A comparative study of the performance of cubic and quadratic models for heptagonal spherical two-factor second-order designs based on the sum of square errors

Abstract

The power of Heptagonal Spherical two-factor second-order designs for a cubic model more than its quadratic counterpart based on the sum of square errors was presented. This study used spherical second-order designs such as; Equiradial designs of the radius or axial distance of 1.0 and 1.414 and Central Composite Designs of Face centred, Inscribed and Circumscribed designs. The Inscribed is a radius of 1.0 and Circumscribed CCD is of a radius of 1.414. These designs were studied with the addition of only one centre point. It was observed that all the designs studied recorded a minimum sum of square errors when the shape is a pentagon, and this occurred only for the quadratic model. The sum of the square error value of Face centred CCD of zero (0) is misleading, since nothing is done humanly that is free of error. The Face centred CCD was found to behave differently from the other designs, this could be because it is not a spherical design. As the shape of these designs increases the sum of square errors increases indiscriminately. The study revealed that all the designs except Face Centered CCD gave the minimum sum of square error for the cubic model, while Face Centered CCD gave a singular matrix for the cubic model in all the shapes. The axial distance affects the sum of square errors for the quadratic model, that is to say, the sum of square errors for an axial distance of 1.0 is minimum, while for an axial distance of 1.414, the sum of square errors is maximized. The study proposed the cubic model as a robust model for second-order designs when the shape is Heptagon (n=7) and the quadratic model as a robust model for second-order designs with a radius or axial distance of 1.0.