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On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs



Polycarp E. Chigbu and Emmanuel U. Ohaegbulem
 
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ABSTRACT

Two variations of N-point Central Composite Designs (CCDs) which are either orthogonally or rotatably restricted are compared. The basis of variation in these designs is the common distance of the axial points from the center of the design. Using the D-optimality criterion, it is concluded that replicated cubes plus one star variation is preferable to the replicated stars plus one cube variation under both restrictions.

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Polycarp E. Chigbu and Emmanuel U. Ohaegbulem, 2011. On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs. Journal of Applied Sciences, 11: 3732-3737.

DOI: 10.3923/jas.2011.3732.3737

URL: https://scialert.net/abstract/?doi=jas.2011.3732.3737
 
Received: August 02, 2011; Accepted: November 12, 2011; Published: December 29, 2011



INTRODUCTION

It is sometimes desirable to replicate the points in a design. This will allow the experimenter to later estimate the pure error in the experiment. Many authors have discussed the analysis of such experiments (Cochran and Cox, 1957; Montgomery, 1991; Atkinson and Donev, 1992). However, it should be clear that, when replicating the design points, the experimenter could compute the variability of measurements within each unique combination of factor levels. This variability will give an indication of the random error in the measurement because the replicated observations are taken under identical conditions. Such an estimate of the pure error can be used to evaluate the size and statistical significance of the variability that can be attributed to manipulated factors (Box and Draper, 1987). It may not be possible to replicate each unique combination of factor levels, that is, the full design, but the experimenter can still gain an estimate of pure error by replicating only some of the runs (points) in the design. In this case of partial replications, the experimenter faces the problem of choosing the points to be replicated and the points not to be replicated in the design.

In this study, specific interest is centered on the Central Composite Designs (CCDs) which are the most practically useful class of second-order designs and one of many experimental designs where partial replication is applicable. Since the best design option is sought, we shall hover around the restricted CCD (which is a CCD with restrictions such as orthogonality, rotatability, etc., imposed on it). It has been shown in Nwobi et al. (2001) that, with designs, restrictions involving orthogonality or rotatability or even both orthogonality and rotatability is better (in the sense of smaller variance) than the unrestricted. A CCD consists of a 2K factorial or a 2K-q fractional factorial portion (run), usually called a cube, with points selected from the 2K points (x1, x2,..., xk) = (±1, ±1,..., ±1) usually of resolution V or higher, plus a set of 2K axial points (runs) at a distance α from the origin, usually called a star, plus one or more center points. In general, the cube and star may be replicated also (Draper, 1982; Draper and Lin, 1990). Thus, we have a total of N (= 2K n1+2Kn2+n0) points where n1 is the number of cubes; n2 is the number of stars and n0 is the number of center points. Typically, the value of α will be chosen to satisfy the property of orthogonality or rotatability (Draper and John, 1998). More so, CCDs are extremely useful for sequential experimentation in which the cube portion is used to allow for estimation of the first-order effects, the later addition of the star points permits second-order terms to be added to the model and estimated. Although cube and star portions are used at different stages and purposes, it is plausible to know the portion of a restricted CCD to be replicated when it is not possible to replicate all points equally, because of the reason mentioned earlier. Box and Hunter (1957) has pointed out that the variance of estimate can be reduced by increasing N, the number of experiments (e.g., by replicating the points); see also Box and Draper (1982). An attempt to break this dilemma is made in this study. The two variations of restricted CCD that will arise which are replicated cubes plus one star and replicated stars plus one cube are compared. The variations are distinguished by the value of α only.

Our approach to this study is to express the information matrix of a restricted CCD in terms of numbers of cubes and stars and then find out the desired portion, which optimizes the design under consideration. The D-optimal design criterion, which maximizes the determinant of the information matrix of a design, is employed for the comparison in this work. This is so because it has been proved by Nalimov et al. (1970) that the D-optimum concept can be used as the theoretical basis for building and comparing response surface designs in use.

CENTRAL COMPOSITE DESIGNS

The CCD is the 2K factorial or a 2K-q fractional factorial design with the levels of each factor coded to the usual -1, +1, augmented by the following points: (±α, 0, ..., 0), (0, ±α, ..., 0) (0, 0, ... ,±α) and (0, 0, ..., 0). The experimenter according to some restrictions such as orthogonality or rotatability selects the value of α. In order to show how these restrictions are made in choosing α, attention will be paid to the expanded design matrix, X and the information matrix, X’X, for the general CCD.

Orthogonal restriction: Consider the response surface, say = φ(x1, x2,..., xk) represented in the experimental area, [±α, ±1], by the quadratic function:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(1)

where, yj and ej are, respectively the response and the random error of the jth observation; β00, βi0, βii and βii are the unknown parameters of the regression model and x1, x2,..., xk are the independent variables. Alternatively, in vector notation, (1) is given by:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(2)

where, Y and e are the, respective (Nx1) response and error column vectors; Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs is an (NxP) matrix of independent variables of rank P; β is the (Px1) column vector of the unknown parameters. From (1), we obtain the average:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(3)

By subtracting Eq. 3 from Eq. 1, we obtain:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(4)

where, Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designsand Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs.

