Composite Designs Based on Orthogonal Arrays and Definitive Screening Designs

ABSTRACT Central composite designs are widely used in practice for factor screening and building response surface models. We study two classes of new composite designs. The first class consists of a two-level factorial design and a three-level orthogonal array; the second consists of a two-level factorial and a three-level definitive screening design. We derive bounds of their efficiencies for estimating all and part of the parameters in a second-order model and obtain some general theoretical results. New composite designs are constructed. They are more efficient than central composite designs and other existing designs. Supplementary materials are available online.


Introduction
Response surface methodology, proposed by Box and Wilson (1951), explores the relationships between several explanatory variables and one or more response variables in many practical experiments, where the forms of the relationships are unknown.It typically has three steps.The first step is to identify important factors from all of the potential factors by using a two-level factorial design; the second step is to find optimum region; the last step is to find a suitable approximation for the true functional relationship between the response and important factors.Typically, there is curvature in the optimum region, and a second-order model is employed because such a model is easy to estimate and apply, even when little is known about the process.For k quantitative factors, denoted by x 1 , . . ., x k , a second-order model is where β 0 , β i , β ii , and β i j are the intercept, linear, quadratic, and bilinear (or interaction) terms, respectively, and is the error term.For the quadratic terms β ii to be estimated, all the factors must have at least three levels.A design is called a second-order design if it allows all parameters in (1) to be estimated.Many second-order designs have been proposed in the literature.The most popular designs are central composite designs (CCDs) introduced by Box and Wilson (1951) and variations such as small composite designs (Draper and Lin 1990).Other second-order designs include Box and Behnken (1960) designs, augmented pairs designs (Morris 2000), and subset designs (Gilmour 2006).For a comprehensive account of response surface methodology, see Box and Draper (2007), Khuri and Cornell (1996), and Myers, Montgomery, and Anderson-Cook (2009).Kiefer (1961), Farrell, Kiefer, and Walbran (1967), Lucas CONTACT Hongquan Xu hqxu@stat.ucla.eduDepartment of Statistics, University of California, Los Angeles, CA .Supplementary materials for this article are available online.Please go to www.tandfonline.com/r/JASA.(1974,1976), Pesotchinsky (1975), and many others studied D-optimal designs for the second-order model.Dette and Grigoriev (2014) recently studied E-optimal designs for the second-order model.
For k factors, a composite design d consists of three parts: (i) n 1 cube points (x 1 , . . ., x k ) with all x i = −1 or 1; (ii) n 2 additional points with all x i = −α, 0, or α; (iii) n 0 center points with all x i = 0.
The cube points have two levels, the additional points have three levels and a composite design has three or five different levels depending on whether α= 1 or not.
Two-level orthogonal arrays (OAs) such as full or fractional factorial designs are often used as the cube points.In a CCD, n 2 = 2k axial points (with one of x i = α or −α and all other x i = 0) are chosen as the additional points.For convenience, we focus on α = 1 in the article, but the basic idea can be extended to general α.Xu, Jaynes, and Ding (2014) introduced a new class of composite designs, called orthogonal-array composite designs (OACDs), which use runs of a three-level OA as the additional points.In other words, an OACD consists of a two-level factorial design, a three-level OA, and some center points.Like a CCD, an OACD can be used in a single experiment or a sequential experiment.The additional points from a three-level OA in an OACD provide information on linear and quadratic terms as well as bilinear terms in (1); in contrast, the axial points in a CCD are used to estimate linear and quadratic terms and do not provide any information on bilinear terms.As a result, an OACD is often more efficient than a CCD in estimating the parameters, especially bilinear terms.The ability of identifying important bilinear terms is the most crucial for some experiments such as combinatory drug experiments with many drugs we encounter in practice.
One appealing feature of an OACD is that it allows us to perform multiple analyses with different parts of the data for cross-validation.Xu, Jaynes, and Ding (2014) suggested to build three models: a model with linear and bilinear terms for the two-level design, a model with linear and quadratic terms for the three-level OA, and a full second-order model for the entire design.Since each linear effect is estimated three times and each bilinear or quadratic effect is estimated twice, we can check the consistency of their estimation and identify possible problems of the data or models.Due to this and other desirable features, researchers at UCLA Micro Systems Lab have used OACDs to investigate several biological systems, including a system of herpes simplex virus type I with five antiviral drugs (Ding et al. 2013), production of lipids of a cell-free system with six chemicals (Jaynes et al. 2016), a system of oral cancer with 11 drugs and others.Xu, Jaynes, and Ding (2014) provided a collection of OACDs based on popular two-level and three-level designs for k = 3-10 factors.Many other OACDs, possibly with better properties, could be constructed.Yet there are no general results on their properties.Further studies on the construction and properties of OACDs are called for to meet the practical needs.One purpose of this article is to develop some theoretical results on OACDs.We derive lower and upper bounds on the efficiency of OACDs and show that OACDs are more efficient than CCDs for estimating the parameters in a second-order model in Section 2.
We further introduce and study a new class of composite designs by using a three-level definitive screening (DS) design as the additional points in Section 3. DS designs were proposed by Jones and Nachtsheim (2011) and have many advantages over the standard two-level or three-level designs.One important property of a DS design is that all linear effects are orthogonal to bilinear and quadratic effects, so that the estimates of linear effects are not biased by the presence of active secondorder effects.Combining a two-level factorial and a three-level DS design, we obtain a definitive screening composite design (DSCD).We compare DSCDs with OACDs and CCDs and show that DSCDs are typically more efficient in estimating linear terms than OACDs and CCDs.Section 4 studies some practical issues regarding OACDs and DSCDs.Some new OACDs are constructed.They are more efficient than CCDs and other existing designs.We give some concluding remarks in Section 5 and proofs in the online supplementary Appendix.

