Factor Mixture Models with Ordered Categorical Outcomes: The Mathematical Relation to Mixture Item Response Theory Models and a Comparison Of Maximum Likelihood and Bayesian Model Parameter Estimation Methods

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A factor mixture model (FMM) is a hybrid of latent class analysis and factor analysis modeling techniques. It can be used to investigate group differences in the absence of known class membership. The current study investigates the relation between FMMs and mixture item response theory (IRT) models. A formal proof of the mathematical equivalence between mixture graded-response models and FMMs with ordered categorical outcomes is presented and conversion formulas between the parameters of the two types of models are provided. More importantly, the current study conducts a Monte Carlo simulation study to compare Bayesian estimation with three different priors and maximum likelihood (ML) approach in fitting FMMs. Parameter recovery and classification accuracy are evaluated and compared. Besides the estimation method, the sample size, the number of outcome indicators, and the magnitude of factor loadings are manipulated in the simulation. It is found that in general that ML and Bayesian estimation with weakly informative priors perform well with a small sample size, and that all estimation methods perform well with a large sample size. The results of this simulation also have implications for mixture IRT models based on its relation to FMMs.