Statistics, 2014 Vol. 48, No. 4, 815–833, http://dx.doi.org/10.1080/02331888.2012.760097
Efficient three-level screening designs using weighing matrices Stelios D. Georgioua , Stella Stylianoua and Manohar Aggarwalb * a Department
of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi, Samos 83200, Greece; b Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA (Received 23 February 2012; final version received 15 December 2012) Screening is the first stage of many industrial experiments and is used to determine efficiently and effectively a small number of potential factors among a large number of factors which may affect a particular response. In a recent paper, Jones and Nachtsheim [A class of three-level designs for definitive screening in the presence of second-order effects. J. Qual. Technol. 2011;43:1–15] have given a class of three-level designs for screening in the presence of second-order effects using a variant of the coordinate exchange algorithm as it was given by Meyer and Nachtsheim [The coordinate-exchange algorithm for constructing exact optimal experimental designs. Technometrics 1995;37:60–69]. Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8] have used conference matrices to construct definitive screening designs with good properties. In this paper, we propose a method for the construction of efficient three-level screening designs based on weighing matrices and their complete foldover. This method can be considered as a generalization of the method proposed by Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8]. Many new orthogonal three-level screening designs are constructed and their properties are explored. These designs are highly D-efficient and provide uncorrelated estimates of main effects that are unbiased by any second-order effect. Our approach is relatively straightforward and no computer search is needed since our designs are constructed using known weighing matrices. Keywords: weighing design; screening; optimal design; foldover; correlation; efficiency AMS Subject Classifications: Primary 62K05, 62K20; Secondary 62K15, 62K25
1.
Introduction
Screening is a process through which experimenters are able to identify the most important factors among the many factors that may affect a particular response. Screening is important in many areas of application including industrial research and development, drug discovery, oncology, genetics, biomedical engineering, computer simulation experiments and machine learning (see e.g. [1]). These experiments are cost-effective as they can estimate a more accurate model. Screening experiments are usually performed when the number of feasible experimental runs is limited. Generally, screening is applied by fitting a multivariate linear model. An example of a linear main effects model is y = Xβ + ε, *Corresponding author. Email:
[email protected] © 2013 Taylor & Francis
ε ∼ Nn (0n , σ 2 In ),
(1)
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where y is the n × 1 response vector and X = [x1 , . . . , xm ] is the n × m model matrix. The jth column of the model matrix is denoted by xj = [x1j , . . . , xnj ] . The experimental error is denoted by ε and is assumed to be i.i.d. multivariate normal with the zero mean vector and a variance matrix = σ 2 In , where In is the identity matrix of order n. The m × m matrices X X and (X X)−1 are called information matrix and dispersion matrix, respectively. The covariance matrix of the linear least-squares estimator βˆ of β is σ 2 (X X)−1 , and for the construction of optimal designs, we seek to minimize a given function of the dispersion matrix (or equivalently, to maximize a function of the information matrix). A mean orthogonal design with all its main effects orthogonal to each other will be called orthogonal design. For quantitative factors, the linear model (1) with a suitably chosen screening design can be used for screening out the main effects in the presence of active second-order terms, such as two-factor interactions or pure-quadratic effects, in the true model. In a second stage of such experiments, we usually try to fit the following second-order model using only the p m main effects retrieved by the screening stage: y = β0 +
p i=1
βi xi +
p−1 p i=1 j=i+1
βij xij +
p
βii xii + ε,
(2)
i=1
where y is the response vector, xi , xii = xi2 and xij = xi xj are columns corresponding to the main, pure-quadratic and two-factor interaction effects, respectively. β0 , βi , βii and βij are unknown constant coefficients corresponding to the intercept, main, pure-quadratic and two-factor interaction effects, respectively, while ε is the error vector with components εj being i.i.d. N(0, σ 2 ). Two-level fractional factorial designs of resolution III or IV are generally used to identify the main effects in model (1) (see [2] or [3]). Designs of resolution III or IV may not be applicable in some situations as designs of resolution III have their main effects and some two-factor interactions fully confounded, while designs of resolution IV, or higher, are much more expensive. Moreover, two-level designs do not have the ability of capturing any curvature or any active pure-quadratic effects that may exist in the underlying true model. The common way that researchers address this problem in the literature is to add some centre points to some known two-level designs. However, applying this method will only give a clue about the existence of curvature but it cannot provide independent estimation of the pure-quadratic effects of each factor. Using this approach, further experimentation is necessary to overcome this problem. Several other traditional three-level screening designs, including the full 3m factorial or 3m−p fractional factorial designs and the three-level orthogonal arrays OA(n, m, s = 3, t), could also be used. Moreover, optimal designs that can be constructed algorithmically by maximizing some defined criterion could also be employed. For example, one may use the D-criterion (see [4,5]), the Q-criterion (see [6,7]) and many more. In a recent paper, Jones and Nachtsheim [5] suggest using an alternative class of quantitative three-level designs of the form described in Table 1. It is quite hard to find orthogonal designs of this form since a computer search is needed. The authors suggested using a variant of the coordinate exchange algorithm as it was given by Meyer and Nachtsheim [8]. A design with all main effects orthogonal to each other might not be found, by the computer search proposed in [5], due to the infeasible amount of time the complete search needs. For example, it was impossible to find an orthogonal design for m > 10 factors by the computer search applied in [5]. Such algorithms have some serious drawbacks such as they may be trapped in local maxima, and in this case, the generated design is not globally optimal. A complete search algorithm has high complexity and it is infeasible even for small m. For example, if m = 12, there are 211×12 cases to be tested for an exhaustive search.
Statistics Table 1.
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Form of the Jones and Nachtsheim design.