Orthogonal-Array Composite Designs
Some notation and background are necessary.An orthogonal array (OA) of N runs, k columns, s levels, and strength t, denoted by OA(N, s k , t ), is an N × k matrix in which all s t level combinations appear equally often in every N × t submatrix.We omit the strength t when t = 2.
Let d be a k-factor composite design that consists of (i) a twolevel design d 1 with n 1 runs, (ii) a three-level design d 2 with n 2 runs and (iii) n 0 center points.The total number of runs of d is N = n 1 + n 2 + n 0 .With proper arrangement of the columns, let X = (1, Q, L, B) be the model matrix of the second-order model (1) for any design d, where 1 is a column of ones, Q, L, and B are quadratic, linear, and bilinear terms, respectively.Let M(d) = X X/N be the information matrix for d.The Doptimal criterion seeks to maximize |M(d)|, the determinant of the information matrix.Let ξ * be an approximate D-optimal design over the cube [−1, 1] k and define MaxD k = |M(ξ * )|.Kiefer (1961) and Farrell, Kiefer, and Walbran (1967) showed where For any k-factor design d, its D-efficiency is defined by where p = (k + 1)(k + 2)/2 is the number of parameters in the second-order model (1).We also compare designs in terms of the precision for estimating a subset of the model parameters.For s, a subset of factors of a design d, define where X s and X (s) are the submatrices of X corresponding to the parameters in s and not in s, respectively, and |s| is the number of parameters in s.Since Following Kiefer (1961), approximate D s -optimal designs on [−1, 1] k can be derived easily for linear, quadratic, and bilinear terms, that is, s = L, Q, B, respectively.Specifically, a D Land D B -optimal design is a product design ξ 1 × • • • × ξ k where ξ i puts equal weights on points x i = −1, 1.The D L -and D Boptimal design has D L and D B value of 1.A D Q -optimal design can be constructed as a product design ξ 1 × • • • × ξ k where ξ i puts weights 0.25, 0.5, 0.25 on points x i = −1, 0, 1.The D Qoptimal design has D Q value of 1/4.Then the D s -efficiency of a design d can be calculated as (7) Next consider a CCD with k factors and n 0 center points.When the two-level portion d 1 is an OA(n 1 , 2 k , 4) (or a full factorial for k < 4), the linear, quadratic, and bilinear terms are orthogonal to each other, that is, Q L = 0, Q B = 0, L B = 0.The information matrix is block diagonal as follows: where 1 k is a column of k ones, I k is the k × k identity matrix, J k is the k × k matrix of ones, and q = k(k − 1)/2.It is easy to obtain that From these equations, we obtain the efficiencies for a CCD.
Lemma 1.For a CCD with k factors and n 0 center points, if the two-level portion d 1 is an OA(n 1 , 2 k , 4), its D-, D L -, D B -, and D Q -efficiencies are, respectively, where In an OACD, the three-level portion d 2 forms a three-level OA.The information matrix X X is no longer block diagonal because its linear, quadratic, and bilinear terms are not necessarily orthogonal to each other.The information matrix and efficiencies for OACDs depend on the specific three-level OA used.So we consider lower and upper bounds in the following.
Theorem 1.Let d 1 be an OA(n 1 , 2 k , 4) and d 2 be an OA(n 2 , 3 k ).Then an OACD consisting of d 1 , d 2 , and n 0 center points is a second-order design.The determinant of its information matrix and D-efficiency have the following lower bounds, respectively, where which can be reached when d 2 is an OA(n 1 , 3 k , 4), where Examining ( 8) and ( 13), we know that an OACD typically has larger |X X| than a CCD provided (6n 1 + 4n 2 )n 2 /27 > 2n 1 + 4, which holds in general because n 2 ≥ 2k + 1 ≥ 9 for an OACD when k ≥ 4.This often leads to larger D-efficiency for an OACD even after adjustment for different run sizes used in the threelevel portion.Here is an example.
Example 1.For k = 4, . . ., 12, we construct OACDs by choosing the smallest OA(n 1 , 2 k , 4) as d 1 and the smallest threelevel OA as d 2 .Specifically, we choose a full factorial 2 k for k = 4 or a regular 2 k−m design with resolution at least V for k = 5-11 and the third column of Table 1 gives the m generators.For k = 12, we use an OA(128, 2 15 , 4) from Xu (2005) since the smallest regular resolution V design for 12 factors has 256 runs.For the three-level OA, we use the first k columns of "oa.9.4.3.2.txt, " "oa.18.7.3.2.txt, " and "oa.27.13.3.2.txt" from Sloane's website http://neilsloane.com/oadir/.The D-efficiency of an OACD depends on the choice of d 2 , but not on d 1 as long as d 1 has strength 4. For each OACD, we compare it with a CCD consisting of the same two-level portion.Table 1 shows the D-efficiency of OACDs and CCDs as well as the lower bound δ eff (oacd) in ( 14) with n 0 = 0 center points.For every k ≥ 5, an OACD has larger D-efficiency than a CCD, and the lower bound δ eff (oacd) is also larger than D eff (ccd) when k ≥ 6.When the number of center points n 0 increases, the D-efficiencies of both OACDs and CCDs decrease, and when n 0 > 0, an OACD has larger D-efficiency than a CCD for every k ≥ 4.
For any three-level OA, we can always permute the levels for one or more factors so that it includes a center point.Permuting levels for the three-level OA lead to many OACDs with different geometrical structures and efficiencies.Our theoretical result is important in this regard because the lower bound guarantees the efficiency of all possible resulting OACDs.We have the following result when an OACD and a CCD have the same number of runs.
Corollary 1. Assume that a CCD and an OACD have the same number of runs and use the same strength 4 OA as the two-level portion.For k ≥ 5, the OACD has larger D-efficiency than the CCD if the CCD has at most seven center points.
An OACD can have larger D-efficiency than a CCD even if the conditions in Corollary 1 do not hold; see Example 1 and Table 1.In other words, the conditions in Corollary 1 are sufficient but not necessary.
For the D s -efficiencies of an OACD, we have the following lower and upper bounds.
Theorem 2. Let d be an OACD satisfying the conditions in Theorem 1.Its D L -, D B -, and D Q -efficiencies have the following lower bounds, respectively, where N = n 1 + n 2 + n 0 and q = k(k − 1)/2.Furthermore, its D L -efficiency has an upper bound and the equality holds when the linear terms of d 2 are orthogonal to the bilinear terms of d 2 .The D B -efficiency has an upper bound which can be reached when d 2 is an OA(n 1 , 3 k , 4).The D Qefficiency has an upper bound The lower and upper bounds in Theorem 2 tell us how the choice of n 0 , n 1 , and n 2 would affect the D s -efficiency.When n 0 increases, the D L -, D B -, and D Q -efficiencies decrease.
When n 1 increases, the D L -and D B -efficiencies increase but the D Q -efficiency decreases.When n 2 increases, the D Q -efficiency increases.
Example 2. Consider the same OACDs and CCDs as in Example 1 for k = 4, . . ., 12, and compare their D s -efficiencies for s = L, B, Q. Table 2 shows that an OACD has larger D L -efficiency than a CCD for every case, although the lower bound is smaller than D L,eff (ccd).An OACD also has larger D B -efficiency than a CCD for every case except k = 8.Furthermore, an OACD has larger D Q -efficiency than a CCD when k ≥ 5 and the lower bound δ Q,eff (oacd) is larger than D Q,eff (ccd) when k ≥ 8.
The lower bound δ L,eff (oacd) in ( 16) is often less than D L,eff (ccd) in (10).On the other hand, the upper bound γ L,eff (oacd) in ( 19) is always larger than D L,eff (ccd) in (10).A design is called mirror-symmetric if its mirror-image is itself, that is, reversing the level order for all factors leads to the same design.Tang and Xu (2014) showed that for a mirror-symmetric design its linear terms are orthogonal to the bilinear terms.Comparing Lemma 1 with Theorem 2, we have the following results.
Corollary 2. An OACD has larger D B -efficiency than a CCD if they have the same number of runs and use the same strength 4 OA as the two-level portion.In addition, the OACD also has larger D L -efficiency than the CCD if the three-level portion of the OACD is mirror-symmetric.Tang and Xu (2014) showed that a regular three-level design is mirror-symmetric if and only if it contains a center point.We can always make a regular three-level design mirror-symmetric by permuting levels for one or more columns.

