A dichotomous behavior of Guttman-Kaiser criterion from equi-correlated normal population
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We consider a p-dimensional, centered normal population such that all variables have a positive variance σ^2 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 Marčenko-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 and EFA. 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)].
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