Mixture models for mixed-type data through a composite likelihood approach

A mixture model is considered to classify continuous and/or ordinal variables. Under this model, both the continuous and the ordinal variables are assumed to follow a heteroscedastic Gaussian mixture model, where, as regards the ordinal variables, it is only partially observed. More specifically, the ordinal variables are assumed to be a discretization of some mixture variables. From a computational point of view, this creates some problems for the maximum likelihood estimation of model parameters. Indeed, the likelihood function involves multidimensional integrals, whose evaluation is computationally demanding as the number of ordinal variables increases. The proposal is to replace this cumbersome likelihood with a surrogate objective function that is easier to maximize. A composite approach is used, in particular the original joint distribution is replaced by the product of three blocks: the marginal distribution of continuous variables, all bivariate marginal distributions of ordinal variables and the marginal distributions given by all continuous variables and only one ordinal variable. This leads to a surrogate function that is the sum of the log contributions for each block. The estimation of model parameters is carried out maximizing the surrogate function within an EM-like algorithm. The effectiveness of the proposal is investigated through a simulation study and two applications to real data.