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Bayesian Inference for Graphical Factor Analysis Models

Published online by Cambridge University Press:  01 January 2025

Paolo Giudici  and
Elena Stanghellini
Paolo Giudici
Affiliation:
Dipartimento di Economia Politica e Metodi Quantitativi, Università di Pavia
Elena Stanghellini*
Affiliation:
Dipartimento di Scienze Statistiche, Università di Perugia
*
Requests for reprints should be sent to Elena Stanghellini, Dipartimento di Scienze Statistiche, Università di Perugia, Via A. Pascoli, 1 - C.P. 1315 Succ. 1, 06100 Perugia, ITALY. E-Mail: stanghel@stat.unipg.it
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Abstract

We generalize factor analysis models by allowing the concentration matrix of the residuals to have nonzero off-diagonal elements. The resulting model is named graphical factor analysis model. Allowing a structure of associations gives information about the correlation left unexplained by the unobserved variables, which can be used both in the confirmatory and exploratory context. We first present a sufficient condition for global identifiability of this class of models with a generic number of factors, thereby extending the results in Stanghellini (1997) and Vicard (2000). We then consider the issue of model comparison and show that fast local computations are possible for this purpose, if the conditional independence graphs on the residuals are restricted to be decomposable and a Bayesian approach is adopted. To achieve this aim, we propose a new reversible jump MCMC method to approximate the posterior probabilities of the considered models. We then study the evolution of political democracy in 75 developing countries based on eight measures of democracy in two different years.

Keywords

factor analysisgraphical Gaussian modelsidentificationmodel comparisonreversible jump MCMC
Type
Articles
Information
Psychometrika , Volume 66 , Issue 4 , December 2001 , pp. 577 - 591
Copyright
Copyright © 2001 The Psychometric Society

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Footnotes

We acknowledge support from M.U.R.S.T. of Italy and from the European Science Foundation H.S.S.S. Network. We are grateful to the referees and the Editor for many useful suggestions and comments which led to a substantial improvement of the paper. We also thank Nanny Wermuth for stimulating discussions and Kenneth A. Bollen for kindly providing us with the data-set.

References

Bartholomew, D.J. (1994). Bayes' theorem in latent variable modelling. In Freeman, P.R., & Smith, A.F.M. (Eds.), Aspects of uncertainty: A tribute to D.V. Lindley (pp. 41–50). New York, NY: Wiley.Google Scholar
Bollen, K.A. (1989). Structural equations with latent variables. New York, NY: Wiley.CrossRefGoogle Scholar
Bozdogan, H., & Shigemasu, K. (1998). Bayesian factor analysis model and choosing the number of factors using a new informational complexity criterion. In Rizzi, A., Vichi, M., & Bock, H. (Eds.), Advances in data science and classification (pp. 335–342). Berlin, Germany: Springer.CrossRefGoogle Scholar
Browne, M.W. (1980). Factor analysis of multiple batteries by maximum likelihood. British Journal of Mathematical and Statistical Psychology, 33, 184–199.CrossRefGoogle Scholar
Cox, D.R., & Wermuth, N. (1993). Linear dependencies represented by chain graphs. Statistical Science, 8, 204–283.CrossRefGoogle Scholar
Cox, D.R., & Wermuth, N. (1994). Tests for linearity, multivariate normality and the adequacy of linear scores. Applied Statistics, 43, 347–335.CrossRefGoogle Scholar
Cox, D.R., & Wermuth, N. (2001). Multivariate dependencies. Models analysis and interpretation. London, U.K.: Chapman and Hall.Google Scholar
Cox, D.R., & Wermuth, N. (2000). A sweep operator for triangular matrices and its statistical applications. Mannheim, Germany: Centre for Survey Research and Methodology.Google Scholar
Dawid, A.P., & Lauritzen, S.L. (1993). Hyper Markow Laws in the statistical analysis of decomposable graphical models. Annals of Statistics, 21, 1272–1317.CrossRefGoogle Scholar
Edwards, D. (1995). Introduction to graphical modelling. New York, NY: Springer-Verlag.CrossRefGoogle Scholar
Edwards, D. (2000). Introduction to graphical modelling 2nd ed., New York, NY: Springer-Verlag.CrossRefGoogle Scholar
Frydenberg, M., & Lauritzen, S.L. (1989). Decomposition of maximum likelihood in mixed graphical interaction models. Biometrika, 76, 539–555.CrossRefGoogle Scholar
Giudici, P., & Green, P.J. (1999). Markov Chain Monte Carlo Bayesian decomposable graphical Gaussian model determination. Biometrika, 84, 785–801.CrossRefGoogle Scholar
Green, P.J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 711–732.CrossRefGoogle Scholar
Lauritzen, S.L. (2001). Graphical models. Oxford, England: Oxford University Press.Google Scholar
Lopes, H., & West, M. (1998). Model uncertainty in factor analysis. Durham, NC: Duke University, Institute of Statistics and Decision Sciences.Google Scholar
Pearl, J. (1998). Graphs, causality and structural equation models. Sociological Methods and Research, 27, 226–284.CrossRefGoogle Scholar
Press, S.J., & Shigemasu, K. (1989). Bayesian Inference in Factor Analysis. In Gleser, L.J., Perlman, M.D., Press, S.J., & Sampson, A.R. (Eds.), Contributions to probability and statistics: Essays in honour of Ingram Olkin (pp. 271–287). New York, NY: Springer Verlag.CrossRefGoogle Scholar
Richardson, S., & Green, P.J. (1997). On Bayesian analysis of mixtures with an unknown number of components. Journal of the Royal Statistical Society, Series B, 59, 731–758.CrossRefGoogle Scholar
Stanghellini, E. (1997). Identification of a single-factor model using graphical Gaussian rules. Biometrika, 84, 241–244.CrossRefGoogle Scholar
Vicard, P. (2000). On the identification of a single-factor model with correlated residuals. Biometrika, 87, 199–205.CrossRefGoogle Scholar
Wright, S. (1923). The Theory of path coefficients: a reply to Niles' criticism. Genetics, 8, 239–255.CrossRefGoogle ScholarPubMed
Wright, S. (1934). The method of path coefficients. Annals of Statistics, 5, 161–215.CrossRefGoogle Scholar