A Study of Slope Rotatability of Rotatable Response Surface Designs

One of good properties that response surface experimental designs may have is rotatability. When we are interested in estimating the rates of change of the value of the response variable resulting from change of the value of the explanatory variable, slope rotatability is a very desirable property. Response surface designs cannot simultaneously have rotatability and slope rotatability. So we often need a design among rotatable designs which is as close to a slope-rotatable design as possible. Based on this necessity, in this paper we studied about the degree of slope rotatability of response surface designs having rotatability. The degree of slope rotatability was measured using Park, Kim’s (1992) measure, Qₖ. Specifically, for various rotatable response surface designs we derived formulas expressing Qₖ as a function only of n₀, the number of center points. For the case of the central composite design of type 2, we obtained formulas expressing Qₖ as a function of n₀ and α₁, one of the design parameters specifying the axial points. We also obtained several results including the fact that when central composite designs of type 1 and 2, regular polygonal design, icosahedral design and dodecahedral design have rotatability, they approach to a slope-rotatable design as n₀ becomes larger.