The GMM for longitudinal data analysis discussed in the previous section is a hybrid model that involves both continuous latent variables (i.e., the latent growth factors) and the categorical latent variable (the latent class variable). An example of cross-sectional hybrid model is the factor mixture analysis (FMA) model, which is a hybrid model of the factor analysis model (either EFA or CFA) and latent class model (Lubke and Muthén, 2005; Muthén, 2006; Muthén and Asparouhov, 2006). In a FMA, the factor analysis model clusters items on factors (continuous latent variables) and generates factor scores; and the LCA model clusters individuals/cases into groups (latent categorical variable) with different factor mean scores (Lubke and Muthén, 2005, 2007>, >). Thus, like the GMM, the FMM is a combination of a variable-centered and person-centered modeling approach.

According to Muthèn (2008), hybrid latent variable models can be categorized into four branches. The first two assume measurement invariance, while the last two assume measurement noninvariance, across classes (Muthén, 2008). In this section, we discuss on FMA models that belong to the first two branches: (1) FMA with measurement invariance and parametric factor distribution, also called mixture factor analysis (MFA); and (2) FMA with measurement invariance and nonparametric factor distribution, also called nonparametric factor analysis or latent class factor analysis (LCFA) (Muthén, 2008). The MFA ...

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