Estimation of Parameters and Optimality of Second-Order Spherical Designs Using Quadratic Function Relative to Non-Spherical Face centered CCD

Abstract

The study presented the estimation of parameters and optimality of second-order spherical designs using quadratic model in comparison to the non-spherical face centered CCD for varying axial distances. The designs considered were equiradial design of axial distances of 1.0 and 1.414, inscribed CCD of axial distance of 1.0 and circumscribed CCD of axial distance of 1.414, the study employed sum of square error, variance estimation, D-, A-, and T-optimality criteria as well as Grand mean of these designs for quadratic model and 1 to 10 center runs were considered. The study observed that the sum of square error of the non-spherical face centred CCD is zero (0) for radial point of n=5 with 1 centre point and this result is seen to be a misleading result, because, no process is 100%. While the sum of square error of the spherical designs with axial distance of 1.0 gave minimal sum of square errors and the spherical designs with axial distance of 1.414 gave very large sum of square error. The Grand mean of the spherical and the non-spherical designs were equal or approximately equal for radial point of n=5 for centre points 1-10 inclusive. But as the radial distance increases above 5, the Grand mean of the non-spherical CCD differs significantly from those of the spherical designs. The study suggests that the non-spherical second order design is inferior to their spherical second order design counterparts. The spherical designs with axial distance of 1.414 (equiradial and circumscribed CCD) have better D-optimality, A-optimality and T-optimality than the non-spherical face centred CCD, while, the spherical designs with axial distance of 1.0 (equiradial and inscribed) has inferior D-optimality, A-optimality and T-optimality compared to the non-spherical face centred CCD with axial distance of 1.0.