Construction of Rotatable Designs Through Balanced Incomplete Block Designs

Abstract

Rotatable designs were introduced by Box and Hunter (1954, 1957) for the exploration of response surfaces. They constructed these designs through geometrical configurations and obtained several second order designs. Afterwards, Gardiner and others (1959) obtained some third order designs through the same technique for two and three factors and a third order design for four factors. Bose and Draper (1959) obtained some second order designs by using a different method. Draper (1960 a) gave a method of construction of an infinite series of second order designs in three and more factors. Recently, Box and Behnken (1960 a) have obtained a class of second order rotatable designs from those of first order. Draper (1960 b) has obtained some third order rotatable designs in three dimensions and a third order rotatable design in four dimensions. Das (1961) has obtained such designs, both second and third orders up to 8 factors as fractional replicates of factorial designs. The method of construction of the designs presented in this paper is essentially based on that presented by Das (1961). After the manuscript of this paper was submitted for publication the authors' attention was drawn to the work of Box and Behnken (1960 b). They have obtained some second order designs by following a procedure which uses balanced incomplete block designs in the same manner as described below. They did not, however, extend the method to include other complementary sets of points which, as will be shown, allow one to obtain rotatable second and third order designs based on any balanced incomplete block design. In the present paper a method has been given by using the properties of balanced incomplete block designs through which second order rotatable designs with any number of factors, with a reasonably small number of points, can be obtained. By extending the method, third order rotatable designs, both sequential and nonsequential, up to 15 factors have been obtained with the help of doubly balanced incomplete block designs and complementary B.I.B. designs.