Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-07T18:48:17.677Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  01 January 2025

Yinqiu He ,
Zi Wang  and
Gongjun Xu Open the ORCID record for Gongjun Xu [Opens in a new window]
Yinqiu He
Affiliation:
University of Michigan
Zi Wang
Affiliation:
University of Michigan
Gongjun Xu*
Affiliation:
University of Michigan
*
Correspondence should be made to Gongjun Xu, Department of Statistics, University of Michigan, 456 West Hall, 1085 South University, Ann Arbor, MI48109, USA. Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

exploratory factor analysislikelihood ratio testChi-square approximation
Type
Theory and Methods (T&M)
Information
Psychometrika , Volume 86 , Issue 2 , June 2021 , pp. 442 - 463
Copyright
Copyright © 2021 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This research is partially supported by NSF CAREER SES-1846747, DMS-1712717, and SES-1659328

References

Ait-Sahalia, Y, Xiu, D (2017). Using principal component analysis to estimate a high dimensional factor model with high-frequency data. Journal of Econometrics, 201 (2384–399CrossRefGoogle Scholar
Akaike, H (1987). Factor analysis and AIC. Psychometrika, 52 (3317–332CrossRefGoogle Scholar
Anderson, T. W. (2003). An introduction to multivariate statistical analysis. Wiley.Google Scholar
Bai, Z, Jiang, D, Yao, J-F, Zheng, S (2009). Corrections to LRT on large-dimensional covariance matrix by RMT. Annals of Statistics, 37 (6B3822–3840CrossRefGoogle Scholar
Bai, J, Li, K (2012). Statistical analysis of factor models of high dimension. The Annals of Statistics, 40 (1436–465CrossRefGoogle Scholar
Bai, J, Ng, S (2002). Determining the number of factors in approximate factor models. Econometrica, 70 (1191–221CrossRefGoogle Scholar
Barendse, M, Oort, F, Timmerman, M (2015). Using exploratory factor analysis to determine the dimensionality of discrete responses. Structural Equation Modeling: A Multidisciplinary Journal, 22 (187–101CrossRefGoogle Scholar
Bartholomew, D. J., Knott, M., & Moustaki, I. (2011). Latent variable models and factor analysis: A unified approach (vol. 904). Wiley.CrossRefGoogle Scholar
Bartlett, MS (1950). Tests of significance in factor analysis. British Journal of Statistical Psychology, 3 (277–85CrossRefGoogle Scholar
Bentler, PM, Yuan, K-H (1998). Tests for linear trend in the smallest eigenvalues of the correlation matrix. Psychometrika, 63 (2131–144CrossRefGoogle Scholar
Cattell, R (1966). The scree test for the number of factors. Multivariate Behavioral Research, 1 (2245–276CrossRefGoogle ScholarPubMed
Chen, Y., & Li, X. (2020). Determining the number of factors in high-dimensional generalised latent factor models. arXiv preprint arXiv:2010.02326Google Scholar
Chen, Y, Li, X, Zhang, S (2019). Joint maximum likelihood estimation for high-dimensional exploratory item factor analysis. Psychometrika, 84 (1124–146CrossRefGoogle ScholarPubMed
Costello, AB, Osborne, J (2005). Best practices in exploratory factor analysis: Four recommendations for getting the most from your analysis. Practical Assessment, Research, and Evaluation, 10 (17Google Scholar
Dobriban, E. (2020). Permutation methods for factor analysis and PCA. The Annals of Statistics,.CrossRefGoogle Scholar
Fabrigar, L. R., & Wegener, D. T. (2011). Exploratory factor analysis. Oxford University Press.CrossRefGoogle Scholar
Fabrigar, LR, Wegener, DT, MacCallum, RC, Strahan, EJ (1999). Evaluating the use of exploratory factor analysis in psychological research. Psychological Methods, 4 (3272–299CrossRefGoogle Scholar
Finch, WH, Finch, MEH (2016). Fitting exploratory factor analysis models with high dimensional psychological data. Journal of Data Science, 14 (3519–537CrossRefGoogle Scholar
Gorsuch, R. L. (1988). Exploratory factor analysis (pp. 231–258). Springer.CrossRefGoogle Scholar
Harlow, LL, Oswald, FL (2016). Big data in psychology: Introduction to the special issue. Psychological Methods, 21 (4447CrossRefGoogle ScholarPubMed
Hayashi, K, Bentler, P, Yuan, K-H (2007). On the likelihood ratio test for the number of factors in exploratory factor analysis. Structural Equation Modeling: A Multidisciplinary Journal, 14 (3505–526CrossRefGoogle Scholar
