Chain Graphs for Multilevel Models

In this paper, we propose a way to incorporate multilevel models within graphical models. We introduce three types of nodes for chain graphs to represent (1) individual within clusters, (2) clusters as latent variables and (3) interactive effects. In this way, the chain graph shows both the associations among individuals introduced by clusters and the random coefficients of a multilevel model. Then, independencies implied by the model can be read off the chain graph as well as the additional independence constraints under which the multilevel model reduces to a fixed effect regression model. The paper focuses on hierarchical Gaussian data structures, considering two-level models.