naginterfaces.library.correg.corrmat_​partial

naginterfaces.library.correg.corrmat_partial(ny, nx, isz, r)

corrmat_partial computes a partial correlation/variance-covariance matrix from a correlation or variance-covariance matrix computed by corrmat().

For full information please refer to the NAG Library document for g02by

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g02/g02byf.html

Parameters

nyint

The number of $Y$ variables, $n_y$, for which partial correlation coefficients are to be computed.

nxint

The number of $X$ variables, $n_x$, which are to be considered as fixed.

iszint, array-like, shape $\left(m\right)$

Indicates which variables belong to set $X$ and $Y$.

$isz[i-1]<0$

The $\text{i}$th variable is a $Y$ variable, for $\text{i}=1,2,\dots ,m$.

$isz[i-1]>0$

The $i$th variable is a $X$ variable.

$isz[i-1]=0$

The $i$th variable is not included in the computations.

rfloat, array-like, shape $(m,m)$

The variance-covariance or correlation matrix for the $\text{m}$ variables as given by corrmat(). 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.

Returns

pfloat, ndarray, shape $(ny,ny)$

The strict upper triangle of $p$ contains the strict upper triangular part of the $n_y\times n_y$ partial correlation matrix. The lower triangle contains the lower triangle of the $n_y\times n_y$ 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.

Raises

NagValueError

(errno $1$)

On entry, $ny=⟨value⟩$, $nx=⟨value⟩$ and $m=⟨value⟩$.

Constraint: $ny+nx\le m$.

(errno $1$)

On entry, $nx=⟨value⟩$.

Constraint: $nx\ge 1$.

(errno $1$)

On entry, $ny=⟨value⟩$.

Constraint: $ny>1$.

(errno $1$)

On entry, $m=⟨value⟩$.

Constraint: $m>2$.

(errno $2$)

On entry, $⟨value⟩$ values of $isz$ are greater than zero and $nx=⟨value⟩$.

Constraint: exactly $nx$ values of $isz$ must be greater than zero.

(errno $2$)

On entry, $⟨value⟩$ values of $isz$ are less than zero and $ny=⟨value⟩$.

Constraint: exactly $ny$ values of $isz$ must be less than zero.

(errno $3$)

On entry, the square root of the correlation matrix of the independent variables is singular.

Try removing some of the $X$ variables by setting the appropriate element of $isz=0$.

(errno $3$)

On entry, the correlation matrix of the independent variables is singular.

Try removing some of the $X$ variables by setting the appropriate element of $isz=0$.

(errno $4$)

On entry, an element of the partial correlation matrix is greater than $1$.

Constraint: $r$ must be positive definite.

(errno $4$)

On entry, a diagonal element of the partial covariance matrix is zero.

Constraint: $r$ must be positive definite.

(errno $4$)

On entry, a diagonal element of the partial covariance matrix is zero and an element of the partial correlation matrix is greater than $1$.

Constraint: $r$ must be positive definite.

Notes

In the NAG Library the traditional C interface for this routine uses a different algorithmic base. Please contact NAG if you have any questions about compatibility.

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:

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

where $r_{ij}$ 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,

$$\begin{array}{c}\left[\begin{array}{cc}{\Sigma }_{xx}& {\Sigma }_{xy}\\ {\Sigma }_{yx}& {\Sigma }_{yy}\end{array}\right]\text{.}\end{array}$$

The variance-covariance of $Y$ conditional on fixed values of the $X$ variables is given by:

$${\Sigma }_{y|x}={\Sigma }_{yy}-{\Sigma }_{yx}{\Sigma }_{xx}^{-1}{\Sigma }_{xy}\text{.}$$

The partial correlation matrix is then computed by standardizing ${\Sigma }_{y|x}$,

$$diag\left({\left({\Sigma }_{y|x}\right)}^{-\frac{1}{2}}\right){\Sigma }_{y|x}diag\left({\left({\Sigma }_{y|x}\right)}^{-\frac{1}{2}}\right)\text{.}$$

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

$$r\sqrt{\frac{n-n_x-2}{1-r^2}}\text{,}$$

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