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Supersaturated designs of projectivity P=3 or near P=3
ARTICLE IN PRESS Journal of Statistical Planning and Inference 140 (2010) 1021–1029
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Supersaturated designs of projectivity P=3 or near P=3 John Tyssedal a,, Oddgeir Samset b a b
Department of Mathematical Sciences, The Norwegian University of Science and Technology, 7491 Trondheim, Norway Elkem Salten, NO-8226 Straumen, Norway
a r t i c l e in fo
abstract
Article history: Received 30 January 2009 Received in revised form 8 October 2009 Accepted 8 October 2009 Available online 23 October 2009
Supersaturated designs offer a potentially useful way to investigate many factors in few experiments i.e. typical screening situations. Their design properties have mainly been evaluated based on their ability to identify and estimate main effects. Projective properties have received little attention. In this paper we show how to construct twolevel supersaturated designs for 2(n2) factors in n runs (n a multiple of four) of projectivity P=3 or near projectivity P=3 from orthogonal non-regular two-level designs. The designs obtained also have favourable properties such as low maximum absolute value of the inner product between a main effect column and a two-factor interaction column and relatively few types of different projections onto subsets consisting of three factor columns. & 2009 Elsevier B.V. All rights reserved.
Keywords: Hadamard matrices Projection properties Screening designs
1. Introduction Supersaturated designs are experimental designs that employ fewer runs than the number of factors studied. Their use are primarily as screening designs. If it can be assumed that only a small number of factors (factor sparsity) or effects (effect sparsity) are active, these designs can provide considerable cost savings. So far few applications of supersaturated designs have been reported, but the potential for various fields of research to benefit from their use has been pointed out. Examples are computer and medical experiments (Lin, 1995), industrial and engineering experiments (Wu, 1993; Nguyen, 1996) and biometric applications (Butler et al., 2001). See also Dejaegher and Heyden (2008). Supersaturated designs were introduced by Satherwaite (1959) through the idea of random balanced designs while Booth and Cox (1962) were the first to consider a systematic construction of supersaturated designs. Lin (1993) used halffractions of Hadamard matrices, Nguyen (1996) showed how to construct supersaturated designs from balanced incomplete block designs and Wu (1993) used Hadamard matrices augmented with interaction columns. Computer generation of designs based on some criteria has also been considered (Lin, 1995; Nguyen, 1996; Li and Wu, 1997) among others. In the following we will only consider two-level supersaturated designs. Supersaturated designs cannot be made orthogonal. Let X be a n k design matrix of a design with n runs and k factors each at two levels with n=2 of þ1’s and n=2 of 1’s. Booth and Cox (1962) proposed as a criterion for comparing designs, the minimization of aveðs2 Þ, normally written as Eðs2 Þ, given by k s2ij = ; 2 ioj
X
Corresponding author.
E-mail address: [email protected] (J. Tyssedal). 0378-3758/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2009.10.004
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where sij is the ði; jÞ element of X0 X. An orthogonal design will have aveðs2 Þ ¼ 0, and the criterion can be considered as a measure of the degree of non-orthogonality among the columns in a supersaturated design. A lot of effort has been put into finding supersaturated designs that minimize Eðs2 Þ. Another suggested criterion for comparing supersaturated designs is the minimax criterion which first order designs by smax :¼ maxioj jsij j, the smaller the better, and then by the frequency of sij ¼ 7smax . Both the criteria above focus on the estimation and identification of first order effects. It is, however, possible to construct supersaturated designs that are optimal to both the criteria above where a main effect column is fully aliased with a two-factor interaction column. For instance the NOA4 algorithm in Ryan and Bulutoglu (2007) provided such a design for n ¼ 12 and k ¼ 20. Hence Eðs2 Þ optimal designs may perform badly when two-factor interactions are present. We believe there are many situations under which you want to use a supersaturated design and where the experimental conditions are not very different from when a saturated design is used, except that the number of factors is larger. Also increasing the number of factors will normally increase the probability of interactions between factors and decrease the probability that the heredity principle applies i.e. an interaction can be excluded from the model unless at least one (weak heredity) or both (strong heredity) of the main effects are included in the model. This principle has also been questioned in several papers (Box and Tyssedal, 2001; Montgomery et al., 2005). Any restrictive assumption on the underlying model will necessarily lower our chances to identify the correct subspace of active factors. In factor screening it is often the case that out of a larger number of tested factors only a small subset, typically two or three, maybe four, are expected to really affect a particular response. This is known as factor sparsity. Therefore when choosing an appropriate design for screening experimentation, it is of importance to consider projections of the design onto small subsets of factors. Box and Tyssedal (1996) defined projectivity of two-level designs as follows: a n k design with n runs and k factors each at two levels is said to be of projectivity P if every subset of P columns contains a complete 2P factorial, possibly with some runs replicated. They denoted such designs ðn; k; PÞ screens. Besides ensuring that all main effects and all interactions of any P factors can be estimated with no bias if the other factors are inert, projectivity P also implies that the ability of a subset of P factors to explain the variation in the data can be evaluated with rather weak assumptions on the underlying model (Tyssedal, 2008). While the projectivity of certain saturated orthogonal arrays is well known (Box and Tyssedal, 1996, 2001; Bulutoglu and Cheng, 2003; Tyssedal, 2008), projective properties of supersaturated designs have received little attention. In this paper we will focus on constructing supersaturated designs with nearly the same projective properties as saturated designs. In Section 2 we describe how to construct such designs and in Section 3, we show some interesting properties. A comparison to some other supersaturated designs are given in Section 4 and concluding remarks are given in Section 5.
