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.

似然比检验广泛用于探索性因子分析中，以评估模型拟合并确定潜在因子的数量。尽管它很受欢迎且具有明确的统计原理，但研究人员发现，当响应数据的维度与样本量相比较大时，似然比检验统计量的经典卡方近似值经常失败。理论上，随着数据维数的增加，出现这种现象一直是一个悬而未决的问题；实际上，在探索性因子分析中很少研究高维的影响，并且缺乏关于传统卡方近似有效性的明确统计指南。为了解决这个问题，保证卡方近似有效性的充要条件。结果产生了简单的定量指南，以在实践中进行检查，并且还将为探索性因素分析的实践提供有用的统计见解。