×

Symmetrical and non-symmetrical variants of three-way correspondence analysis for ordered variables. (English) Zbl 07473935

Summary: In the framework of multi-way data analysis, this paper presents symmetrical and non-symmetrical variants of three-way correspondence analysis that are suitable when a three-way contingency table is constructed from ordinal variables. In particular, such variables may be modelled using general recurrence formulae to generate orthogonal polynomial vectors instead of singular vectors coming from one of the possible three-way extensions of the singular value decomposition. As we shall see, these polynomials, that until now have been used to decompose two-way contingency tables with ordered variables, also constitute an alternative orthogonal basis for modelling symmetrical, non-symmetrical associations and predictabilities in three-way contingency tables. Consequences with respect to modelling and graphing will be highlighted.

MSC:

62-XX Statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agresti, A. and Gottard, A. (2007). Independence in multi-way contingency tables: S. N. Roy’s breakthroughs and later developments. J. Statist. Plann. Inference 137 3216-3226. · Zbl 1119.62052 · doi:10.1016/j.jspi.2007.03.006
[2] Alwin, D. F. and Krosnick, J. A. (1985). The measurement of values in surveys: A comparison of ratings and rankings. Public Opin. Q. 49 535-552.
[3] Beh, E. J. (1997). Simple correspondence analysis of ordinal cross-classifications using orthogonal polynomials. Biom. J. 39 589-613. · Zbl 1127.62370
[4] Beh, E. J. (1998). A comparative study of scores for correspondence analysis with ordered categories. Biom. J. 40 413-429. · Zbl 1008.62612
[5] Beh, E. J. and Davy, P. J. (1998). Partitioning Pearson’s chi-squared statistic for a completely ordered three-way contingency table. Aust. N. Z. J. Stat. 40 465-477. · Zbl 1127.62364 · doi:10.1111/1467-842X.00050
[6] Beh, E. J. and Davy, P. J. (1999). Partitioning Pearson’s chi-squared statistic for a partially ordered three-way contingency table. Aust. N. Z. J. Stat. 41 233-246. · Zbl 1045.62519 · doi:10.1111/1467-842X.00077
[7] Beh, E. J. and Lombardo, R. (2014). Correspondence Analysis: Theory, Practice and New Strategies. Wiley Series in Probability and Statistics. Wiley, Chichester. · Zbl 1304.62001 · doi:10.1002/9781118762875
[8] Beh, E. J. and Lombardo, R. (2019). Multiple and multiway correspondence analysis. Wiley Interdiscip. Rev.: Comput. Stat. 11 e1464. · doi:10.1002/wics.1464
[9] Beh, E. J. and Lombardo, R. (2021). Features of the polynomial biplot of ordered contingency tables. J. Comput. Graph. Statist. To appear.
[10] Beh, E. J., Simonetti, B. and D’Ambra, L. (2007). Partitioning a non-symmetric measure of association for three-way contingency tables. J. Multivariate Anal. 98 1391-1411. · Zbl 1178.62063 · doi:10.1016/j.jmva.2007.01.011
[11] Benzécri, J.-P. (1973). L’analyse des Données. Vols. I, II. Dunod, Paris. · Zbl 0297.62038
[12] Benzécri, J. P. (1977). Histoire et préhistoire de l’analyse des données. Partie V: L’analyse des correspondances. Cah. Anal. Données 2 9-40.
[13] Böckenholt, U. and Böckenholt, I. (1990). Canonical analysis of contingency tables with linear constraints. Psychometrika 55 633-649.
[14] Breiman, L. (2001). Statistical modeling: The two cultures. Statist. Sci. 16 199-231. · Zbl 1059.62505 · doi:10.1214/ss/1009213726
[15] Carlier, A. and Kroonenberg, P. M. (1996). Biplots and decompositions in two-way and three-way correspondence analysis. Psychometrika 61 355-373. · Zbl 0876.62048
[16] Carlier, A. and Kroonenberg, P. M. (1998). Three-way correspondence analysis: The case of the French cantons. In Visualization of Categorical Data (J. Blasius and M. Greenacre, eds.) 253-275. Academic Press, New York.
