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A note on likelihood ratio tests for models with latent variables

Chen, Yunxiao ORCID: 0000-0002-7215-2324, Moustaki, Irini and Zhang, H (2020) A note on likelihood ratio tests for models with latent variables. Psychometrika, 85 (4). 996 - 1012. ISSN 0033-3123

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Identification Number: 10.1007/s11336-020-09735-0


The likelihood ratio test (LRT) is widely used for comparing the relative fit of nested latent variable models. Following Wilks’ theorem, the LRT is conducted by comparing the LRT statistic with its asymptotic distribution under the restricted model, a χ 2 distribution with degrees of freedom equal to the difference in the number of free parameters between the two nested models under comparison. For models with latent variables such as factor analysis, structural equation models and random effects models, however, it is often found that the χ 2 approximation does not hold. In this note, we show how the regularity conditions of Wilks’ theorem may be violated using three examples of models with latent variables. In addition, a more general theory for LRT is given that provides the correct asymptotic theory for these LRTs. This general theory was first established in Chernoff (J R Stat Soc Ser B (Methodol) 45:404–413, 1954) and discussed in both van der Vaart (Asymptotic statistics, Cambridge, Cambridge University Press, 2000) and Drton (Ann Stat 37:979–1012, 2009), but it does not seem to have received enough attention. We illustrate this general theory with the three examples.

Item Type: Article
Official URL:
Additional Information: © 2020 The Authors
Divisions: Statistics
Subjects: H Social Sciences > HA Statistics
Date Deposited: 23 Nov 2020 14:45
Last Modified: 20 Oct 2021 01:05

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