Congruence

The congruence between two configurations of points quantifies their similarity. The configurations to be compared are, in general, produced by factor analytic methods that decompose an “observations by variables” data matrix and produce one set of factor scores for the observations and one set of factor scores (i.e., the loadings) for the variables. The congruence between two sets of factor scores collected on the same units (which can be observations or variables) measures the similarity between these two sets of scores. If, for example, two different types of factor analysis are performed on the same data set, the congruence between the two solutions is evaluated by the similarity of the configurations of the factor scores produced by these two techniques.

This entry presents three coefficients used to evaluate congruence. The first coefficient is called the coefficient of congruence. It measures the similarity of two configurations by computing a cosine between matrices of factor scores. The second and third coefficients are the RVcoefficient and the Mantel coefficient. These two coefficients evaluate the similarity of the whole configuration of units. In order to do so, the factor scores of the units are first transformed into a units-by-units square matrix, which reflects the configuration of similarity between the units; and then the similarity between the configurations is measured by a coefficient. For the RV coefficient, the configuration between the units is obtained by computing a matrix of scalar products between the units, and a cosine between two scalar product matrices evaluates the similarity between two configurations. For the Mantel coefficient, the configuration between the units is obtained by computing a matrix of distance between the units, and a coefficient of correlation between two distance matrices evaluates the similarity between two configurations.

The congruence coefficient was first defined by C. Burt under the name unadjusted correlation as [Page 223]a measure of the similarity of two factorial configurations. The name congruence coefficient was later tailored by Ledyard R. Tucker. The congruence coefficient is also sometimes called a monotonicity coefficient.

The RV coefficient was introduced by Yves Escoufier as a measure of similarity between squared symmetric matrices (specifically: positive semidefinite matrices) and as a theoretical tool to analyze multivariate techniques. The RV coefficient is used in several statistical techniques, such as statis and distatis. In order to compare rectangular matrices with the RV or the Mantel coefficients, the first step is to transform these rectangular matrices into square matrices.

The Mantel coefficient was originally introduced by Nathan Mantel in epidemiology but it is now widely used in ecology.

The congruence and the Mantel coefficients are cosines (recall that the coefficient of correlation is a centered cosine), and as such, they take values between − 1 and +1. The RV coefficient is also a cosine, but because it is a cosine between two matrices of scalar products (which, technically speaking, are positive semidefinite matrices), it corresponds actually to a squared cosine, and therefore the RV coefficient takes values between 0 and 1.

The computational formulas of these three coefficients are almost identical, but their usage and theoretical foundations differ because these coefficients are applied to different types of matrices. Also, their sampling distributions differ because of the types of matrices on which they are applied.

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