Definitive Screening Composite Designs
A DS design (Jones and Nachtsheim 2011) for k factors has 2k + 1 runs and can be represented as where C = (c i j ) is a k × k matrix with c ii = 0 and c i j ∈ {−1, 1}, i = j, and 0 is a column of zeros.It is necessary to require that C has full column rank so that all linear effects are estimable.When k is even, it is possible to further choose C such that columns of C are orthogonal to each other.A DS design is mirror-symmetric so all of its linear effects are orthogonal to all quadratic and bilinear effects.Jones and Nachtsheim (2011) listed some DS designs for k = 4-12 factors.Xiao, Lin, and Bai (2012) and Phoa and Lin (2015) proposed to choose a conference matrix as C in ( 22) for even k.A k × k matrix C is a conference matrix if it satisfies (i) c ii = 0 and c i j ∈ {−1, 1}, i = j; (ii) C C = (k − 1)I k .Xiao, Lin, and Bai (2012) listed some conference matrices for k = 2, 4, . . ., 18.When k = 2v + 1 is odd, C C cannot be diagonal.Phoa and Lin (2015) considered a class of DS designs with structure They proposed a systematic construction method and gave a list of C matrices satisfying ( 23) for v = 3-9 and 11-15.We obtain a DSCD by combining a two-level factorial design with a three-level DS design and some center points.Since the DS design already includes a center point, we can take one less additional center point for a DSCD than a CCD so that both designs have the same number of runs.
Example 3. We construct DSCDs using three-level DS designs from Jones and Nachtsheim (2011) for k = 4-12 and compare them with OACDs and CCDs.We choose the twolevel portions as in Example 1 and n 0 = 0 for all designs.Figure 1 shows that DSCDs have smaller D-efficiency, larger D L -efficiency, and smaller D Q -efficiency than both CCDs and OACDs.In terms of D B -efficiency, DSCDs are similar to CCDs and OACDs.Moreover, with the increase of n 0 , DSCDs and CCDs have similar D-efficiencies, especially when n 0 ≥ 3.
We have the following general result on the D-efficiency of DSCDs.
Theorem 3.For k ≥ 4, let d 1 be an OA(n 1 , 2 k , 4) and d 2 be a kfactor DS design.A DSCD consisting of d 1 , d 2 , and n 0 center points is a second-order design.
(i) The D-efficiency has an upper bound where where   We have the following results regarding the D s -efficiency of DSCDs.
Theorem 4. Let d be a DSCD satisfying the conditions in Theorem 3.
When C is a conference matrix for even k or satisfies (23) for odd k, the D L -efficiency is (26) (ii) Its D B -efficiency has, respectively, a lower and an upper bound (iii) Its D Q -efficiency has, respectively, a lower and an upper bound Comparing ( 10) and ( 11) with ( 26) and ( 27), we have the following results.
Corollary 3. A DSCD has larger D B -efficiency than a CCD if they have the same number of runs and use the same strength 4 OA as the two-level portion.In addition, the DSCD has larger D Lefficiency than the CCD if the matrix C is a conference matrix or satisfies ( 23).
Next we compare the D L -efficiency between an OACD and a DSCD.Assume that both designs use the same OA(n 1 , 2 k , 4) as the two-level portion and have n 0 center points.The OACD has N 1 = n 1 + n 2 + n 0 runs and the DSCD has Then the lower bound ( 26) of D L -efficiency of a DSCD for even k is larger than the upper bound ( 19) of an OACD.In general, n 2 ≥ 2k + 1 and If k ≥ 6, we have n 1 ≥ 32 and n 2 ≥ 18 to fulfill the strength requirement, then When n 0 ≤ 10, n 1 + 4(2 − n 0 ) ≥ 0, and > 0, then the DSCD has larger D Lefficiency than the OACD.In summary, we have the following result.
Corollary 4. Assume that a DSCD and an OACD have the same number of center points n 0 and use the same strength 4 OA as the two-level portion.If the matrix C is a conference matrix for even k > 4 and n 0 ≤ 10, then the DSCD has larger D L -efficiency than the OACD.