Run
x1
x2
···
xm
1 2 3 4 .. . 2m − 1 2m 2m + 1
0 0 ±1 ∓1 .. . ±1 ∓1 0
±1 ∓1 0 0 .. . ±1 ∓1 0
··· ··· ··· ··· .. . ··· ··· ···
±1 ∓1 ±1 ∓1 .. . 0 0 0
Xiao et al. [9] showed that definitive designs, as those suggested by Jones and Nachtsheim [5], can be constructed using known conference matrices. The designs constructed by this method have their main effects orthogonal to each other, orthogonal to any quadratic effects and orthogonal to any two-factor interactions. They comment that their method can be generalized to use weighing matrices but they concluded that such approach deserves further study. Another alternative was suggested by Elster and Neumaier [10]. In that paper, they introduced a new class of experimental designs called edge designs. These designs allow a model-independent estimate of the set of relevant variables, thus providing more robustness than traditional designs. They use a special case of weighing matrices, the conference matrices, to obtain a family of designs and an edge strategy for screening the active effects in the presence of quadratic terms. Even though their designs are two level, the suggested method and their edge structure seem to have some desirable robustness properties concerning quadratic terms. For example, the minimal edge design they proposed is of the form j Cm + Im D= , (3) j Cm − I m where j is the m × 1 vector with all its entries equal to 1, Im is the identity matrix of order m and Cm is a conference matrix of order m satisfying Cii = 0. The design consists of n = 2m runs and m disjoint edges and can examine up to m factors. To perform a screening process, they defined a set of pairs E = {(i, i + s), i = 1, . . . , m}, and the corresponding edges (ti , tj ), where t is the th run (row) of the design matrix D. Independent of any particular model, data collected with edge designs may be evaluated using the assumption that only a few, say p, of the m factors are active, that is, contribute to the variability in the observations. This so-called factor sparsity assumption, mentioned, for example, by Lenth [11], is very natural in screening experiments and implies that almost all differences zij = yi − yj , (i, j) ∈ E, consist of noise only. If we assume that the noise in the data is additive, normally distributed with zero mean and variance σ 2 , then m − p of the zi are normally distributed with zero mean and variance 2σ 2 . Because of the unknown number of outliers, the variance must be estimated in a√robust way. For example, we can use the median estimate σˆ = median{|zi | : (i, j) ∈ E}/(0.675 2) which is consistent when p = 0 [11], and hence, it is expected to give reliable results when p m. Outliers then determine the active factors. For more details, the reader can refer to [10]. In this paper, we have introduced a method for constructing efficient three-level quantitative designs by taking foldover of weighing matrices. This method can be considered as a generalization of the method suggested by Xiao et al. [9] since the conference matrices they used in their method is a special case of the weighing matrices we use in this paper. Moreover, weighing matrices can exist in many cases where conference matrices do not exist. The proposed method is a straightforward
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construction and no computer search is needed. In Section 2, we give the definition of weighing matrices and discuss their important properties. In Section 3, we present the proposed construction method and give some illustrating examples of such designs. In Section 4, the properties of the constructed designs are investigated. In particular, we have proved general formulas in closed form for the correlation between the main effects, quadratic effects and interaction effects. The D-efficiency of the designs constructed by the proposed method is also given in a closed form. In the same section, the space exploration of the constructed designs is investigated. In Section 5, we apply our general results to some very interesting special cases and derive some simplified expressions of their correlation structures and their D-efficiencies. In Section 6, we present some practical aspects of the new designs.
2. Weighing matrices Suppose that W is the design matrix of a chemical balance weighing experiment. Then, W = W (m, k) is a square matrix of order m, has entries from the set {0, ±1}, has k non-zero entries per row and column, satisfies WW = W W = kIm and is universally optimal in the class of all m-observation chemical balance weighing designs, with m objects, such that at most k objects are used in each weighing (see [12]). Parameter k is called the weight of W . Example 1 ⎛
+ ⎜0 ⎜ =⎜ ⎝−
0 + 0
+ 0 +
⎞ 0 +⎟ ⎟ ⎟, 0⎠
0
−
0
+
0 ⎜0 ⎜ W4c = ⎜ ⎝−
0 0
+ 0
0 −
0 0
0 +⎟ ⎟ ⎟, 0⎠
W4a
⎛
⎛
W6a
0
0
−
W4b
⎞
+
+
+ ⎜− ⎜ =⎜ ⎝−
+ + 0
− 0 −
⎞ 0 −⎟ ⎟ ⎟, +⎠
0
−
−
−
+ ⎜− ⎜ =⎜ ⎝−
+ +
− +
− −
− +
⎛
W4d
0
+ + − − 0 ⎜+ − − + − ⎜ ⎜ ⎜+ − + 0 + =⎜ ⎜− − 0 − − ⎜ ⎜ ⎝0 − − − + −
⎛
⎞
+ 0⎟ ⎟ ⎟ +⎟ ⎟, +⎟ ⎟ ⎟ −⎠ +
+
W6b
⎛
⎞ − −⎟ ⎟ ⎟, −⎠ −
+ ⎜+ ⎜ ⎜ ⎜− =⎜ ⎜0 ⎜ ⎜ ⎝+
0 + 0 − −
0 + 0 + −
+ − − 0 −
+ 0 + + 0
⎞ − 0⎟ ⎟ ⎟ −⎟ ⎟. +⎟ ⎟ ⎟ 0⎠
0
−
+
0
−
−
W4a = W (4, 2) is a weighing design of order 4 and weight 2, W4b = W (4, 3), W4c = W (4, 1), W4d = W (4, 4), W6a = W (6, 5) and W6b = W (6, 4), where − stands for −1 and + denotes 1. A weighing matrix C = W (n, n − 1) of order n and weight n − 1 is called the conference matrix (in the literature, it can also be found as C-matrix or n-type matrix). This square matrix C can always be transformed to have 0s on the diagonal and ±1s off the diagonal, such that C C = CC is a multiple of the identity matrix I. Thus, if the matrix has order n, C C = (n − 1)In . Matrices W4b and W6a , given in Example 1, are conference matrices of order 4 and 6, respectively.
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For more results and a survey on the existence of weighing and conference matrices, the reader can refer to [13,14] or [15]. In the next section, we present our main construction method for efficient three-level screening designs by using weighing matrices and their full foldover designs.
