Abstract
Replication of the factorial (cube) and/or axial (star) portions of the central composite design (CCD in response surface exploration has gained great attention recently. Some well known metrics (called single-value functions or criteria) and graphical methods are utilized in evaluating the regression based response surface design. The single-value functions considered here are the A-efficiency, and the D-efficiency, , where , k is number of factors, is the kth design measure, is the design’s information matrix, is its inverse and N is the total number of experimental runs. These two functions are very popular in parameter estimation in response surface methodology. The exact measures of these two design criteria will be developed analytically in this work to account for partial replication of the cube and/or star components of the CCD. This will alleviate the burden of manual computation of these metrics when there are partial replications and reduce over reliance on software values which, often, are approximate values and maybe inaccurate.
Highlights
Introduction[8] proposed exact versions of the A − , D − and G − efficiencies as well as the IV − criterion for the classical central composite design (CCD) that is based on replication of the centre point alone
In many industrial experiments, relationship between a response of interest, y, and k independent design variables, x1, x2,..., xk, is often adequately described by second-order response surface model y = β0 + x′b + x′Bx + ε, (1)where y is the N ×1 vector of responses, N being the number of experimental runs, β0 is a constant, x is a point in the design space spanned by the design, b is a k ×1 vector of first-order regression coefficients
Where y is the N ×1 vector of responses, N being the number of experimental runs, β0 is a constant, x is a point in the design space spanned by the design, b is a k ×1 vector of first-order regression coefficients
Summary
[8] proposed exact versions of the A − , D − and G − efficiencies as well as the IV − criterion for the classical CCD that is based on replication of the centre point alone The inability of these exact functions to accommodate the replication of the cube or star or both portions of the CCD is a major drawback. We propose modified versions of the exact A − and D − efficiencies for the CCD that accommodate the replication of the cube, star or both portions of the CCD for any given axial distance, α , for any number of centre points and in any given design region (spherical or cuboidal). Doubts that surround the approximate results of some statistical packages are completely eliminated by using the exact forms of the efficiency criteria
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