SECOND ORDER ROTATABLE DESIGNS OF SECOND TYPE USING BALANCED INCOMPLETE BLOCK DESIGNS

In this paper second order rotatable designs of second type using balanced incomplete block designs is suggested. This design is compared with second order rotatable designs of first type using balanced incomplete block designs (cf. Das and Narasimham [6]) on the basis of efficiency.


INTRODUCTION
Response surface designs is a collection of mathematical and statistical techniques useful for analyzing problems where several independent variables influence a dependent variable or response. Box and Hunter [1] introduced designs having spherical variance function are called rotatable designs. Das and Narasimham [6] constructed rotatable designs using balanced incomplete block designs (BIBD). Raghavarao [21] constructed second order rotatable designs (SORD) using incomplete block designs. Draper and Guttman [7] suggested an index of rotatability. Khuri [14] introduced a measure of rotatability for response surface designs. Draper and Pukelshein [8] developed another look at rotatability. Park et al. [16] introduced new measure of rotatability for second order response surface designs. Das et al. [5] developed modified response surface designs. Kim [13] introduced extended central composite designs (CCD) with the axial points are indicated by two numbers. Victorbabu and Vasundharadevi [25] suggested modified second order response surface designs using BIBD. Victorbabu [23] constructed modified SORD and second order slope rotatable designs using a pair of BIBD.
Victorbabu et al. [26] studied modified second order response surface designs using pairwise balanced designs (PBD). Victorbabu [22] suggested a review on SORD. Victorbabu et al. [27] suggested modified second order response surface designs using CCD. Victorbabu and Vasundharadevi [28] studied second order response surface designs using SUBA with two unequal block sizes. Victorbabu [24] constructed modified SORD using a pair of SUBA with two unequal block sizes. Park and Park [17] suggested the extension of CCD for second order response surface models. Victorbabu and Surekha [29] suggested measure of rotatability for second order response surface designs using incomplete block designs. Victorbabu and Surekha [30] developed measure of rotatability for second order response surface designs using BIBD.
Victorbabu et al. [31][32] studied measure of rotatability for second order response surface designs using a pair of SUBA with two unequal block sizes and a pair of BIBD. Jyostna et al. [9] suggested measure of rotatability for second degree polynomial using CCD. Jyostna and Victorbabu [10][11][12] studied measure of modified rotatability for second degree polynomials using BIBD, PBD and SUBA with two unequal block sizes. Chiranjeevi et al. [2] extended the work of Kim [13] and suggested SORD of second type using CCD for 9≤v≤17 (v: number of factors).
Chiranjeevi and Victorbabu [3][4] studied SORD of second type using SUBA with two unequal block sizes and PBD.
In this paper, second order rotatable designs of second type using balanced incomplete block designs is suggested. This design is compared with second order rotatable designs of first type using balanced incomplete block designs (Das and Narasimham [6]) on the basis of efficiency. 2343 SORD OF SECOND TYPE USING BIBD

Stipulations and formulas for second order rotatable designs
Suppose we want use the second order polynomial response surface design D = ((xiu)) to fit where xiu represents the level of i th factor (i=1,2,…,v) in the u th run (u=1,2,…,N) of the  of the point (x1, x2, …, xv) from the origin (center) of the design such a spherical variance function for estimation of responses in the second order polynomial model is achieved if the design points satisfy the following the following conditions (cf. Box and Hunter [1]).
All odd order moments are must be zero. In their words when at least one odd power x's equal to zero. 5.
The variances and covariances of the estimated parameters are and other covariances vanish. (2.7) The variance of the estimated response at the point (x10, x20,…, xv0) is The coefficient of 2

MAIN RESULTS
3.1. SORD of first type using balanced incomplete block designs (cf. Das and Narasimham [6]) Balanced incomplete block design: The parameters of BIBD denote by (v, b, r, k, λ) is an arrangement of v-treatments in b blocks 2345 SORD OF SECOND TYPE USING BIBD each block contains k (<v) treatments, if (i) Every treatment occurs at most once in each block (ii) Each treatment occurs in exactly r blocks and (iii) Every pair of treatments occurs together in λ times.

Incidence matrix:
Let (v, b, r, k, λ) denote parameters of a BIBD. Associated with any design D is the incidence For example the plan and incidence matrix of BIBD is given as following design (v=3, b=3, r=2, k=2, λ=1).
Let (v, b, r, k, λ) denote parameters of BIBD, 2 t(k) denote a fractional replicate of 2 k in +1 or -1 levels in which no interaction with less than five factors are confounded.


The condition for the design becomes an orthogonal design.
From equation 2 (i) of (2.3) and (3) of (2.4), we have For the convenience N is replaced by M By using the orthogonality condition we have and the condition for the design become rotatability.
From equation 2 (ii) of (2.3) and 3 of (2.4), we have For the convenience N is replace by M Then the rotatability condition equation (2.6), we have

Proposed method of SORD of second type using balanced incomplete block design
Following methods construction of Das and Narasimham [6] and Kim [13], Here we studied SORD of second type using BIBD is given bellow.
In the same design (v=3, b=3, r=2, k=2, λ=1) generation of design points in BIBD X1 X2 X3 Let (v, b, r, k, λ) denote parameters of BIBD, 2 t(k) denote a fractional replicate of 2 k in +1 or -1 levels in which no interaction with less than five factors are confounded. [1-(v, b, r, k, λ)] 2349 SORD OF SECOND TYPE USING BIBD denote the design points generated from transpose of the incidence matrix of BIBD. Let [1-(v, b, r, k, λ)]2 t(k) are the b2 t(k) design points generated from BIBD by "multiplication" (cf. Raghavarao [18], pp 298-300). We use the additional set of points like of construction of SORD of second type (cf. Kim [13]) using BIBD is given in the following theorem.
Theorem (1): The design points, Hence the non singularity condition is also satisfied.
The variance and covariance of the estimated parameters are given as fallows The variance of the estimated response at the point (x10, x20,…,xv0) is  'a +a ' makes orthogonal second order response surface designs by using SORD of second type using BIBD.