3. Three-level quantitative screening designs from weighing matrices The method presented in this section is a generalization of the method proposed by Xiao et al. [9]. Our main construction is given in the following. Construction: Let W = W (m, k) be a weighing matrix of order m and weight k. Set ⎛ ⎞ W ⎜ ⎟ D = ⎝0s×m ⎠ , (4) −W where by 0s×m we denote the s × m zero matrix (the centre points). Then, matrix D can be looked upon as the design matrix of a three-level screening design with 2m + s runs that can examine up to m factors. Lemma 1 The columns of the design matrix D generated by the construction given in Equation (4) are orthogonal to each other. Proof
⎛
D D = (W
0s×m
⎞ W ⎜ ⎟ − W ) ⎝0s×m ⎠ = W W + 0s×m 0s×m + W W = 2kIm . −W
The following remark describes some straightforward properties of the design matrix given in Equation (4). Remark 1 The design matrix D, as constructed by Equation (4), has m mean orthogonal threelevel quantitative factors and n = 2m + s runs (including s centre points). Excluding the centre points, there are 2(m − k) zeros, k positive and k negative elements in each factor and there are m − k zeros in each run. The factors of D are pairwise orthogonal and D D = 2kIm , where Im is the identity matrix of order m. The following examples show the proposed construction in detail. Example 2 Let W4a = W (4, 2) be the weighing design of order 4 and weight 2 as it was defined in Example 1. Set ⎛ ⎞ + 0 + 0 ⎜ ⎟ ⎜ 0 + 0 +⎟ ⎜ ⎟ ⎜− 0 + 0 ⎟ ⎜ ⎟ ⎞ ⎜ ⎛ ⎟ W4a ⎜ 0 − 0 +⎟ ⎜ ⎟ ⎟ ⎜ ⎟ D = ⎝ 01×4 ⎠ = ⎜ ⎜0 0 0 0⎟. ⎜ ⎟ −W4a ⎜− 0 − 0 ⎟ ⎜ ⎟ ⎜ 0 − 0 −⎟ ⎜ ⎟ ⎜ ⎟ ⎝+ 0 − 0 ⎠ 0 + 0 −
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Design matrix D can be used to examine 4 quantitative three-level factors with 9 runs. If, instead of one, we include s centre points, then the obtained design will be able to examine 4 quantitative three-level factors with 8 + s runs. Using W6a = W (6, 5), as it was given in Example 1, and s = 1, we end up with the same design as the one in [5] or [9] for 6 factors. Remark 2 If we have 3k = 2m + s, then each of the three levels appears equally often in each column of the design constructed by Equation (4). If 2m ≤ 3k, then this condition can always be satisfied if we add s = 3k − 2m centre points, otherwise the resulted design cannot be balanced. The following example illustrates this result. Example 3 Let W = W (m, k) = W (13, 9). Examples of such matrices can be found in [16] or [17]. Define ⎞ ⎛ 0 + + 0 + + + + 0 − 0 − − ⎜ ⎟ ⎜ + 0 0 0 − + + + − 0 − + + ⎟ ⎟ ⎜ ⎜ − + + 0 − 0 0 0 + − + + + ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 + − + + + − 0 + 0 − + 0 ⎟ ⎜ ⎟ ⎜ + 0 + + − + − 0 0 + + 0 − ⎟ ⎟ ⎜ ⎛ ⎞ ⎜ ⎟ W ⎜ − − − 0 0 + + + + + + 0 0 ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ D = ⎝0s×13 ⎠ , with W = ⎜ ⎜ 0 0 + + 0 0 + − + + − − + ⎟, ⎟ ⎜ −W ⎜ − 0 + + + − 0 + − + 0 + 0 ⎟ ⎟ ⎜ ⎜ 0 + 0 − + + 0 − − + + 0 + ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ + − 0 + + 0 + − 0 − + + 0 ⎟ ⎜ ⎟ ⎜ + + 0 − 0 − + 0 + + 0 + − ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ + − + − + 0 − + + 0 0 0 + ⎠ + + − + 0 − 0 + 0 0 + − + where 0s is the column vector with s zeros. Matrix D can examine up to 13 quantitative (or qualitative) three-level factors with 26 + s runs. Note that for s = 1, we have 3k = 2m + s and thus from Remark 2 the above design D has, in addition, the equal occurrence property (each of the three levels, on each factor, appears the same number of times).
4.
Some properties of the proposed designs
In this section, we investigate some properties of the constructed designs. In particular, we study their alias structures, their correlation properties, their D-efficiencies and their space exploration ability. For the remaining of this paper, the following notation will be used. Notation 1 We use d to denote a mean orthogonal design with 2m + s runs and m factors and D = (d1 , . . . , dm ) for its corresponding design matrix constructed by Equation (4). X1 denotes the model matrix of the first-order model, including a column of 1s and the m columns of D. X denotes the model matrix of the full second-order model.
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4.1. Alias structure for fitting a first-order model Let Xint be the n × m(m − 1)/2 matrix with all the possible two-factor interactions and Xquad be the n × m matrix with all the pure-quadratic terms. The alias matrices for the first-order model associated with the two-factor interactions and the pure-quadratic terms are given by Aint = (X1 X1 )−1 X1 Xint
(5)
Aquad = (X1 X1 )−1 X1 Xquad ,
(6)
and respectively. Designs that are suitable for screening are expected to have relatively small absolute values in these biased matrices. Theorem 1 Let D be as above. Then, (i) any quadratic effect of a factor is orthogonal to all the main effects in the constructed design; (ii) any two-factor interaction is orthogonal to all the main effects in the constructed design. Proof (i) Without loss of generality, we may exclude the centre points from D and denote the remaining matrix by G. Thus, G = (g1 , . . . , gm ), where gj , j = 1, . . . , m, is the jth column vector of length 2m. Due to the full foldover construction of G, the bottom half of each of its columns is the negative of the top half of the same column. Thus, if z is any column of G, then zi = −zi+m for all i = 1, . . . , m. The quadratic effect z2 of z is the elementwise product of z by itself, that is, z2 = [z12 , . . . , zm2 , z12 , . . . , zm2 ] . Let w be any column of G (w can be the z column or w might be any other column of G). The column w has the same property as any column of G, that is, the bottom half of w is the negative of the top half of w. Thus, w = [w1 , . . . , wm , −w1 , . . . , −wm ] and so the Euclidean inner product of z2 and w is z2 , w =
2m
zi2 wi =
i=1
m i=1
zi2 wi −
m
zi2 wi = 0.
i=1
The result follows. (ii) The proof is similar and omitted. From Theorem 1, we have the following. Corollary 1 The alias matrix (5) for the first-order model associated with the two-factor interactions is a zero matrix Aint = 0. The alias matrix (6) for the first-order model associated with the pure-quadratic terms is ⎛
Aquad
⎞ 2k(2m + s) . . . 2k(2m + s) ⎜ ⎟ 0 ... 0 ⎜ ⎟ ⎟. =⎜ .. .. .. ⎜ ⎟ ⎝ ⎠ . . . 0 ... 0
Theorem 1 is very general. Actually, the only property needed for its proof is the foldover structure of the constructed designs. So, it is straightforward that any design of this form, with a foldover structure, ensures the properties proved in Theorem 1.
