Abstract
Lazraq and Cléroux (Psychometrika, 2002, 411–419) proposed a test for identifying the number of significant components in redundancy analysis. This test, however, is ill-conceived. A major problem is that it regards each redundancy component as if it were a single observed predictor variable, which cannot be justified except for the rare situations in which there is only one predictor variable. Consequently, the proposed test leads to drastically biased results, particularly when the number of predictor variables is large, and it cannot be recommended for use. This is shown both theoretically and by Monte Carlo studies.
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References
Anderson, T.W. (1951). Estimating linear restrictions on regression coefficients for multivariate normal distributions. Annals of Mathematical Statistics 22, 327–351.
Bartlett, M.S. (1951). The goodness of fit of a single hypothetical discriminant function in the case of several groups. Annals of Eugenics 16, 199–214.
ter Braak, C.J.F., Šmilauer, P. (1998). CANOCO Reference Manual and User’s Guide to Canoco for Windows: Software for Canonical Community Ordination (version 4).Ithaka N. Y: Microcomputer Power.
Horn, J.L. (1965). A rationale and the test of the number of factors in factor analysis. Psychometrika, 30, 179–185.
Lambert, Z.V., Wildt, A.R., Durand, R.M. (1988). Redundancy analysis: An alternative to canonical correlation and multivariate multiple regression in exploring interset association. Psychological Bulletin, 104, 282–289.
Lancaster, H.O. (1963). Canonical correlations and partitions of χ2. Quarterly Journal of Mathematics, 14, 220–224.
Lazraq, A., Cléroux, R. (1988). Un algorithme pas à pas de sélection de variables en régression linéaire multivariée [A stepwise variable selection algorithm in multivariate linear regression]. Statistique et Analyse des donnes, 13, 39–58.
Lazraq, A.., Cléroux, R. (2002). Testing the significance of the successive components in redundancy analysis. Psychometrika, 67, 411–419.
Legendre, P., Legendre, L. (1998). Numerical Ecology. Second English edition. Oxford: Elsevier.
Legendre, P., & ter Braak, C.J.F. (in preparation). Partial canonical analysis, variation partitioning, and tests of significance of eigenvalues in redundancy analysis.
Rao, C.R. (1964). The use and interpretation of principal component analysis in applied research. Sankhya A, 26, 329–358.
Reinsel, G.C., Velu, R.P. (1998). Multivariate Reduced-rank Regression. New York, Springer.
Takane, Y., Hwang, H. (2002). Generalized constrained canonical correlation analysis. Multivariate Behavioral Research, 37, 163–195.
Takane, Y., van der Heijden, P.G.M., Browne, M.W. (2003). On likelihood ratio tests for dimensionality selection. In T. Higuchi, Y. Iba, M. Ishiguro (Eds). Proceedings of Science of Modeling: The 30th Anniversary Meeting of the Information Criterion (AIC), (pp. 348–349).
van den Wollenberg, A.L. (1977). Redundancy analysis An alternative for canonical correlation analysis. Psychometrika, 42, 207–219.
Wilks, S.S. (1938). The large-sample distriution of the likelihood ratio for testing composite hypotheses. Annals of Mathematical Statistics, 9, 60–62.
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The work reported in this paper was supported by Grant A6394 to the first author from the Natural Sciences and Engineering Research Council of Canada.
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Takane, Y., Hwang, H. On a test of dimensionality in redundancy analysis. Psychometrika 70, 271–281 (2005). https://doi.org/10.1007/s11336-003-1089-x
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DOI: https://doi.org/10.1007/s11336-003-1089-x