Blackwell Publishing
Oxford, UK
RSSB Journal of the Royal Statistical Society Series B 1369-7412 2004 Royal Statistical Society
August 2004 66 3 687 717 Original Articles Joint Response Graphs N. Wermuth and D. R. Cox

[Received October 2001. Final revision December 2003]

Nanny Wermuth, Department of Mathematical Statistics, Chalmers, Gothenburg University, Eklandagatan 86, 41296 Göteborg, Sweden.
E-mail: nanny.wermuth@math.chalmers.se
Joint response graphs and separation induced by triangular systems Nanny Wermuth 1 and D. R. Cox 2
1 Gothenburg University, Sweden 2 Nuffield College, Oxford, UK
Summary.

We consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions. The independences are captured by missing edges in a directed graph. A matrix form of such a graph, called the generating edge matrix, is triangular so the distributions that are generated over such graphs are called triangular systems. We study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models, i.e. for alternative recursive factorizations of the given density using joint conditional distributions. For this we introduce families of linear triangular equations which do not require assumptions of distributional form. The strength of the associations that are implied by such linear families for chain graph models is derived. The edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix. It is shown how induced independences and dependences can be studied by graphs, by edge matrix calculations and via the properties of densities. Some ways of using the results are illustrated.

Keywords:  Chain graphs; Concentration graphs; Covariance graphs; Directed acyclic graphs; Graphical Markov models; Univariate recursive regressions