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 secondorder 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 2^{K} factorial
or a 2^{Kq} fractional factorial portion (run), usually called a cube,
with points selected from the 2^{K} points (x_{1}, x_{2},...,
x_{k}) = (±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 (=
2^{K} n_{1}+2Kn_{2}+n_{0}) points where n_{1}
is the number of cubes; n_{2} is the number of stars and n_{0}
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 firstorder
effects, the later addition of the star points permits secondorder 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 Doptimal 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 Doptimum concept can be used as the theoretical
basis for building and comparing response surface designs in use.
CENTRAL COMPOSITE DESIGNS
The CCD is the 2^{K} factorial or a 2^{Kq} 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(x_{1}, x_{2},..., x_{k}) represented in the experimental area, [±α, ±1], by the quadratic function:
where, y_{j} and e_{j} are, respectively the response and the random error of the j^{th} observation; β_{00}, β_{i0}, β_{ii} and β_{ii} are the unknown parameters of the regression model and x_{1}, x_{2},..., x_{k} are the independent variables. Alternatively, in vector notation, (1) is given by:
where, Y and e are the, respective (Nx1) response and error column vectors;
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:
By subtracting Eq. 3 from Eq. 1, we obtain:
where, and
.
Recall from Eq. 1, that:
hence, .
Then, (2) becomes:
after subtracting Eq. 3 from it, where
and:
is the {Nx(P1)} design matrix; N = 2^{K}n_{1}+2Kn_{2}+n_{0},
1 = (1, 1, ..., 1)' is a 2^{K}n_{1}/4 component vector,
is of n_{2} components, α^{2} = (α^{2, }α^{2,...,
}α^{2}) 0_{1} = (0, 0, ..., 0) is of n_{2}
components 0_{2} = (0, 0, ..., 0) is of (K2)n_{2} components,
0_{3} = (0, 0,...., 0) is of n_{0} components. The information
matrix of the design is obtained as:
where, M_{1} is the sum of squares of the elements in the vectors associated with the firstorder terms while M_{2} is the sum of squares of the elements in the vectors associated with the twoway crossproduct terms. However, M_{3} 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 secondorder terms and also, whose offdiagonal elements (denoted by q) are the sums of crossproducts of the elements in the vectors associated with the adjusted secondorder terms. Using notations accordingly, M_{1} = 2^{K} n_{1}+2n_{2}α^{2}, M_{2} = 2^{K} n_{1} and M_{3} = (pq)I_{K}+qJ_{K}. Note that, I_{K} is an (KxK) identity matrix, I_{t} is an (txt) identity matrix; where:
and J_{k} = 11, 1 is a column vector of K components. Using Eq. 6, we obtain;
where:
Having obtained
and ,
the definition of orthogonallyrestricted CCD is given below.
OrthogonallyRestricted CCD: A CCD is orthogonallyrestricted if
has a diagonal structure, that is, if and only if q = 0 in M_{3} 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 .
These elements are known as design moments although, originally, the elements
of:
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 (1^{δ1}
2^{δ2} ... K^{δK}) and is given by:
where,
are nonnegative integers. The sum:
is called the order of the design moment and is denoted by δ (δ =
0, 1, ...,2d). For example, (1^{1} 3^{1} 5^{3}) is a
design moment of order δ = 5Δ and is equal to
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:
where, γ_{δ} is a quantity that depends on d, δ and
N(Box and Hunter, 1957). According to Myers
(1991), a secondorder design with moments, which have the following conditions:
• 
All moments that have at least one δ_{i} to be
odd are zero 
• 
Pure fourth moments, which is equal to ,
are three times the mixed fourth moments, that is 
is rotatable. Notice from the portion of a typical design matrix, X for general CCD containing the secondorder terms that:
and
Of the two conditions given, which must be met in order that a secondorder
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.
