Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-07T18:53:25.812Z Has data issue: false hasContentIssue false
  • English
  • Français

Bayesian Model Comparison for the Order Restricted RC Association Model

Published online by Cambridge University Press:  01 January 2025

G. Iliopoulos ,
M. Kateri  and
I. Ntzoufras
G. Iliopoulos
Affiliation:
University of Piraeus
M. Kateri*
Affiliation:
University of Piraeus
I. Ntzoufras
Affiliation:
Athens University of Economics and Business
*
Requests for reprints should be sent to M. Kateri, Department of Statistics and Insurance Science, University of Piraeus, Piraeus, Greece. E-mail: mkateri@unipi.gr
Get access Rights & Permissions [Opens in a new window]

Abstract

Association models constitute an attractive alternative to the usual log-linear models for modeling the dependence between classification variables. They impose special structure on the underlying association by assigning scores on the levels of each classification variable, which can be fixed or parametric. Under the general row-column (RC) association model, both row and column scores are unknown parameters without any restriction concerning their ordinality. However, when the classification variables are ordinal, order restrictions on the scores arise naturally. Under such restrictions, we adopt an alternative parameterization and draw inferences about the equality of adjacent scores using the Bayesian approach. To achieve that, we have constructed a reversible jump Markov chain Monte Carlo algorithm for moving across models of different dimension and estimate accurately the posterior model probabilities which can be used either for model comparison or for model averaging. The proposed methodology is evaluated through a simulation study and illustrated using actual datasets.