[17] Carroll, J. D. and Chang, J. J. (1970). Analysis of individual differences in multidimensional scaling via an \(n\)-way generalization of Eckart-Young decomposition. Psychometrika 35 283-319. · Zbl 0202.19101
[18] Ceulemans, E. and Kiers, H. A. L. (2006). Selecting among three-mode principal component models of different types and complexities: A numerical convex hull based method. Br. J. Math. Stat. Psychol. 59 133-150. · doi:10.1348/000711005X64817
[19] Ceulemans, E. and Kiers, H. A. L. (2009). Discriminating between strong and weak structures in three-mode principal component analysis. Br. J. Math. Stat. Psychol. 62 601-620. · doi:10.1348/000711008X369474
[20] Ceulemans, E., Timmerman, M. and Kiers, H. A. L. (2011). The CHULL procedure for selecting among multilevel component solutions. Chemom. Intell. Lab. Syst. 106 12-20.
[21] Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Mathematics and Its Applications 13. Gordon and Breach Science Publishers, New York. · Zbl 0389.33008
[22] Chihara, T. S. (1990). The three term recurrence relation and spectral properties of orthogonal polynomials. In Orthogonal Polynomials (Columbus, OH, 1989). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 294 99-114. Kluwer Academic, Dordrecht. · Zbl 0705.42019
[23] Choulakian, V. (1988). Exploratory analysis of contingency tables by loglinear formulation and generalizations of correspondence analysis. Psychometrika 53 235-250. · Zbl 0718.62117 · doi:10.1007/BF02294135
[24] Clogg, C. C. (1982). Some models for the analysis of association in multiway cross-classifications having ordered categories. J. Amer. Statist. Assoc. 77 803-815.
[25] D’Ambra, L. and Lauro, N. (1989). Nonsymmetrical analysis of three-way contingency tables. In Multiway Data Analysis (Rome, 1988) 301-315. North-Holland, Amsterdam.
[26] Emerson, P. L. (1968). Numerical construction of orthogonal polynomials from general recurrence formula. Biometrics 24 696-701.
[27] Favard, J. (1935). Sur les polynomes de Tchebicheff. C. R. Acad. Sci. 200 2052-2053. · Zbl 0012.06205
[28] Gini, C. (1912). Variabilitá e Mutuabilitá [Variability and Mutability]. Contributo Allo Studio delle Distribuzioni e delle Relazioni Statistiche. Cuppini Press, Bologna, Italy.
[29] Goodman, L. A. and Kruskal, W. H. (1954). Measures of association for cross classifications. J. Amer. Statist. Assoc. 49 732-764. · Zbl 0056.12801
[30] Gower, J. C., Le Roux, N. J. and Gardner-Lubbe, S. (2016). Biplots: Qualititive data. Wiley Interdiscip. Rev.: Comput. Stat. 8 82-111. · doi:10.1002/wics.1377
[31] Greenacre, M. J. (1984). Theory and Applications of Correspondence Analysis. Academic Press [Harcourt Brace Jovanovich, Publishers], London. · Zbl 0555.62005
[32] Greenacre, M. J. (1990). Some limitations of multiple correspondence analysis. Comput. Stat. Q. 3 249-256. · Zbl 0726.62087
[33] Greenacre, M. J. (2017). Correspondence Analysis in Practice, 3rd ed. CRC Press/CRC, Barcelona. · Zbl 1352.62003
[34] Greenacre, M. J. and Blasius, J. (2006). Multiple Correspondence Analysis and Related Methods. Chapman & Hall/CRC, Boca Raton. · Zbl 1277.62156
[35] Harshman, R. A. (1970). Foundation of the PARAFAC procedure: Models and conditions for an explanatory multi-modal factor analysis. UCLA Work. Pap. Phon. 16 1-84.
[36] Kahle, L. R. (1983). Social Values and Social Change: Adaptation to Life in America. Praeger, New York.