STUDY OF ORTHOGONALITY IN SORD OF SECOND TYPE USING BIBD
Let (v, b, r, k, λ) denote parameters of BIBD, 2 t(k) denote a fractional replicate of 2 k in +1 or -1 levels in which no interaction with less than five factors are confounded. [1-(v, b, r, k, λ)] denote the design points generated from transpose of the incidence matrix of BIBD. Let [1-(v, b, r, k, λ)]2 t(k) are the b2 t(k) design points generated from BIBD by "multiplication" (cf. Raghavarao [18], pp 298-300). We use the additional set of points like

EFFICIENCY COMPARISON FOR SORD OF SECOND TYPE USING BIBD WITH SORD OF FIRST TYPE USING BIBD
In this section, SORD of second type using BIBD is used as the basis for estimating specific coefficient in the response surface model, SORD of second type using BIBD is compared with SORD of first type using BIBD. This comparison criterion is based on the precision at which the coefficient is estimated. It is consider that the numbers of experimental plots are required at same way.
For example in terms of estimating mixed quadratic coefficient bij (i≠j), two experimental designs, lets try to compare D1 and D2. The number of experimental plots required in D1 and D2 are M and N respectively. The relative efficiency of D1 and D2 is given by the following equation (see Myers [15], section 7.2).
N=b2 +4v+n (Design points in SORD of second type using BIBD) t(k) 0 M=b2 +2v+m (Design points in SORD of first type using BIBD) In this case, in order to compare fairly, the experimental system should make the second product equal to value of 2 iu x N  . It must be scaled and for this the following scaling criteria is used.
However the ij V(b ) is multiplied by with the scaling factor 'g' than the 2 ij 4 4 According to equation (5.1) the relative efficiency SORD of second type using BIBD versus SORD of first type using BIBD in the mixed quadratic coefficient of bij is obtained as follows σ r2 +2a (b2 +2v+m ) Nμ b2 +2v+m SORD of second type using BIBD E = SORD of first type using BIBD σ r2 +2a +2a (b2 +4v+n ) Nμ b2 +4v+n From equation (5.4) the condition that SORD of second type using BIBD SORD of first type using BIBD E       >1, than the SORD of second type using BIBD is more efficient than SORD of first type using BIBD t(k) 2 2 t(k) 2 t(k) 0 1 2 t(k) 0 1 b2 +4v+n a +a < (r2 +2a ) -r2 2 b2 +2v+m From the values of (3.1) and (4.1) substitute in (5.4) and then we get the value of greater than 1.
From this orthogonal SORD of second type using BIBD has the same degree of efficiency as orthogonal SORD of first type using BIBD, and consider the efficiency of SORD of second type using BIBD is giving the better efficiency than SORD of first type using BIBD. Now, the efficiency comparison of SORD of second type using BIBD versus SORD of first type using BIBD with rotatability. Substituting the values of (3.2) into (5.5) and we evaluated that the SORD of second type using BIBD will be more efficient than the SORD of first type using BIBD with rotatability.

Comparison in the pure quadratic coefficient bii
Now this time in terms of estimating the pure quadratic coefficient bii, the efficiency of SORD of second type using BIBD is comparing with SORD of first type using BIBD, here the scaling factor the equation (5.2) is applied. The relative efficiency SORD of second type using BIBD versus SORD of first type using BIBD is as follows based on the equation (5.1) r2 +2a σ e b2 +2v+m b2 +2v+m SORD of second type using BIBD E = SORD of first type using BIBD r2 +2a +2a σ e b2 +4v+n b2 +4v+n For example, in SORD of second type using BIBD of design (v=3,b=3,r=2,k=2,λ=1) , no=1, a1=1, a2=1, e2=0.125 and in SORD of first type using BIBD m0=1, a=1.1892, e1= 0.1947, lets us compare the relative efficiency of SORD of second type using BIBD versus SORD of first type using BIBD, if you get the equation (5.8) as 1.6688, then we conclude that the SORD of second type using BIBD is more efficient than SORD of first type using BIBD.

Comparison in terms estimating the first order coefficient bi
It can be developed in the same process of the V(bi) by multiplying the scaling factor then 2 i t(k) 2 2 1 2 σ V(b )= (r2 +2a +2a ) is to be multiplied by 1/g 2 then we get, and it obtained 1, then the efficiency of SORD of second type using BIBD is more efficient than SORD of first type using BIBD.

CONCLUSION
In this paper, SORD of second type using BIBD is developed. The variance covariance of the estimated parameters are studied and we evaluated for the SORD of second type using BIBD is most orthogonal for second order response surface designs and the results of the orthogonality are also provided in the paper.
The comparison between the SORD of second type using BIBD versus SORD of first type using BIBD for different coefficients are studied then we conclude that the SORD of second type using BIBD is more efficient than SORD of first type using BIBD. It is convenient to use the practical situations and give the more efficiency when compared to SORD of first type using BIBD.