Summary
In the log-linear model for bivariate probability functions the conditional and joint probabilities have a simple form. This property make the log-linear parametrization useful when modeling these probabilities is the focus of the investigation. On the contrary, in the log-linear representation of bivariate probability functions, the marginal probabilities have a complex form. So the log-linear models are not useful when the marginal probabilities are of particular interest. In this paper the previous statements are discussed and a model obtained from the log-linear one by imposing suitable constraints on the marginal probabilities is introduced.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aitchison, J. (1986),The Statistical Analysis of Compositional Data, Chapman and Hall, London.
Agresti, A. (1990),Categorical Data Analysis, Wiley, New York.
Andersen, E. (1991),The Statistical Analysis of Categorical Data, Springer Verlag, Berlin.
Birch, M. W. (1964), A new proof of the Pearson-Fisher Theorem,Ann. Math. Statist., 35, 718–824.
Colombi, R. (1994a), Alcune Alternative ai Modelli Log-Lineari nella analisi delle tabelle di contingenza,Atti della XXXVII Riunione Scientifica della S. I. S., 357–368, CISU, Roma.
Colombi, R. (1994b), A class of Log-linear Models with parameters subject to non-linear constraints,9th International Workshop on Statistical Modelling, Exeter 1994.
Dale, J. R. (1986), Global Cross-Ratio Models for Biariate, Discrete, Ordered Responses.Biometrics, 42, 909–917.
Darroch, J. N., Ratcliff, D. (1972), Generalized Iterative Scaling for Log-Linear models,The Annals of mathematical Statistics, 43, 5, p. 1470–1480.
Darroch, J. N., Speed, T. (1983), Additive and multiplicative models and interactions,The Annals of Statistics, 11, 3, pp. 724–738.
Fienberg, S., Meyer, M. (1983), Interative Proportional Fitting Procedure, inEncyclopedia of Statistical Sciences, vol. 4, John Wiley, New York.
Fitzmaurice, G. M., Laird, N. M. (1993), A likelihood-based method for analysing longitudinal binary responses,Biometrika, 90, p. 141–151.
Franconi, L. (1991),Modelli Probabilistici per l'Analisi, di Misurazioni Ripetute Dicotomiche, Tesi di Dottorato, IV Ciclo, Università degli Studi di Trento.
Hagenaars, A. (1990),Categorical Longitudinal Data, Sage Publications, London.
Ireland, C. T., Kullback, S. (1968), Contingency tables with given Marginals,Biometrika, 55, pp. 179–188.
Lang, J., Agresti, A. (1994), Simultaneously Modeling Joint and Marginal Distributions of Multivariate Categorical Responses,J.A.S.A., 89, pp. 625–632.
Liang, K. Y., Zeger, S. L., Qaqish B. (1992) Multivariate Regression Analyses for Categorical Data,J. R. Statist. Soc. B, 54, 1, p. 3–40.
Madsen, M. (1976), Statistical Analysis of Multiple Contingency Table-Two Examples,Scand. J. Statist., 3, 97–106.
Magnus, J. (1988),Linear Structures, Oxford University Press, London.
McCullagh, P., Nelder, J. A. (1989),Generalized Linear Models, Chapman and Hall, London.
McCullagh, P., (1989), Models for Discrete Multivariate Responses,Bull. Int. Statist. Inst., 53, 3, p. 407–417.
Molenberghs, G. (1992),A Full Maximum Likelihood Method for the Analysis of Multivariate Ordered Categorical Data, Universiteit Antwerpen, Departement Wiskunde en Informatica.
Molenberghs, G., Lesaffre, E. (1994), Marginal Modelling of Correlated Ordinal Data Using a Multivariate Plackett Distribution,J.A.S.A., 89, pp. 633–644.
Santner, T., Duffy, D. (1989),The Statistical Analysis of Discrete Data, Springer-Verlag, Berlin.
Author information
Authors and Affiliations
Additional information
This work was supported by a M.U.R.S.T. grant.
Rights and permissions
About this article
Cite this article
Colombi, R. A class of log-linear models with constrained marginal distributions. J. It. Statist. Soc. 4, 147–165 (1995). https://doi.org/10.1007/BF02589099
Issue Date:
DOI: https://doi.org/10.1007/BF02589099