An Empirical Study of Second Order Rotatable Designs under Tri-Diagonal Correlated Structure of Errors using Incomplete Block Designs

In this paper, an empirical study of second order rotatable designs under tri-diagonal correlated structure of errors using incomplete block designs like pairwise balanced designs (PBD) and symmetrical unequal block arrangements (SUBA) with two unequal block sizes are suggested. Further we study the variance function of the estimated response for different values of tri-diagonal correlated coefficient (ρ) and distance from center (d) for 6≤v≤15 (vnumber of factors)).


Introduction
introduced rotatable designs for the exploration of response surface designs. Das and Narasimham (1962) constructed rotatable designs through balanced incomplete block designs (BIBD). Raghavarao (1962) constructed symmetrical unequal block arrangements (SUBA) with two unequal block sizes. Raghavarao (1963) constructed second order rotatable designs (SORD) using incomplete block designs. Tyagi (1964) constructed SORD using pairwise balanced designs (PBD). Panda and Das (1994) studied first order rotatable designs with correlated errors. Das (1997Das ( , 1999Das ( , 2003 introduced and studied robust second order rotatable designs (RSORD). Rajyalakshmi and Victorbabu (2014a) suggested an study of SORD under tri-diagonal correlated structure of errors empirical using central composite designs. Rajyalakshmi  The variances and covariances of the estimated parameters under the tri-diagonal correlated structure of errors are as follows: and other covariances are zero.
An inspection of the variance shows that a necessary condition for the existence of a non-singular second order design is The variance of the response ̂ at any point estimated through the surface comes out as, Hence the variance of estimate of ̂ becomes, * +

An empirical study on SORD under tri-diagonal correlated structure of errors using PBD
Following Tyagi (1964) and Das (2003) methods of constructions of SORD and RSORD, an empirical study on SORD under tri-diagonal correlated structure of errors using PBD is studied.
denotes a resolution V fractional factorial of 2 k in ±1 levels, in which no interaction with less than five factors is confounded and n 0 denote the number of central points. Consider a SORD using PBD having "n" non-central design points. The set of "n"-non central design points are extended to 2n design points by adding "n" (n 0 =n) central points just below or above the "n" non-central design points. Hence be the total number of design points (N) of SORD under tri-diagonal correlated structure of errors using PBD.

Case (ii): when r=3λ
The design points [1-(v, b, r, k 1 , k 2 ,…, k p , λ)] 2 t(k) (n 0 ) will give a vdimensional SORD under tri-diagonal correlated structure of errors in N(=2n, n= b ) design points. Proof: For the design points generated from the PBD, simple symmetry conditions are true. Further we have values. The variance of the estimated response for a given "v" is tabulated for ρ=-0.9 (0.1) 0.9.

A study of dependence of the variance function of the response at different design points
Here, dependence of variance function of response at different design points for SORD under tri-diagonal correlated structure of errors using PBD with parameters (v=8, b=15, r=6, k 1 =4, k 2 =3, k 3 =2, λ=2) with the tri-diagonal correlated coefficient "ρ" and distance from centre "d" is studied. The variance function is given by (by taking d=0.1, ρ=-0.9 and σ=1). For a given v, the study of variance function of response at different design points for SORD under tri-diagonal correlated structure of errors using PBD for "v =8 factor" and distance from centre d for d=0.1 (0.1) 1 are tabulated.
In the first row of table-I, we calculate the variance of the estimated response for different factors of "v" at "ρ=-0.9".
In the first row of table-II, we study the dependence of variance function at different design points for d=0.1(0.1)1 at "ρ=-0.9".

An empirical study on SORD under tri-diagonal correlated structure of errors using SUBA with two unequal block sizes
Following Raghavarao (1962,1963) and Das (2003) methods of constructions of SORD and RSORD, an empirical study on SORD under tridiagonal correlated structure of errors using SUBA with two unequal block sizes is studied. Let (v, b, r, k 1 , k 2, b 1 , b 2 λ) is an equi-replicated SUBA with two unequal block sizes, k = sup [k 1 , k 2 ] and b=b 1 +b 2 . denotes a resolution V fractional factorial of 2 k in ±1 levels, in which no interaction with less than five factors is confounded and n 0 denote the number of central points. Consider a SORD using SUBA with two unequal block sizes having "n" noncentral design points. The set of "n"-non central design points are extended to 2n design points by adding "n" (n 0 =n) central points just below or above the "n" non-central design points. Hence be the total number of design points of the SORD under tri-diagonal correlated structure of errors using SUBA with two unequal block sizes. ISSN 2424-6271 IASSL

A study of dependence of the variance function of the response at different design points
Here, dependence of variance function of response at different design points for SORD under tri-diagonal correlated structure of errors using SUBA with two unequal block sizes with parameters (v=8, b=12, r=4, k 1 =2, k 2 =3, b 1 =4, b 2 =8, λ=1) with the tri-diagonal correlated coefficient "ρ" and distance from centre "d" is studied.
The variance function is given by (by taking d=0.1, ρ=-0.9 and σ=1). For a given v, the study of variance function of response at different design points for SORD under tri-diagonal correlated structure of errors using SUBA with two unequal block sizes for "v = 8 factor" and distance from centre d for d=0.1 (0.1) 1 are given in table-III and IV. In the first row of table-III, we calculate the variance of the estimated response for different factors of "v" at "ρ=-0.9". In the first row of table-IV, we study the dependence of variance function at different design points for d=0.1(0.1)1 at "ρ=-0.9".

Results and Conclusions:
From the