relative efficiency and dt- optimality criteria for the existing six specific second order rotatable designs in three dimensions

Abstract

Experimentation plays an important role in Science, Engineering and Industry. This is an application of treatments to experimental units, and then measurement of one or more responses. It is part of scientific method which requires observing and gathering information about how process and system work where some input variable x’s transform into an output that has one or more observable response variables y. Therefore, useful results and conclusions can be drawn. In order to obtain an objective conclusion there is need to plan and design an experiment and analyze the results. The approximation of the response function y is called Response Surface Methodology. This study focused on the existing six specific second order rotatable designs in three dimensions. These designs were denoted by M 1 , M 2 , M 3 , M 4 , M 5 and M 6 . A design matrix X was developed from the designs, further their information matrices C 1 , C 2 , C 3 , C 4 , C 5 and C 6 were obtained from which the alphabetic optimal values for these designs were evaluated, the optimum values obtained were used to calculate the A-, D-, E- and T- relative efficiencies for both calculus optimum and unit value designs, for instance to evaluate E- relative efficiency the formula λ min (ε ) where λ min ( ε ¿ ) ¿ λ min (ε ) is the least E- optimum valuewhile λ min (ε ) is the respective least design optimal value. Finally the compound optimality criterion (DT-) for all the six designs was also evaluated. In this study optimal values already evaluated were used to evaluate efficiencies and the DT- optimality criterion. From the results Calculus optimum values designs are generally more efficient than Unit Value Designs, on checking both the calculus optimum designs and unit value designs D- efficiency was found to be the best as it gave a higher efficiency than the rest relative efficiency criteria. On comparison of all designs M 1 was found to be most efficient as compared to the rest. For DT- optimality M 2 is DT- Optimal.