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Optimal scaling for arbitrarily ordered categories

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Abstract

The methods of optimal scaling are usually formulated as the maximization problem of a ratio of quadratic forms, and the optimal scores are obtained by solving an eigenequation. However, there sometimes exist order relations among categories. For such cases, Bradley, Katti and Coons [2] proposed an algorithm to maximize the criterion under complete order restrictions. Nishisato and Arri [7] discussed the case of partial order and proposed an algorithm using separable programming. Their method is, however, limited to a special type of partial order. Avoiding this limitation, the present paper gives a generalized formulation applicable to arbitrary order restrictions and proposes an efficient algorithm using Wolfe's reduced gradient method. Numerical examples are provided to show the validity, the rapidness of convergence and the stability of the procedure.

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Authors
  1. Yutaka Tanaka
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Takeda Chemical Industries, Ltd.

Now at Okayama University.

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Tanaka, Y. Optimal scaling for arbitrarily ordered categories. Ann Inst Stat Math 31, 115–124 (1979). https://doi.org/10.1007/BF02480269

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  • DOI: https://doi.org/10.1007/BF02480269

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