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4.2. Correlation between two quadratic effects Quadratic effects cannot be estimated using two-level designs since they are fully confounded with each other and with the intercept. In this section, we provide a general formula, in closed form, for the correlation between two pure-quadratic terms of the constructed designs. Theorem 2 Suppose that the used weighing matrix W = W (m, k) includes two columns wj and w having t zeros, 0 ≤ t ≤ m − k, in the same positions (rows). Then, the Pearson correlation between the corresponding two pure-quadratic effects of X is rjj, = r(xjj , x ) =
k−m+t+a , a
(7)
where a = k − 2k 2 /(2m + s). Proof The Pearson correlation between the pure-quadratic effects of columns xj and x of X is given by 2m+s (xi,jj − x¯ jj )(xi, − x¯ ) rjj, = i=1 . (8) 2m+s 2m+s 2 2 ¯ jj ) ¯ ) i=1 (xi,jj − x i=1 (xi, − x For any j = 1, . . . , m, the column xjj (that corresponds to the pure-quadratic effect of xj ) consists of 2k ones and 2m − 2k + s zeros. So, x¯ jj = 2k/(2m + s), j = 1, . . . , m. Thus, 2m+s
(xi,jj − x¯ jj )2 = (2m − 2k + s)¯xjj2 + 2k(1 − x¯ jj )2 = 2k −
i=1
4k 2 = 2a. 2m + s
From Equation (9), we have that
2m+s 2m+s √ (xi,jj − x¯ jj )2 (xi, − x¯ )2 = 2a2a = 2a. i=1
(9)
(10)
i=1
Moreover, we have that 2m+s
(xi,jj − x¯ jj )(xi, − x¯ ) =
i=1
2m+s i=1
=
2m+s i=1
xi,jj xi, −
2m+s
xi,jj x¯ −
i=1
2m+s
x¯ jj xi, +
2m+s
i=1
x¯ jj x¯
i=1
2 2k 2k 2k xi,jj xi, − 2k − 2k + (2m + s) 2m + s 2m + s 2m + s
and so 2m+s i=1
(xi,jj − x¯ jj )(xi, − x¯ ) =
2m+s i=1
xi,jj xi, −
4k 2 . 2m + s
(11)
Since there are t coincidence of zeros in the rows of columns wj and w , there will be 2t + s coincidence of zeros in the rows of columns xj and x and thus the column xjj = xjj x = xj xj x x will have 4(m − k) − 2t + s zeros and 4k − 2m + 2t ones. From Equation (11), we obtain 2m+s i=1
(xi,jj − x¯ jj )(xi, − x¯ ) = 4k − 2m + 2t −
4k 2 = 2k − 2m + 2t + 2a. 2m + s
(12)
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Using Equations (10) and (12) in Equation (8), we obtain rjj, = r(xjj , x ) =
k−m+t+a . a
In the next corollary, we give some alternative forms of the correlation between two purequadratic effects. Corollary 2 Suppose that the used weighing matrix W = W (m, k) includes two columns wj and w having t zeros, 0 ≤ t ≤ m − k, in the same positions (rows). Then, the Pearson correlation between the two corresponding pure-quadratic effects of X can also be written as (t − (m − k))(2m + s) , k(s + 2(m − k)) 2t + s (m − k) − t = r(xjj , x ) = − . s + 2(m − k) k
rjj, = r(xjj , x ) = 1 +
(13)
rjj,
(14)
Remark 3 The correlation properties, of the derived design, depend on the structure of the used weighing matrix. (1) Suppose that the used weighing matrix W = W (m, k) includes two columns wj and w having all zeros in the same positions (rows). Theorem 2 with t = m − k implies that the purequadratic effects, corresponding to columns xj and x , will have correlation rjj, = 1. This is not surprising since the columns xj and x can be considered as columns from an orthogonal two-level design with 2k runs (with all elements ±1) and 2m − 2k + s centre points. (2) Suppose that a pair of columns (wj and w ) in the used weighing matrix W = W (m, k) have no zeros in the same positions (rows). Theorem 2 with t = 0 implies that the pure-quadratic effects, corresponding to columns xj and x , will have correlation rjj,, = (k − m + a)/a. A straightforward condition for a weighing matrix to have this property, for all pairs of columns, is to have weight m − 1. So, we can conclude that a weighing matrix W = W (m, m − 1) has no coincidence of zeros in any two columns and vice versa. In the following lemma, we describe the structure of a pair of columns of a weighing matrix. This will be particularly useful in the sequel. Lemma 2 Suppose that wj and w are two columns of the weighing matrix W = W (m, k), having t zeros, in the same positions (rows). t can take any value from the set t ∈ {0, . . . , m − k} for which m + t is even. Proof These two columns can be written to the form t
wj = (0 · · · 0 w = (0 · · · 0
m−k−t
0 ··· 0 ±1 · · · ±1
m−k−t
2k−m+t
±1 · · · ±1 ±1 · · · ±1), 0 ··· 0 ±1 · · · ±1).
Due to the orthogonality of the weighing matrix, 2k − m + t has to be even and thus m + t has to be even (both m and t have to be either odd or even).
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4.3. Correlation between a quadratic effect and a two-factor interaction Due to the orthogonality of the used weighing matrix, we can conclude that any two-factor interaction is mean orthogonal. The correlation between a quadratic effect xqq and a two-factor interaction xq , where q = , is the same as the correlation between the main effects xq and x and thus is equal to zero. So, rqq,q = r(xqq , xq ) = 0, for any q = . So we need to investigate the correlation between a quadratic effect xjj and a two-factor interaction xq , where = q = j = . The next theorem provides a closed formula for the correlation between a pure-quadratic term and a two-factor interaction in the model matrix. Theorem 3 Suppose that the used weighing matrix W = W (m, k) includes two columns wj and w having t zeros, 0 ≤ t ≤ m − k, in the same positions (rows) and = q = j = . Then, the Pearson correlation between a quadratic effect xqq of X and the two-factor interaction effect xj , corresponding to xj and x , can be written as √ ( 2m + s) 2m+s i=1 xi,qq xi, j rqq, j = r(xqq , xj ) = . (15) (4k − 2m + 2t)(2k(2m + s) − 4k 2 ) Proof The proof is similar to the proof of Theorem 2 and thus is omitted.
