In
multivariate statistics, 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 observe ...
. It was introduced in 1948 by
Cyril Burt who referred to it as ''unadjusted correlation''. It is also called ''Tucker's congruence coefficient'' after
Ledyard Tucker 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 elements is an m \times 1 matrix consisting of a single column of 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 , 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
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
of the angle between factor axes based on the same set of variables (e.g., tests) obtained for two samples (see
Cosine similarity). 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
References
Factor analysis