[37] Kateri, M. (2014). Contingency Table Analysis: Methods and Implementation Using R. Statistics for Industry and Technology. Birkhäuser/Springer, New York. · Zbl 1291.62012 · doi:10.1007/978-0-8176-4811-4
[38] Kiers, H. A. L. (1989). Three-Way Methods for the Analysis of Qualitative and Quantitative Two-Way Data. DSWO Press, Leiden, NL.
[39] Kiers, H. A. L. (2000). Towards a standardized notation and terminology in multiway analysis. Chemom. Intell. Lab. Syst. 14 105-122.
[40] Kroonenberg, P. M. (1983). Three-Mode Principal Component Analysis: Theory and Applications. DSWO Press, Leiden, NL. (Errata, 1989; available from the author).
[41] Kroonenberg, P. M. (1992). Multilinear models: Applications in spectroscopy: Comment: PARAFAC in three-way land. Statist. Sci. 7 312-314.
[42] Kroonenberg, P. M. (2008). Applied Multiway Data Analysis. Wiley Series in Probability and Statistics. Wiley Interscience, Hoboken, NJ. · Zbl 1160.62002 · doi:10.1002/9780470238004
[43] Kroonenberg, P. M. (2014). History of multiway component analysis and three-way correspondence analysis. In Visualization and Verbalization of Data (M. Greenacre and J. Blasius, eds.) 78-93. CRC Press, Boca Raton, FL.
[44] Kroonenberg, P. M. (2020). Multiway extensions of the SVD. In Advanced Studies in Behaviormetrics and Data Science (T. Imaizumi, A. Nakayama and S. Yokoyama, eds.) 141-157. Springer, Singapore.
[45] Kroonenberg, P. M. and Anderson, C. J. (2006). Additive and multiplicative models for three-way contingency tables: Darroch (1974) revisited. In Multiple Correspondence Analysis and Related Methods (M. Greenacre and J. Blasius, eds.) 455-486. Chapman & Hall, London. · Zbl 1277.62152
[46] Kroonenberg, P. M. and de Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika 45 69-97. · Zbl 0431.62035 · doi:10.1007/BF02293599
[47] Kroonenberg, P. M. and Lombardo, R. (1999). Non-symmetric correspondence analysis: A tool for analysing contingency tables with a dependence structure. Multivar. Behav. Res. 34 367-397.
[48] Kroonenberg, P. M. and Oort, F. J. (2003). Three-mode analysis of multimode covariance matrices. Br. J. Math. Stat. Psychol. 56 305-335. · doi:10.1348/000711003770480066
[49] Lancaster, H. O. (1951). Complex contingency tables treated by the partition of \[{\chi^2}\]. J. Roy. Statist. Soc. Ser. B 13 242-249. · Zbl 0045.22902
[50] Lauro, N. and D’Ambra, L. (1984). L’analyse non symétrique des correspondances. In Data Analysis and Informatics, III (Versailles, 1983) 433-446. North-Holland, Amsterdam.
[51] Light, R. J. and Margolin, B. H. (1971). An analysis of variance for categorical data. J. Amer. Statist. Assoc. 66 534-544. · Zbl 0222.62035
[52] Lin, Y. and Zhang, H. H. (2006). Component selection and smoothing in multivariate nonparametric regression. Ann. Statist. 34 2272-2297. · Zbl 1106.62041 · doi:10.1214/009053606000000722
[53] Loisel, S. and Takane, Y. (2016). Partitions of Pearson’s chi-square statistic for frequency tables: A comprehensive account. Comput. Statist. 31 1429-1452. · Zbl 1348.65034 · doi:10.1007/s00180-015-0619-1
[54] Lombardo, R. (1994). Modelli di Decomposizione per L’analisi della Dipendenza Nelle Tabelle di Contingenza a Tre-Vie. [Decomposition Models for the Analysis of Three-Way Contingency Tables]. Tesi di Dottorato di Ricerca in Statistica Computazionale e Applicazioni VI Cicio. Università di Napoli, Italy.
[55] Lombardo, R. and Beh, E. J. (2017). Three-way correspondence analysis for ordinal-nominal variables. In SIS 2017 Statistics and Data Science: New Challenges, New Generations, \(28-30 June 2017 Florence ( Italy )\). Proceedings of the Conference of the Italian Statistical Society 613-620. ISBN:978-88-6453-521-0.
