Abstract
Most multivariate tests are based on the hypothesis of multinormality. But often this hypothesis fails, or we have variables that are non quantitative. On the other hand we can deal with a large number of variables. Defining probabilistic models with mixed data is not easy. However, it is always possible to define a measure of distance between two observations. We prove that the use of distances can provide alternative tests for comparing several populations when the data are of general type. This approach is illustrated with three real data examples. We also define and study a measure of association between two data sets and make a Bayesian extension of the so-called distance-based discriminant rule.
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References
Anderson, M. J. (2006). Distance-based tests for homogeneity of multivariate dispersions. Biometrics, 62:245–253.
Arenas, C. and Cuadras, C. M. (2002). Recent statistical methods based on distances. Contributions to Science, 2:183–191.
Arenas, C. and Cuadras, C. M. (2004). Comparing two methods for joint representation of multivariate data. Communications in Statistics, Simulation and Computation, 33:415–430.
Cox, T. F. and Cox, M. A. A. (1994). Multidimensional Scaling. Chapman and Hall, London.
Cuadras, C. M. (1989). Distance analysis in discrimination and classification using both continuous and categorical variables. In Statistical Data Analysis and Inference, (Ed. Y. Dodge), pp. 459–473. Elsevier Science Publishers B. V. (North–Holland), Amsterdam.
Cuadras, C. M. (1992). Some examples of distance based discrimination. Biometrical Letters, 29:3–20.
Cuadras, C. M. and Arenas, C. (1990). A distance based regression model for prediction with mixed data. Communications in Statistics, Theory and Methods, 19:2261–2279.
Cuadras, C. M. and Fortiana, J. (1995). A continuous metric scaling solution for a random variable. Journal of Multivariate Analysis, 52:1–14.
Cuadras, C. M. and Fortiana, J. (2004). Distance-based multivariate two sample tests. In Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis and Quality of Life, (Eds. M. S. Nikulin, N. Balakrishnan, M. Mesbah, N. Limnios), 273–290. Birkhauser, Boston.
Cuadras, C. M., Arenas, C., and Fortiana, J. (1996). Some computational aspects of a distance-based model for prediction. Communications in Statistics, Simulation and Computation, 25:593–609.
Cuadras, C. M., Atkinson, R. A., and Fortiana, J. (1997a). Probability densities from distances and discriminant analysis. Statistics and Probability Letters, 33:405–411.
Cuadras, C. M., Fortiana, J., and Oliva, F. (1997b). The proximity of an individual to a population with applications in discriminant analysis. Journal of Classification, 14:117–136.
Cuadras, C. M., Cuadras, D., and Lahlou, Y. (2006). Principal directions of the general Pareto distribution with applications. Journal of Statistical Planning and Inference, 136:2572–2583.
Cuadras, C. M. and Lahlou, Y. (2000). Some orthogonal expansions for the logistic distribution. Communications in Statistics, Theory and Methods, 29:2643–2663.
Escoufier, Y. (1973). Le trataiment des variables vectorielles. Biometrics, 29:751–760.
Flury, B. (1997). A First Course in Multivariate Statistics. Springer-Verlag, New York.
Gower, J. C. (1966). Some distance properties of latent roots and vector methods in multivariate analysis. Biometrika, 53:315–328.
Gower, J. C. and Legendre, P. (1986). Metric and Euclidean properties of dissimilarity coefficients. Journal of Classification, 3:5–48.
Liu, Z. J. and Rao, C. R. (1995). Asymptotic distribution of statistics based on quadratic entropy and bootstrapping. Journal of Statistical Planning and Inference, 43:1–18.
Mardia, K. V, Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
Rao, C. R. (1982). Diversity: its measurement, decomposition, apportionment and analysis. Sankhya A, 44:1–21.
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Cuadras, C.M. (2008). Distance Based Association and Multi-Sample Tests for General Multivariate Data. In: Advances in Mathematical and Statistical Modeling. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4626-4_5
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DOI: https://doi.org/10.1007/978-0-8176-4626-4_5
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