Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design

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, ( ) ( ) { } 1 100 k A p N trace M ξ − = and the D-efficiency, ( ) { } 1 p k D M N ξ = , where ( )( ) 1 2 2 p k k = + + , k is number of factors, k ξ is the k design measure, ( ) k M ξ is the design’s information matrix, ( ) 1 k M ξ − 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.

( )( ) ξ is the k th design measure, ( ) k M ξ is the design's information matrix, ( )

Introduction
In many industrial experiments, relationship between a response of interest, y , and k independent design variables, 1 2 , ,..., k x x x , is often adequately described by second-order response surface model 0 where y is the 1 N × vector of responses, N being the number of experimental runs, 0 β is a constant, x is a point in the design space spanned by the design, b is a 1 k × vector of first-order regression coefficients.In addition, B is a k k × symmetric matrix whose main diagonal entries are the coefficients of the pure quadratic terms and the offdiagonal entries are coefficients of one-half the mixed quadratic (interaction) terms, and ε is the random error term associated with the responses (see [1]).The p number of model parameters consists of one constant, k first-order terms, k quadratic terms and ( ) Designs for fitting second-order response surface models are called second-order response surface designs.The central composite design (CCD) of [2] is the most popular and commonly used second-order response surface designs.The design has three major components: the factorial portion (the cube) with coordinates, , , . .., 1, 1, . .., 1 the factors; the axial portion (the star) with coordinates,

( ) ( )
, 0, 0, . .., 0 , . .., 0, 0, . .., α α ± ± , α is the distance of the star point (also called the axial distance) from the centre of the design space, and the centre point with coordinates, ( ) 0, 0, . .., 0 .The number of design runs for the CCD is , where f is the factorial or cube run, c n is the replication of f, 2k is the star run and s n is the replication of 2k, 0 n is the number of centre points, c is the cube, s is the star and o is zero (see, for example, [3]).
The A-efficiency and D-efficiency are the most commonly used single-value functions for estimating parameters in response surface exploration using response surface designs like the CCD (see [4]).[5] have extended the use of these optimality criteria to evaluating and selecting optimal designs for functional magnetic resonance imaging (fMRI).Further extension of the use of these efficiency criteria comes in the area of blocking.[6] considered partial replications of the components of the CCD which are arranged in orthogonal blocks and the emanating designs evaluated using the A-and D-efficiencies among other criteria.Statistical packages have made the computation of these single-value functions very easy.However, and as indicated by [7], the measures of these criteria provided by some statistical softwares could be approximated values and often misleading.Hence, it is imperative to have exact forms of these criteria that could be more reliable in design evaluation.[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.In [9], exact functions have been provided for the G-efficiency and IV-criterion which compensate for partial replications of the components of the CD.In this work, 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).The proposed modified versions will make it easy to evaluate the parameters of the regression model in response surface exploration involving the CCD where the components of the CCD are replicated to enhance the design's performance.Doubts that surround the approximate results of some statistical packages are completely eliminated by using the exact forms of the efficiency criteria.

Exact A-and D-Efficiency for the Partial Replication of CCD
The D-efficiency is derived from the determinant of the information matrix, ( ) , X is the design matrix expanded to model form.According to [8] and [10], the A − and D − efficiency measures represent the percentage number of runs required by a particular orthogonal design to achieve the same determinant and trace.
To obtain the D-efficiency, we first, derive the information matrix of the partially replicated CCD, then the determinant of the information matrix.Consider the design matrix of equation ( 2) for the k factor CCD, where k is a fixed positive integer and k>1.Let and , , be any two variables of the CCD.The CCD has (q is an integer) support points at the vertices, 2k support points at the axes and support points at the centre of the design space.
Let f have c n multiplicity, 2k have s n multiplicity and with 0 n centre points, with the subscripts, c, s and 0 representing cube, star and zero, respectively.Also, let k X be the N p × expanded design matrix from the N k × design matrix of the partially replicated k-factor CCD and k X ′ , its transpose for all k>1.The information matrix, ( ) is determined directly by matrix multiplication.The elements of ( ) , be an element of the u th column of k X ′ , then the inner product of k X ′ and k X is given by (see, for example, [11]) ...
Hence, the information matrix for the k-factor partially replicated CCD in block form, containing sub-matrices and vectors, is given by .
where 1 ( ) F as the entries.For example, for k = 2, the information matrix of the CCD with partial replication of the cube and star is given by The derivation of the determinant of the information matrix, ( ) follows.Let ju a , j,u = 1, 2,..., p, be the element of ( ) k M ξ and ju W be the ( ) ( ) matrix obtained by deleting the j th row and u th column containing ju a .Then, the determinant of ( ) M ξ is given by (see [11]), ( ) ( ) for any u, where ( ) obtained by expanding ju W to the second-order determinant.
After performing the tedious matrix algebra, the determinant of the information matrix of the partially replicated CCD is given by ( ) ( ) ( ) ( ) Multiplying the p th root of the determinant by 100 and dividing by N gives the percentage D-efficiency.That is,

(
) ( ) To obtain the A-efficiency, the inverse of the information matrix will be derived first.To derive the inverse of ( ) k M ξ , the cofactors of the elements of ( ) k M ξ are obtained as described in the case of the determinant to form the matrix of the cofactors, ( ) k M C .This matrix is transposed to obtain the adjugate or adjoint which is multiplied by the reciprocal of the determinant to obtain the inverse, ( ) and therefore, equal to the adjugate.The inverse of the information matrix for the partially replicated CCD is given by ( . where ( ) ( ) For example, the matrix of cofactors for two-factor partially replicated CCD is given by Therefore, the inverse of the information matrix of the two-factor partially replicated CCD is obtained by multiplying Let be the diagonal elements of : see, for example, [12].Then the trace of ( ) Hence, the percentage A-efficiency is the product of p and 100 divided by the product of the trace of ( ) These mathematical expressions for the exact Aand Defficiencies will reduce to the [8] type of exact Aand efficiencies if 1 c s n n = = .That is when there are no partial replications of the components of the CCD.

Conclusion
The central composite design is a popular second-order response surface design.By replicating the cube and star portions of the CCD analytical functions have been developed for obtaining the A − and D − efficiencies for the partially replicated design options for any given number of centre points and for any axial distance.The higher the values of Aand D-efficiencies, the better the model's parameter estimation with minimum variance which corresponds to maximizing the information.With these results, parameter estimation in response surface exploration using the partially replicated regression-based CCD becomes less tedious and there is no risk of making inference based on sometimes misleading approximate values.
Ukaegbu and Polycarp E. Chigbu: Modified Exact Single-Value Criteria for Partial Replicationsof the Central Composite Design zero matrices and = 2, the determinant is given by