Advertisement

Statistics and Computing

, Volume 18, Issue 1, pp 1–13 | Cite as

Bayesian parsimonious covariance estimation for hierarchical linear mixed models

  • Sylvia Frühwirth-SchnatterEmail author
  • Regina Tüchler
Article

Abstract

We consider a non-centered parameterization of the standard random-effects model, which is based on the Cholesky decomposition of the variance-covariance matrix. The regression type structure of the non-centered parameterization allows us to use Bayesian variable selection methods for covariance selection. We search for a parsimonious variance-covariance matrix by identifying the non-zero elements of the Cholesky factors. With this method we are able to learn from the data for each effect whether it is random or not, and whether covariances among random effects are zero. An application in marketing shows a substantial reduction of the number of free elements in the variance-covariance matrix.

Keywords

Covariance selection Random-effects models Markov chain Monte Carlo Fractional prior Variable selection 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Albert, J.H., Chib, S.: Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts. J. Bus. Econ. Stat. 11, 1–15 (1993) CrossRefGoogle Scholar
  2. Chen, Z., Dunson, D.: Random effects selection in linear mixed models. Biometrics 59, 762–769 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  3. Frühwirth-Schnatter, S., Tüchler, R., Otter, T.: Bayesian analysis of the heterogeneity model. J. Bus. Econ. Stat. 22, 2–15 (2004) CrossRefGoogle Scholar
  4. Gelfand, A.E., Sahu, S.K., Carlin, B.P.: Efficient parametrisations for normal linear mixed models. Biometrika 82, 479–488 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  5. George, E.I., McCulloch, R.: Approaches for Bayesian variable selection. Stat. Sinica 7, 339–373 (1997) zbMATHGoogle Scholar
  6. Lindstrom, M., Bates, D.: Newton-Raphson and the EM-Algorithm for linear mixed-effects models for repeated measures data. J. Am. Stat. Assoc. 83, 1014–1022 (1988) zbMATHCrossRefMathSciNetGoogle Scholar
  7. Meng, X.-L., van Dyk, D.: Fast EM-type implementations for mixed effects models. J. Roy. Stat. Soc. B 60, 559–578 (1998) zbMATHCrossRefGoogle Scholar
  8. O’Hagan, A.: Fractional Bayes factors for model comparison. J. Roy. Stat. Soc. B 57, 99–118 (1995) zbMATHMathSciNetGoogle Scholar
  9. Otter, T., Tüchler, R., Frühwirth-Schnatter, S.: Capturing consumer heterogeneity in metric conjoint analysis using Bayesian mixture models. Int. J. Mark. Res. 21, 285–297 (2004) CrossRefGoogle Scholar
  10. Papaspiliopoulos, O., Roberts, G., Skold, M.: Non-centered parameterizations for hierarchical models and data augmentation. In: Bernardo, J.M., Bayarri, M.J., Berger, J.O., Dawid, A.P., Heckerman, D., Smith, A.F.M., West, M. (eds.) Bayesian Statistics 7, pp. 307–326. Oxford University Press, Oxford (2003) Google Scholar
  11. Pinheiro, J., Bates, D.: Unconstrained parameterizations for variance-covariance matrices. Stat. Comput. 6, 289–296 (1996) CrossRefGoogle Scholar
  12. Rossi, P.E., Allenby, G.M., McCulloch, R.: Bayesian Statistics and Marketing. Wiley Series in Probability and Statistics. Wiley, New York (2005) zbMATHGoogle Scholar
  13. Smith, M., Kohn, R.: Parsimonious covariance matrix estimation for longitudinal data. J. Am. Stat. Assoc. 97, 1141–1153 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  14. Tüchler, R., Frühwirth-Schnatter, S.: Bayesian parsimonious estimation of observed and unobserved heterogeneity. In: Verbeeke, G., Molenberghs, G., Aerts, M., Fieuws, S. (eds.) Proceedings of the 18th International Workshop on Statistical Modelling, pp. 427–431. Leuven, Belgium (2003) Google Scholar
  15. Van Dyk, D., Meng, X.-L.: The art of data augmentation. J. Comput. Graph. Stat. 10, 1–50 (2001) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Applied Statistics and EconometricsJohannes Kepler Universität LinzLinzAustria
  2. 2.Department of Statistics and MathematicsVienna University of Economics and Business AdministrationViennaAustria

Personalised recommendations