On the Normal Linear Regression Model with Recursive Graphical Markov Structure

Abstract

Graphical models use graphs, either undirected, directed, or mixed, to represent multivariate statistical dependencies. Statistical variables are represented by nodes of the graph, and local dependencies are specified by postulating that each variable is conditional independent of other variables given the neighbor variables. The terms "other" and "neighbor" are determined by the graphical structure. Graphical Markov models given by an acyclic digraph (ADG), also called Recursive Markov Models, or simple ADG models, have especially amenable statistical properties. In this paper we study normal multivariate statistical models that combine the Markov property with respect to an ADG with multivariate linear regression. The complete solution to likelihood inference will be described. This includes simultaneous inference for expectation [Image62.gif]and covariance [Image63.gif]. The tools are the recursive factorization of the likelihood function induced by the ADG, where each factor will represent a standard MANOVA problem. For Example, by imposing the Markov covariance restrictions determined by a suitable ADG, it is possible to formulate and solve the following normal multivariate model: a 4 variate two-way MANOVA with no interaction, no row or column effects for variable 1, no column effects for variable 2, and no row effects for variable 3.(This is joint work with Michael Perlman).