On the construction of second and third order rotatable 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 other (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) gave a method of construction of an infinite series of second order designs in three and more factors. Recently, Box and Behnken (1960) have obtained a class of second order rotatable designs from those of first order. Draper (1960) has obtained some third order rotatable designs in three dimensions and a sequential third order rotatable design in four dimensions. Das (1960) has obtained such designs, both second and third order up to 8 factors as fractional replicates of factorial designs. Thaker (1960) has obtained series of second and third order rotatable designs by a different method. In the present work a method of constructing second order rotatable designs with any number of factors, by using balanced incomplete block designs has been presented in chapter II. Another method of constructing such designs through a particular class of balanced incomplete block designs with unequal block sizes has been presented in chapter III. These two methods have been found to be useful for constructing second order rotatable designs with reasonably small number of design points. Second order rotatable designs up to eleven factors with the minimum number of design points obtainable through these two methods have been tabulated below.