Abstract
A family of solutions for linear relations amongk sets of variables is proposed. It is shown how these solutions apply fork=2, and how they can be generalized from there tok≥3.
The family of solutions depends on three independent choices: (i) to what extent a solution may be influenced by differences in variances of components within each set; (ii) to what extent the sets may be differentially weighted with respect to their contribution to the solution—including orthogonality constraints; (iii) whether or not individual sets of variables may be replaced by an orthogonal and unit normalized basis.
Solutions are compared with respect to their optimality properties. For each solution the appropriate stationary equations are given. For one example it is shown how the determinantal equation of the stationary equations can be interpreted.
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Reference notes
Dauxois, J., and Pousse, A. (1976).Les analyses factorielles en calcul des probabilités et en statistique: essai d'étude synthétique. Toulouse; Ph.D. Thesis, Université de Toulouse.
Ten Berge, J. M. F. (1977).Optimizing factorial invariance. Groningen; Ph.D. Thesis, University of Groningen.
References
DeSarbo, W. S. Canonical/redundancy factorial analysis. (1981).Psychometrika, 46, 307–329.
Guttman, L. (1953). Image theory and the structure of quantitative variables.Psychometrika, 18, 277–296.
Hoerl, A. E., and Kennard, R. W. (1970). Ridge regression. Biased estimation for nonorthogonal problems.Technometrics, 12, 55–67.
Hoerl, A. E., and Kennard, R. W. (1970). Ridge regression: applications to nonorthogonal problems.Technometrics, 12, 67–82.
Horst, P. (1961). Relations amongm sets of measures.Psychometrika, 26, 129–149.
Horst, P. (1965).Factorial analysis of data matrices. New York, Holt, Rinehart and Winston.
Johansson, J. K. (1981). An extension of Wollenberg's redundancy analysis.Psychometrika, 46, 93–104.
Kettenring, J. R. (1971). Canonical analysis of several sets of variables.Biometrika, 58, 433–451.
Kristof, W. (1971). Orthogonal inter-battery factor analysis.Psychometrika, 32, 199–227.
Tucker, L. R. (1958). An inter-battery method of factor analysis.Psychometrika, 23, 111–136.
Van de Geer, J. P. (1971).Introduction to multivariate analysis for the social sciences. San Francisco: Freeman and Company.
Van den Wollenberg, A. L. (1977). Redundancy analysis: an alternative for canonical correlation analysis.Psychometrika, 42, 297–220.
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Van de Geer, J.P. Linear relations amongk sets of variables. Psychometrika 49, 79–94 (1984). https://doi.org/10.1007/BF02294207
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DOI: https://doi.org/10.1007/BF02294207