Graphical log-linear models for homologous factors

In this paper I propose an extension of quasi-symmetric graphical log-linear models, introduced by Gottard et al. (2009), to allow for different kinds of symmetry constraint on main effects concerning homologous factors. This class of models can be associated to a graph with coloured edges and nodes. In quasi-symmetric graphical log linear models, coloured edges are associated to a set of equality constraints on interaction parameters, extending the class of quasi-symmetry models to the presence of conditional independence and non-homologous factors. Introducing coloured nodes, quasisymmetric graphical log linear models are here extended to include symmetry models. The proposed models apply with contingency tables in which some conditional margins are symmetric and show marginal homogeneity.