Recall from Eq. 1, that:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

hence, Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs. Then, (2) becomes:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(5)

after subtracting Eq. 3 from it, where Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order DesignsImage for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs and:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(6)

is the {Nx(P-1)} design matrix; N = 2Kn1+2Kn2+n0,

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

1 = (1, 1, ..., 1)' is a 2Kn1/4 component vector, Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs is of n2 components, α2 = (α2, α2,..., α2) 01 = (0, 0, ..., 0) is of n2 components 02 = (0, 0, ..., 0) is of (K-2)n2 components, 03 = (0, 0,...., 0) is of n0 components. The information matrix of the design is obtained as:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(7)

where, M1 is the sum of squares of the elements in the vectors associated with the first-order terms while M2 is the sum of squares of the elements in the vectors associated with the two-way cross-product terms. However, M3 is a (KxK) matrix whose diagonal elements (denoted by p) are the sums of squares of the elements in the vectors associated with the adjusted second-order terms and also, whose off-diagonal elements (denoted by q) are the sums of cross-products of the elements in the vectors associated with the adjusted second-order terms. Using notations accordingly, M1 = 2K n1+2n2α2, M2 = 2K n1 and M3 = (p-q)IK+qJK. Note that, IK is an (KxK) identity matrix, It is an (txt) identity matrix; where:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

and Jk = 11, 1 is a column vector of K components. Using Eq. 6, we obtain;

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(8)

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(9)

where:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

Having obtained Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs and Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs, the definition of orthogonally-restricted CCD is given below.

Orthogonally-Restricted CCD: A CCD is orthogonally-restricted if Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs has a diagonal structure, that is, if and only if q = 0 in M3 of Eq. 7.

Rotatable restriction: The concept of rotatability was first introduced by Box and Hunter (1957) and has since become an important design criterion. The important feature of rotatability is that the quality of variance of prediction of the response denoted by V[í(x)] is invariant to any rotation of the coordinate axes in the space of the input variables (Khuri, 1988). A succinct characterization of ratability is given in terms of the elements of Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs. These elements are known as design moments although, originally, the elements of:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

are referred to as design moments (Khuri, 1988). One the whole, a design moment for a model such as the one given in Eq. 1 of order d (d = 2)and in K input variables is denoted by (1δ1 2δ2 ... KδK) and is given by:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(10)

where, Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs are nonnegative integers. The sum:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

is called the order of the design moment and is denoted by δ (δ = 0, 1, ...,2d). For example, (11 31 53) is a design moment of order δ = 5Δ and is equal to

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

A necessary and sufficient condition for a design for fitting a model such as that of (1) to be rotatable is that the design moments of δ (δ = 0, 1, ...,2d) be of the form:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(11)

where, γδ is a quantity that depends on d, δ and N(Box and Hunter, 1957). According to Myers (1991), a second-order design with moments, which have the following conditions:

All moments that have at least one δi to be odd are zero
Pure fourth moments, which is equal to Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs, are three times the mixed fourth moments, that is
Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(12)

is rotatable. Notice from the portion of a typical design matrix, X for general CCD containing the second-order terms that:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(13)

and

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(14)

Of the two conditions given, which must be met in order that a second-order design is rotatable, the first is automatically met by mere inspection of X for the CCD (Myers, 1991). Thus, it only remains to find the value of α for which the second condition holds. Using Eq. 12-14 accordingly, we obtain:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(15)

Now for the information matrix in Eq. 7:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

where, q = 2K n1 and p = 2K n1+2n2α4

Rotatably-restricted CCD: A CCD is rotatably-restricted if and only if 2K n1+2α4n2 = 3(2Kn1).

CUBE REPLICATIONS VERSUS STAR REPLICATIONS IN RESTRICTED CCD

For the sake of the comparison, denote the replicated cubes plus one star variation of restricted CCD by ξ. Also, denote the corresponding information matrix for this variation by M(ξ). Similarly, let η denote the one cube plus replicated stars variation of restricted CCD while M (η) denotes its information matrix. The criterion for comparison employed in this papers is the D-optimal design criteria, which is defined thus: given any two designs, ξ and η, with information matrices, M(ξ) and M(η), respectively, then ξ is preferred to η if the difference, M(ξ)-M(η) is positive definite, that is:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(16)

where, φ is the D-optimality criterion function which maximizes the determinant of the information matrix; for instance (Pazman, 1986; Fedorov, 1972; Onukogu, 1997):

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(17)

The first restriction to be illustrated here is that for which the design is orthogonal. Recall that for orthogonality to be ensured in the CCD, the condition q = 0 must be satisfied, which implies that:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

Hence,

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

is the value that always gives an orthogonal CCD, where a = 2Kn1. Consequently, the information matrix in Eq. 7

becomes m0 = Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs = diagonal (M1-IK, M2I1, p0IK) for: orthogonally-restricted CCD, where, p0 = 2n2α4. Putting

