The Empirical Kaiser Criterion (EKC; Auerswald & Moshagen, 2019; Braeken & van Assen, 2017)
refines Kaiser-Guttman Criterion
by accounting for random sample variations in eigenvalues. At the population level, the EKC is
equivalent to the original Kaiser-Guttman Criterion, extracting all factors whose eigenvalues
from the correlation matrix are greater than one. However, at the sample level, it adjusts for
the distribution of eigenvalues in normally distributed data. Under the null model, the eigenvalue
distribution follows the Marčenko-Pastur distribution (Marčenko & Pastur, 1967) asymptotically.
The upper bound of this distribution serves as the reference eigenvalue for the first eigenvalue \(\lambda\), so
$$\lambda_{1,ref} = \left( 1 + \sqrt{\frac{I}{N}} \right)^2$$
, which is determined by N individuals and I items. For subsequent eigenvalues, adjustments are
made based on the variance explained by previous factors. The j-th reference eigenvalue is:
$$\lambda_{j,ref} = \max \left[ \frac{I - \sum_{i=0}^{j-1} \lambda_i}{I - j + 1} \left( 1 + \sqrt{\frac{I}{N}} \right)^2, 1 \right]$$
The j-th reference eigenvalue is reduced according to the magnitude of earlier eigenvalues
since higher previous values mean less unexplained variance remains. As in the original
Kaiser-Guttman Criterion, the reference eigenvalue cannot drop below one.
$$F = \sum_{i=1}^{I} I(\lambda_i > \lambda_{i,ref})$$
Here, \( F \) represents the number of factors determined by the EKC, and \(I(\cdot)\) is the
indicator function, which equals 1 when the condition is true, and 0 otherwise.