Choice of n 0 , n 1 , and n 2
First consider how the choice of n 0 affects the D-efficiency and the bounds.
Example 5. Figure 3 plots the D-efficiency of CCDs and bounds of OACDs and DSCDs against the number of center points, n 0 , for k = 6 factors, with n 1 and n 2 as in Example 1.The Defficiency of CCDs and the bounds of OACDs decrease slowly as the number of center points increases.A similar pattern holds for DSCDs when n 0 ≥ 1.The lower bound of a DSCD is close to the D-efficiency of a CCD whereas the lower bound of an OACD is always larger than the D-efficiency of a CCD.
As Figure 3 shows, the D-efficiency of a composite design typically decreases as the number of center points increases.To achieve high D-efficiency, we should choose n 0 = 0 for OACDs and n 0 = 0 or 1 for DSCDs.On the other hand, one of the primary purposes of adding center points is to provide an independent estimate of pure error variance.To balance the efficiency and estimation of pure error variance, we recommend three to five center points for both OACDs and DSCDs, following the common practice for CCDs.
To study the effect of choice of n 1 and n 2 on the D-efficiency of an OACD, let n 0 = 0 and consider the ratio r = n 1 /n 2 .Then N = (1 + r)n 2 and the lower and upper bounds of Defficiencies in Theorem 1 become , and MaxD k is defined in (2).Both δ eff (oacd) and γ eff (oacd) have a unique global maximum.Let r * and r * be their global maximum points, respectively.Table 3 shows ratios r * and r * and corresponding maximum values of δ eff (oacd) and γ eff (oacd) for k = 3-12.These ratios can be used as a guideline to choose n 1 and n 2 to achieve high D-efficiency.For example, when k = 10 and n 1 = 128, we have n 1 /r * = 128/4.874= 26.3 and n 1 /r * = 128/2.457= 52.1.A choice of n 2 around 26 to 52 would typically lead to a high D-efficiency.Some good choices of n 2 would be 27, 36, and 54 to have d 2 as an OA(n 2 , 3 k ); see Section 4.3.

