Distinguishing between latent classes and continuous factors with categorical outcomes: Class invariance of parameters of factor mixture models.

Factor mixture models (FMM's) are latent variable models with categorical and continuous latent variables which can be used as a model-based approach to clustering. A previous paper covered the results of a simulation study showing that in the absence of model violations, it is usually possible to choose the correct model when fitting a series of models with different numbers of classes and factors within class. The response format in the first study was limited to normally distributed outcomes. The current paper has two main goals, firstly, to replicate parts of the first study with 5-point Likert scale and binary outcomes, and secondly, to address the issue of testing class invariance of thresholds and loadings. Testing for class invariance of parameters is important in the context of measurement invariance and when using mixture models to approximate non-normal distributions. Results show that it is possible to discriminate between latent class models and factor models even if responses are categorical. Comparing models with and without class-specific parameters can lead to incorrectly accepting parameter invariance if the compared models differ substantially with respect to the number of estimated parameters. The simulation study is complemented with an illustration of a factor mixture analysis of ten binary depression items obtained from a female subsample of the Virginia Twin Registry.