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

TL;DR

This paper analyzes the asymptotic behavior of the Guttman-Kaiser criterion in high-dimensional normal populations with equi-correlation, revealing a dichotomous outcome depending on the correlation coefficient.

Contribution

It provides a rigorous theoretical foundation for the behavior of the Guttman-Kaiser criterion in large, equi-correlated normal populations, connecting spectral distribution limits to factor retention rules.

Findings

For $ ho>0$, the criterion retains few variables, aligning with classical expectations.

For $ ho=0$, the criterion retains about half of the variables, matching simulation results.

The spectral distribution converges to the Marčenko-Pastur law under the specified conditions.

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 Mar\v{c}enko-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)].