The Annals of Statistics

Prior induction in log-linear models for general contingency table analysis.

S.P. Brooks and R. King

Full-text: Open access

Abstract

Log-linear modelling plays an important role in many statistical applications, particularly in the analysis of contingency table data.With the advent of powerful new computational techniques such as reversible jump MCMC, Bayesian analyses of these models, and in particular model selection and averaging, have become feasible. Coupled with this is the desire to construct and use suitably flexible prior structures which allow efficient computation while facilitating prior elicitation. The latter is greatly improved in the case where priors can be specified on interpretable parameters about which relevant experts can express their beliefs.

In this paper, we show how the specification of a general multivariate normal prior on the log-linear parameters induces a multivariate log- normal prior on the corresponding cell counts of a contingency table. We derive the parameters of this distribution in an explicit practical form and state the corresponding mean and covariances of the cell counts. We discuss the importance of these results in terms of applying both uninformative and informative priors to the model parameters and provide an illustration in the context of the analysis of a 23 contingency table.

Article information

Source
Ann. Statist., Volume 29, Issue 3 (2001), 715-747.

Dates
First available in Project Euclid: 24 December 2001

Permanent link to this document
https://projecteuclid.org/euclid.aos/1009210687

Digital Object Identifier
doi:10.1214/aos/1009210687

Mathematical Reviews number (MathSciNet)
MR1865338

Zentralblatt MATH identifier
1041.62050

Subjects
Primary: 62F15: Bayesian inference 62H99: None of the above, but in this section

Keywords
Bayesian analysis contingency table multivariate normal prior elicitation

Citation

King, R.; Brooks, S.P. Prior induction in log-linear models for general contingency table analysis. Ann. Statist. 29 (2001), no. 3, 715--747. doi:10.1214/aos/1009210687. https://projecteuclid.org/euclid.aos/1009210687


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References

  • Aitchison, J. and Brown, J. A. C. (1957). The Lognormal Distribution: With Special Reference to Its Uses in Economics. Cambridge Univ. Press.
  • Dawid, A. P. and Lauritzen, S. L. (1993). Hyper Markov laws in the statistical analysis of decomposable graphical models. Ann. Statist. 21 1272-1317.
  • Dellaportas, P. and Forster, J. J. (1999). Markov chain Monte Carlo model determination for hierarchical and graphical log-linear models. Biometrika 86 615-633.
  • Evans, M., Gilula, Z. and Guttman, I. (1993). Computational issues in the Bayesian analysis of categorical data: log-linear and Goodman's RC model. Statist. Sinica 3 391-406.
  • Forster, J. J. (1992). Models and marginal densities for multiway contingency tables. Ph.D. thesis, Univ. Nottingham.
  • Giudici, P., Green, P. J. and Tarantola, C. (1999). Efficient model determination for discrete graphical models. Technical report, Univ. Pavia, Italy.
  • Hook, E. B., Albright, S. G. and Cross, P. K. (1980). Use of binomial census and log-linear methods for estimating the prevalence of spina bifida in livebirths and the completeness of vital record reports in New York State. J. Amer. Epidemiology 112 750-758.
  • Hook, E. B. and Regal, R. R. (1995). Capture-recapture methods in epidemiology: methods and limitations. Epidemiologic Rev. 17 243-264.
  • King, R. and Brooks, S. P. (2001). On the Bayesian analysis of population size. Biometrika 88 317-336.
  • Knuiman, M. W. and Speed, T. P. (1988). Incorporating prior information into the analysis of contingency tables. Biometrics 44 1061-1071.
  • Madigan, D. and York, J. C. (1997). Bayesian methods for estimation of the size of a closed population. Biometrika 84 19-31.