2. Construction of supersaturated designs of projectivity P=3 or near P=3 Most supersaturated designs can only be of projectivity P=2 for modest to small number of runs, n. Evangelaras and Koukouvinos (2004) introduced the criterion of generalized projectivity. A n k design with n runs and k factors each at two levels is said to be of generalized projectivity Pa if for any selection of P columns out of possible k all main effects and interactions of order ra can be estimated. While projectivity P=3 for instance is a desirable property for a saturated design, projectivity P ¼ 32 i.e. all main effects and two-factor interactions for any three factors can be estimated with no bias if the other factors are inert, might be an attractive property of a supersaturated design. Algorithms designed to identify the active factor space for a projectivity P orthogonal two-level design (Box and Meyer, 1993; Tyssedal and Samset, 1997) can easily be modified to also take care of projectivity Pa designs. We now propose a new class of supersaturated designs constructed as follows: let Dn be a n k orthogonal two-level design with balanced columns. Further let c be a column in Dn and define Dcþ n=2 as the half fraction corresponding to the þ1 entries in c and Dc n=2 the one corresponding to the 1 entries in c. The following design: 2 cþ 3 Dcþ Dn=2 n=2 c;2ðk1Þ 4 5 Dn ¼ ð1Þ Dc Dc n=2 n=2 will, provided k4n=2 be a supersaturated design which can accommodate 2ðk 1Þ factors in n runs. An example of such a design constructed from the 12 run Plackett–Burman (Plackett and Burman, 1946) design is given in Fig. 1. In general designs obtained in this way are not Eðs2 Þ optimal designs, but as shown in the next section they have other desirable properties. 3. Properties of Dn2ðk1Þ designs The properties derived in this section do not depend on the choice of c. The superscript c will therefore be omitted. is obviously a balanced design. For any branching column Dþ D2ðk1Þ n n=2 and Dn=2 has exactly n=4 þ 1’s and n=4 1’s entries does not change that fact. and interchanging the signs in D n=2
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Fig. 1. A Da;20 the first column, column a as the branching column. Columns 12 design constructed from the 12 run Plackett and Burman (PB) design with " aþ # D6 b,y,k are columns from the PB designs. Columns l,y, u are the constructed ones in . Da 6
For the following discussion let i be a vector of m þ 1’s where m ¼ n=4. Then three arbitrary columns in Dn can be written as 3 2 i i x 6 i i x 7 7 6 ð2Þ 7 6 4 i i x 5 i
i
x
where x is a m 1 vector of pþ1’s and ðm pÞ 1’s. A fourth column will be of the form ½y1 y2 y3 y4 0 where the number of þ10 s in y2 and y3 are both equal to m q where q is the number of þ10 s in y1 and y4 . One particular case is worth mentioning. For m even and p ¼ q ¼ m=2 it is possible that there exists a defining relation between four columns. Let four such columns be written as 3 2 i i x x 6 i i x x 7 7 6 7: 6 4 i i x x 5 i
i
x
x
Further let c ¼ ½i i i i0 be the branching column. It is clear that the level combinations in ½x x and ½x x are equal. Also the level combinations in ½x x and ½x x must be the same. Hence interchanging the signs in ½i x x and when projected onto the three last columns above has only four distinct runs, an undesirable situation. ½i x x, D2ðk1Þ n We will now show the following result: Let Dn be an orthogonal two-level design of projectivity P ¼ 3 and assume that no defining relation exists between any four factors. Then " þ # Dn=2 D n=2 is of projectivity P=3. Proof. With reference to the discussion above we need to show that 3 2 i x y1 7 6 6 i x y2 7 7 6 6 i x y3 7 5 4 i
x
ð3Þ
y4
is of projectivity P=3. Without lack of generality we can assume pZqZm=2. Otherwise we can just multiply the corresponding column with 1 or interchange columns. Case 1: p4q. If p4q, then ½i x y1 contains the level combinations ! þ þ þ ; þ þ
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½i x y2 contains ! ; þ ½i x y3 contains ! þ þ þ and ½i x y4 contains ! þ þ : þ Together we now have all the eight possible level combinations in a 23 design. Since the ones in ½i x y4 are just the mirror images of the ones in ½i x y3 , we still have a complete 23 experiment in the three columns after all the signs in the n=2 last runs are interchanged. Case 2: p ¼ q; m odd. Then both p and q are greater than m=2. It is possible that x0 yi ¼ m for one i, but not for two since this will violate that the columns are assumed to be orthogonal. Let us assume that x0 y1 ¼ m. Then ½i x y1 will only contain the level combinations ! þ þ þ þ
and the level combination ðþ þ Þ is missing. To avoid that one more x0 yi ¼ m, ½i x y2 must contain 0 1 þ B C @ A; þ ½i x y3 must contain 0 1 þ Bþ þC @ A þ
þ
and ½i x y4 must contain 0 1 þ þ B þ C @ A: þ The two last triplets of level combinations are just mirror images and interchanging the signs in the string ð þÞ gives us the one that is missing. A similar discussion can be carried through if x0 yi ¼ m for i=2, 3 and 4. If all x0 yi am, the discussion is the same as for case 1. Case 3: p ¼ q ¼ m=2, m even. In this case orthogonality does not forbid that x0 yi ¼ m for one i and m for another. Let us assume x0 y1 ¼ m and x0 y4 ¼ m or vice versa. Then ½i x y1 and ½i x y4 together contains four different level combinations also after the signs in ½i x y4 are interchanged. Either one of ½i x y2 and ½i x y3 also contain four additional runs and hence a complete 23 experiment is obtained. If x0 yi ¼ x0 yi0 ¼ m for some other choices of i and i0 , the discussion is similar. If x0 yi ¼ m (or m) for just one i, two of the three others x0 yi will contain the level combinations 3 2 þ þ 6þ þ þ7 7 6 7 6 4þ þ5 þ
and the third will contain 3 2 þ 6 þ þ7 7 6 7 6 4 þ5
