A Note on the Likelihood Ratio Test in High-Dimensional Exploratory Factor Analysis

TL;DR

This paper investigates the failure of the classical chi-square approximation for the likelihood ratio test in high-dimensional exploratory factor analysis and provides conditions to ensure its validity.

Contribution

It derives necessary and sufficient conditions for the validity of the chi-square approximation in high-dimensional settings, offering practical guidelines.

Findings

Identifies when the chi-square approximation fails in high dimensions

Provides quantitative criteria for the validity of the likelihood ratio test

Offers statistical insights into high-dimensional exploratory factor analysis

Abstract

The likelihood ratio test is widely used in exploratory factor analysis to assess the model fit and determine the number of latent factors. Despite its popularity and clear statistical rationale, researchers have found that when the dimension of the response data is large compared to the sample size, the classical chi-square approximation of the likelihood ratio test statistic often fails. Theoretically, it has been an open problem when such a phenomenon happens as the dimension of data increases; practically, the effect of high dimensionality is less examined in exploratory factor analysis, and there lacks a clear statistical guideline on the validity of the conventional chi-square approximation. To address this problem, we investigate the failure of the chi-square approximation of the likelihood ratio test in high-dimensional exploratory factor analysis, and derive the necessary and sufficient condition to ensure the validity of the chi-square approximation. The results yield simple quantitative guidelines to check in practice and would also provide useful statistical insights into the practice of exploratory factor analysis.