More detailed results will be given in the sequel for some interesting special cases. 4.4. Correlation of a pair of two-factor interactions In this section, we examine the correlation of a pair of two-factor interactions (xqp , xj ). The next theorem deals with the case when all q, p, j, are distinct since any other cases were investigated in some previous sections. Theorem 4 Suppose that the used weighing matrix W = W (m, k) includes two columns wj and w having t1 zeros, 0 ≤ t1 ≤ m − k, in the same positions (rows), two columns wp and wq having t2 zeros, 0 ≤ t2 ≤ m − k, in the same positions (rows), and all indices j, , p, q are distinct. Then, the Pearson correlation of the pair of two-factor interactions (xj , xqp ) corresponding to columns xj , x , xq and xp can be written as 2m+s i=1 xi,qp xi, j . (16) rqp, j = r(xqp , xj ) = √ (4k − 2m + 2t1 )(4k − 2m + 2t2 ) Proof The proof is similar to the proof of Theorem 2 and thus is omitted.
Remark 4 Note that the correlation of a pair of two-factor interactions is not affected by the number of centre points. Also, it is obvious that if t1 = t2 = t, then the correlation is simplified to 2m+s xi,qp xi, j . rqp, j = r(xqp , xj ) = i=1 (4k − 2m + 2t) When the two-factor interactions consist of non-distinct columns, then the non-distinct columns can be considered as a pure-quadratic effect and the correlation can be calculated as in the previous sections. One such example is given in Example 4.
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Example 4 If q = j, then the correlation rqp, j = r(xqp , xj ) of the two-factor interaction xqp and xj (corresponding to the pairs of columns (xp , xq ) and (xj , x )) is exactly the same as the correlation rqq,p = r(xqq , xp ) of the pure-quadratic effect xqq (of xq ) and the two-factor interaction xp . So, this correlation can be calculated by Theorem 3. 4.5. Design efficiency In this section, we investigate the D-efficiency of the derived designs based on the weight of the weighing matrix used for the construction. It is well known that the D-efficiency of any design d1 , relative to a non-singular design d2 , is given by |X(d1 ) X(d1 )| 1/ Deff (d1 , d2 ) = , |X(d2 ) X(d2 )| where X(di ) is the model matrix of design di for some specified model, i = 1, 2, and is the number of terms in the model (see [18]). For screening itself, the efficiency measure is immaterial. However, when a linear relation happens to be the correct model, perhaps after suitable transformations of the data, a high efficiency implies that good use was made of the given number of experiments. The model of interest here consists of the intercept term and all m linear effects. The (absolute) D-efficiency of any design d is given by Deff (d) = Deff (d, dD ), where dD is the D-optimal design. In order to obtain the D-efficiencies of the proposed designs, it is necessary to obtain the D-optimal main effects design for n = 2m + s runs. For consistency and comparability, we have used the D-optimal designs for n = 2m runs and augmented this design with s centre points. When m is even, orthogonal main effects plans for n = 2m, such as Plackett−Burman designs, are readily available and are known to be D-optimal with determinant (2m)m . Theorem 5 Suppose m be even. The D-efficiency of the constructed design d1 = d associated with the model matrix X1 is Deff = (k/m)m/(m+1) . Proof From the orthogonality of the used weighing matrix, we have that D D = 2kIm . The determinant of the information matrix D D is (2k)m . Then, the determinant of the corresponding information matrix |X1 X1 | = 2m(2k)m . Since m is even, there exists an orthogonal design d2 = dD with the first-order model matrix Z1 = Z1 (dD ) of size 2m × (m + 1), elements ±1 and determinant of the corresponding information matrix |Z1 (dD ) Z1 (dD )| = (2m)m+1 . Thus, Deff (d1 , d2 ) =
|X(d1 ) X(d1 )| |X(d2 ) X(d2 )|
1/ =
|X1 X1 | |Z1 Z1 |
1/ =
2m(2k)m (2m)m+1
1/(m+1) =
m/(m+1) k . m
Remark 5 The D-efficiency of the designs d, with design matrices D constructed by Equation (4), depends on the weight of the used weighing matrix and it is independent of the number of centre points used in this design. The largest determinant is obtained when the used weighing matrix W (n, k) has weight k = n. This is the case where the obtained design matrix D will be the full foldover of a saturated two-level design with s centre points. As mentioned above, this construction can give a clue for the existence of curvature but it cannot estimate the pure-quadratic effects. The next best D-efficiency is obtained for k = n − 1. In this case, the weighing matrix is the well-known conference matrix of order n. So, if we use only one centre point (to make the design as economical as possible) and a weighing matrix W (n, n − 1) (conference matrix), then the derived design is identical to the designs constructed in [9]. Using other weighing matrices W (n, k), with k < n − 1, results to new designs.
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Screening is usually performed using a large number of potentially active factors. The aim of screening is to reduce the large number of potentially active factors (sometimes this number can be in the order of some hundreds or even some thousands) using as few experiments as possible and fitting a first-order linear model. Then, in a second stage of experimentation, a more complex model (e.g. polynomial) can be employed, using only the few active main effects retrieved in the screening stage. Thus, finding economical screening designs with many factors and high D-efficiency is of great importance. When second-order terms are present in the true model, screening with a first-order model is accomplished with high efficiency using the designs constructed from weighing matrices, while it is misleading when using a traditional two-level design. The designs d, with design matrices D constructed using Equation (4), have high Defficiency and their D-efficiency increase with the number of factors m. Actually the following holds. Corollary 3 Suppose that W = W (m, k) is the weighing matrix used in Equation (4) to construct the (2m + s) × m design matrix D. Suppose that m is even and k = m − c, for some fixed constant c ∈ {0, 1, . . . , m − 1}. Then, the D-efficiency of the constructed design d, with the corresponding model matrix X1 , tends to 1 as m tends to infinity (limm→∞ Deff (d) = 1). Proof
Since c is a fixed constant, the D-efficiency of d is m − c m/(m+1) c m/(m+1) Deff (d) = Deff (d, dD ) = = 1− , m m
and the result follows.
Corollary 4 Suppose that W = W (m, k) is the weighing matrix used in Equation (4) to construct the (2m + s) × m design matrix D. Suppose that m is even and k = m − c, for some constant c ∈ {0, 1, . . . , m − 1}. For fixed m, the D-efficiency of the constructed design d, with the corresponding model matrix X1 , reduces as c increases. Equivalently, the D-efficiency of the constructed design d, with the corresponding model matrix X1 , increases as the weight k of W increases. Proof
Deff (d) =
and the result follows.
m−c m
m/(m+1)
c m/(m+1) = 1− , m
4.6. Space exploration For the unknown underlying model, the shape of the response surface may be different in different regions of the box of interest. Since this may affect the screening process and the decision on the right set of variables, it is important that the design points are located so as to explore the whole design space to get a good global view of the response surface and to guard against unexpected nonlinearities. In the next theorem, we examine the distance between the inner points of the screening designs constructed by Equation (4). In several papers, the authors use the design inner point distances as a natural criterion to investigate the space coverage and exploration (see e.g. [19] or [10]). Theorem 6 Let D be the design matrix constructed by Equation (4) using a weighing matrix i j W = W (m, k). The Euclidean distance of any two, √ non-centre, points (rows) d and d of D is √ 2k if j = i + m (or equivalently d i = −d j ) or 4k if j = i + m (or equivalently d i = −d j ).
