A GRAPHICAL CHARACTERIZATION OF LATTICE CONDITIONAL-INDEPENDENCE MODELS

Lattice conditional independence (LCI) models for multivariate normal data recently have been introduced for the analysis of non-monotone mi ssing data patterns and of nonnested dependent linear regression model s (equivalent to seemingly unrelated regressions). It is shown here th at the class of LCI models coincides with a subclass of the class of g raphical Markov models determined by acyclic digraphs (ADGs), namely, the subclass of transitive ADG models. An explicit graph-theoretic cha racterization of those ADGs that are Markov equivalent to some transit ive ADG is obtained. This characterization allows one to determine whe ther a specific ADG D is Markov equivalent to some transitive ADG, hen ce to some LCI model, in polynomial time, without an exhaustive search of the (possibly superexponentially large) equivalence class [D]. The se results do not require the existence or positivity of joint densiti es.