A Note on Likelihood Ratio Tests for Models with Latent Variables

TL;DR

This paper examines the limitations of the traditional likelihood ratio test for models with latent variables and introduces a more general asymptotic theory to address these issues.

Contribution

It highlights violations of Wilks' theorem in latent variable models and applies a broader asymptotic framework from Chernoff, 1954, to improve test accuracy.

Findings

Wilks' theorem often fails for latent variable models

A more general asymptotic theory corrects the LRT distribution

Illustrations with three latent variable models

Abstract

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 $\chi^2$-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 $\chi^2$ 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 (1954) and discussed in both van der Vaart (2000) and Drton (2009), but it does not seem to have received enough attention. We illustrate this general theory with the three examples.