On Horn's approximation to the sampling distribution of eigenvalues from random correlation matrices in parallel analysis

Parallel analysis proposed by Horn (Psychometrika, 30(2), 179-185, 1965) has been recommended for determining the number of factors. Horn suggested using the eigenvalues from several generated correlation matrices with uncorrelated variables to approximate the theoretical distribution of the eigenvalues from random correlation matrices lacking at that time. The distribution of eigenvalues and the distribution of L-1, the rescaled largest eigenvalue, obtained from a random correlation matrix, were finally proved in this century and are shown to asymptotically converge to the Marcenko-Pastur distribution and the Tracy-Widom distribution, respectively. Nonetheless, the sample sizes and the numbers of variables commonly encountered in factor analysis were oftentimes limited. We showed that under finite datasets (a) the distribution of eigenvalues closely resembled the Marcenko-Pastur distribution; and (b) the distribution of L-1 departed from the Tracy-Widom distribution, though the original largest random eigenvalues before rescaling from these two distributions were very close. Our findings support Horn's idea of using average eigenvalues from generated random data to reflect sampling fluctuations under finite datasets. Implications of the results for newly developed number-of-factor-determining methods based on these asymptotic distributions are discussed.