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On the interpretation of inflated correlation path weights in concentration graphs

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  • Alberto Roverato

    (University of Padova)

Abstract

Statistical models associated with graphs, called graphical models, have become a popular tool for representing network structures in many modern applications. Relevant features of the model are represented by vertices, edges and other higher order structures. A fundamental structural component of the network is represented by paths, which are a sequence of distinct vertices joined by a sequence of edges. The collection of all the paths joining two vertices provides a full description of the association structure between the corresponding variables. In this context, it has been shown that certain pairwise association measures can be decomposed into a sum of weights associated with each of the paths connecting the two variables. We consider a pairwise measure called an inflated correlation coefficient and investigate the properties of the corresponding path weights. We show that every inflated correlation weight can be factorized into terms, each of which is associated either to a vertex or to an edge of the path. This factorization allows one to gain insight into the role played by a path in the network by highlighting the contribution to the weight of each of the elementary units forming the path. This is of theoretical interest because, by establishing a similarity between the weights and the association measure they decompose, it provides a justification for the use of these weights. Furthermore we show how this factorization can be exploited in the computation of centrality measures and describe their use with an application to the analysis of a dietary pattern.

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  • Alberto Roverato, 2021. "On the interpretation of inflated correlation path weights in concentration graphs," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(5), pages 1485-1505, December.
    • Handle: RePEc:spr:stmapp:v:30:y:2021:i:5:d:10.1007_s10260-021-00587-1
    DOI: 10.1007/s10260-021-00587-1
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    References listed on IDEAS

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    1. Beatrix Jones & Mike West, 2005. "Covariance decomposition in undirected Gaussian graphical models," Biometrika, Biometrika Trust, vol. 92(4), pages 779-786, December.
    2. Alberto Roverato & Robert Castelo, 2020. "Path weights in concentration graphs," Biometrika, Biometrika Trust, vol. 107(3), pages 705-722.
    3. Steffen L. Lauritzen & Thomas S. Richardson, 2002. "Chain graph models and their causal interpretations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 64(3), pages 321-348, August.
    4. Alberto Roverato & Robert Castelo, 2017. "The networked partial correlation and its application to the analysis of genetic interactions," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 66(3), pages 647-665, April.
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