g02bc 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.

Syntax

C#
public static void g02bc( int n, int m, double[,] x, int[] miss, double[] xmiss, double[] xbar, double[] std, double[,] ssp, double[,] r, out int ncases, double[,] cnt, out int ifail )

Visual Basic
Public Shared Sub g02bc ( _ n As Integer, _ m As Integer, _ x As Double(,), _ miss As Integer(), _ xmiss As Double(), _ xbar As Double(), _ std As Double(), _ ssp As Double(,), _ r As Double(,), _ <OutAttribute> ByRef ncases As Integer, _ cnt As Double(,), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub g02bc ( _
	n As Integer, _
	m As Integer, _
	x As Double(,), _
	miss As Integer(), _
	xmiss As Double(), _
	xbar As Double(), _
	std As Double(), _
	ssp As Double(,), _
	r As Double(,), _
	<OutAttribute> ByRef ncases As Integer, _
	cnt As Double(,), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void g02bc( int n, int m, array<double,2>^ x, array<int>^ miss, array<double>^ xmiss, array<double>^ xbar, array<double>^ std, array<double,2>^ ssp, array<double,2>^ r, [OutAttribute] int% ncases, array<double,2>^ cnt, [OutAttribute] int% ifail )

Visual C++

public:
static void g02bc(
	int n, 
	int m, 
	array<double,2>^ x, 
	array<int>^ miss, 
	array<double>^ xmiss, 
	array<double>^ xbar, 
	array<double>^ std, 
	array<double,2>^ ssp, 
	array<double,2>^ r, 
	[OutAttribute] int% ncases, 
	array<double,2>^ cnt, 
	[OutAttribute] int% ifail
)

F#
static member g02bc : n : int * m : int * x : float[,] * miss : int[] * xmiss : float[] * xbar : float[] * std : float[] * ssp : float[,] * r : float[,] * ncases : int byref * cnt : float[,] * ifail : int byref -> unit

static member g02bc : 
        n : int * 
        m : int * 
        x : float[,] * 
        miss : int[] * 
        xmiss : float[] * 
        xbar : float[] * 
        std : float[] * 
        ssp : float[,] * 
        r : float[,] * 
        ncases : int byref * 
        cnt : float[,] * 
        ifail : int byref -> unit

Parameters

n: Type: System..::..Int32
On entry: $n$ , the number of observations or cases.

Constraint: $n \geq 2$ .

m: Type: System..::..Int32
On entry: $m$ , the number of variables.

Constraint: $m \geq 2$ .

x: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, m]
Note: dim1 must satisfy the constraint: $dim1 \geq n$
On entry: $x [i - 1, j - 1]$ 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 $j = 1, 2, \dots, m$ .

miss: Type: array<System..::..Int32>[]()[][]
An array of size [m]
On entry: $miss [j - 1]$ 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.

xmiss: Type: array<System..::..Double>[]()[][]
An array of size [m]
On entry: $xmiss [j - 1]$ 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 [Accuracy]).

xbar: Type: array<System..::..Double>[]()[][]
An array of size [m]
On exit: the mean value, ${\bar{x}}_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .

std: Type: array<System..::..Double>[]()[][]
An array of size [m]
On exit: the standard deviation, $s_{j}$ , of the $j$ th variable, for $j = 1, 2, \dots, m$ .

ssp: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, m]
Note: dim1 must satisfy the constraint: $dim1 \geq m$
On exit: $ssp [j - 1, k - 1]$ is the cross-product of deviations $S_{j k}$ , for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .

r: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, m]
Note: dim1 must satisfy the constraint: $dim1 \geq m$
On exit: $r [j - 1, k - 1]$ is the product-moment correlation coefficient $R_{j k}$ between the $j$ th and $k$ th variables, for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .

ncases: Type: System..::..Int32%
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).

cnt: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, m]
Note: dim1 must satisfy the constraint: $dim1 \geq m$
On exit: $cnt [j - 1, k - 1]$ is the number of cases, $c_{j k}$ , actually used in the calculation of $S_{j k}$ , and $R_{j k}$ , the sum of cross-products and correlation coefficient for the $j$ th and $k$ th variables, for $j = 1, 2, \dots, m$ and $k = 1, 2, \dots, m$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

The input data consist 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. 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.

