Measure of slope rotatability for second order response surface designs using a pair of symmetrical unequal block arrangements with two unequal block sizes

Introduction

Response surface methodology is used in experiments in which the main interests are to determine the relationship between the response and the setting of a group of experimental factors. The concept of rotatability, which is very important in response surface designs, was proposed by Box and Hunter (1957) [1] . Das and Narasimham (1962) [3] constructed rotatable designs through balanced incomplete block designs. Narasimham et al. (1983) [5] constructed second order rotatable designs through a pair of balanced incomplete block designs. A design is said to be rotatable if the variance of the response estimate is a function only of the distance of the point from the design center. The study of rotatable designs is mainly emphasized on the estimation of differences of yields and its precision. Estimation of differences in responses at two points in the factor space will often be of great importance. If differences in responses at two points close together are of interest then estimation of local slope (rate of change) of the response is required. Estimation of slopes occurs frequently in practical situations. For instance, there are cases in which we want to estimate rate of reaction in chemical experiment, rate of change in the yield of a crop to various fertilizer doses and rate of disintegration of radioactive material in an animal, etc. (Park, 1987) [6] . Hader and Park (1978) [4] proposed slope rotatability for second order response surface designs and constructed slope rotatable central composite designs. Victorbabu and Narasimham (1991a) [13] suggested conditions for slope rotatability in any general second order response surface designs and constructed second order slope rotatable designs (SOSRD) using balanced incomplete block designs (BIBD). Victorbabu and Narasimham (1991b) [14] constructed a new unified method of SOSRD through a pair of incomplete block designs. Victorbabu and Narasimham (1993a) [15] constructed three level SOSRD using balanced incomplete block designs. Victorbabu and Narasimham (1993b) [16] studied SOSRD using pairwise balanced designs (PBD). Victorbabu (2002) [8] constructed SOSRD using symmetrical unequal block arrangements (SUBA) with two unequal block sizes. Victorbabu (2005, 06) [9] constructed modified SOSRD using central composite designs (CCD) and BIBD. Victorbabu and Narayanarao (2006) [10,17] constructed SOSRD through a pair of suitably chosen SUBA with two unequal block sizes. Victorbabu (2007) [11] suggested review on second order slope rotatable designs. Victorbabu (2019) [12] suggested a new method of SOSRD using PBIBD. Park and Kim (1992) [7] introduced measure of slope rotatability for second order response surface experimental designs. Victorbabu and Surekha (2011, 12a, 12b, 12c, 13, and 16) [18][19][20][21][22][23] studied measure of slope rotatability for second order response surface designs using CCD, BIBD, PBD, SUBA with two unequal block sizes and partially balanced incomplete block type designs, pair of BIBD respectively. Victorbabu and Chiranjeevi (2018) [24] studied measure of rotatability for second order response surface designs using SUBA with two unequal block sizes when r >3λ is suggested. Chiranjeevi and Victorbabu (2019) [2] suggested measure of slope rotatability for second order response surface designs using supplementary difference sets. In this paper, a new measure of slope rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes is suggested.

Conditions for second order slope rotatable designs

Suppose we want to use the second order response surface designs D= ((x iu )) to fit the surface,

where x iu denotes the level of the i th factor (i=1,2,…,v) in the u th run (u=1,2,…,N) of the experiment and the e u 's are uncorrelated random errors with mean zero and variance σ 2 . A second order response surface design D is said to be SOSRD if the variance of estimated of first order partial derivative of u 1 2 v Y (x , x , ...,x ) with respect to each of independent variable (x i ) is only a function of the distance  

 of the point 1 2 v (x , x , ...,x ) from the origin (center) of the design. Such a spherical variance function for estimation of slopes in the second order response surface is achieved. If the design points satisfy the following conditions. (cf. Hader and Park (1978) [4] , Victorbabu and Narasimham (1991a) [13] ).

x =constant=cN ; for all i  

x x =constant=N ;for alli j

where c, λ 2 and λ 4 are constants and the summation is over the design points. The variances and co-variances of the estimated parameters are,

and other covariances vanish.

Therefore the conditions (2) to (7) give a set of conditions for slope rotatability in any general second order response surface design.

3. Second order slope rotatable designs using a pair of SUBA with two unequal block sizes (cf. Victorbabu and Narayanarao (2006) [10,17] )

The method of construction of SOSRD using a pair of SUBA is given the following result. If k =sup(k ,k ), 11

N=b 2 +b 2 +n design points, where a 2 is a positive real root of the biquardratic equation

If at least one positive real root exist for a 2 in equation ( 8) then the design exists. Note: Value of SOSRD using a pair of SUBA with two unequal block sizes can be obtained by solving the above equation (8).

Conditions of Measure of Slope Rotatability for Second Order Response Surface Designs

(cf. Park and Kim 1992) [7] Following Hader and Park (1978) [4] , Victorbabu and Narasimham (1991a) [13] and Park and Kim (1992) [7] conditions (2) to (7) give the necessary and sufficient conditions for a measure of slope rotatability for any general second order response surface designs. Further we have, V(b i ) are equal for i, V(b ii ) are equal for i, V(b ij ) are equal for i, j, where i≠j,

Park and Kim (1992) [7] proposed that if the conditions in (2) to ( 7) and ( 9) are met, then the following measure (Q v (D)) given below can be used to assess the degree slope of rotatability for any general second order response surface design D with v independent variables.

where Q v (D) is the proposed measure of slope rotatability. Further, it is greatly simplified to

Study of measure of slope rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes

The proposed method of measure of slope rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes is suggested in this section. Let The design points achieved from the transpose of incidence matrix of design D 1 by multiplication (see Das and Narasimham 1962) [3] are denoted by   1 give a v-dimensional measure of slope rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes in x x =λ 2 +λ 2 a =Nλ  (13) Solving equations ( 12) and ( 13), we get c.

Measure of slope rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes can be obtained by solving the given equation (cf. Park and Kim 1992) [7] . http://www.mathsjournal.com

will give a measure of slope of rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes in N=417 design points for 12-factors. Here equations ( 11), ( 12) and ( 13) are,

From equations ( 15) and ( 16) we get 265536a -786432a +90880a +786432a -649984=0

Alternatively from equation ( 8) we obtain equation ( 17) directly. Solving (17), we get a 2 = 2.5100, substituting this value of a 2 in ( 14), ( 15) and ( 16) and on simplification we obtained λ 2 =0.4424, λ 4 = 0.1592 and c=5.5181. Non singularity condition ( 5) is also satisfied as 0.8135 > 0.7265. We exemplify the measure of slope rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes for v =12 factors the design points N=417 hence we get slope rotatability value a=1.5843, then the v Q (D)=0.Instead of taking a=1.5843, suppose we take a=1.

Conclusions

In this paper measure of slope rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes have been proposed which enable us to assess the degree of slope rotatability for second order response surface designs. It may be pointed out here that the measure of slope rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes with parameters of Victorbabu and Surekha (2016) [23] need 881 design points. Similarly, for the measure of slope rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes with parameters of Victorbabu and Surekha (2016) [23] need 1241 design points. Thus this new method sometimes lead to measure of slope rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes with lesser number of design points than the measure of slope rotatability for second order response surface design using a pair of BIBD in some cases.

References

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