congruence {rrcov3way}

Coefficient of factor congruence (phi)

Description

The function congruence(x, y) computes the Tucker's congruence (phi) coefficients among two sets of factors.

Usage

congruence(x, y = NULL)

Arguments

x

A vector or matrix of factor loadings.

y

A vector or matrix of factor loadings (may be NULL).

Details

Find the Tucker's coefficient of congruence between two sets of factor loadings. Factor congruences are the cosines of pairs of vectors defined by the loadings matrix and based at the origin. Thus, for loadings that differ only by a scaler (e.g. the size of the eigen value), the factor congruences will be 1.

For factor loading vectors of X and Y the measure of factor congruence, phi, is

\phi = \frac{\sum X Y}{\sqrt{\sum(X^2)\sum(Y^2)}} .

If y=NULL and x is a numeric matrix, the congruence coefficients between the columns of the matrix x are returned. The result is a symmetric matrix with ones on the diagonal. If two matrices are provided, they must have the same size and the result is a square matrix containing the congruence coefficients between all pairs of columns of the two matrices.

Value

A matrix of factor congruences.

Author(s)

Valentin Todorov, [e-mail]

References

L.R Tucker (1951). A method for synthesis of factor analysis studies. Personnel Research Section Report No. 984. Department of the Army, Washington, DC.

Examples

 require(rrcov)

 data(delivery, package="robustbase")
 X <- getLoadings(PcaClassic(delivery))
 Y <- getLoadings(PcaHubert(delivery, k=3))
 round(congruence(X,Y),3)