A likelihood ratio test for MTP2 within binary variables

Abstract

Multivariate Totally Positive (MTP2) binary distributions have been studied in many fields, such as statistical mechanics, computer storage and latent variable models. We show that MTP2 is equivalent to the requirement that the parameters of a saturated log-linear model belong to a convex cone, and we provide a Fisher-scoring algorithm for maximum likelihood estimation. We also show that the asymptotic distribution of the log-likelihood ratio is a mixture of chi-squares (a distribution known as chi-bar-squared in the literature on order restricted inference); for this we derive tight bounds which turn out to have very simple forms. A potential application of this method is for Item Response Theory (IRT) models, which are used in educational assessment to analyse the responses of a group of subjects to a collection of questions (items): an important issue within IRT is whether the joint distribution of the manifest variables is compatible with a single latent variable representation satisfying local independence and monotonicity which, in turn, imply that the joint distribution of item responses is MTP2.