Finite Mixture EFA in Mplus

In this document we describe the Mixture EFA model estimated in Mplus. Four types of dependent variables are possible in this model: normally distributed, ordered categorical with logit or probit link, Poisson distributed with the exponential link function, and censored variables. Inflation is not available for the Censored and Poisson variables. Suppose that we estimate a K class model with M factors and P dependent variables. Denote the variables by Y1, ..., YP and the normally distributed factors by η1, ..., ηM . Let η be the vector of all latent factors η = (η1, ..., ηM). The Mixture model is based on a single categorical latent class variable C. For a normally distributed variable Yp we estimate the following model in class k Yp = νkp + λkpη + εp where νkp is the intercept parameter, λkp is a vector of loadings of dimension M , and εp is a zero mean normally distributed residual with variance θkp. For an ordered categorical variable Yp we estimate the following model in class k P (Yp = j) = F (τkpj − λkpη)− F (τkpj−1 − λkpη) for j = 1, ..., rp where rp is the number of categories that the variable Yp takes. The parameters τkpj are monotonically increasing for j and for identification purposes τkprp = ∞ and τkp0 = −∞. The function F is either the standard normal distribution function, for probit link, or the logit distribution function F (x) = 1/(1 + Exp(−x)), for logit link. Alternatively we can specify the model as follows Yp = j ⇐ τkpj−1 ≤ Y ∗ p < τkpj