}\mathit{k}}$, between the variables specified in ${\mathbf{kvar}}\left(\mathit{j}\right)$ and ${\mathbf{kvar}}\left(\mathit{k}\right)$, for $\mathit{j}=1,2,\dots ,p$ and $\mathit{k}=1,2,\dots ,p$.
14:   $\mathbf{ldrz}$ – IntegerInput
On entry: the first dimension of the array rz as declared in the (sub)program from which g02bmf is called.
Constraint: ${\mathbf{ldrz}}\ge {\mathbf{nvars}}$.
15:   $\mathbf{ncases}$ – 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).
16:   $\mathbf{cnt}\left({\mathbf{ldcnt}},{\mathbf{nvars}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{cnt}}\left(\mathit{j},\mathit{k}\right)$ is the number of cases, ${c}_{\mathit{j}\mathit{k}}$, actually used in the calculation of the sum of cross-product and correlation-like coefficient for the variables specified in ${\mathbf{kvar}}\left(\mathit{j}\right)$ and ${\mathbf{kvar}}\left(\mathit{k}\right)$, for $\mathit{j}=1,2,\dots ,p$ and $\mathit{k}=1,2,\dots ,p$.
17:   $\mathbf{ldcnt}$ – IntegerInput
On entry: the first dimension of the array cnt as declared in the (sub)program from which g02bmf is called.
Constraint: ${\mathbf{ldcnt}}\ge {\mathbf{nvars}}$.
18:   $\mathbf{ifail}$ – 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, because for this routine the values of the output arguments may be useful even if ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{​ or ​}1$ 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).

## 6Error 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).
Note: g02bmf may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{nvars}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nvars}}\ge 2$ and ${\mathbf{nvars}}\le {\mathbf{m}}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ldcnt}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nvars}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldcnt}}\ge {\mathbf{nvars}}$
On entry, ${\mathbf{ldrz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nvars}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldrz}}\ge {\mathbf{nvars}}$.
On entry, ${\mathbf{ldsspz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nvars}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldsspz}}\ge {\mathbf{nvars}}$.
On entry, ${\mathbf{ldx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=4$
On entry, $\mathit{i}=〈\mathit{\text{value}}〉$, ${\mathbf{kvar}}\left(\mathit{i}\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{kvar}}\left(\mathit{i}\right)\le {\mathbf{m}}$.
${\mathbf{ifail}}=5$
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 ${\mathbf{ifail}}={\mathbf{5}}$.
${\mathbf{ifail}}=-99$
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.

## 7Accuracy

g02bmf 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. g02bmf treats all values in the inclusive range $\left(1±{0.1}^{\left({\mathbf{x02bef}}-2\right)}\right)×{xm}_{j}$, 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.

## 8Parallelism and Performance

g02bmf is not threaded in any implementation.

The time taken by g02bmf depends on $n$ and $p$, and the occurrence of missing values.
The routine uses a two-pass algorithm.

## 10Example

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 about zero, and correlation-like 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 variable, and cases $3$ and $4$ for the fourth and second variables, etc.

### 10.1Program Text

Program Text (g02bmfe.f90)

### 10.2Program Data

Program Data (g02bmfe.d)

### 10.3Program Results

Program Results (g02bmfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017