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or vice versa. In any case we have a complete 23 experiment also after the last n=2 signs are interchanged. If all x0 yi am, the discussion is the same as for case 1. Case 4: p ¼ q4m=2, m even. The arguments are the same as for case 2. The number of distinct runs directly relates to how many effects it is possible to estimate. For instance in a P=3 design all main effects and interactions effects can be estimated for any three factors. Now if all projections onto three columns have seven distinct runs, all main effects and two-factor interactions can be estimated for any three factors. Paley matrices are a family of Hadamard matrices constructed by Paley (1933). Designs derived from these matrices are called Paley designs. For nZ12 these matrices have no defining relation of length 4 (Bulutoglu and Cheng, 2003). Deng and Tang (2002) introduced the J-characteristic. Let s ¼ fc1 ; . . . ; ck g be a k-subset in Dn . The value jk ðsÞ ¼ Pn i¼1 ci1 ; ci2 ; . . . ; cik is called the J-characteristic of Dn associated with the factors in s. For k=3 and 4 the following bounds for jk ðsÞ for Paley designs with nZ12 derived by Bulutoglu and Cheng (2003), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ jjk ðsÞjrk þ 1 þ ðk 1Þ ðn 1Þ for k odd and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjk ðsÞ þ 1jrk þ 1 þ ðk 2Þ ðn 1Þ
for k odd even
ð5Þ
are useful for proving the next result. contains at least seven If Dn is of projectivity P=3 with m odd or a Paley design, every subset of three columns from D2ðk1Þ n of the eight possible level combinations in a 23 experiment. Proof. The only thing left to consider is when the subset of three columns have columns from both " þ # " þ # Dn=2 Dn=2 and : D D n=2 n=2 Let us consider one column from " þ # Dn=2 D n=2 and two columns from " þ # Dn=2 : D n=2
Let these three columns be denoted a; b and d . Let the corresponding columns to b and d in " þ # Dn=2 D n=2
be b and d If b ¼ a the projection ½a b d can be written as 2 3 i i i i 6 7 i 50 : 4 i i i x x x x Since x cannot contain m plusses or minuses, the projection contains a complete 23 experiment. Assume b and daa. Let nþ be the number of distinct level combinations in the first n=2 rows of ½a b d . If nþ ¼ 4 or 5, we can do the same discussion as in cases 2–4 in the proof of the preceding result and find that as a minimum we will have 3 or 2 additional level P combinations in the last n=2 rows respectively. Let us assume nþ ¼ 6 and define Si ¼ im ði1Þmþ1 bi di ; i ¼ 1; 2; 3; 4. Since b and P4 d are orthogonal we have that i¼1 Si ¼ 0. In order for the number of distinct level-combinations in a; b and d not to be increased by one when the last n=2 rows are taken into account, the level combinations in the third and fourth m-tuples of ½b d must be mirror images of the ones in the first and the second respectively. This implies that S1 ¼ S3 and S2 ¼ S4 which PN P4 further gives, since i¼1 Si ¼ 0 that S1 ¼ S2 and S3 ¼ S4 . If m is odd this implies i¼1 bi ci di ¼ 0, which according to Cheng (1995, p. 1226) is not possible. For the rest of the discussion we then assume that m is even. The j3 ðsÞ characteristic for the columns a; b and d equals 4S1 . Also without lack of generality we can assume that S1 Z0 since otherwise we can investigate the possible level combinations in ½x y2 with the same argumentation as below. Since Dn is of projectivity P ¼ 3, the restrictions on Si, i ¼ 1; 2; 3; 4 implies x0 y1 am and ½x y1 must contain the level combinations ðþ þÞ and ðþ Þ. There must be one more level combination. Let us assume it is ð Þ. Then S1 ¼ m=2 ðp qÞ þ m p and from the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð2p qÞZ3m 2 4m 1. Now since 2p q will increase if q bounds on j3 ðsÞ we obtain that p and q must satisfy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi decreases we set q ¼ m=2 and obtain pZm ð1=2 4m 1=4Þ. Also the bounds for j3 ðsÞ must be fulfilled for the columns
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c; a and b. For these three columns j3 ðsÞ ¼ 4ð2p mÞ and we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! m 1 4m 1 þ pr þ 4 2 2 Thereby p must satisfy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 m 4m 1 rpr þ m þ 4 2 2
1 þ 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4m 1 : 4
ð6Þ
Now for mZ8, the right hand side is in fact smaller than the left hand side, and for m ¼ 6, we get lower bound E4.3 and upper bound E4.7, which means that no such p can be found for Paley designs. If the third level combination instead is ð þÞ, in order for S1 to be Z0 we must have pZ3m=4 and j3 ðsÞ ¼ 2m. Again for mZ8, this does not satisfy the bounds on j3 ðsÞ and for m ¼ 6 we obtain pZ4:5 which implies p ¼ 5, a larger value than allowed. Hence we must have four different level combinations in ½i x y1 and thereby at least seven different level combinations in " # i x y1 i
x
y2
since ½i x y2 must have at least three different level combinations to avoid that x0 y2 ¼ m or m. If one column is in " þ # Dn=2 D n=2 and two in " þ # Dn=2 D n=2 we may write the column in " þ # Dn=2 D n=2 0
as a ¼ ½i i i i. The same arguments as above can be used with a little modification. For nþ ¼ 6, in order for the number of distinct level-combinations in a ; b and d not to be increased by one the level combinations in the third and fourth m-tuples of ½b d must be equal to the ones in the second and the first respectively. We now get that j4 ðsÞ ¼ 4S1 for the columns c; a; b and d. Using the bound in (5) for j4 ðsÞ and the bound in (4) for j3 ðsÞ for the columns c; a and b, we arrive at exactly the same bounds for p as we obtained above. A case where we have exactly seven runs is for instance given by the columns d, l and m in Fig. 1. Plackett–Burman designs for n ¼ 12; 20; 24; 44; 48; 60; 68; 72; 80 and 84 are all Paley designs. Thereby all non-regular Plackett–Burman designs, except the ones for n ¼ 40; 56; 64; 88 and 96 which are obtained by doubling, can be used as base with the desired properties. We also mention that the 32 run Paley design, see Neil Sloane’s design for constructing a D2ðk1Þ n web site: www.research.att.com/~njas/index.html#TABLES, is an excellent design with projectivity P ¼ 3 and generalized projectivity P ¼ 43. . Two columns from Let us consider the inner products between main-effects columns in D2ðk1Þ n " þ # Dn=2 D n=2 are obviously orthogonal since multiplying half of the entries with 1 in both columns does not change their inner product. Let a and b be two arbitrary columns from " þ # Dn=2 D n=2
and let a and b be the corresponding columns from " þ # Dn=2 D n=2
We notice that a is the entry-wise product between a and c and b the entry-wise product between b and c. Thus the inner product between a and b is the same as the inner product between a and the two-factor interaction column bc. Hence the
