A note on Variance-Sum Third Order Slope Rotatable Designs

Abstract

Response Surface Methodology (RSM) is a collection of mathematical and statistical techniques useful for analyzing experiments where the yield is believed to be influenced by one or more controllable factors. Box and Hunter (1957) introduced rotatable designs in order to explore the response surfaces. The analogue of Box-Hunter rotatability criterion is a requirement that the variance of be constant on circles (v=2), spheres (v=3) or hyperspheres (v ?4) at the design origin. These estimates of the derivatives would then be equally reliable for all points equidistant from the design origin. This property is called as slope rotatability (Hader and Park (1978)).Anjaneyulu et al (1995 &2000) introduced Third Order Slope Rotatable Designs. Anjaneyulu et al(2004) introduced and established that TOSRD(OAD) has the additional interesting property that the sum of the variance of estimates of slopes in all axial directions at any point is a function of the distance of the point from the design origin. In this paper we made an attempt to construct Variance-Sum Third Order Slope Rotatable in four levels.

Introduction

1.  Introduction

Box and Hunter (1957) proved that a necessary and sufficient condition for a design of order d (d=1, 2…) to be rotatable.  Gardiner, Grandage and Hader (1959) constructed some third order rotatable designs for two and three factors. Das and Narasimham(1962) constructed TORDs both sequential and non-sequential, uptofifteen factors, using doubly balanced incomplete block designs and complementary BIB designs. Hader and Park (1978) introduced slope rotatability for central composite second order designs analogous to Box and Hunter (1957) central composite second order rotatable designs several authors gave various methods to construct second order slope rotatable designs. Anajaneyulu et al (1993; 1998) have introduced embedded type SOSRDs, Group-Divisible SOSRDs respectively. Anjaneyulu (1995, 2000) introduced Third Order Slope Rotatable Designs over an axial direction analogous to Third Order Rotatable Designs and constructed TOSRDs using Central Composite type Design points and Doubly Balanced Incomplete Block Designs.

Park (1987) studied the necessary and sufficient conditions for Second Order Slope RotatabilityOver All Directions.Anjaneyulu et al (2004) introduced TOSRD (OAD).Slope rotatability in axial directions introduced by Hader and Park (1978) requires that the variance of the estimated slope in every axial direction be constant at points equidistant from the design origin. Sloperotatability over all directions requires that the variance of the estimated slope averaged over all directions through a uniforms distribution be constant at points equidistant from the origin.

Anjaneyulu et al (1997) established that SOSRD (OAD) has the additional interesting property that the sum of the variances of estimates of slopes in all axial directions at any point is a function of the distance of the point from the design origin.

In this paper an attempt is made to introduce Variance – Sum Third Order Slope Rotatable Designs.

References

[1] ANJANEYULU G. V. S. R., DATTATREYA RAO, A.V and NARASIMHAM, V. L. (1993): Embeeding in Second-Order-Slope-Rotatable Designs, Reports of Statistical Application Research, JUCE, Vol.40, Issue 4, pp:1-11. [2] ANJANEYULU G. V. S. R., DATTATREYA RAO, A.V and NARASIMHAM, V. L. (1995): Construction of Third Order Slope Rotatable Designs through Doubly Balanced Incomplete Block Designs, Gujarat Statistical Review, Vol.(21-22), pp:49-54. [3] ANJANEYULU G. V. S. R., DATTATREYA RAO, A.V and NARASIMHAM, V. L. (1997): A note On Second Order Slope Rotatable Designs Over All Directions, Communications in Statistics: Theory and Methods , Vol.(26), Issue 6, pp:1477-1479. [4] ANJANEYULU G. V. S. R., DATTATREYA RAO, A.V and NARASIMHAM, V. L. (1998): Group-Divisible Second Order Slope Rotatable Designs, Gujarat Statistical Review,Vol.(25), pp:1-2, 3-8. [5] ANJANEYULU G. V. S. R., DATTATREYA RAO, A.V and NARASIMHAM, V. L. (2000): Third Order Slope Rotatable Designs, J. Ind. Soc. For Probablity And Statistics, Vol. (5), pp: 65-74. [6] ANJANEYULU G. V. S. R.,NAGABHUSHANAM. P.,and NARASIMHAM. V. L.(2004): On Third Order Slope Rotatable Design (OAD), Journal of ISPS, Vol.8, pp:92-102. [7] BOX.G.E.P,and HUNTER, J.S. (1957): Multi-Factor Experimental Designs for Exploring Response Surfaces, Ann. Math. Statist. Vol.28,Issue 1, pp: 195-241. [8] DAS,M. N. and NARASIMHAM, V. L.(1962): Construction of Rotatable Designs Through Balanced Incomplete Block Designs, Ann. Math. Statist. Vol.33, Issue 4, pp: 1421-1439. [9] GARDINER, D. A. . GRANDAGE, A. H. E and HADER, R. J.(1959): Third OrderRotatable Designs for Exploring Response Surfaces,Ann. Math. Statist. Vol.30,Issue 4, pp: 1082-1096. [10] HADER, R.J. and PARK, S.H. (1978): Slope-Rotatable central composite designs, Technometrics, Vol.20, pp:413-417. [11] PARK.SUNG H (1987): A class of multi factor designs for estimating the slope of response surfaces, Technometrics, Nov.1987.

Copyright

Copyright © 2021 R. Md. Mastan Shareef, B. Guravaiah, G. V. S. R. Anjaneyulu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.