1214 accordingly, we obtain:
Now for the information matrix in Eq. 7:
where, q = 2^{K} n_{1} and p = 2^{K} n_{1}+2n_{2}α^{4}
Rotatablyrestricted CCD: A CCD is rotatablyrestricted if and only if 2^{K} n_{1}+2α^{4}n_{2} = 3(2^{K}n_{1}).
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 Doptimal 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:
where, φ is the Doptimality criterion function which maximizes the determinant
of the information matrix; for instance (Pazman, 1986;
Fedorov, 1972; Onukogu, 1997):
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:
Hence,
is the value that always gives an orthogonal CCD, where a = 2^{K}n_{1}. Consequently, the information matrix in Eq. 7
becomes m_{0} =
= diagonal (M_{1}I_{K}, M_{2}I_{1}, p_{0}I_{K})
for: orthogonallyrestricted CCD, where, p_{0} = 2n_{2}α^{4}.
Putting
in M_{1} and p_{0}, we obtain:
and
Obviously, the eigenvalues of M_{0} are its diagonal elements. Hence,
the ordered eigenvalues of M_{0} are:
where:
and
each has K multiplicities in the above ordering while α has:
multiplicities in the same ordering. We then have for the orthogonallyrestricted CCD that the determinant of the information matrix is given by:
Table 1: 
Comparison of Doptimality values of the orthogonallyRestricted
variations for Selected Npoint CCD 

Table 2: 
Comparison of Doptimality Values of the RotatablyRestricted
Variations for Selected Npoint CCD 

Therefore, its Doptimality value becomes:
Using Eq. 19, the numerical values of φ[M_{o}(η)] and φ[M_{o}(ξ)] for each variation of orthogonallyrestricted 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 rotatablyrestricted if and only if 2^{K} n_{1}+2α^{4}n_{2}
= 3(2^{K}n_{1}). Therefore:
always gives a rotatable CCD. Putting
in M_{1} and M_{3} of (7), we obtain M_{1}a+2(αn_{2})^{1/2}
and M_{3} = 2al_{K}+aJ_{K}. Thus, the information matrix,
M_{R} for the rotatable CCD could be written as .
It can easily be seen that the eigenvalues of M_{R} has a regular pattern,
hence, the ordered eigenvalues of this matrix are:
where, a has:
multiplicities, (a+2(an_{2})^{1/2}) has K multiplicities, 2a has (K1) multiplicities, and (K+2)a occurs only once in the ordering. We therefore have for the rotatablyrestricted CCD that the determinant of the information matrix is given by:
Therefore, its Doptimality value becomes:
Using Eq. 21, the numerical values of φ[M_{R}(ξ)] and φ[M_{R}(η)] for each variation of rotatablyrestricted CCD given in Table 2 are obtained.
RESULTS AND DISCUSSION
The following variations of the designs have been examined for illustration:
one cube (n_{1} = 1) plus two replicated stars (n_{2} = 2);
one cube (n_{1} = 1) plus three replicated stars (n_{2} = 3);
one cube (n_{1} = 1) plus four replicated stars (n_{2} = 4);
two replicated cubes (n_{1} = 2) plus one star (n_{2} = 1);
three replicated cubes (n_{1} = 3) plus one star (n_{2} = 1)
and four replicated cubes (n_{1} = 4) plus one star (n_{2} =
1). In Table 1 and 2, the results of the
comparison are shown. We have obtained the numerical values of φ[M_{R}(ξ)]
and φ[M_{R}(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 Npoint 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 Doptimality values. We can note from the table that in all the three cases and for all the values of K considered, maximum Doptimality values were obtained when replicated cubes plus one star variation of both orthogonallyrestricted CCD and rotatablyrestricted CCD was used. That is, for the two conditions of restrictions, the strict inequality φ[M_{R}(ξ)] ≥ φ[M_{R}(ξ)] holds. A tTest: Paired Two Sample for Means conducted afterwards between the Doptimality values of M(ξ) and M(η) revealed pvalues of 0.02 and 0.04, respectively, for onetailed and twotailed 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 Doptimality 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 Doptimality criterion.