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Proof The proof follows by noting that ||d i ||2 = k, ∀i = 1, . . . , m, d i , −d i = −k, d i , d j = 0, ∀i = j + m, i, j = 1, . . . , m, where ·, · is the Euclidean inner product, and ||d i − d j ||2 = ||d i ||2 + ||d j ||2 − 2 < d i , d j >, ||d i + d j ||2 = ||d i ||2 + ||d j ||2 + 2 < d i , d j >. The following theorem shows that a foldover structure of the design matrix D ensures high average of the squared Euclidian distances of the points in the design matrix. Theorem 7 The average squared Euclidean distance of all pairs of points (rows) (d i , d j ), G of a design matrix D2m×m = ( −G ), with exactly k non-zero elements (±1) in each row is 2km/(2m − 1). Proof The squared Euclidean distance of any two points d i and d j is given by ||d i − d j ||2 = ||d i ||2 | + ||d j ||2 − 2 d i , d j . It is obvious that ||d i ||2 = ||d j ||2 = || − d i ||2 = || − d j ||2 = k and suppose d i , d j = a, for some d i = ±d j . The pairs of points (−d i , d j ), (d i , −d j ) and (−d i , −d j ) also exist in the design matrix and their inner product will be −a, −a and a, respectively. When d i = −d j (equivalently j = i + m), the inner product d i , d j = −k. There are in total 2m(2m − 1) pairs of the 2m points of the design matrix D; 2m pairs will have inner product equal to −k, while all the other 4m(m − 1) pairs will have inner products that sum to zero. The sum of squared Euclidean distance of all pairs of distinct points is 2m
||d i − d j ||2 =
i, j=1, j =i
2m
(||d i ||2 + ||d j ||2 − 2 d i , d j )
i, j=1, j =i
=
2m i, j=1, j =i
(||d i ||2 + ||d j ||2 ) − 2
m
( d i , −d i + −d i , d i )
i=1
= 2m(m − 1)k + 2m(m − 1)k + 4km = 4km2 . Thus, the average squared Euclidean distance of all pairs of points (rows) (d i , d j ), of a design G matrix D2m×m = ( −G ), with exactly k non-zero elements (±1) in each row is 4km2 /2m(2m − 1) = 2km/(2m − 1). Remark 6 The above theorem shows that the foldover construction ensures that the average of the squared distances of the inner design points will be high. This fact does not necessarily result to large minimum distances of all pairs of the design points. Following the same arguments as the proof of the theorem, it is easy to show that if a design has some√inner products of points d i , d j = a = 0, then the minimum distance of these pairs will be 2k − 2|a| which is less than the minimum distance in the points of the design matrix constructed by using weighing matrices (see Theorem 6). Thus, the foldover structure together with the orthogonality of the weighing matrix ensure that the design constructed by Equation (4) will be optimal in terms of this minimum distance criterion. 5.
Some special cases
The results presented in Section 4 are general. In this section, we further investigate the properties of the constructed designs when weighing matrices of some large weights are used.
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5.1. Using weighing matrices W = W(m, m − 1) with one zero (conference matrices) Some results on this special case were recently published by Xiao et al. [9]. In this subsection, we present further results concerning the correlation between the model effects for any number of centre points s. Corollary 5 Suppose that the used weighing matrix is W = W (m, m − 1) (a conference matrix of order m), then (i) the correlation between any two pure-quadratic effects of X is rjj, =
s 1 − ; s+2 m−1
(ii) the correlation between a pure-quadratic effect and a two-factor interaction of X is rqq, j =
√ ± 2m + s (m − 1)(m − 2)(s + 2)
;
(iii) the D-efficiency of the constructed design d, with the corresponding model matrix X1 , is Deff (d) = ((m − 1)/m)m/(m+1) = (1 − 1/m)m/(m+1) . Proof This follows from the results of Section 4.
Some observations concerning Corollary 5 are presented in the following remark. Remark 7 It is straightforward from the correlation formula, given in Corollary 5, that the correlation of two-factor interactions is independent of the number of added centre points. Also, the correlation of the pure-quadratic effects increases while the correlation of the two-factor interactions and the correlation between a pure-quadratic effect and a two-factor interaction decrease with the number of centre points added to the design. (1) For s = 0 (no centre point), the correlation between any two pure-quadratic effects of X is rjj, = −1/(m − 1), while the √correlation between a pure-quadratic effect and a two-factor interaction of X is rqq, j = ± m/(m − 1)(m − 2). (2) For s = 1 (one centre point), the correlation between any two pure-quadratic effects is rjj, = 13 − 1/(m − 1), while√ the correlation between a pure-quadratic effect and a two-factor interaction of X is rqq, j = ± (2m + 1)/3(m − 1)(m − 2). Note that in this case the results are the same as the results given in [9]. (3) For s = 2 (two centre points), the correlation between any two pure-quadratic effects is rjj, = 21 − 1/(m − 1), while √ the correlation between a pure-quadratic effect and a two-factor interaction of X is rqq, j = ± (m + 1)/2(m − 1)(m − 2). (4) For s = 3 (three centre points), the correlation between any two pure-quadratic effects is rjj, = 35 − 1/(m − 1), while √ the correlation between a pure-quadratic effect and a two-factor interaction of X is rqq, j = ± (2m + 3)/5(m − 1)(m − 2). 5.2. Using weighing matrices W = W(m, m − 2) with two zeros The next highest weight is m − 2. In this section, we investigate the case of W (m, m − 2). Such cases are also important and may be employed when a conference matrix does not exist.