Choice of the Two-Level Portions
So far we assume that the two-level portion d 1 is an OA(n 1 , 2 k , 4), which may have too many runs for some applications when k > 5. To reduce the number of runs, one can choose a strength 2 or 3 OA as the two-level portion.The following theorem shows a sufficient condition for an OACD or a DSCD to be a second-order design.
Theorem 5. Let d 1 be an OA(n 1 , 2 k ).Let L 1 and B 1 be the linear and bilinear terms of d 1 in the second-order model (1), respectively.If the matrix (L 1 , B 1 ) has full column rank, then the resulting OACD or DSCD is a second-order design.
The condition in Theorem 5 is also sufficient for a CCD to be a second-order design.It is well known that a regular resolution IV design cannot be used as the two-level portion for a CCD as this does not lead to a second-order design.In contrast, we can use a regular resolution IV design as the two-level portion to construct a small OACD with high D-efficiency.Examples are given in Xu, Jaynes, and Ding (2014) and below.In this regard, the OACD approach provides a broader class of composite designs than the CCD approach.
The upper bounds of D-and D s -efficiencies in Theorems 1-4 still hold when d 1 is an OA of strength 2 or 3.However, using strength 2 or 3 OAs does not lead to high D-efficiency compared with strength 4 OAs.For this reason, we do not pursue lower bounds when d 1 is an OA of strength 2 or 3. Kiefer (1961) and Farrell, Kiefer, and Walbran (1967) constructed approximate D-optimal designs for the second-order model.Galil and Kiefer (1977) listed approximate p -optimal (including D-, A-, and E-optimal) designs for k = 2-10.Dette and Grigoriev (2014) studied approximate E-optimal designs in general.Lim and Studden (1988) studied approximate D soptimal designs for estimating all second-order (i.e., both bilinear and quadratic) terms.Approximate D-optimal designs are not unique and may have different support points and different weights, but they have the same D-and D s -efficiencies.The same holds for approximate A-, E-, and D s -optimal designs.Approximate optimal designs cannot be used directly in practice and may have a large number of support points, but they are useful as comparison benchmark.Lucas (1976) compared various exact designs for the secondorder model and concluded that optimal CCDs from Lucas (1974) and designs from Pesotchinsky (1975) are the best.The optimal CCD consists of a full factorial or a resolution V design, 2k axial points, and 0 center points.There are two Pesotchinsky   4 are either full factorials or regular designs with resolution IV or higher.The three-level portions are chosen from some popular three-level OAs with runs ranging from 9 to 81.In all cases we use n 0 = 0.These OACDs, given in the supplementary material of this article, are constructed to have relatively high D-efficiency.We can construct many other OACDs to have smaller number of runs or larger D s -efficiency by using different two-level and three-level designs.

