A dichotomous behavior of Guttman-Kaiser criterion from equi-correlated normal population

Abstract

We consider a $p$-dimensional, centered normal population such that all variables have a positive variance ${\sigma }^2$ and any correlation coefficient between different variables is a given nonnegative constant $\rho <1$. Suppose that both the sample size $n$ and population dimension $p$ tend to infinity with $p/n\to c>0$. We prove that the limiting spectral distribution of a sample correlation matrix is Marčenko-Pastur distribution of index $c$ and scale parameter $1-\rho$. 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 $\rho >0$, 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 $\rho =0$, 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)].

arXiv commentThis manuscript has appeared in J. Indones. Math. Soc. Volume 28 Number 3 2022