g02by computes a partial correlation/variance-covariance matrix from a correlation or variance-covariance matrix computed by g02bx.

Syntax

C#
public static void g02by( int m, int ny, int nx, int[] isz, double[,] r, double[,] p, out int ifail )

Visual Basic
Public Shared Sub g02by ( _ m As Integer, _ ny As Integer, _ nx As Integer, _ isz As Integer(), _ r As Double(,), _ p As Double(,), _ <OutAttribute> ByRef ifail As Integer _ )

Visual C++
public: static void g02by( int m, int ny, int nx, array<int>^ isz, array<double,2>^ r, array<double,2>^ p, [OutAttribute] int% ifail )

F#
static member g02by : m : int * ny : int * nx : int * isz : int[] * r : float[,] * p : float[,] * ifail : int byref -> unit

Parameters

m: Type: System..::..Int32
On entry: the number of variables in the variance-covariance/correlation matrix given in r.

Constraint: $m \geq 3$ .

ny: Type: System..::..Int32
On entry: the number of $Y$ variables, $n_{y}$ , for which partial correlation coefficients are to be computed.

Constraint: $ny \geq 2$ .

nx

Type: System..::..Int32

On entry: the number of

X

variables,

n_{x}

, which are to be considered as fixed.

Constraints:

$nx \geq 1$ ;
$ny + nx \leq m$ .

isz

Type: array<System..::..Int32>[]()[][]

An array of size [m]

On entry: indicates which variables belong to set

X

and

Y

$isz [i - 1] < 0$: The $i$ th variable is a $Y$ variable, for $i = 1, 2, \dots, m$ .
$isz [i - 1] > 0$: The $i$ th variable is a $X$ variable.
$isz [i - 1] = 0$: The $i$ th variable is not included in the computations.

Constraints:

exactly ny elements of isz must be $< 0$ ;
exactly nx elements of isz must be $> 0$ .

r: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, m]
Note: dim1 must satisfy the constraint: $dim1 \geq m$
On entry: the variance-covariance or correlation matrix for the m variables as given by g02bx. Only the upper triangle need be given.
Note: the matrix must be a full rank variance-covariance or correlation matrix and so be positive definite. This condition is not directly checked by the method.

p: Type: array<System..::..Double,2>[,](,)[,][,]
An array of size [dim1, ny]
Note: dim1 must satisfy the constraint: $dim1 \geq ny$
On exit: the strict upper triangle of p contains the strict upper triangular part of the $n_{y}$ by $n_{y}$ partial correlation matrix. The lower triangle contains the lower triangle of the $n_{y}$ by $n_{y}$ partial variance-covariance matrix if the matrix given in r is a variance-covariance matrix. If the matrix given in r is a partial correlation matrix then the variance-covariance matrix is for standardized variables.

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

Partial correlation can be used to explore the association between pairs of random variables in the presence of other variables. For three variables,

y_{1}

y_{2}

and

x_{3}

, the partial correlation coefficient between

y_{1}

and

y_{2}

given

x_{3}

is computed as:

\frac{r_{12} - r_{13} r_{23}}{\sqrt{(1 - r_{13}^{2}) (1 - r_{23}^{2})}},

where

r_{i j}

is the product-moment correlation coefficient between variables with subscripts

i

and

j

. The partial correlation coefficient is a measure of the linear association between

y_{1}

and

y_{2}

having eliminated the effect due to both

y_{1}

and

y_{2}

being linearly associated with

x_{3}

. That is, it is a measure of association between

y_{1}

and

y_{2}

conditional upon fixed values of

x_{3}

. Like the full correlation coefficients the partial correlation coefficient takes a value in the range (

- 1, 1

) with the value

0

indicating no association.

In general, let a set of variables be partitioned into two groups

Y

and

X

with

n_{y}

variables in

Y

and

n_{x}

variables in

X

and let the variance-covariance matrix of all

n_{y} + n_{x}

variables be partitioned into,

[\begin{array}{l} Σ_{x x} & Σ_{x y} \\ Σ_{y x} & Σ_{y y} \end{array}] .

The variance-covariance of

Y

conditional on fixed values of the

X

variables is given by:

Σ_{y ∣ x} = Σ_{y y} - Σ_{y x} Σ_{x x}^{- 1} Σ_{x y} .

The partial correlation matrix is then computed by standardizing

Σ_{y ∣ x}

diag {(Σ_{y ∣ x})}^{- \frac{1}{2}} Σ_{y ∣ x} diag {(Σ_{y ∣ x})}^{- \frac{1}{2}} .

To test the hypothesis that a partial correlation is zero under the assumption that the data has an approximately Normal distribution a test similar to the test for the full correlation coefficient can be used. If

r

is the computed partial correlation coefficient then the appropriate

t

statistic is

r \sqrt{\frac{n - n_{x} - 2}{1 - r^{2}}},

which has approximately a Student's

t

-distribution with

n - n_{x} - 2

degrees of freedom, where

n

is the number of observations from which the full correlation coefficients were computed.

References

Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press

Morrison D F (1967) Multivariate Statistical Methods McGraw–Hill

Osborn J F (1979) Statistical Exercises in Medical Research Blackwell

Snedecor G W and Cochran W G (1967) Statistical Methods Iowa State University Press

Error Indicators and Warnings

Errors or warnings detected by the method:

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

$ifail = 1$

On entry,	$m < 3$ ,
or	$ny < 2$ ,
or	$nx < 1$ ,
or	$ny + nx > m$ ,

$ifail = 2$

On entry,	there are not exactly ny elements of $isz < 0$ ,
or	there are not exactly nx elements of $isz > 0$ .

$ifail = 3$: On entry, the variance-covariance/correlation matrix of the $X$ variables, $Σ_{x x}$ , is not of full rank. Try removing some of the $X$ variables by setting the appropriate element of $isz = 0$ .

$ifail = 4$: Either a diagonal element of the partial variance-covariance matrix, $Σ_{y ∣ x}$ , is zero and/or a computed partial correlation coefficient is greater than one. Both indicate that the matrix input in r was not positive definite.

$ifail = -9000$: 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

g02by computes the partial variance-covariance matrix,

Σ_{y ∣ x}

, by computing the Cholesky factorization of

Σ_{x x}

. If

Σ_{x x}

is not of full rank the computation will fail. For a statement on the accuracy of the Cholesky factorization see (F07GDF not in this release).

Parallelism and Performance

None.

Further Comments

Models that represent the linear associations given by partial correlations can be fitted using the multiple regression method g02da.

Example

Example program (C#): g02bye.cs

Example program data: g02bye.d

Example program results: g02bye.r