In
multivariate statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.
Multivariate statistics concerns understanding the different aims and background of each of the dif ...
, the congruence coefficient is an index of the similarity between factors that have been derived in a
factor analysis
Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observed ...
. It was introduced in 1948 by
Cyril Burt
Sir Cyril Lodowic Burt, FBA (3 March 1883 – 10 October 1971) was an English educational psychologist and geneticist who also made contributions to statistics. He is known for his studies on the heritability of IQ. Shortly after he died, his s ...
who referred to it as ''unadjusted correlation''. It is also called ''Tucker's congruence coefficient'' after
Ledyard Tucker
Ledyard R. Tucker (19 September 1910 – 16 August 2004) was an American mathematician who specialized in statistics and psychometrics. His Ph.D. advisor at the University of Chicago was Louis Leon Thurstone. He was a lecturer in psychology at Pr ...
who popularized the technique. Its values range between -1 and +1. It can be used to study the similarity of extracted factors across different samples of, for example, test takers who have taken the same test.
[Lorenzo-Seva, U. & ten Berge, J.M.F. (2006). Tucker’s Congruence Coefficient as a Meaningful Index of Factor Similarity. ''Methodology, 2,'' 57–64.][Jensen, A.R. (1998). ''The ''g'' factor: The science of mental ability''. Westport, CT: Praeger, pp. 99–100.][Abdi, H. (2007)]
RV Coefficient and Congruence Coefficient.
In Neil Salkind (Ed.), ''Encyclopedia of Measurement and Statistics.'' Thousand Oaks (CA): Sage.
Definition
Let ''X'' and ''Y'' be
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
s of factor loadings for two different samples. The formula for the congruence coefficient, or ''r
c'', is then
:
Interpretation
Generally, a congruence coefficient of 0.90 is interpreted as indicating a high degree of factor similarity, while a coefficient of 0.95 or higher indicates that the factors are virtually identical. Alternatively, a value in the range 0.85–0.94 has been seen as corresponding to a fair similarity, with values higher than 0.95 indicating that the factors can be considered to be equal.
The congruence coefficient can also be defined as the
cosine of the angle between factor axes based on the same set of variables (e.g., tests) obtained for two samples (see
Cosine similarity
In data analysis, cosine similarity is a measure of similarity between two sequences of numbers. For defining it, the sequences are viewed as vectors in an inner product space, and the cosine similarity is defined as the cosine of the angle betwe ...
). For example, with perfect congruence the angle between the factor axes is 0 degrees, and the cosine of 0 is 1.
Comparison with Pearson's ''r''
The congruence coefficient is preferred to
Pearson's ''r'' as a measure of factor similarity, because the latter may produce misleading results. The computation of the congruence coefficient is based on the deviations of factor loadings from zero, whereas ''r'' is based on the deviations from the mean of the factor loadings.
See also
*
RV coefficient In statistics, the RV coefficient
is a multivariate generalization of the ''squared'' Pearson correlation coefficient (because the RV coefficient takes values between 0 and 1).
It measures the closeness of two set of points that may each be repres ...
References
Factor analysis