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maximum absolute value of the inner product between two main effect columns in D2ðk1Þ can never exceed the maximum n absolute value of the inner product between a main effect column and a two-factor interaction column in Dn . We can also investigate the inner product between main-effect columns and two-factor interaction columns. If one column is in " þ # Dn=2 D n=2 and two in " þ # Dn=2 D n=2 this inner product would be the same as between the corresponding columns in Dn . If two columns are in " þ # Dn=2 D n=2 and one is in " þ # Dn=2 D n=2 or if all three columns are in " þ # Dn=2 ; D n=2 the inner product may be expressed as the inner product between ac and bd or the inner product between two two-factor interaction columns in Dn where again a; b and d are three arbitrary columns in " þ # Dn=2 : D n=2 From the bounds on the J-characteristics, it follows that the maximum absolute value of the inner product between a maineffect column and a two-factor interaction column in Dn and D2ðk1Þ is the same if Dn is a Paley design with nZ12. n Computational investigations also reveal that this is true for all the others Plackett–Burman designs, except for those with n ¼ 40; 56; 64; 88 and 96. As a result these designs will normally have a low maximum absolute value of the inner product between a main effect column and a two-factor interaction column. design is not needed; will it then be possible to construct a A natural question is the following. Suppose a full D2ðk1Þ n 2ðk1Þ design with better projection properties than the one for which k ¼ n 1? With reference to Fig. 1 we may want to Dn investigate 15 factors in 12 runs for instance. We leave this as a topic for future research, but point out the following. Separately both " þ # Dn=2 D n=2 and "
Dþ n=2
#
D n=2 have the same good projection properties as Dn . Therefore if the active factors most likely are contained in a subset of t factors, trn 2, it will be wise to assign these t factors to columns in one of " þ # Dn=2 D n=2 or "
Dþ n=2 D n=2
#
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If very little is known about which factors are active it seems natural to remove about equally many columns from both of " þ # Dn=2 D n=2 and "
Dþ n=2
#
D n=2 either from the left or from the right in both of them. An argument for this is that the three columns a; b and d always contain a full 23 experiment whenever d ¼ a or b . 4. Comparison with other designs Lin (1993) constructed supersaturated designs as half fractions of Hadamard matrices restructured such that their first column consisted of only þ1’s. In the terminology above these would be written as Dþ n=2 or Dn=2. Nguyen (1996) constructed his designs from balanced incomplete block designs. He provided designs for 2n 2 factors in n runs for n ¼ 2t, where 3rtr15. Compared with these designs, our designs will in general have lower maximum absolute value of the inner product between main effect columns and two-factor interaction columns. For instance for n=12, 20, 24, 28 and 32 these inner products divided by n are (0.33, 0.60, 0.33, 0.43, 0.25) while they are (0.67, 0.60, 0.67, 0.57, 0.67) for Nguyen’s designs of size 12, 20, 24, 28 and 30. It has to be noted that Nguyen’s designs can accommodate two more factors for the same number of runs. The results for Lin’s designs are very similar to the ones for Nguyen’s designs. Due to the way the designs are constructed there will be a few two-factor interactions that are fully aliased. The percentage is very low. For the five designs mentioned above it is from 0.75% to 0.08%, decreasing with n. An even more striking difference is obtained when the number of different projections into three dimensions is considered. For the designs given above these are 3, 7, 6, 9 and 10 for our design while the number for Nguyen’s designs are 6, 18, 22, 27, 20. Again the numbers for Nguyen’s design and Lin’s designs are very similar. Finally if we look at the minimum number of distinct runs in a projection onto three factors these are 7, 7, 7, 7 and 8 for our designs and 6, 6, 7, 7 and 7 for Nguyen’s designs. Thereby, using the Paley design for n ¼ 32, our construction method provides us with a ð32; 60; 3Þ screen where the absolute value of the inner product between main effect columns and twofactor interaction columns does not exceed 0.25. In addition the Plackett–Burman designs for n=44, 48, 60, 68 72, 80 and 84 provide us with (44, 84, 3), (48, 92, 3), (60, 116, 3), (68, 132, 3), (72, 140, 3), (80, 156, 3) and (84, 164, 3) screens respectively. 5. Concluding remarks We have presented a simple way of constructing supersaturated designs from orthogonal non-regular two-level arrays. For an arbitrary column c in a n k orthogonal two-level design with balanced columns, Dn , let Dþ n=2 be the half fraction corresponding to þ1 values in c and D n=2 the one corresponding to 1 values in c. The corresponding supersaturated design is then given by " þ # Dn=2 Dþ n=2 2ðk1Þ : ¼ Dn D D n=2 n=2 The designs obtained are projectivity P ¼ 3 designs or near projectivity P ¼ 3 designs. They have also low maximum absolute value of the inner product between main effect columns and two-factor interaction columns. In addition their number of different projections on to a subset of three factor columns is much lower than for Nguyen’s and Lin’s designs with a similar number of runs. Hence they are expected to be well suitable screening designs when factor sparsity applies and only a few two-factor interactions are active.
Acknowledgement We are thankful to an anonymous referee for very valuable remarks. References Booth, K.H.V., Cox, D.R., 1962. Some systematic supersaturated designs. Technometrics 4, 489–495. Box, G.E.P., Meyer, R.D., 1993. Finding the active factors in fractionated screening experiments. Journal of Quality Technology 25, 94–105. Box, G.E.P., Tyssedal, J., 1996. Projective properties of certain orthogonal arrays. Biometrika 83 (4), 950–955. Box, G.E.P., Tyssedal, J., 2001. Sixteen run designs of high projectivity for factor screening. Communication of Statistics—Simulations and Computation 30 (2), 217–228.
