g02by: FL CL CPP AD PY MB

NAG CL Interface
g02byc (corrmat_partial)

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NAG Library Manual, Mark 30.1

Interfaces: FL CL CPP AD PY MB

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G02 (Correg) Chapter Introduction

g02by: FL CL CPP AD PY MB

▸▿ 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

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1 Purpose

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

2 Specification

#include <nag.h>

void	g02byc (Integer m, Integer ny, Integer nx, const Integer sz[], const double r[], Integer tdr, double p[], Integer tdp, NagError *fail)

The function may be called by the names: g02byc, nag_correg_corrmat_partial or nag_partial_corr.

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: $m$ – Integer Input

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

Constraint:

m \geq 3

2: $ny$ – Integer Input

On entry: the number of

Y

variables,

n_{y}

, for which partial correlation coefficients are to be computed.

Constraint:

ny \geq 2

3: $nx$ – Integer Input

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 Integer Input

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 double Input

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 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: $tdr$ – Integer Input

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

Constraint:

tdr \geq m

7: $p [ny \times tdp]$ – double Output

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

8: $tdp$ – Integer Input

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 7 in the Introduction to the NAG Library CL Interface).

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 $1$ . 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

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

Background information to multithreading can be found in the Multithreading documentation.

g02byc is not threaded in any implementation.

9 Further Comments

Models that represent the linear associations given by partial correlations can be fitted using the multiple regression function 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 g02bxc and the partial correlation between deaths and smoke given sulphur dioxide is computed using g02byc.

g02by: FL CL CPP AD PY MB

NAG CL Interfaceg02byc (corrmat_​partial)

▸▿ 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 CL Interface
g02byc (corrmat_partial)