  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.
