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 and 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. It is the unstandardized version of the Pearson correlation and takes the difference in loadings into account as well as consistency of ranking Barrett does not recommend its use although he concedes that it has probably given good results in the past (further details are in the pdf file here.)
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 here. There is a help guide in the download. This software needs at least two factors to compute similarity coefficients including congruences. According to Barrett (1986) congruences of at least 0.80, preferably over 0.90, are indicative of 'useful conceptual similarity' between loadings.
References
Barrett, P. (1986). Factor comparison: An examination of three methods. Personality and Individual Differences, 7(3), 327-340.
Wrigley, C & O, Neuhaus, J (1955). The matching of two sets of factors. American Psychologist, 10, 418-419.