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Joint response graphs and separation induced by triangular systems

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  • Nanny Wermuth
  • D. R. Cox

Abstract

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.

Suggested Citation

  • Nanny Wermuth & D. R. Cox, 2004. "Joint response graphs and separation induced by triangular systems," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(3), pages 687-717, August.
    • Handle: RePEc:bla:jorssb:v:66:y:2004:i:3:p:687-717
    DOI: 10.1111/j.1467-9868.2004.b5161.x
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    Cited by:

    1. Loperfido, Nicola, 2010. "A note on marginal and conditional independence," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1695-1699, December.
    2. Nanny Wermuth & Kayvan Sadeghi, 2012. "Sequences of regressions and their independences," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 215-252, June.
    3. Marchetti, Giovanni M., 2006. "Independencies Induced from a Graphical Markov Model After Marginalization and Conditioning: The R Package ggm," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 15(i06).
    4. repec:jss:jstsof:15:i06 is not listed on IDEAS
    5. Marchetti, Giovanni M. & Stanghellini, Elena, 2008. "A note on distortions induced by truncation with applications to linear regression systems," Statistics & Probability Letters, Elsevier, vol. 78(6), pages 824-829, April.
    6. Nanny Wermuth & Kayvan Sadeghi, 2012. "Rejoinder on: Sequences of regressions and their independences," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 274-279, June.

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