Some OACDs and Comparison of Designs
Most OACDs given in Table 4 have similar D-, D L -, D B -, and D Q -efficiencies as approximate D-optimal designs and approximate D s -optimal designs (for estimating second-order terms).These OACDs in general have larger D-, D L -, D B -efficiencies but smaller D Q -efficiency than approximate A-and E-optimal designs.
Compared to exact designs, several OACDs have larger Defficiency than the best CCD for every k.For k = 7, 9, 10, OACDs using regular two-level resolution IV designs have smaller number of runs yet larger D-efficiency than the best CCD.We also find many OACDs having larger D-efficiency than the Pesotchinsky designs except for one case with k = 6.
OACDs have good projection properties.Any two-factor projection of an OACD is a combination of a 2 2 and a 3 2 full factorial, each replicated n 1 /4 and n 2 /9 times, respectively.When an OACD has high D-efficiency, its projection designs also have high D-efficiency.Consider two OACDs in Table 4, the 9-factor 209-run and 10-factor 182-run OACD.We compute the mean, minimum, and maximum D-efficiencies among all possible projection designs.Table 5 shows that all of the projection designs have high D-efficiencies.For k = 5−7, the D-efficiency of any projection design from the 9-factor 209-run OACD exceeds the largest D-efficiency given in Table 4.