[56] Lombardo, R., Beh, E. J. and Kroonenberg, P. M. (2016). Modelling trends in ordered correspondence analysis using orthogonal polynomials. Psychometrika 81 325-349. · Zbl 1345.62157 · doi:10.1007/s11336-015-9448-y
[57] Lombardo, R., Carlier, A. and D’Ambra, L. (1996). Nonsymmetric correspondence analysis for three-way contingency tables. Methodologica 4 59-80.
[58] Lombardo, R., Takane, Y. and Beh, E. J. (2020). Familywise decompositions of Pearson’s chi-square statistic in the analysis of contingency tables. Adv. Data Anal. Classif. 14 629-649. · Zbl 1499.62186 · doi:10.1007/s11634-019-00374-7
[59] Lorenzo-Seva, U., Timmerman, M. E. and Kiers, H. AL. (2011). The Hull method for selecting the number of common factors. Multivar. Behav. Res. 46 340-364. 2011. · doi:10.1080/00273171.2011.564527
[60] Marcotorchino, F. (1985). Utilisation des Comparaisons Par Paires en Statistique des Contingencies: Partie III. [Use of Paired Comparisons in Contingency Statistics. Part III.] Étude du Centre Scientifique No F 081. IBM, Paris, France.
[61] Nishisato, S. (1980). Analysis of Categorical Data: Dual Scaling and Its Applications. Mathematical Expositions 24. Univ. Toronto Press, Toronto, Canada. · Zbl 0487.62001
[62] Rayner, J. C. W. and Beh, E. J. (2009). Towards a better understanding of correlation. Stat. Neerl. 63 324-333. · doi:10.1111/j.1467-9574.2009.00425.x
[63] Rayner, J. C. W. and Best, D. J. (1996). Smooth extensions of Pearsons’s product moment correlation and Spearman’s rho. Statist. Probab. Lett. 30 171-177. · Zbl 0861.62043 · doi:10.1016/0167-7152(95)00216-2
[64] Rodrigue, N., Guillet, M., Fortin, J. and Martin, J. F. (2000). Comparing information obtained from ranking and descriptive tests of four sweet corn products. Food Qual. Prefer. 11 47-54.
[65] Smilde, A. K., Bro, R. and Geladi, P. (2004). Multi-Way Analysis: Applications in the Chemical Sciences. Wiley, Chichester.
[66] Takane, Y. and Jung, S. (2009). Tests of ignoring and eliminating in nonsymmetric correspondence analysis. Adv. Data Anal. Classif. 3 315-340. · Zbl 1306.62136 · doi:10.1007/s11634-009-0054-7
[67] Takane, Y., Yanai, H. and Mayekawa, S. (1991). Relationships among several methods of linearly constrained correspondence analysis. Psychometrika 56 667-684. · Zbl 0760.62057 · doi:10.1007/BF02294498
[68] Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika 31 279-311. · doi:10.1007/BF02289464
[69] Van Herk, H. and Van de Velden, M. (2007). Insight into the relative merits of rating and ranking in a cross-national context using three-way correspondence analysis. Food Qual. Prefer. 18 1096-1105.
[70] van der Heijden, P. G. M., de Falguerolles, A. and de Leeuw, J. (1989). A combined approach to contingency table analysis using correspondence analysis and log-linear analysis. J. R. Stat. Soc. Ser. C. Appl. Stat. 38 249-292. · Zbl 0707.62114 · doi:10.2307/2348058
[71] Wahba, G. (1990). Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics 59. SIAM, Philadelphia, PA. · Zbl 0813.62001 · doi:10.1137/1.9781611970128
[72] Wahba, G., Wang, Y., Gu, C., Klein, R. and Klein, B. (1995). Smoothing spline ANOVA for exponential families, with application to the Wisconsin Epidemiological Study of Diabetic Retinopathy. Ann. Statist. 23 1865-1895 · Zbl 0854.62042 · doi:10.1214/aos/1034713638
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.