  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.
