Testing Lattice Conditional Independence Models

The lattice conditional independence (LCI) model N(K) is defined to be the set of all normal distributions N(0, Σ) on RI such that for every pair L, M ∈ K, xL and xM are conditionally independent given xL ∩ 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, xK is the coordinate projection of x ∈ RI onto RK. 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.