Third Order Rotatable Designs for Exploring Response Surfaces

Some third-order rotatable designs in three dimensions

“…Thus nonnegative p, q, r and hence, nonnegative a ~, #~, r ~ satisfying (3) exist. Therefore, when N'=7, the 36 points described in (1) It can readily be seen from (7), (8) and (9) …”

“…Gardiner, Grandage and Hader [8], Draper [4], [5], [7], Thaker and Das [15], Das and Narasimham [3], Herzberg [9], Tyagi [17], Nigam [14] and Huda [13] have considered the problem of constructing thirdorder rotatable designs. In particular, Gardiner, Grandage and Hader [8], Draper [4], [7] and Das and Narasimham [3] provided a large number of three-dimensional third-order rotatable designs.…”

“…In particular, Gardiner, Grandage and Hader [8], Draper [4], [7] and Das and Narasimham [3] provided a large number of three-dimensional third-order rotatable designs. However, none of these designs can be performed 'sequentially' in the factors.…”

“…It is known from Box and Hunter [2], Bose and Carter [1] and Gardiner, Grandage and Hader [8] that a set of N I (>=7) points equally spaced on a circle centred at the origin satisfies the moment requirements of a third-order rotatable set and hence, two-dimensional thirdorder rotatable designs may be constructed by combining such pointsets associated with two or more distinct circles. Suppose such a design is given by the points ~/x(~)l~, x (~)~ / (i=l, 2", u=l,... , N 1) where for each i the points are equally spaced on the circle of radius p~ (i=l, 2).…”

“…The necessary and sufficient conditions for a kdimensional point-set to be a non-singular third-order rotatable design were obtained by Gardiner, Grandage and Hader [8] and the geometrical interpretation of these conditions was derived by Draper [4]. For the sake of brevity, these conditions are not to be restated here.…”