Tucker's Congruence Coefficient $\phi$ (Factor Similarity)

Computes Tucker's (1951) congruence coefficient $\phi$, a measure of similarity between two factor-loading patterns (typically the standardized loadings of the same factor estimated on two different samples or with different methods), together with a permutation-based p-value testing the null hypothesis of unrelated loading patterns. $\phi$ is the standard tool for factor-replication studies (Lorenzo-Seva & ten Berge, 2006).

Usage

procrustes_phi(loadings_1, loadings_2, n_perm = 10000L)

Arguments

loadings_1, loadings_2: Numeric vectors of factor loadings on the same indicator set, of equal length. Either standardized or raw loadings work; the coefficient is scale-invariant.
n_perm: Number of permutations for the significance test. Default 10000. Set to 0 to skip the test.

Value

A data.frame (class dmar_tbl) in term / value layout with the row tucker_phi, the point estimate of $\phi$. When n_perm > 0 the table also carries p_value_perm, the two-sided permutation p-value, and n_perm, the number of permutations requested.

Details

Definition. For two vectors of loadings $\bm\lambda_1, \bm\lambda_2$ on a shared set of $p$ indicators, Tucker's congruence coefficient is $$\phi(\bm\lambda_1, \bm\lambda_2) \;=\; \frac{\sum_{i=1}^{p} \lambda_{1i} \lambda_{2i}} {\sqrt{\sum_{i=1}^{p} \lambda_{1i}^2 \cdot \sum_{i=1}^{p} \lambda_{2i}^2}}.$$ $\phi$ is the cosine of the angle between the two loading vectors and ranges over $[-1, 1]$; values near $\pm 1$ indicate high (anti-)congruence, values near 0 indicate orthogonality.

Permutation test. Under the null hypothesis that the two loading patterns are unrelated, randomly permuting one of the loading vectors and recomputing $\phi$ produces a sampling distribution against which the observed $\phi$ can be evaluated. The two-sided p-value is $(r + 1) / (m + 1)$, where r counts the permuted $|\phi|$ values at least as large as the observed $|\phi|$ and m is n_perm. Adding one to each part counts the observed arrangement, which is itself a legitimate permutation; without it a p-value of exactly zero could be reported, a value a sampled permutation test cannot support (Phipson & Smyth, 2010). The smallest reportable p-value is therefore $1 / (m + 1)$.

Interpretation. Benchmark values for $\phi$ have been proposed in the literature (Lorenzo-Seva & ten Berge, 2006), but context always matters; this package reports the coefficient with its uncertainty and leaves interpretation to the context of the application.

References

Lorenzo-Seva, U., & ten Berge, J. M. F. (2006). Tucker's congruence coefficient as a meaningful index of factor similarity. Methodology, 2(2), 57–64. doi:10.1027/1614-2241.2.2.57

Phipson, B., & Smyth, G. K. (2010). Permutation p-values should never be zero: Calculating exact p-values when permutations are randomly drawn. Statistical Applications in Genetics and Molecular Biology, 9(1), Article 39. doi:10.2202/1544-6115.1585

Tucker, L. R. (1951). A method for synthesis of factor analysis studies (Personnel Research Section Report No. 984). Department of the Army.

Author

Ken Kelley kkelley@nd.edu

Examples

set.seed(113)
# 1. Two highly similar loading patterns:
l1 <- c(0.72, 0.65, 0.81, 0.55, 0.69)
l2 <- c(0.70, 0.62, 0.83, 0.58, 0.66)
procrustes_phi(l1, l2)
#>  term         value 
#>  tucker_phi   0.999 
#>  p_value_perm 0.0080
#>  n_perm       10000 

# 2. Loadings on different factors should show low congruence:
l3 <- c(0.10, 0.05, 0.20, 0.85, 0.78)
procrustes_phi(l1, l3, n_perm = 5000)
#>  term         value 
#>  tucker_phi   0.702 
#>  p_value_perm 0.8244
#>  n_perm       5000

Tucker's Congruence Coefficient \(\phi\) (Factor Similarity)