Testing Lattice Conditional Independence Models

Abstract

The lattice conditional independence (LCI) model N( K ) is defined to be the set of all normal distributions N(0, Σ) on R I such that for every pair L, M ∈ K , x L and x M are conditionally independent given x L ∩ M . Here K is a ring of subsets (hence a distributive lattice) of the finite index set I such that ∅ I ∈ K , while for K ∈ K , x K is the coordinate projection of x ∈ R I onto R K . These LCI models have especially tractable statistical properties and arise naturally in the analysis of non-monotone multivariate missing data patterns and non-nested dependent linear regression models ≡ seemingly unrelated regressions. The present paper treats the problem of testing one LCI model against another, i.e., testing N( K ) vs N( M ) when M is a subring of K . The likelihood ratio test statistic is derived, together with its central distribution, and several examples are presented.