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Bulutoglu, D.A., Cheng, C.S., 2003. Hidden projection properties of some nonregular factorial designs and their applications. The Annals of Statistics 31 (3), 1012–1026. Butler, N.A., Mead, R., Eskridge, K., Gilmour, S.G., 2001. A general method of constructing E(s2)—optimal supersaturated designs. Journal of the Royal Statistical Society: Series B (Statistical Methods) 63 (Part 3), 621–632. Cheng, C.S., 1995. Some projection properties of orthogonal arrays. Annals of Statistics 23, 1223–1233. Dejaegher, B., Heyden, Y.V., 2008. Supersaturated designs: set-ups, data interpretation, and analytical applications. Analytical and Bioanalytical Chemistry 390, 1227–1240. Deng, L., Tang, B., 2002. Design selection and classification for Hadamard matrices using generalized minimum aberration criteria. Technometrics 44 (2), 173–184. Evangelaras, H., Koukouvinos, C., 2004. On generalized projectivity of two-level screening designs. Statistics and Probability Letters 68 (4), 429–434. Li, W.W., Wu, C.F.J., 1997. Columnwise-pairwise algorithms with applications to the construction of supersaturated designs. Technometrics 39 (2), 171–179. Lin, D.K.J., 1993. A new class of supersaturated designs. Technometrics 35, 28–31. Lin, D.K.J., 1995. Generating systematic supersaturated designs. Technometrics 37, 213–225. Montgomery, D.C., Myers, R.H., Carter, W.H., Vining, G.G., 2005. The hierarchy principle in designed industrial experiments. Quality and Reliability Engineering International 21, 197–2001. Nguyen, N.K., 1996. An algorithmic approach to constructing supersaturated designs. Technometrics 38, 69–73. Plackett, R.L., Burman, J.P., 1946. The design of optimum multifactorial experiments. Biometrika 33, 305–325. Paley, R.E.A.C., 1933. On orthogonal matrices. Journal of Mathematics and Physics 12, 311–320. Ryan, K.J., Bulutoglu, D.A., 2007. E(s2)—optimal supersaturated designs with good minimax properties. JSPI 137, 2250–2262. Satherwaite, F.E., 1959. Random balance experimentation (with discussion). Technometrics 1, 111–137. Tyssedal, J., 2008. Projectivity in Experimental Designs. Encyclopedia of Statistics in Quality and Reliability. Wiley, pp. 1516–1520. Tyssedal, J., Samset, O., 1997. Analysis of the 12 run Plackett and Burman design. Technical Report 8, Department of Mathematical Sciences, The Norwegian University of Science and Technology, Norway. Wu, C.F.J., 1993. Construction of supersaturated designs through partially aliased interactions. Biometrika 80, 661–669.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Supersaturated designs of projectivity P=3 or near P=3 John Tyssedal a,, Oddgeir Samset b a b
Department of Mathematical Sciences, The Norwegian University of Science and Technology, 7491 Trondheim, Norway Elkem Salten, NO-8226 Straumen, Norway
a r t i c l e in fo
abstract
Article history: Received 30 January 2009 Received in revised form 8 October 2009 Accepted 8 October 2009 Available online 23 October 2009
Supersaturated designs offer a potentially useful way to investigate many factors in few experiments i.e. typical screening situations. Their design properties have mainly been evaluated based on their ability to identify and estimate main effects. Projective properties have received little attention. In this paper we show how to construct twolevel supersaturated designs for 2(n2) factors in n runs (n a multiple of four) of projectivity P=3 or near projectivity P=3 from orthogonal non-regular two-level designs. The designs obtained also have favourable properties such as low maximum absolute value of the inner product between a main effect column and a two-factor interaction column and relatively few types of different projections onto subsets consisting of three factor columns. & 2009 Elsevier B.V. All rights reserved.
Keywords: Hadamard matrices Projection properties Screening designs
1. Introduction Supersaturated designs are experimental designs that employ fewer runs than the number of factors studied. Their use are primarily as screening designs. If it can be assumed that only a small number of factors (factor sparsity) or effects (effect sparsity) are active, these designs can provide considerable cost savings. So far few applications of supersaturated designs have been reported, but the potential for various fields of research to benefit from their use has been pointed out. Examples are computer and medical experiments (Lin, 1995), industrial and engineering experiments (Wu, 1993; Nguyen, 1996) and biometric applications (Butler et al., 2001). See also Dejaegher and Heyden (2008). Supersaturated designs were introduced by Satherwaite (1959) through the idea of random balanced designs while Booth and Cox (1962) were the first to consider a systematic construction of supersaturated designs. Lin (1993) used halffractions of Hadamard matrices, Nguyen (1996) showed how to construct supersaturated designs from balanced incomplete block designs and Wu (1993) used Hadamard matrices augmented with interaction columns. Computer generation of designs based on some criteria has also been considered (Lin, 1995; Nguyen, 1996; Li and Wu, 1997) among others. In the following we will only consider two-level supersaturated designs. Supersaturated designs cannot be made orthogonal. Let X be a n k design matrix of a design with n runs and k factors each at two levels with n=2 of þ1’s and n=2 of 1’s. Booth and Cox (1962) proposed as a criterion for comparing designs, the minimization of aveðs2 Þ, normally written as Eðs2 Þ, given by k s2ij = ; 2 ioj
X
Corresponding author.
