How do I compare a pair of factor loadings?

We are often interested in comparing pairs of factor loadings to assess versions of a test or performances of the same test on different samples such as Russian or British people.

The Coefficient of Congruence (Wrigley & Neuhaus, 1955) is used to measure the similarity between factor loadings and may be used in R 2.10 or above using the syntax below which compares two sets of loadings from a single factor.

install.packages("psych")
library(psych)
factor.congruence

# RUSSIAN VS VERSION 1 RBMT
# COEFFICIENT OF CONGRUENCE = 0.95

fa <- matrix(c(.307,.466,.479,.277,.797,.792,.234,.384,.438,.422,.274,.709,.707,.540),14,1)
fc <- matrix(c(.520,.330,.540,.100,.710,.710,.540,.560,.640,.360,.530,.460,.750,.740),14,1)
factor.congruence(fa,fc)

A free-standing program called ORTHOSIM2 created by Paul Barrett may be downloaded for free from [http://www.pbarrett.net/orthosim2.htm here.] There is a help guide in the download. This software needs at least two factors to compute similarity coefficients including congruences. According to sources quoted by Paul Barrett congruences of at least 0.80, preferably over 0.90, are indicative of a close association between loadings.

Reference

Wrigley, C & O, Neuhaus, J (1955). The matching of two sets of factors. American Psychologist, 10, 418-419.