Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bollen K (1989)Structural equations with latent variables, Wiley, New York
Browne MW (1980) Factor analysis of multiple batteries by maximum likelihood,British J. of Math. and Statist. Psychology 33: 184–199
Cox DR, Wermuth N (1996)Multivariate dependencies: models, analysis and interpretation, Chapman and Hall, London
Foulds LR (1992)Graphs theory applications, Springer
Frydenberg M, Lauritzen SL (1989) Decomposition of maximum likelihood in mixed interaction models.Biometrika 76: 539–555
Giudici P, Green PJ (1999) Markov Chain Monte Carlo Bayesian decomposable graphical Gaussian model determination.Biometrika 86: 785–801
Giudici P, Stanghellini E (2001) Bayesian inference for graphical factor analysis models.Psycometrica 66(4): 577–592
Lauritzen SL (1996)Graphical models. Oxford university Press
Poli I, Roverato A (2000) A genetic algorithm for graphical model selection.Journ. of Italian Stat. Soc. 7(2): 197–208
Schervish MJ (1995)Theory of statistics. Springer, New York
Speed TP, Kiiveri H (1986) Gaussian Markov distributions over finite graphs.Ann. Statist. 14(1): 138–150
Stanghellini E (1997) Identification of a single-factor model using graphical Gaussian rules.Biometrika 84: 241–4
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: 566–579
Thulasiraman K, Swamy MNS (1992)Graphs: theory and algorithms. Wiley
Vicard P (2000) On the identification of a single-factor model with correlated residuals.Biometrika 87: 199–205
Vicard P (2001) Detecting the identifiability of a single factor model using bipartite graphs. manuscript
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Tarantola, C., Vicard, P. Spanning trees and identifiability of a single-factor model. Statistical Methods & Applications 11, 139–152 (2002). https://doi.org/10.1007/BF02511482
Issue Date:
DOI: https://doi.org/10.1007/BF02511482