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

Syntax

[p, ifail] = g02by(ny, nx, isz, r, 'm', m)

[p, ifail] = nag_correg_corrmat_partial(ny, nx, isz, r, 'm', m)

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

Parameters

Compulsory Input Parameters

1: $ny$ – int64int32nag_int scalar

The number of

Y

variables,

n_{y}

, for which partial correlation coefficients are to be computed.

Constraint:

ny \geq 2

2: $nx$ – int64int32nag_int scalar

The number of

X

variables,

n_{x}

, which are to be considered as fixed.

Constraints:

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

3: $isz (m)$ – int64int32nag_int array

Indicates which variables belong to set

X

and

Y

$isz (i) < 0$: The $i$ th variable is a $Y$ variable, for $i = 1, 2, \dots, m$ .
$isz (i) > 0$: The $i$ th variable is a $X$ variable.
$isz (i) = 0$: 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 $> 0$ .

4: $r (ldr, m)$ – double array

ldr, the first dimension of the array, must satisfy the constraint

ldr \geq m

The variance-covariance or correlation matrix for the m variables as given by nag_correg_corrmat (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 function.

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the dimension of the array isz and the first dimension of the array r and the second dimension of the array r. (An error is raised if these dimensions are not equal.)
The number of variables in the variance-covariance/correlation matrix given in r.

Constraint: $m \geq 3$ .

Output Parameters

1: $p (ldp, ny)$ – double array: 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.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

$ifail = 1$

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

$ifail = 2$

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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

nag_correg_corrmat_partial (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 nag_lapack_dpptrf (f07gd).

Further Comments

Models that represent the linear associations given by partial correlations can be fitted using the multiple regression function nag_correg_linregm_fit (g02da).

Example

Data, given by Osborn (1979), on the number of deaths, smoke (

mg / m^{3}

) and sulphur dioxide (parts/million) during an intense period of fog is input. The correlations are computed using nag_correg_corrmat (g02bx) and the partial correlation between deaths and smoke given sulphur dioxide is computed using nag_correg_corrmat_partial (g02by). Both correlation matrices are printed using the function nag_file_print_matrix_real_gen (x04ca).

Open in the MATLAB editor: g02by_example

function g02by_example


fprintf('g02by example results\n\n');

x = [ 112 0.30 0.09;
      140 0.49 0.16;
      143 0.61 0.22;
      120 0.49 0.14;
      196 2.64 0.75;
      294 3.45 0.86;
      513 4.46 1.34;
      518 4.46 1.34;
      430 1.22 0.47;
      274 1.22 0.47;
      255 0.32 0.22;
      236 0.29 0.23;
      256 0.50 0.26;
      222 0.32 0.16;
      213 0.32 0.16];

% Calculate correlation matrix
[xbar, std, v, r, ifail] = g02bx(x);

% Calculate partial correlation matrix
nx = int64(1);
ny = int64(2);
isz = [int64(-1); -1; 1];

[p, ifail] = g02by(ny, nx, isz, r);

mtitle = 'Correlation matrix:';
matrix = 'Upper';
diag   = 'Non-unit';

[ifail] = x04ca( ...
                 matrix, diag, r, mtitle);

fprintf('\n');
mtitle = 'Partial Correlation matrix:';
diag   = 'Unit';

[ifail] = x04ca( ...
                 matrix, diag, p, mtitle);

g02by example results

 Correlation matrix:
          1       2       3
 1   1.0000  0.7560  0.8309
 2           1.0000  0.9876
 3                   1.0000

 Partial Correlation matrix:
          1       2
 1   1.0000 -0.7381
 2           1.0000

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox