Fourth order rotatable designs

Abstract

The technique of response surface designs 1S not new. Useful designs in this kind of technique are known as rotatable designs1 Several authors have made contributions in this area. Box [1 J gave the concept of rotatability and he also gave the designs of first degree. Box and Hunter [4J worked out second order rotatable designs and Gardiner, Grandage and Hader[lS] worked out the moment conditions and non-singularity conditions for ~ set of experimental points to form a third order rotatable design. Some rotatable designs of second and third orders have been obtained by Bose and Draper [7J and Norman Draper [11, 12, 13, 14J respectively. This thesis deals with the fourth order response surface designs. Chapter 1 gives basic ideas in response surface designs and describes briefly some relevant work that has been done by various authors. The chapter also sets out notations which,are used in the subsequent development 6f the theory in the later chapters. These notations~are similar 10 those used by the authors sited above. In chapter 2, the moment conditions for a set of experimental points to form a fourth order arrangement are obtained. These are given in-s~ction 2.2.4. To obtain these-conditions, a few ~nown results and definitions have been given in the !lrs( sections. This is done for clarity and easy reference. In chapter 3, the non-singularity conditions for a set of experimental points to form a fourth order rotatable design have been obtained.To obtain these conditions, the moment matrix for tile fourth order response surface is obtained and hence its determinant ~ is worked out using the known methods. The determinant is assumed, to be greater than zero and then the conditions