E-mail address: [email protected] (J. Tyssedal). 0378-3758/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2009.10.004
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where sij is the ði; jÞ element of X0 X. An orthogonal design will have aveðs2 Þ ¼ 0, and the criterion can be considered as a measure of the degree of non-orthogonality among the columns in a supersaturated design. A lot of effort has been put into finding supersaturated designs that minimize Eðs2 Þ. Another suggested criterion for comparing supersaturated designs is the minimax criterion which first order designs by smax :¼ maxioj jsij j, the smaller the better, and then by the frequency of sij ¼ 7smax . Both the criteria above focus on the estimation and identification of first order effects. It is, however, possible to construct supersaturated designs that are optimal to both the criteria above where a main effect column is fully aliased with a two-factor interaction column. For instance the NOA4 algorithm in Ryan and Bulutoglu (2007) provided such a design for n ¼ 12 and k ¼ 20. Hence Eðs2 Þ optimal designs may perform badly when two-factor interactions are present. We believe there are many situations under which you want to use a supersaturated design and where the experimental conditions are not very different from when a saturated design is used, except that the number of factors is larger. Also increasing the number of factors will normally increase the probability of interactions between factors and decrease the probability that the heredity principle applies i.e. an interaction can be excluded from the model unless at least one (weak heredity) or both (strong heredity) of the main effects are included in the model. This principle has also been questioned in several papers (Box and Tyssedal, 2001; Montgomery et al., 2005). Any restrictive assumption on the underlying model will necessarily lower our chances to identify the correct subspace of active factors. In factor screening it is often the case that out of a larger number of tested factors only a small subset, typically two or three, maybe four, are expected to really affect a particular response. This is known as factor sparsity. Therefore when choosing an appropriate design for screening experimentation, it is of importance to consider projections of the design onto small subsets of factors. Box and Tyssedal (1996) defined projectivity of two-level designs as follows: a n k design with n runs and k factors each at two levels is said to be of projectivity P if every subset of P columns contains a complete 2P factorial, possibly with some runs replicated. They denoted such designs ðn; k; PÞ screens. Besides ensuring that all main effects and all interactions of any P factors can be estimated with no bias if the other factors are inert, projectivity P also implies that the ability of a subset of P factors to explain the variation in the data can be evaluated with rather weak assumptions on the underlying model (Tyssedal, 2008). While the projectivity of certain saturated orthogonal arrays is well known (Box and Tyssedal, 1996, 2001; Bulutoglu and Cheng, 2003; Tyssedal, 2008), projective properties of supersaturated designs have received little attention. In this paper we will focus on constructing supersaturated designs with nearly the same projective properties as saturated designs. In Section 2 we describe how to construct such designs and in Section 3, we show some interesting properties. A comparison to some other supersaturated designs are given in Section 4 and concluding remarks are given in Section 5.
2. Construction of supersaturated designs of projectivity P=3 or near P=3 Most supersaturated designs can only be of projectivity P=2 for modest to small number of runs, n. Evangelaras and Koukouvinos (2004) introduced the criterion of generalized projectivity. A n k design with n runs and k factors each at two levels is said to be of generalized projectivity Pa if for any selection of P columns out of possible k all main effects and interactions of order ra can be estimated. While projectivity P=3 for instance is a desirable property for a saturated design, projectivity P ¼ 32 i.e. all main effects and two-factor interactions for any three factors can be estimated with no bias if the other factors are inert, might be an attractive property of a supersaturated design. Algorithms designed to identify the active factor space for a projectivity P orthogonal two-level design (Box and Meyer, 1993; Tyssedal and Samset, 1997) can easily be modified to also take care of projectivity Pa designs. We now propose a new class of supersaturated designs constructed as follows: let Dn be a n k orthogonal two-level design with balanced columns. Further let c be a column in Dn and define Dcþ n=2 as the half fraction corresponding to the þ1 entries in c and Dc n=2 the one corresponding to the 1 entries in c. The following design: 2 cþ 3 Dcþ Dn=2 n=2 c;2ðk1Þ 4 5 Dn ¼ ð1Þ Dc Dc n=2 n=2 will, provided k4n=2 be a supersaturated design which can accommodate 2ðk 1Þ factors in n runs. An example of such a design constructed from the 12 run Plackett–Burman (Plackett and Burman, 1946) design is given in Fig. 1. In general designs obtained in this way are not Eðs2 Þ optimal designs, but as shown in the next section they have other desirable properties. 3. Properties of Dn2ðk1Þ designs The properties derived in this section do not depend on the choice of c. The superscript c will therefore be omitted. is obviously a balanced design. For any branching column Dþ D2ðk1Þ n n=2 and Dn=2 has exactly n=4 þ 1’s and n=4 1’s entries does not change that fact. and interchanging the signs in D n=2
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Fig. 1. A Da;20 the first column, column a as the branching column. Columns 12 design constructed from the 12 run Plackett and Burman (PB) design with " aþ # D6 b,y,k are columns from the PB designs. Columns l,y, u are the constructed ones in . Da 6
For the following discussion let i be a vector of m þ 1’s where m ¼ n=4. Then three arbitrary columns in Dn can be written as 3 2 i i x 6 i i x 7 7 6 ð2Þ 7 6 4 i i x 5 i
i
x
where x is a m 1 vector of pþ1’s and ðm pÞ 1’s. A fourth column will be of the form ½y1 y2 y3 y4 0 where the number of þ10 s in y2 and y3 are both equal to m q where q is the number of þ10 s in y1 and y4 . One particular case is worth mentioning. For m even and p ¼ q ¼ m=2 it is possible that there exists a defining relation between four columns. Let four such columns be written as 3 2 i i x x 6 i i x x 7 7 6 7: 6 4 i i x x 5 i
