Congruence coefficient

Last updated March 21, 2019

In multivariate statistics, the congruence coefficient is an index of the similarity between factors that have been derived in a factor analysis. 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. [1] [2] [3]

Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. The application of multivariate statistics is multivariate 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 variables mainly reflect the variations in two unobserved (underlying) variables. Factor analysis searches for such joint variations in response to unobserved latent variables. The observed variables are modelled as linear combinations of the potential factors, plus "error" terms. Factor analysis aims to find independent latent variables.

Sir Cyril Lodowic Burt, FBA was an English educational psychologist and geneticist who made contributions also to statistics. He is known for his studies on the heritability of IQ. Shortly after he died, his studies of inheritance and intelligence were discredited after evidence emerged indicating he had falsified research data, inventing correlations in separated twins which did not exist.

Definition

Let X and Y be column vectors of factor loadings for two different samples. The formula for the congruence coefficient, or r_c, is then [2]

r_{c}={\frac {\sum {XY}}{\sqrt {\sum {X^{2}}\sum {Y^{2}}}}}

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. [1] [2]

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). For example, with perfect congruence the angle between the factor axes is 0 degrees, and the cosine of 0 is 1. [2]

Cosine similarity is a measure of similarity between two non-zero vectors of an inner product space that measures the cosine of the angle between them. The cosine of 0° is 1, and it is less than 1 for any angle in the interval $(0,π]$ radians. It is thus a judgment of orientation and not magnitude: two vectors with the same orientation have a cosine similarity of 1, two vectors oriented at 90° relative to each other have a similarity of 0, and two vectors diametrically opposed have a similarity of -1, independent of their magnitude. The cosine similarity is particularly used in positive space, where the outcome is neatly bounded in $. The name derives from the term "direction cosine": in this case, unit vectors are maximally "similar" if they're parallel and maximally "dissimilar" if they're orthogonal. This is analogous to the cosine, which is unity when the segments subtend a zero angle and zero when the segments are perpendicular.$

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. [2]

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References

1 2 Lorenzo-Seva, U. & ten Berge, J.M.F. (2006). Tucker’s Congruence Coefficient as a Meaningful Index of Factor Similarity. Methodology, 2, 57–64.
1 2 3 4 5 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.

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[lorenzo-1] 1 2 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-2] 1 2 3 4 5 Jensen, A.R. (1998). The g factor: The science of mental ability. Westport, CT: Praeger, pp. 99–100.

[herve-3] Abdi, H. (2007). RV Coefficient and Congruence Coefficient. In Neil Salkind (Ed.), Encyclopedia of Measurement and Statistics. Thousand Oaks (CA): Sage.