The Annals of Mathematical Statistics

Complex Representation in the Construction of Rotatable Designs

R. C. Bose and R. L. Carter

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Response surface techniques are discussed as a generalization of factorial designs, emphasizing the concept of rotatability. It is shown that the necessary and sufficient conditions for a configuration of sample points to be a rotatable arrangement of a specified order are greatly simplified if, in the case of two factors, the factor space is considered as the complex plane. A theorem giving these conditions is proved, with an application to the conditions governing the combination of circular rotatable arrangements into configurations possessing a higher order of rotatability. This is done by showing that certain coefficients must vanish in the "design equation" whose roots are the (complex) values of the various sample points. A method is presented by which any configuration of sample points (for example, some configuration fixed by extra-statistical conditions) may be completed into a rotatable design of the first order by the addition of only two properly chosen further sample points.

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Ann. Math. Statist., Volume 30, Number 3 (1959), 771-780.

First available in Project Euclid: 27 April 2007

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Bose, R. C.; Carter, R. L. Complex Representation in the Construction of Rotatable Designs. Ann. Math. Statist. 30 (1959), no. 3, 771--780. doi:10.1214/aoms/1177706206.

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