i
x
x
Further let c ¼ ½i i i i0 be the branching column. It is clear that the level combinations in ½x x and ½x x are equal. Also the level combinations in ½x x and ½x x must be the same. Hence interchanging the signs in ½i x x and when projected onto the three last columns above has only four distinct runs, an undesirable situation. ½i x x, D2ðk1Þ n We will now show the following result: Let Dn be an orthogonal two-level design of projectivity P ¼ 3 and assume that no defining relation exists between any four factors. Then " þ # Dn=2 D n=2 is of projectivity P=3. Proof. With reference to the discussion above we need to show that 3 2 i x y1 7 6 6 i x y2 7 7 6 6 i x y3 7 5 4 i
x
ð3Þ
y4
is of projectivity P=3. Without lack of generality we can assume pZqZm=2. Otherwise we can just multiply the corresponding column with 1 or interchange columns. Case 1: p4q. If p4q, then ½i x y1 contains the level combinations ! þ þ þ ; þ þ
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½i x y2 contains ! ; þ ½i x y3 contains ! þ þ þ and ½i x y4 contains ! þ þ : þ Together we now have all the eight possible level combinations in a 23 design. Since the ones in ½i x y4 are just the mirror images of the ones in ½i x y3 , we still have a complete 23 experiment in the three columns after all the signs in the n=2 last runs are interchanged. Case 2: p ¼ q; m odd. Then both p and q are greater than m=2. It is possible that x0 yi ¼ m for one i, but not for two since this will violate that the columns are assumed to be orthogonal. Let us assume that x0 y1 ¼ m. Then ½i x y1 will only contain the level combinations ! þ þ þ þ
and the level combination ðþ þ Þ is missing. To avoid that one more x0 yi ¼ m, ½i x y2 must contain 0 1 þ B C @ A; þ ½i x y3 must contain 0 1 þ Bþ þC @ A þ
þ
and ½i x y4 must contain 0 1 þ þ B þ C @ A: þ The two last triplets of level combinations are just mirror images and interchanging the signs in the string ð þÞ gives us the one that is missing. A similar discussion can be carried through if x0 yi ¼ m for i=2, 3 and 4. If all x0 yi am, the discussion is the same as for case 1. Case 3: p ¼ q ¼ m=2, m even. In this case orthogonality does not forbid that x0 yi ¼ m for one i and m for another. Let us assume x0 y1 ¼ m and x0 y4 ¼ m or vice versa. Then ½i x y1 and ½i x y4 together contains four different level combinations also after the signs in ½i x y4 are interchanged. Either one of ½i x y2 and ½i x y3 also contain four additional runs and hence a complete 23 experiment is obtained. If x0 yi ¼ x0 yi0 ¼ m for some other choices of i and i0 , the discussion is similar. If x0 yi ¼ m (or m) for just one i, two of the three others x0 yi will contain the level combinations 3 2 þ þ 6þ þ þ7 7 6 7 6 4þ þ5 þ
and the third will contain 3 2 þ 6 þ þ7 7 6 7 6 4 þ5
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or vice versa. In any case we have a complete 23 experiment also after the last n=2 signs are interchanged. If all x0 yi am, the discussion is the same as for case 1. Case 4: p ¼ q4m=2, m even. The arguments are the same as for case 2. The number of distinct runs directly relates to how many effects it is possible to estimate. For instance in a P=3 design all main effects and interactions effects can be estimated for any three factors. Now if all projections onto three columns have seven distinct runs, all main effects and two-factor interactions can be estimated for any three factors. Paley matrices are a family of Hadamard matrices constructed by Paley (1933). Designs derived from these matrices are called Paley designs. For nZ12 these matrices have no defining relation of length 4 (Bulutoglu and Cheng, 2003). Deng and Tang (2002) introduced the J-characteristic. Let s ¼ fc1 ; . . . ; ck g be a k-subset in Dn . The value jk ðsÞ ¼ Pn i¼1 ci1 ; ci2 ; . . . ; cik is called the J-characteristic of Dn associated with the factors in s. For k=3 and 4 the following bounds for jk ðsÞ for Paley designs with nZ12 derived by Bulutoglu and Cheng (2003), pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4Þ jjk ðsÞjrk þ 1 þ ðk 1Þ ðn 1Þ for k odd and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jjk ðsÞ þ 1jrk þ 1 þ ðk 2Þ ðn 1Þ
for k odd even
ð5Þ
are useful for proving the next result. contains at least seven If Dn is of projectivity P=3 with m odd or a Paley design, every subset of three columns from D2ðk1Þ n of the eight possible level combinations in a 23 experiment. Proof. The only thing left to consider is when the subset of three columns have columns from both " þ # " þ # Dn=2 Dn=2 and : D D n=2 n=2 Let us consider one column from " þ # Dn=2 D n=2 and two columns from " þ # Dn=2 : D n=2
Let these three columns be denoted a; b and d . Let the corresponding columns to b and d in " þ # Dn=2 D n=2
be b and d If b ¼ a the projection ½a b d can be written as 2 3 i i i i 6 7 i 50 : 4 i i i x x x x Since x cannot contain m plusses or minuses, the projection contains a complete 23 experiment. Assume b and daa. Let nþ be the number of distinct level combinations in the first n=2 rows of ½a b d . If nþ ¼ 4 or 5, we can do the same discussion as in cases 2–4 in the proof of the preceding result and find that as a minimum we will have 3 or 2 additional level P combinations in the last n=2 rows respectively. Let us assume nþ ¼ 6 and define Si ¼ im ði1Þmþ1 bi di ; i ¼ 1; 2; 3; 4. Since b and P4 d are orthogonal we have that i¼1 Si ¼ 0. In order for the number of distinct level-combinations in a; b and d not to be increased by one when the last n=2 rows are taken into account, the level combinations in the third and fourth m-tuples of ½b d must be mirror images of the ones in the first and the second respectively. This implies that S1 ¼ S3 and S2 ¼ S4 which PN P4 further gives, since i¼1 Si ¼ 0 that S1 ¼ S2 and S3 ¼ S4 . If m is odd this implies i¼1 bi ci di ¼ 0, which according to Cheng (1995, p. 1226) is not possible. For the rest of the discussion we then assume that m is even. The j3 ðsÞ characteristic for the columns a; b and d equals 4S1 . Also without lack of generality we can assume that S1 Z0 since otherwise we can investigate the possible level combinations in ½x y2 with the same argumentation as below. Since Dn is of projectivity P ¼ 3, the restrictions on Si, i ¼ 1; 2; 3; 4 implies x0 y1 am and ½x y1 must contain the level combinations ðþ þÞ and ðþ Þ. There must be one more level combination. Let us assume it is ð Þ. Then S1 ¼ m=2 ðp qÞ þ m p and from the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ð2p qÞZ3m 2 4m 1. Now since 2p q will increase if q bounds on j3 ðsÞ we obtain that p and q must satisfy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi decreases we set q ¼ m=2 and obtain pZm ð1=2 4m 1=4Þ. Also the bounds for j3 ðsÞ must be fulfilled for the columns
