Heterogeneity among contingency tables diagnosed by hierarchical log-linear models and their effect on Biplots





simualate, biplots, model, heterogeneity, tables


The theory of singular value decomposition of matched matrices is used to verify the heterogeneity of rows, columns and between matched two-way tables. An exploratory analysis that can be visualized in biplots and through simulations studies with the hierarchical log-linear model using ordinary residuals and the components of residual deviance. The effect of heterogeneity was studied generating different sample sizes and their behavior was checked by adjusting Poisson’s model. We concluded that the model of ordinary residuals is the one that best reflects the degree of heterogeneity among the matched tables. Finally, na illustrative example is presented in order to guide the researcher to interpret the relationship between the results of the log-linear models with the biplots considering the effects between the sum and difference between the tables.

Author Biographies

Carla Regina Guimarães Brighenti, Universidade Federal de São João del-Rei- UFSJ

Prof. Dr.,  Department of Zootechnics, UFSJ, São João del Rei, MG, Brazil.

Daniela Aparecida Mafra, Universidade Federal de Lavras - UFLA

PhD student in Agricultural Statistics and Experimentation, UFLA, MG, Brazil.

Marcelo Angelo Cirillo, Universidade Federal de Lavras - UFLA

Prof. Dr., Department of Statistics, UFLA, Lavras, MG, Brazil.


ABDI, H.; WILLIANS, L.; VALENTIN, D. Multiple factor analysis: principal component analysis for multitable and multiblock data sets. Wiley Interdisciplinary reviews: computational statistics, Hoboken, v. 5, n. 2, p. 149-179, 2013. DOI: https://doi.org/10.1002/wics.1246.

AITCHISON, J.; GREENACRE, M. Biplots of compositional data. Journal of the Royal Statistical Society. Series C (Applied Statistics), Oxford, v. 51, n. 4, p. 375-392, 2001.

BÉCUE-BERTAUT, M.; PAGÉS, J. Multiple factor analy sis and clustering of a mixture of quantitative, categorical and frequency data. Computational Statistics & Data Analysis, Amsterdam, v. 52, n. 6, p. 3255-3268, 2008. DOI: https://doi.org/10.1016/j.csda.2007.09.023.

BEH, E. J. Simple correspondence analysis using adjusted residuals. Journal of Statistical Planning and Inference, Amsterdam, v. 142, n. 4, p. 965-973, 2012. DOI: https://doi.org/10.1016/j.jspi.2011.11.004.

BRZEZINSKA, J. Hierarquical log-linear models for con- tingency tables. Acta Universitatis Lodziensis Folia Oeconomica, ?ód?, v. 269, p. 123-129, 2012.

CARLIER, A.; KROONENBERG, P. M. The case of the French cantons: an application of three-way correspon- dence analysis. In: BLASIUS, J.; GREENACRE, M. Visualization of categorical data. Cambridge: Academic Press, 1998. p. 253-275.

DOSSOU-GBÉTÉ, S.; GRORUD, A. Biplots for matched two-way tables. Annales de la Faculté des sciences de Toulouse: Mathématiques, Toulouse, v. 11, n. 4, p. 469-483, 2002.

FALGUEROLLES, A. GBMs: GLMs with bilinear terms. In: BETHLEHEM, J. G.; VAN DER HEIJDEN; P.G.M. Compstat 2000. Heidelberg: Physica, 2000. p. 53-64.

FALGUEROLLES, A.; FRANCIS, B. An algorithmic approach to bilinear models for two-way contingency tables. In: DIDAY, E.; LECHEVALLIER, Y.; SCHADER, M.; BERTRAND, P.; BURTSCHY, B. New approaches in classification and data analysis. Berlim: Springer Berlin Heidelberg, 1994. p. 518-524.

GREENACRE, M. Biplots in correspondence analysis. Journal of Applied Statistics, London, v. 20, n. 2, p. 251 - 269, 1993. DOI: https://doi.org/10.1080/02664769300000021.

GREENACRE, M. Contribution biplots. Journal of Computational and Graphical Statistics, Alexandria, v. 22, n. 1, p. 107-122, 2013.

GREENACRE, M. Singular value decomposition of matched matrices. Journal of Applied Statistics, London, v. 30, n. 10, p. 1101-1113, 2003. DOI: https://doi.org/10.1080/0266476032000107132.

POWERS, D. A; XIE, Y. Statistical methods for categorical data analysis. San Diego: Academic Press, 2000.

R CORE TEAM. R: a language and environment for statistical computing. Vienna: R Foundation for Statistical Computing, 2015. Available from: http://www.R-project.org. Acess in: Nov. 10, 2022.

VAN DER HEIJDEN, P. G. M.; DE FALGUEROLLES, A.; DE LEEUW, J. A combined approach to contingency table analysis using correspondence analysis (with Discussion). Journal of the Royal Statistical Society: Series C (Applied Statistics), Oxford, v. 38, n. 249-292, 1989.

VAN DER HEIJDEN, P. G. M.; MOOIJAART, A. S. Some new-bilinear models for the analysis of asymmetry in a square contingency table. Sociological Methods and Research, Beverly Hills, v. 24, n. 1, p. 7-29, 1995. DOI: https://doi.org/10.1177/0049124195024001002.




How to Cite

Brighenti, C. R. G., Mafra, D. A., & Cirillo, M. A. (2022). Heterogeneity among contingency tables diagnosed by hierarchical log-linear models and their effect on Biplots. Semina: Ciências Exatas E Tecnológicas, 43(2), 135–146. https://doi.org/10.5433/1679-0375.2022v43n2p135



Original Article

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