A likelihood ratio test for $MTP_2$ within binary variables

Abstract

Multivariate Totally Positive $(MTP_2)$ binary distributions have been studied in many fields, such as statistical mechanics, computer storage and latent variable models. We show that $MTP_2$ 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 $MTP_2$.