There is a lot of literature written on the subject of choosing the correct number of classes for a latent class model (I mean "latent class model" in a very, very general sense, which encompasses a lot of different types of models, such as finite mixture models, Hidden Markov Models, Bayesian latent class models, etc.; although the details of many of these models differ, in a broad sense this question applies to all of them). So far as I can tell, however, all of the literature on the subject is dedicated to choosing the number of classes $k$, where $k\geq2$. That is, testing for the appropriate number of latent classes conditional on the fact that there are at least two latent groups in the data. This is usually done via some form of information criterion, a variant of the likelihood ratio test (e.g. the bootstrap LRT), or graphical measures (e.g. scree plots). There are also Bayesian methods that involve treating $k$ as random and giving it a prior distribution.

However, one thing that I am having a much harder time finding are discussions of how to compare a model with latent classes (of any number) to one without. This could be considered equivalent to testing whether or not it is necessary to assume the presence of latent classes at all. The traditional methods for choosing $k$ are likely not appropriate for this type of comparison.

So, what is the most principled method for comparing a model with latent classes to one without (or, equivalently, testing for the presence of ANY latent classes, regardless of $k$)?

I'm aware that the goal of the analysis (e.g. inference vs. prediction) will impact what methods may or may not be appropriate; I am also aware that substantive knowledge of whether latent classes are reasonable to assume is more important than finding a test. But the fact remains that there does not seem to be, that I can tell, a standardized or recommended method for making this sort of comparison, which seems significant.

(A similar question was asked here, but has no answer. I don't view this as a duplicate since this other question is more specific, while my question is more general: Comparing a model with a latent variable to one without)


I don't see the reason why one cannot compare one-class model with 2+ class models. One can compute likelihood score for 1-class model and then compare it to the likelihood of 2-class model, and then make their decision which one fits data better. So if 2-class model is rejected you may conclude there are no classes in the data and all of the observed variables are independent. As far as you have likelihood values, you can apply information criteria (AIC, BIC, etc.) and appropriate likelihood ratio tests.

See classic McCutcheon (1987, p.32), where he directly compares 1-class with 2-class solutions; or Goodman (1974, p.220), where he considered rejection of 2-class model in favor of 1-class model. For a more recent example, see how LRT performs for detecting 2-class against 1-class solution (Nylund & Muthen, 2007, p.559). Many other papers actually consider 1-class solution when deciding on a number of classes.

I guess, the reason why people sometimes don't discuss 1-class solution is their assumption of a latent variable presence, which implies at least 2 classes.


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