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Corollary 6 Suppose that the used weighing matrix is W = W (m, m − 2), and t is the number of zeros coincidence in columns wj and w of W . (i) The correlation between any two pure-quadratic effects of X is rjj, = 1 +
(t − 2)(2m + s) . (s + 4)(m − 2)
(ii) The correlation between a pure-quadratic effect xqq of X and the two-factor interaction xj is dependent on t. When t = 0, we have either rqq, j = 0 or √ ±2 2m + s rqq, j = √ , (m − 4)(m − 2)(s + 4) and if t = 2, we have either rqq, j = 0 or rqq, j
±2 = m−2
2m + s . s+4
(iii) The D-efficiency of the constructed design d, with the corresponding model matrix X1 , is Deff (d) = ((m − 2)/m)m/(m+1) = (1 − 2/m)m/(m+1) . Some observations concerning Corollary 6 are presented in the following remark. Remark 8 It is straightforward, from the correlation formula given in Corollary 6, that the correlation between two pure-quadratic effects increases, while the correlation between a purequadratic effect and a two-factor interaction decreases with the increase in the number of centre points added to the design. Also, it is obvious that there will always be some pairs of columns with t = 2 coincidence of zeros and thus the corresponding pure-quadratic effects will have correlation 1. This is obvious since the definition of weighing matrices implies that there will be exactly two zeros in each column and so there exists some pairs of columns having a coincidence of at least one zero. Due to the orthogonality of the weighing matrix, m + t cannot be odd (see Lemma 2). Since m can be odd only for m = 3, we have that t will be equal to 2 for some pairs of columns (for all m > 3). Some examples of the correlation between a pure-quadratic effect and a two-factor interaction are given below. Example 5 When k = m − 2, the correlation between a pure-quadratic effect and a two-factor interaction takes the following values for s = 0, 1, 2, 3. (1) For s = 0 (no centre point), the correlation between √ a pure-quadratic effect and a two-factor interaction of X is either √ rqq, j = 0 or rqq, j = ± 2m/(m − 1)(m − 2) when t = 0 and either rqq, j = 0 or rqq, j = ± 2m/(m − 2) when t = 2. (2) For s = 1 (one centre point), the correlation between √ a pure-quadratic effect and a two-factor interaction of X is either rqq, j = 0 or rqq, j = √ ±2 (2m + 1)/5(m − 2)(m − 4) when t = 0 and either rqq, j = 0 or rqq, j = (±2/(m − 2)) (2m + 1)/5 when t = 2. (3) For s = 2 (two centre points), the correlation between √ a pure-quadratic effect and a two-factor interaction of X is either rqq, j = 0 or rqq, j =√±2 (m + 1)/3(m − 2)(m − 4) when t = 0 and either rqq, j = 0 or rqq, j = (±2/(m − 2)) (m + 1)/3 when t = 2. (4) For s = 3 (three centre points), the correlation between√a pure-quadratic effect and a twofactor interaction of X is either rqq, j = 0 or rqq, j = √ ±2 (2m + 3)/7(m − 2)(m − 4) when t = 0 and either rqq, j = 0 or rqq, j = (±2/(m − 1)) (2m + 3)/7 when t = 2.
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5.3. Using weighing matrices W = W(m, m − 3) with three zeros Designs with weight equal to m − 2 are useful but, since some of their quadratic effects are fully confounded with each other, one may select the weighing matrix with the next available weight. One such case would be a W (m, m − 3). Remark 9 Suppose that the used weighing matrix is W = W (m, m − 3), then the D-efficiency of the constructed design d, with the corresponding model matrix X1 , is Deff (d) = ((m − 3)/m)m/(m+1) = (1 − 3/m)m/(m+1) . Corollary 7 Suppose that the used weighing matrix is W = W (m, m − 3), then the correlation between any pure-quadratic effects of X is rjj, = 1 +
(t − 3)(2m + s) , (s + 6)(m − 3)
where t is the number of zeros coincidence in columns wj and w of W . Some observations concerning Corollary 7 are presented in the following remark. Remark 10 It is straightforward from the correlation formula, given in Corollary 7, that the correlation of the pure-quadratic effects increases with the increase in the number of centre points added to the design. It is known that if W = W (m, m − 3) exists, then m = 7 or m = 4(i + 1) for some non-negative integer i. For m = 7, we have that t = 1 for all pairs of columns. For m > 7 and using Lemma 2, we have that there might be some pairs of columns with t = 0 coincidence of zeros and some others with t = 2. The complete structure of a W = W (m, m − 3) is ⎛ ⎞ I4 ⎜ ⎟ . ⎟ W = W (m, m − 3) = ⎜ ⎝(±1) . . (±1)⎠ , I4 with all elements outside the 4 × 4 diagonal blocks being all non-zero, and this structure was revealed in [20]. So, for the correlation of the pure-quadratic effects, there are 6(p + 1) pairs of corresponding columns with t = 2 and (p + 1)(16p + 6) pairs with t = 0. (1) For s = 0 (no centre point), there are 6(p + 1) pairs of columns with the corresponding purequadratic effects having correlation 23 − 1/(m − 3) and (p + 1)(16p + 6) pairs of columns with the corresponding pure-quadratic effects having correlation −3/(m − 3). (2) For s = 1 (one centre point), there are 6(p + 1) pairs of columns with the corresponding purequadratic effects having correlation 57 − 1/(m − 3) and (p + 1)(16p + 6) pairs of columns with the corresponding pure-quadratic effects having correlation 71 − 3/(m − 3). (3) For s = 2 (two centre points), there are 6(p + 1) pairs of columns with the corresponding purequadratic effects having correlation 43 − 1/(m − 3) and (p + 1)(16p + 6) pairs of columns with the corresponding pure-quadratic effects having correlation 41 − 3/(m − 3). (4) For s = 3 (two centre points), there are 6(p + 1) pairs of columns with the corresponding purequadratic effects having correlation 79 − 1/(m − 3) and (p + 1)(16p + 6) pairs of columns with the corresponding pure-quadratic effects having correlation 31 − 3/(m − 3). Corollary 8 Suppose that the used weighing matrix is W = W (m, m − 3) and wq , wj , w are three distinct columns of W (i.e. q = j = = q). Moreover, if wj , w have t zeros in the same
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positions, then the correlation between the pure-quadratic effect xqq and the two-factor interaction xj of X takes the following eight values: rqq, j = ±ρ or rqq, j = ±3ρ, where 2m + s ρ= , t = 0, 2. (m − 3)(m − 6 + t)(s + 6) Proof A weighing matrix W = W (m, m − 3) has two zeros in each row and column (by definition). From Lemma 2 and the structure of W , as described in Remark 10, we have either t = 0 or t = 2. For k = m − 3, we have that 4k − 2m + 2t = 2(m − 6 + t) and 2k(2m + s) − 4k 2 = 2(m − 3)(s + 6). Employing the structure of the used weighing matrix W (m, m − 3), as described in Remark 10, and the mean orthogonality of the interaction xj , we see that 2m+s i=1 xi,qq xi, j can take only one of the values ±2, ±6 depending on the structure of the quadratic effect xqq . The result follows from Theorem 2. Some observations concerning Corollary 8 are presented in the following remark. Remark 11 It is straightforward from the correlation formula, given in Corollary 8, that the correlation between the pure-quadratic effect xqq and a two-factor interaction xj of X decreases with the increase of the number of centre points added to the design. For example, the correlation between the pure-quadratic effect xqq and a two-factor interaction xj of X takes the following eight values: rqq, j = ±ρ or rqq, j = ±3ρ, where √ (1) √ for s = 0 (no centre point), we have that ρ = m/3(m − 3)(m − 6) for t = 0 and ρ = m/3(m − 3)(m − 4) for t = 2. √ (2) for s √ = 1 (one centre point), we have that ρ = (2m + 1)/7(m − 3)(m − 6) for t = 0 and ρ = (2m + 1)/7(m − 3)(m − 4) for t = 2. √ (3) for s √ = 2 (two centre points), we have that ρ = (m + 1)/4(m − 3)(m − 6) for t = 0 and ρ = (m − 1)/4(m − 3)(m − 4) for t = 2. √ (4) for s √ = 3 (three centre points), we have that ρ = (2m + 3)/9(m − 3)(m − 6) for t = 0 and ρ = (2m + 3)/9(m − 3)(m − 4) for t = 2. Since the formulas given in Section 4 are general, similar results can be computed for W (m, m − 4), W (m, m − 5), . . . etc.