Let

w_{i j} = 0

if the

i

th observation for the

j

th variable is a missing value, i.e., if a missing value,

{xm}_{j}

, has been declared for the

j

th variable, and

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

(see also [Accuracy]); and

w_{i j} = 1

otherwise, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

The quantities calculated are:

(a)

Means:

{\bar{x}}_{j} = \frac{\sum_{i = 1}^{n} w_{i j} x_{i j}}{\sum_{i = 1}^{n} w_{i j}}, j = 1, 2, \dots, m .

(b)

Standard deviations:

s_{j} = \sqrt{\frac{\sum_{i = 1}^{n} w_{i j} {(x_{i j} - {\bar{x}}_{j})}^{2}}{(\sum_{i = 1}^{n} w_{i j}) - 1}}, j = 1, 2, \dots, m .

(c)

Sums of squares and cross-products of deviations from means:

S_{j k} = \sum_{i = 1}^{n} w_{i j} w_{i k} (x_{i j} - {\bar{x}}_{j (k)}) (x_{i k} - {\bar{x}}_{k (j)}), j, k = 1, 2, \dots, m,

where

{\bar{x}}_{j (k)} = \frac{\sum_{i = 1}^{n} w_{i j} w_{i k} x_{i j}}{\sum_{i = 1}^{n} w_{i j} w_{i k}} and {\bar{x}}_{k (j)} = \frac{\sum_{i = 1}^{n} w_{i k} w_{i j} x_{i k}}{\sum_{i = 1}^{n} w_{i k} w_{i j}},

(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.)

(d)

Pearson product-moment correlation coefficients:

R_{j k} = \frac{S_{j k}}{\sqrt{S_{j j (k)} S_{k k (j)}}}, j, k, = 1, 2, \dots, m,

where

S_{j j (k)} = \sum_{i = 1}^{n} w_{i j} w_{i k} {(x_{i j} - {\bar{x}}_{j (k)})}^{2}

and

S_{k k (j)} = \sum_{i = 1}^{n} w_{i k} w_{i j} {(x_{i k} - {\bar{x}}_{k (j)})}^{2}

and

{\bar{x}}_{j (k)}

and

{\bar{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).

S_{j j (k)}

S_{k k (j)}

is zero,

R_{j k}

is set to zero.

(e)

The number of cases used in the calculation of each of the correlation coefficients:

c_{j k} = \sum_{i = 1}^{n} w_{i j} w_{i k}, j, k = 1, 2, \dots, m .

(The diagonal terms,

c_{j j}

, for

j = 1, 2, \dots, m

, also give the number of cases used in the calculation of the means,

{\bar{x}}_{j}

, and the standard deviations,

s_{j}

References

None.

Error Indicators and Warnings

Note: g02bc may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the method:

Some error messages may refer to parameters that are dropped from this interface (LDX, LDSSP, LDR, LDCNT) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.

$ifail = 1$

On entry,

n < 2

$ifail = 2$

On entry,

m < 2

$ifail = 4$: 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 method even if $ifail = 4$ .

$ifail = -9000$: An error occured, see message report.
$ifail = -6000$: Invalid Parameters $〈value〉$
$ifail = -4000$: Invalid dimension for array $〈value〉$
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

Accuracy

g02bc 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. g02bc treats all values in the inclusive range

(1 \pm {0.1}^{(x02be - 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.

Parallelism and Performance

None.

Further Comments

The time taken by g02bc depends on

n

and

m

, and the occurrence of missing values.

The method uses a two-pass algorithm.

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