Abstract
Seven generalized criteria are proposed with the corresponding formulations for optimal scaling of multiple responses. Then, on the basis of a natural probabilistic model, the asymptotic theories are derived concerning the test statistics on factor-response relationships as well as the distributions of sample criteria (eigenvalues of the determinantal equation) and optimal scores (eigenvectors) by means of socalled δ-method. A numerical example is provided for illustrations.
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References
Hayashi, C. (1952). On the prediction of phenomena from qualitative data on the quantification of qualitative data from the mathematico-statistical point of view,Ann. Inst. Statist. Math.,3, 69–98.
Hayashi, C. (1961). Sample survey and theory of quantification,Bull. Inst. Statist. Inst.,38, 505–514.
Kendall, M. G. and Stuart, A. (1967).The Advanced Theory of Statistics Vol. 2, 2nd edition, Griffin, London.
Rao, C. R. (1965).Linear Statistical Inference and Its Applications, Wiley, New York.
Searle, S. R. (1971).Linear Models, Wiley, New York.
Tanaka, Y. (1978). Some generalized methods of optimal scaling and their asymptotic theories: The case of single response-multiple factors, to appear. Also see, Tanaka, Y. and Asano, C. (1978),Research Report No. 77, Research Institute of Fundamental Information Science, Kyushu Univ.
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Tanaka, Y. Some generalized methods of optimal scaling and their asymptotic theories: The case of multiple responses-multiple factors. Ann Inst Stat Math 30, 329–348 (1978). https://doi.org/10.1007/BF02480223
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DOI: https://doi.org/10.1007/BF02480223