Parsimonious Latent Class Models
Latent class model or multivariate multinomial mixture is expected to be useful to represent nonhomogeneous discrete populations. It uses a conditional independence assumption given the latent class to which a statistical unit is belonging. Latent class models differ according to the number of latent classes and to the assumptions regarding the multivariate multinomial distribution. In this contribution, five parsimonious models based on a particular representation of the multinomial parameters will be presented. Maximum likelihood estimation of the five different model parameters will be presented through the EM algorithm. Stochastic and classification variants will be introduced. Then, full Bayesian analysis with Jeffreys non informative prior distribution of the five models will be considered through Gibbs sampling and a reversible jump Markov chain Monte Carlo method capable of jumping between parameter subspaces corresponding to different numbers of classes or to different assumptions on the multinomial distributions will be presented. In a third part, the problem of choosing a reliable latent class model will be considered. In addition of the full Bayesian approach derived from the reversible jump algorithm, penalized likelihood criteria as AIC, BIC or ICL will be considered. Practical aspects of latent class analysis will be emphasized. Multiple maxima of the likelihood function, slow convergence situations for EM, convergence issues for the full Bayesian methodology, practical behavior of penalized likelihood criteria to select a useful model according to the modelling purpose will be discussed. In that aim, numerical experients on both simulated and real data sets will be thoroughfully analysed.