Congruences are the cosines of pairs of vectors defined by a matrix and based at the origin. Thus, for values that differ only by a scaler the congruence will be 1.
For two matrices, F1 and F2, the measure of congruence, phi, is
$$
\phi = \frac{\sum F_1 F_2}{\sqrt{\sum(F_1^2)\sum(F_2^2)}}
.$$
It is an interesting exercise to compare congruences with the correlations of the respective values. Congruences are based upon the raw cross products, while correlations are based upon centered cross products. That is, correlations of items are cosines of the vectors based at the mean loading for each column.
$$
\phi = \frac{\sum (F_1-a) (F_2 - b)}{\sqrt{\sum((F_1-a)^2)\sum((F_2-b)^2)}}
.$$.
For congruence coefficients, a = b= 0. For correlations a=mean F1, b= mean F2.
Input may either be one or two matrices.
For congruences of factor or component loading matrices, use factor.congruence.
Normally, all factor loading matrices should be complete (have no missing loadings). In the case where some loadings are missing, if the use option is specified, then variables with missing loadings are dropped.
If the data are zero centered, this is the correlation, if the data are centered around the scale midpoint (M), this is Cohen's Similarity coefficient. See examples. If M is not specified, it is found as the midpoint of the items in x and y.
cohen.profile applies the congruence function to data centered around M. M may be specified, or found from the data. The last example is taken from Cohen (1969).
distance finds the generalized distance as a function of r. City block (r=1), Euclidean (r=2) or weighted towards maximimum (r >2).