He, Y., Meng, B., Zeng, Z., & Xu, G. (2020b). On the phase transition of Wilks’ phenomenon. Biometrika, to appear.CrossRefGoogle Scholar
He, Y, Jiang, T, Wen, J, Xu, G (2020). Likelihood ratio test in multivariate linear regression: From low to high dimension. Statistica Sinica, 46, 8479–8492Google Scholar
Horn, JL (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30 (2179–185CrossRefGoogle ScholarPubMed
Jiang, T, Qi, Y (2015). Likelihood ratio tests for high-dimensional normal distributions. Scandinavian Journal of Statistics, 42 (4988–1009CrossRefGoogle Scholar
Jiang, T, Yang, F (2013). Central limit theorems for classical likelihood ratio tests for high-dimensional normal distributions. Annals of Statistics, 41 (42029–2074CrossRefGoogle Scholar
Kaiser, HF (1960). The application of electronic computers to factor analysis. Educational and Psychological Measurement, 20 (1141–151CrossRefGoogle Scholar
Keeling, KB (2000). A regression equation for determining the dimensionality of data. Multivariate Behavioral Research, 35 (4457–468CrossRefGoogle ScholarPubMed
Koran, J (2020). Indicators per factor in confirmatory factor analysis: More is not always better. Structural Equation Modeling: A Multidisciplinary Journal, 27, 1–8CrossRefGoogle Scholar
Lawley, DN, Maxwell, AE (1962). Factor analysis as a statistical method. Journal of the Royal Statistical Society. Series D (The Statistician), 12 (3209–229Google Scholar
Luo, L, Arizmendi, C, Gates, KM (2019). Exploratory factor analysis (EFA) programs in R. Structural Equation Modeling: A Multidisciplinary Journal, 26 (5819–826CrossRefGoogle Scholar
MacCallum, RC, Widaman, KF, Zhang, S, Hong, S (1999). Sample size in factor analysis. Psychological Methods, 4 (184CrossRefGoogle Scholar
Muirhead, R. J. (2009). Aspects of multivariate statistical theory (vol. 197). Wiley.Google Scholar
Mukherjee, BN (1970). Likelihood ratio tests of statistical hypotheses associated with patterned covariance matrices in psychology. British Journal of Mathematical and Statistical Psychology, 23 (289–120CrossRefGoogle Scholar
Mundfrom, DJ, Shaw, DG, Ke, TL (2005). Minimum sample size recommendations for conducting factor analyses. International Journal of Testing, 5 (2159–168CrossRefGoogle Scholar
Preacher, KJ, MacCallum, RC (2002). Exploratory factor analysis in behavior genetics research: Factor recovery with small sample sizes. Behavior Genetics, 32 (2153–161CrossRefGoogle ScholarPubMed
Reckase, M. (2009). Multidimensional item response theory. Statistics for social and behavioral sciences. Springer.CrossRefGoogle Scholar
Schwarz, G (1978). Estimating the dimension of a model. The Annals of Statistics, 6 (2461–464CrossRefGoogle Scholar
Steiger, JH (2016). Notes on the Steiger–Lind (1980) handout. Structural Equation Modeling: A Multidisciplinary Journal, 23 (6777–781CrossRefGoogle Scholar
Sundberg, R, Feldmann, U (2016). Exploratory factor analysis—parameter estimation and scores prediction with high-dimensional data. Journal of Multivariate Analysis, 148, 49–59CrossRefGoogle Scholar
Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications. American Psychological Association.CrossRefGoogle Scholar
Tucker, LR, Lewis, C (1973). A reliability coefficient for maximum likelihood factor analysis. Psychometrika, 38 (11–10CrossRefGoogle Scholar
Velicer, WF, Fava, JL (1998). Affects of variable and subject sampling on factor pattern recovery. Psychological Methods, 3 (2231CrossRefGoogle Scholar
Winter, JCF, Dodou, D (2012). Factor recovery by principal axis factoring and maximum likelihood factor analysis as a function of factor pattern and sample size. Journal of Applied Statistics, 39 (4695–710CrossRefGoogle Scholar
Winter, JCF, Dodou, D, Wieringa, PA (2009). Exploratory factor analysis with small sample sizes. Multivariate Behavioral Research, 44 (2147–181CrossRefGoogle ScholarPubMed
Yuan, K-H, Marshall, LL, Bentler, PM (2002). A unified approach to exploratory factor analysis with missing data, non-normal data, and in the presence of outliers. Psychometrika, 67 (195–121CrossRefGoogle Scholar