Concluding Remarks
We have studied properties of OACDs and DSCDs and compared them with the popular CCDs and other types of designs.OACDs are more effective in estimating the parameters in a second-order model than CCDs, in terms of the overall Defficiency, as well as D L -, D B -, and D Q -efficiencies.DSCDs are comparable with CCDs in terms of D-efficiency, and have larger D L -and D B -efficiency than CCDs.DSCDs have smaller Defficiency but larger D L -efficiency than OACDs.Following the analysis strategy proposed by Xu, Jaynes, and Ding (2014), we can use either an OACD or a DSCD to perform multiple analyses with different parts of the data for cross-validation.
OACDs have a unique feature that CCDs and DSCDs do not have.When combining a two-level and a three-level design, we often have some repeat runs in an OACD.We can always permute levels for one or more factors for the three-level OA to have more or less repeat runs.We may want to minimize the number of repeat runs and have more distinct points in some situations.On the other hand, we may want to maximize the number of repeat runs for estimating the pure error efficiently or to obtain a small OACD by deleting repeat runs when runs are costly.We can also permute levels for a three-level OA to include some center points and decrease the number of additional center points.
In summary, OACDs and DSCDs have many appealing properties such as allowing multiple analyses for cross-validation, permitting the use of resolution IV designs for the two-level portion and having high efficiency for estimating a secondorder model.They are useful alternatives to the popular CCDs for response surface modeling.If the primary interest focuses on linear terms, DSCDs are preferred; otherwise, OACDs are recommended.

Figure  .
Figure .Bounds of D-efficiencies of composite designs against number of factors with n 0 = 3. Composite design (symbol): CCD (c), lower bound of DSCD (d), upper bound of DSCD (D), lower bound of OACD (o), upper bound of OACD (O).

Example 4 .
We compare the lower and upper bounds of the Defficiency of DSCDs with the lower and upper bounds of OACDs and the D-efficiency of CCDs.We use n 1 and n 2 as in Example 1 and n 0 = 3. Figure 2 shows that the lower bound of the Defficiency of a DSCD is close to the D-efficiency of a CCD for each k.The lower bound of the D-efficiency of an OACD is larger than the D-efficiency of a CCD for k = 6-12 and is larger than the upper bound of D-efficiency of a DSCD for k = 9-12.For k = 4, an OACD and a DSCD have the same lower and upper bounds.

Figure  .
Figure .Bounds of D-efficiencies of composite designs against number of center points for k =  factors.Composite design (symbol): CCD (c), lower bound of DSCD (d), upper bound of DSCD (D), lower bound of OACD (o), upper bound of OACD (O).

Table  .
Comparison of D-efficiencies between OACDs and CCDs.

Table  .
Comparison of D s -efficiencies between OACDs and CCDs.
and MaxD k is defined in (2).(ii) If the matrix C in d 2 is a conference matrix for even k or satisfies (23) for odd k, then the D-efficiency has a lower bound

Table  .
Optimal ratios to maximize the lower and upper bounds of D-efficiencies.

Table  .
Comparison of designs.

Table  .
The D-efficiencies of projection designs of two OACDs in Table.We can construct many OACDs with varying run sizes and efficiencies by combining available two-level and three-level OAs for each k.Table4lists some OACDs and compares them with other designs for k = 4 − 10.The two-level portions of OACDs in Table