Computes partial correlation/variance-covariance matrix from correlation/variance-covariance matrix computed by g02bxc (nag_corr_cov)

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NAG[g02byc] NAG[nag_partial_corr] - Computes partial correlation/variance-covariance matrix from correlation/variance-covariance matrix computed by g02bxc (nag_corr_cov)

Calling Sequence

g02byc(nx, sz, r, p, 'm'=m, 'ny'=ny, 'tdr'=tdr, 'tdp'=tdp, 'fail'=fail)

nag_partial_corr(. . .)

Parameters

nx - integer;

On entry: the number of variables, , which are to be considered as fixed.

Constraints:

;

sz - Vector(1..m, datatype=integer[kernelopts('wordsize')/8]);

On entry: indicates which variables belong to set and .

sz The th variable is a variable, for .

sz The th variable is a variable.

sz The th variable is not included in the computations.

Constraints:

exactly ny elements of sz must be ,

exactly nx elements of sz must be .

r - Matrix(1..m, 1..tdr, datatype=float[8], order=C_order);

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

p - Matrix(1..ny, 1..tdp, datatype=float[8], order=C_order);

On exit: the strict upper triangle of p contains the strict upper triangular part of the by partial correlation matrix. The lower triangle contains the lower triangle of the by 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.

'm'=m - integer; (optional)

Default value: the first dimension of the arrays sz, r and the second dimension of the arrays sz, rthe array r.

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

Constraint: . .

'ny'=ny - integer; (optional)

Default value: the dimension of the array p.

On entry: the number of variables, , for which partial correlation coefficients are to be computed.

Constraint: . .

'tdr'=tdr - integer; (optional)

On entry: the second dimension of the array r as declared in the function from which nag_partial_corr (g02byc) is called.

Constraint: . .

'tdp'=tdp - integer; (optional)

On entry: the second dimension of the array p as declared in the function from which nag_partial_corr (g02byc) is called.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

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

Description

Partial correlation can be used to explore the association between pairs of random variables in the presence of other variables. For three variables, , and the partial correlation coefficient between and given is computed as:

(r[12]-r[13]r[23])/(sqrt((1,-,r[13]^2)(1,-,r[23]^2)))

where is the product-moment correlation coefficient between variables with subscripts and . The partial correlation coefficient is a measure of the linear association between and having eliminated the effect due to both and being linearly associated with . That is, it is a measure of association between and conditional upon fixed values of . Like the full correlation coefficients the partial correlation coefficient takes a value in the range with the value 0 indicating no association.

In general, let a set of variables be partitioned into two groups and with variables in and variables in and let the variance-covariance matrix of all variables be partitioned into,

$Matrix(2, 2, {(1, 1) = Sigma[x^2], (1, 2) = Sigma[x*y], (2, 1) = Sigma[x*y], (2, 2) = Sigma[y^2]})$

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

The partial correlation matrix is then computed by standardizing ,

(diag(Sigma[y|x]))^(-1/2)Sigma[y|x](diag(Sigma[y|x]))^(-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 is the computed partial correlation coefficient then the appropriate statistic is

r*sqrt((n-n[x]-2)/(1-r^2))

which has approximately a Student's -distribution with degrees of freedom, where is the number of observations from which the full correlation coefficients were computed.

Error Indicators and Warnings

"NE_3_INT_ARG_CONS"

On entry, , and . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_NX_SET"

On entry, and there are not exactly nx values of .

"NE_BAD_NY_SET"

On entry, and there are not exactly ny values of . Number of values of .

"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 must not be less than 3: .

"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 consult NAG for assistance.

	Accuracy
	nag_partial_corr (g02byc) computes the partial variance-covariance matrix, , by computing the Cholesky factorization of . If is not of full rank the computation will fail.

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

Examples

m := 3:
ny := 2:
nx := 1:
tdr := 3:
tdp := 3:
sz := Vector([-1, -1, 1], datatype=integer[kernelopts('wordsize')/8]):
r := Matrix([[17053.69523809524, 152.8584285714286, 45.94742857142857], [152.8584285714286, 2.397368571428571, 0.6475057142857145], [45.94742857142857, 0.6475057142857145, 0.1793171428571429]], datatype=float[8], order='C_order'):
p := Matrix(2, 3, datatype=float[8], order='C_order'):
NAG:-g02byc(nx, sz, r, p, 'm' = m, 'ny' = ny, 'tdr' = tdr, 'tdp' = tdp):

	See Also
	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 g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.