RV and Congruence Coefficients

Notations and Computational Formulas

Let X be an I × J matrix and Y be an I × K matrix. The vec operation transforms a matrix into a vector whose entries are the elements of the matrix. The trace operation applies to square matrices and gives the sum of the diagonal elements.

The congruence coefficient is defined when both matrices have the same number of rows and columns (i.e., J = K). These matrices can represent factor loadings (i.e., factors by items) or factor projections (i.e., observations by factors). The congruence coefficient is denoted φ or sometimes rc, and it can be computed with three different equivalent formulas:

The RV coefficient was defined by Escoufier as a similarity coefficient between positive semi-definite matrices. Escoufier and Robert and Escoufier pointed out that the RV coefficient had important mathematical properties because it can be shown that most multivariate analysis techniques amount to maximizing this coefficient with suitable constraints. Recall, at this point, that a matrix S is called positive semi-definite when it can be obtained as the product of a matrix by its transpose. Formally, we say that S is positive semi-definite when there exists a matrix X such that

Note that as a consequence of the definition, positive semi-definite matrices are square and symmetric, and their diagonal elements are always larger or equal to zero.

[Page 850]

If we denote by S and T two positive semi-definite matrices of same dimensions, the RV coefficient between them is defined as

This formula is computationally equivalent to

For rectangular matrices, the first step is to transform the matrices into positive semi-definite matrices by multiplying each matrix by its transpose. So, in order to compute the value of the RV coefficient between the I × J matrix X and the I × K matrix Y, the first step is to compute