
The congruence coefficient is a measure of similarity between two Configurations.
The congruence coefficient c(X, Y) for the configurations X and Y is defined as:
c(X, Y) = ∑_{i<j} w_{ij} d_{ij}(X) d_{ij}(Y) / ([∑_{i<j} w_{ij} d^{2}_{ij}(X)]^{1/2} [∑_{i<j} w_{ij} d^{2}_{ij}(Y)]^{1/2}), 
where d_{ij}(X) is the distance between the points i and j in configuration X and w_{ij} are nonnegative weights (default: w_{ij} = 1).
Since distances are nonnegative, the congruence coefficient has a value between 0 and 1.
The congruence coefficient is a better measure of the similarity between configurations than the correlation coefficient of the distances. Borg & Groenen (1997) give a simple example where things go wrong with correlation coefficients: two configurations X and Y with three points each, have distances d_{12}(X) = 1, d_{13}(X) = 2, d_{23}(X) = 3 and d_{12}(Y) = 2, d_{13}(Y) = 3, d_{23}(Y) = 4. These distances have a correlation coefficient of 1. However, in X the three points lie on a straight line and in Y the points form a triangle. This unwanted situation occurs because in the calculation of the correlation coefficient the mean is subtracted from the distances and the resulting values are no longer distances (they may become negative). In calculating the correlation between the distances we should not subtract the mean. In fact, the congruence coefficient is exactly this correlation coefficient calculated with respect to the origin and not with respect to the centroid position (the "mean").
For further information on how well one number can assess the similarity between two configurations see Borg & Groenen (1997) section 19.7.
© djmw, April 7, 2004