Keywords

contingency tablesordinal variablesreversible jump MCMC algorithmequality of oddsBayesian model averaging
Type
Theory and Methods
Information
Psychometrika , Volume 74 , Issue 4 , December 2009 , pp. 561 - 587
Copyright
Copyright © 2009 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Agresti, A., Chuang, C., Kezouh, A. (1987). Order-restricted score parameters in association models for contingency tables. Journal of the American Statistical Association, 82, 619–623.CrossRefGoogle Scholar
American Psychiatric Association (1987).. DSM-III-R: Diagnostic and statistical manual of mental disorders, (3rd ed.). Washington: APA.Google Scholar
Anderson, C.J., Yu, H.T. (2007). Log-multiplicative association models as item response models. Psychometrika, 72, 5–23.CrossRefGoogle Scholar
Anderson, J.A. (1984). Regression and ordered categorical variables. Journal of the Royal Statistical Society B, 46, 1–30.CrossRefGoogle Scholar
Bartolucci, F., Forcina, A. (2002). Extended RC association models allowing for order restrictions and marginal modeling. Journal of the American Statistical Association, 97, 1192–1199.CrossRefGoogle Scholar
Brooks, S.P. (2002). Discussion of the paper by Spiegelhalter, Best, Carlin and van der Linde. Journal of the Royal Statistical Society B, 64, 616–618.Google Scholar
Burman, P. (2004). On some testing problems for sparse contingency tables. Journal of Multivariate Analysis, 88, 1–18.CrossRefGoogle Scholar
Chen, M.H., Ibrahim, J.G., Shao, Q.M. (2000). Power prior distributions for generalized linear models. Journal of Statistical Planning and Inference, 84, 121–137.CrossRefGoogle Scholar
Chuang, C. (1982). Empirical Bayes methods for a two-way multiplicative-interaction model. Communications in Statistics: Theory and Methods, 11, 2977–2989.CrossRefGoogle Scholar
Clogg, C.C. (1982). Some models for the analysis of association in multi-way cross-classifications having ordered categories. Journal of the American Statistical Association, 77, 803–815.CrossRefGoogle Scholar
De Rooij, M. (2007). The Distance Perspective of Generalized Biadditive Models: Scalings and Transformations. Journal of Computational and Graphical Statistics, 16, 210–227.CrossRefGoogle Scholar
Dellaportas, P., Forster, J.J., Ntzoufras, I. (2002). On Bayesian model and variable selection using MCMC. Statistics and Computing, 12, 27–36.CrossRefGoogle Scholar
Dong, J., Simonoff, J.S. (1995). A geometric combination estimator for d-dimensional ordinal sparse contingency tables. The Annals of Statistics, 23, 1143–1159.CrossRefGoogle Scholar
Evans, M., Gilula, Z., Guttman, I. (1993). Computational issues in the Bayesian analysis of categorical data: log-linear and Goodman’s RC model. Statistica Sinica, 3, 391–406.Google Scholar
Galindo-Garre, F., Vermunt, J.K. (2004). The order-restricted association model: two estimation algorithms and issues in testing. Psychometrika, 69, 641–654.CrossRefGoogle Scholar
George, E.I., McCulloch, R.E. (1993). Variable selection via Gibbs sampling. Journal of the American Statistical Association, 88, 881–889.CrossRefGoogle Scholar
Goodman, L.A. (1979). Simple models for the analysis of association in cross-classifications having ordered categories. Journal of the American Statistical Association, 74, 537–552.CrossRefGoogle Scholar
Goodman, L.A. (1981). Association models and canonical correlation in the analysis of cross-classifications having ordered categories. Journal of the American Statistical Association, 76, 320–334.Google Scholar
Goodman, L.A. (1986). Some useful extensions of the usual correspondence analysis approach and the usual log-linear models approach in the analysis of contingency tables (with a discussion and reply by the author). International Statistical Review, 54, 243–309.CrossRefGoogle Scholar
Green, P. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika, 82, 711–732.CrossRefGoogle Scholar
Hoeting, J.A., Madigan, D., Raftery, A.E., Volinski, C.T. (1999). Bayesian model averaging: a tutorial. Statistical Science, 14, 382–417.Google Scholar
Iliopoulos, G., Kateri, M., Ntzoufras, I. (2007). Bayesian estimation of unrestricted and order-restricted association models for a two-way contingency table. Computational Statistics and Data Analysis, 51, 4643–4655.CrossRefGoogle Scholar
Iliopoulou, K. (2004). Schizotypy and consumer behavior (in Greek). M.Sc. Thesis, Department of Business Administration. University of the Aegean, Chios, Greece, http://stat-athens.aueb.gr/~jbn/courses/diplomatikes/business/Iliopoulou(2004).pdf.Google Scholar
Kass, R.E., Raftery, A.E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.CrossRefGoogle Scholar
Kateri, M., Nicolaou, A., Ntzoufras, I. (2005). Bayesian inference for the RC(m) association model. Journal of Computational and Graphical Statistics, 14, 116–138.CrossRefGoogle Scholar
Kuo, L., Mallick, B. (1998). Variable selection for regression models. Sankhyã B, 60, 65–81.Google Scholar
Lee, R.D., Carter, L.R. (1992). Modeling and forecasting US mortality. Journal of the American Statistical Association, 87, 659–671.Google Scholar
Maxwell, A.E. (1961). Analyzing Qualitative Data, London: Methuen.Google Scholar
Maydeu-Olivares, A., Joe, H. (2006). Limited information goodness-of-fit testing in multidimensional contingency tables. Psychometrika, 71, 713–732.CrossRefGoogle Scholar
Raftery, A.E. (1995). Bayesian model selection in social research. Sociological Methodology, 25, 111–163.CrossRefGoogle Scholar
Richardson, S., Green, P.J. (1997). On Bayesian analysis of mixtures with an unknown number of components (with discussion). Journal of the Royal Statistical Society B, 59, 731–792.CrossRefGoogle Scholar
Ritov, Y., Gilula, Z. (1991). The order-restricted RC model for ordered contingency tables: estimation and testing of fit. The Annals of Statistics, 19, 2090–2101.CrossRefGoogle Scholar
Ritov, Y., Gilula, Z. (1993). Analysis of contingency tables by correspondence models subject to order constraints. Journal of the American Statistical Association, 88, 1380–1387.CrossRefGoogle Scholar
Simonoff, J.S. (1983). A penalty function approach to smoothing large sparse contingency tables. The Annals of Statistics, 11, 208–218.CrossRefGoogle Scholar
Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A. (2002). Bayesian measures of model complexity and fit (with discussion). Journal of the Royal Statistical Society B, 64, 583–639.CrossRefGoogle Scholar
Tarantola, C., Consonni, G., Dellaportas, P. (2008). Bayesian clustering of row effects models. Journal of Statistical Planning and Inference, 138, 2223–2235.CrossRefGoogle Scholar