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

in M1 and p0, we obtain:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

and

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

Obviously, the eigenvalues of M0 are its diagonal elements. Hence, the ordered eigenvalues of M0 are:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

where:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

and Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs each has K multiplicities in the above ordering while α has:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

multiplicities in the same ordering. We then have for the orthogonally-restricted CCD that the determinant of the information matrix is given by:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(18)


Table 1: Comparison of D-optimality values of the orthogonally-Restricted variations for Selected N-point CCD
Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

Table 2: Comparison of D-optimality Values of the Rotatably-Restricted Variations for Selected N-point CCD
Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

Therefore, its D-optimality value becomes:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(19)

Using Eq. 19, the numerical values of φ[Mo(η)] and φ[Mo(ξ)] for each variation of orthogonally-restricted CCD given in Table 1 is obtained.

Another interesting and important restriction in this article is that of readability. A design is rotatable when the variance of the estimated response is a function of only the distance from the center of the design and not on the direction. Recall a CCD is rotatably-restricted if and only if 2K n1+2α4n2 = 3(2Kn1). Therefore:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

always gives a rotatable CCD. Putting

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

in M1 and M3 of (7), we obtain M1a+2(αn2)1/2 and M3 = 2alK+aJK. Thus, the information matrix, MR for the rotatable CCD could be written as Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs. It can easily be seen that the eigenvalues of MR has a regular pattern, hence, the ordered eigenvalues of this matrix are:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

where, a has:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs

multiplicities, (a+2(an2)1/2) has K multiplicities, 2a has (K-1) multiplicities, and (K+2)a occurs only once in the ordering. We therefore have for the rotatably-restricted CCD that the determinant of the information matrix is given by:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(20)

Therefore, its D-optimality value becomes:

Image for - On the Preference of Replicating Factorial Runs to Axial Runs in Restricted Second-order Designs
(21)

Using Eq. 21, the numerical values of φ[MR(ξ)] and φ[MR(η)] for each variation of rotatably-restricted CCD given in Table 2 are obtained.

RESULTS AND DISCUSSION

The following variations of the designs have been examined for illustration: one cube (n1 = 1) plus two replicated stars (n2 = 2); one cube (n1 = 1) plus three replicated stars (n2 = 3); one cube (n1 = 1) plus four replicated stars (n2 = 4); two replicated cubes (n1 = 2) plus one star (n2 = 1); three replicated cubes (n1 = 3) plus one star (n2 = 1) and four replicated cubes (n1 = 4) plus one star (n2 = 1). In Table 1 and 2, the results of the comparison are shown. We have obtained the numerical values of φ[MR(ξ)] and φ[MR(η)] for two, three, four and five factors. For each of these factors, three cases have been considered namely, Case one: two cubes plus one star versus one cube plus two stars: Case two: three cubes plus one star versus one cube plus three stars and Case three: four cubes plus one star versus one cube plus four stars. For each variation of N-point restricted CCD, at least two points are taken from the center in order to make up the required N points and also, to get a minimum of three degrees of freedom for pure error. Observe that the basis of variation in each case is the value of α which differs from each other for the two variations.

Table 1 and 2 summarize the comparison using the D-optimality values. We can note from the table that in all the three cases and for all the values of K considered, maximum D-optimality values were obtained when replicated cubes plus one star variation of both orthogonally-restricted CCD and rotatably-restricted CCD was used. That is, for the two conditions of restrictions, the strict inequality φ[MR(ξ)] ≥ φ[MR(ξ)] holds. A t-Test: Paired Two Sample for Means conducted afterwards between the D-optimality values of M(ξ) and M(η) revealed p-values of 0.02 and 0.04, respectively, for one-tailed and two-tailed tests for both orthogonal and rotatable restrictions of the CCD (indicating significant difference). Hence, this variation is preferred to one cube plus replicated stars variation of restricted CCD. We conjecture that this is true for any number of factors.

In general, the one cube plus replicated stars variation does not do well in this comparison.

CONCLUSION

The computational results in the tables show that the replicated cubes plus one star variation is better than the one cube plus replicated stars variation when any of the two restrictions considered in this work is imposed on a CCD. Since, in general, the D-optimality values of M(ξ) are significantly greater than those of M(η), we conclude that the replicated cubes plus one star variation is better than the one cube plus replicated stars variation in the sense of D-optimality criterion.

REFERENCES

1:  Atkinson, A.C. and A.N. Donev, 1992. Optimum Experimental Designs. Clarendon, Oxford

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13:  Nalimov, V.V., T.I. Golikova and N.G. Mikeshina, 1970. On practical use of the concept of D-optimality. Technometrics, 12: 799-812.
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14:  Nwobi, F.N., A.C. Okoroafor and I.B. Onukogu, 2001. Restricted second-order designs on one and two concentric balls. Statistica, 61: 103-112.

15:  Onukogu, I.B., 1997. Foundations of Optimal Exploration of Response Surfaces. 1st Edn., Ephrata Press, Nsukka, Nigeria, pp: 187

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