6.
Some practical issues and designs evaluation
In this section, we give some practical issues of the constructed designs. We present a simulated example to show how these designs can be employed in certain cases. So, we conclude our work with the application of a conference design to a simulated screening scenario. The data are simulated from the noisy quadratic model y = −x1 + x4 + x9 − x92 + x1 x9 + ε, where ε ∼ N(0, σ 2 ) and σ 2 = 41 . A conference matrix Cm of order m = 12 was employed Cm m ) and XE = ( jj CCmm +I and the design matrices DC = ( −C −Im ) are constructed. Note that DC is m the design matrix obtained by Equation (4) with s = 0 centre points. XE is the edge design obtained by Equation (3). XC = [j DC ] and XE are the used model matrices, respectively. More details on edge designs can be found in [10]. The simulated data obtained are yE = (−3.22, −0.65, −4.68, 1.05, −1.14, 0.72, −1.28, −1.43, −0.95, 0.77, −0.86, −1.29, 0.54, −1.00,
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−4.90, −0.77, −1.22, 0.88, −1.38, −0.96, −4.76, 0.52, −1.00, −1.33) , yC = (−1.22, −0.65, −4.68, 0.05, −1.14, 0.72, −1.28, −1.43, −1.95, 0.77, −0.86, −1.29, −1.46, −3.00, 1.10, −3.77, 0.78, −1.12, 0.62, 1.04, 2.24, −5.48, 1.00, −3.33) , while matrix C used for the construction of XE and XC is ⎞ ⎛ 0 + − + − − + − − + + + ⎜ − 0 + − − + − − + + + + ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ + − 0 − + − − + − + + + ⎟ ⎜ ⎟ ⎜ − + + 0 + − + + + − + + ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ + + − − 0 + + + + + + − ⎟ ⎟ ⎜ ⎜ + − + + − 0 + + + + − + ⎟ ⎟ ⎜ C=⎜ ⎟. ⎜ − + + − − − 0 + − + − − ⎟ ⎟ ⎜ ⎜ + + − − − − − 0 + − − + ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ + − + − − − + − 0 − + − ⎟ ⎟ ⎜ ⎜ − − − + − − − + + 0 + − ⎟ ⎟ ⎜ ⎟ ⎜ ⎝ − − − − − + + + − − 0 + ⎠ −
−
−
−
+
−
+
−
+
+
−
0
An analysis of the data using XC and yC in regression analysis revealed three significant main effects (marked in bold-face in Table 2) and gave an estimated linear model yˆ = −1.01 − 1.06x1 + x4 + 0.96x9 + ε, with ε of mean zero and standard deviation σ = 0.96. On the other hand, using the screening process of Elster and Neumaier [10] with edge design XE and data yE , we obtain the 12 edges with their absolute values given in Table 2. √ The median estimate σˆ = median{|zi | : (i, j) ∈ E}/(0.675 2) of σ is σˆ = 0.49. Since, |zi | > 3σ , for i = 1, 4, 9, one may assume that the active set of variables is {x1 , x4 , x9 }. Even though both designs discover correctly the same set of active factors, the edge design requires further experimentation to estimate an appropriate quadratic model that fits the data since the model matrix used is a two-level matrix and thus does not have the ability to estimate quadratic effects. On the other hand, the proposed design can be projected onto the three active factors and try to fit the full quadratic model using the full second-order model of the three active factors (retrieved by the screening stage) without any further experimentation. So, if one constructs the full quadratic model matrix X3 , consisting of the columns x1 , x4 , x9 , x12 , x42 , x92 , x1 x4 , x1 x9 , x4 x9 , and uses the same data set yC with linear regression, he/she will obtain the model yˆ = −1.06x1 + x4 + 0.96x9 − 1.27x92 + 0.95x1 x9 + ε, with ε of mean zero and standard deviation σ = 0.20. In this example, the use of a design matrix constructed by Equation (4) not only provides enough robustness to screen out the active main effects, but also achieves the estimation of the true underlying quadratic model without requiring any further experimentation. In screening experiments, there might be hundreds of potentially active factors, but only very few of them are expected to be important. When second-order terms are present in the true Table 2.
Model-independent check by Elster and Neumaier [10].
z1 |y1 − y13 | 3.76
z2 |y2 − y14 | 0.35
z3 |y3 − y15 | 0.22
z4 |y4 − y16 | 1.82
z5 |y5 − y17 | 0.09
z6 |y6 − y18 | 0.17
z7 |y7 − y19 | 0.10
z8 |y8 − y20 | 0.47
z9 |y9 − y21 | 3.81
z10 |y10 − y22 | 0.25
z11 |y11 − y23 | 0.14
z12 |y12 − y24 | 0.04
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model, screening with a first-order model is accomplished with high efficiency using the designs constructed from weighing matrices, while it is misleading using a traditional two-level design. The proposed method can generate a number of new and efficient screening designs that requires further research. Acknowledgements The authors are grateful to the anonymous referees and the associate editor for their valuable comments and suggestions that substantially improved the results and the appearance of the paper.
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