Optimization in latent class analysis

In latent class analysis (LCA) one seeks a clustering of categorical data, such as patterns of symptoms of a patient, in terms of locally independent stochastic models. This leads to practical definitions of criteria, e.g., whether to include patients in further diagnostic examinations. The clustering is often determined by parameters that are estimated by the maximum likelihood method. The likelihood function in LCA has in many cases – especially for sparse data sets – a complicated shape with many local extrema, even for small-scale problems. Hence a global optimization must be attempted. This paper describes an algorithm and software for the global optimization of the likelihood function constrained by the requirement of a good fit of the data with a minimal number of classes. The problem is formulated in the algebraic modeling language AMPL and solved via state of the art optimization solvers. The approach is successfully applied to three real-life problems. Remarkably, the goodnessof-fit constraint makes one of the three problems identifiable by eliminating all but one of the local minimizers.