Statistical Methods and Applications

, Volume 11, Issue 2, pp 139–152 | Cite as

Spanning trees and identifiability of a single-factor model

  • Claudia Tarantola
  • Paola Vicard
Statistical Methods


The aim of this paper is to propose conditions for exploring the class of identifiable Gaussian models with one latent variable. In particular, we focus attention on the topological structure of the complementary graph of the residuals. These conditions are mainly based on the presence of odd cycles and bridge edges in the complementary graph. We propose to use the spanning tree representation of the graph and the associated matrix of fundamental cycles. In this way it is possible to obtain an algorithm able to establish in advance whether modifying the graph corresponding to an identifiable model, the resulting graph still denotes identifiability.

Key words

Bridge edge graphical Gaussian models identifiability legal moves odd cycle spanning tree 


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  1. 1.
    Bollen K (1989)Structural equations with latent variables, Wiley, New YorkzbMATHGoogle Scholar
  2. 2.
    Browne MW (1980) Factor analysis of multiple batteries by maximum likelihood,British J. of Math. and Statist. Psychology 33: 184–199zbMATHMathSciNetGoogle Scholar
  3. 3.
    Cox DR, Wermuth N (1996)Multivariate dependencies: models, analysis and interpretation, Chapman and Hall, LondonzbMATHGoogle Scholar
  4. 4.
    Foulds LR (1992)Graphs theory applications, SpringerGoogle Scholar
  5. 5.
    Frydenberg M, Lauritzen SL (1989) Decomposition of maximum likelihood in mixed interaction models.Biometrika 76: 539–555zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Giudici P, Green PJ (1999) Markov Chain Monte Carlo Bayesian decomposable graphical Gaussian model determination.Biometrika 86: 785–801zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Giudici P, Stanghellini E (2001) Bayesian inference for graphical factor analysis models.Psycometrica 66(4): 577–592MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lauritzen SL (1996)Graphical models. Oxford university PressGoogle Scholar
  9. 9.
    Poli I, Roverato A (2000) A genetic algorithm for graphical model selection.Journ. of Italian Stat. Soc. 7(2): 197–208CrossRefGoogle Scholar
  10. 10.
    Schervish MJ (1995)Theory of statistics. Springer, New YorkzbMATHGoogle Scholar
  11. 11.
    Speed TP, Kiiveri H (1986) Gaussian Markov distributions over finite graphs.Ann. Statist. 14(1): 138–150zbMATHMathSciNetGoogle Scholar
  12. 12.
    Stanghellini E (1997) Identification of a single-factor model using graphical Gaussian rules.Biometrika 84: 241–4zbMATHCrossRefGoogle Scholar
  13. 13.
    Tarjan RE, Yannakakis M (1984) Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs.SIAM Journal of Computing 13: 566–579zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Thulasiraman K, Swamy MNS (1992)Graphs: theory and algorithms. WileyGoogle Scholar
  15. 15.
    Vicard P (2000) On the identification of a single-factor model with correlated residuals.Biometrika 87: 199–205zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Vicard P (2001) Detecting the identifiability of a single factor model using bipartite graphs. manuscriptGoogle Scholar

Copyright information

© Springer-Verlag 2002

Authors and Affiliations

  • Claudia Tarantola
    • 1
  • Paola Vicard
    • 2
  1. 1.Dipartimento di Economia Politica e Metodi QuantitativiUniversità di PaviaPaviaItaly
  2. 2.Dipartimento di EconomiaUniversità Roma TreRomaItaly

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