NAG Library Function Document

nag_partial_corr (g02byc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_partial_corr (g02byc) computes a partial correlation/variance-covariance matrix from a correlation or variance-covariance matrix computed by nag_corr_cov (g02bxc).

2 Specification

#include <nag.h>

#include <nagg02.h>

void	nag_partial_corr (Integer m, Integer ny, Integer nx, const Integer sz[], const double r[], Integer tdr, double p[], Integer tdp, NagError *fail)

3 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{matrix} Σ_{x x} & Σ_{x y} \\ Σ_{y x} & Σ_{y y} \end{matrix}]

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.

4 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

5 Arguments

1: m – IntegerInput

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

Constraint:

m \geq 3

2: ny – IntegerInput

On entry: the number of

Y

variables,

n_{y}

, for which partial correlation coefficients are to be computed.

Constraint:

ny \geq 2

3: nx – IntegerInput

On entry: the number of

X

variables,

n_{x}

, which are to be considered as fixed.

Constraints:

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

4: sz[m] – const IntegerInput

On entry: indicates which variables belong to set

X

and

Y

sz $(i) < 0$: The $i$ th variable is a $Y$ variable, for $i = 1, 2, \dots, m$ .
sz $(i) > 0$: The $i$ th variable is a $X$ variable.
sz $(i) = 0$: The $i$ th variable is not included in the computations.

Constraints:

exactly ny elements of sz must be $< 0$ ,
exactly nx elements of sz must be $> 0$ .

5: r[ $m \times tdr$ ] – const doubleInput

Note: the

(i, j)

th element of the matrix

R

is stored in

r [(i - 1) \times tdr + j - 1]

On entry: the variance-covariance or correlation matrix for the m variables as given by nag_corr_cov (g02bxc). Only the upper triangle need be given.

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.

6: tdr – IntegerInput

On entry: the stride separating matrix column elements in the array r.

Constraint:

tdr \geq m

7: p[ $ny \times tdp$ ] – doubleOutput

Note: the

(i, j)

th element of the matrix

P

is stored in

p [(i - 1) \times tdp + j - 1]

On exit: the strict upper triangle of p contains the strict upper triangular part of the

n_{y}

n_{y}

partial correlation matrix. The lower triangle contains the lower triangle of the

n_{y}

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 correlation matrix then the variance-covariance matrix is for standardized variables.

8: tdp – IntegerInput

On entry: the stride separating matrix column elements in the array p.

Constraint:

tdp \geq ny

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tdp = ⟨value⟩$ while $ny = ⟨value⟩$ . These arguments must satisfy $tdp \geq ny$ .
On entry, $tdr = ⟨value⟩$ while $m = ⟨value⟩$ . These arguments must satisfy $tdr \geq m$ .
NE_3_INT_ARG_CONS: On entry, $ny = ⟨value⟩$ , $nx = ⟨value⟩$ and $m = ⟨value⟩$ . These arguments must satisfy $ny + nx \leq m$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_NX_SET: On entry, $nx = ⟨value⟩$ and there are not exactly nx values of $sz < 0$ .
NE_BAD_NY_SET: On entry, $ny = ⟨value⟩$ and there are not exactly ny values of $sz < 0$ .
Number of values of $sz < 0 = ⟨value⟩$ .
NE_COR_MAT_POSDEF: Either a diagonal element of the partial variance-covariance matrix 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.
NE_COR_MAT_RANK: On entry, either the variance-covariance matrix or the correlation matrix is not of full rank. Try removing some of the nx variables by setting the appropriate elements of sz to zero.
NE_INT_ARG_LT: On entry, $m = ⟨value⟩$ .
Constraint: $m \geq 3$ .
On entry, $nx = ⟨value⟩$ .
Constraint: $nx \geq 1$ .
On entry, $ny = ⟨value⟩$ .
Constraint: $ny \geq 2$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

7 Accuracy

nag_partial_corr (g02byc) 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.

8 Parallelism and Performance

Not applicable.

9 Further Comments

Models that represent the linear associations given by partial correlations can be fitted using the multiple regression function nag_regsn_mult_linear (g02dac).

10 Example

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

(m g / m^{3})

and sulphur dioxide (parts/million) during an intense period of fog is input. The correlations are computed using nag_corr_cov (g02bxc) and the partial correlation between deaths and smoke given sulphur dioxide is computed using nag_partial_corr (g02byc).

NAG Library Function Documentnag_partial_corr (g02byc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_partial_corr (g02byc)