A Dichotomous Behavior of Guttman-Kaiser Criterion from Equi-Correlated Normal Population - Yohji Akama; Atina Husnaqilati (Journal of the Indonesian Mathematical Society, 2022)
We consider a p-dimensional, centered normal population such that all variables have a positive variance σ2 and any correlation coefficient between different variables is a given nonnegative constant ρ < 1. Suppose that both the sample size n and population dimension p tend to infinity with p/n → c > 0. We prove that the limiting spectral distribution of a sample correlation matrix is the Marcenko-Pastur distribution of index c and scale parameter 1 − ρ. By the limiting spectral distributions, we rigorously show the limiting behavior of widespread stopping rules Guttman-Kaiser criterion and cumulative-percentage-of-variation rule in PCA andEFA. As a result, we establish the following dichotomous behavior of Guttman-Kaiser criterion when both n and p are large, but p/n is small: (1) the criterion retains a small number of variables for ρ > 0, as suggested by Kaiser, Humphreys, and Tucker [Kaiser, H. F. (1992). On Cliff’s formula, the Kaiser-Guttman rule and the number of factors. Percept. Mot. Ski. 74]; and (2) the criterion retains p/2 variables for ρ = 0, as in a simulation study [Yeomans, K. A. and Golder, P. A. (1982). The Guttman-Kaiser criterion as a predictor of the number of common factors. J. Royal Stat. Soc. Series D. 31(3)].