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c; a and b. For these three columns j3 ðsÞ ¼ 4ð2p mÞ and we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! m 1 4m 1 þ pr þ 4 2 2 Thereby p must satisfy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 1 m 4m 1 rpr þ m þ 4 2 2
1 þ 2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4m 1 : 4
ð6Þ
Now for mZ8, the right hand side is in fact smaller than the left hand side, and for m ¼ 6, we get lower bound E4.3 and upper bound E4.7, which means that no such p can be found for Paley designs. If the third level combination instead is ð þÞ, in order for S1 to be Z0 we must have pZ3m=4 and j3 ðsÞ ¼ 2m. Again for mZ8, this does not satisfy the bounds on j3 ðsÞ and for m ¼ 6 we obtain pZ4:5 which implies p ¼ 5, a larger value than allowed. Hence we must have four different level combinations in ½i x y1 and thereby at least seven different level combinations in " # i x y1 i
x
y2
since ½i x y2 must have at least three different level combinations to avoid that x0 y2 ¼ m or m. If one column is in " þ # Dn=2 D n=2 and two in " þ # Dn=2 D n=2 we may write the column in " þ # Dn=2 D n=2 0
as a ¼ ½i i i i. The same arguments as above can be used with a little modification. For nþ ¼ 6, in order for the number of distinct level-combinations in a ; b and d not to be increased by one the level combinations in the third and fourth m-tuples of ½b d must be equal to the ones in the second and the first respectively. We now get that j4 ðsÞ ¼ 4S1 for the columns c; a; b and d. Using the bound in (5) for j4 ðsÞ and the bound in (4) for j3 ðsÞ for the columns c; a and b, we arrive at exactly the same bounds for p as we obtained above. A case where we have exactly seven runs is for instance given by the columns d, l and m in Fig. 1. Plackett–Burman designs for n ¼ 12; 20; 24; 44; 48; 60; 68; 72; 80 and 84 are all Paley designs. Thereby all non-regular Plackett–Burman designs, except the ones for n ¼ 40; 56; 64; 88 and 96 which are obtained by doubling, can be used as base with the desired properties. We also mention that the 32 run Paley design, see Neil Sloane’s design for constructing a D2ðk1Þ n web site: www.research.att.com/~njas/index.html#TABLES, is an excellent design with projectivity P ¼ 3 and generalized projectivity P ¼ 43. . Two columns from Let us consider the inner products between main-effects columns in D2ðk1Þ n " þ # Dn=2 D n=2 are obviously orthogonal since multiplying half of the entries with 1 in both columns does not change their inner product. Let a and b be two arbitrary columns from " þ # Dn=2 D n=2
and let a and b be the corresponding columns from " þ # Dn=2 D n=2
We notice that a is the entry-wise product between a and c and b the entry-wise product between b and c. Thus the inner product between a and b is the same as the inner product between a and the two-factor interaction column bc. Hence the
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maximum absolute value of the inner product between two main effect columns in D2ðk1Þ can never exceed the maximum n absolute value of the inner product between a main effect column and a two-factor interaction column in Dn . We can also investigate the inner product between main-effect columns and two-factor interaction columns. If one column is in " þ # Dn=2 D n=2 and two in " þ # Dn=2 D n=2 this inner product would be the same as between the corresponding columns in Dn . If two columns are in " þ # Dn=2 D n=2 and one is in " þ # Dn=2 D n=2 or if all three columns are in " þ # Dn=2 ; D n=2 the inner product may be expressed as the inner product between ac and bd or the inner product between two two-factor interaction columns in Dn where again a; b and d are three arbitrary columns in " þ # Dn=2 : D n=2 From the bounds on the J-characteristics, it follows that the maximum absolute value of the inner product between a maineffect column and a two-factor interaction column in Dn and D2ðk1Þ is the same if Dn is a Paley design with nZ12. n Computational investigations also reveal that this is true for all the others Plackett–Burman designs, except for those with n ¼ 40; 56; 64; 88 and 96. As a result these designs will normally have a low maximum absolute value of the inner product between a main effect column and a two-factor interaction column. design is not needed; will it then be possible to construct a A natural question is the following. Suppose a full D2ðk1Þ n 2ðk1Þ design with better projection properties than the one for which k ¼ n 1? With reference to Fig. 1 we may want to Dn investigate 15 factors in 12 runs for instance. We leave this as a topic for future research, but point out the following. Separately both " þ # Dn=2 D n=2 and "
Dþ n=2
#
D n=2 have the same good projection properties as Dn . Therefore if the active factors most likely are contained in a subset of t factors, trn 2, it will be wise to assign these t factors to columns in one of " þ # Dn=2 D n=2 or "
Dþ n=2 D n=2
#
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If very little is known about which factors are active it seems natural to remove about equally many columns from both of " þ # Dn=2 D n=2 and "
Dþ n=2
#
D n=2 either from the left or from the right in both of them. An argument for this is that the three columns a; b and d always contain a full 23 experiment whenever d ¼ a or b . 4. Comparison with other designs Lin (1993) constructed supersaturated designs as half fractions of Hadamard matrices restructured such that their first column consisted of only þ1’s. In the terminology above these would be written as Dþ n=2 or Dn=2. Nguyen (1996) constructed his designs from balanced incomplete block designs. He provided designs for 2n 2 factors in n runs for n ¼ 2t, where 3rtr15. Compared with these designs, our designs will in general have lower maximum absolute value of the inner product between main effect columns and two-factor interaction columns. For instance for n=12, 20, 24, 28 and 32 these inner products divided by n are (0.33, 0.60, 0.33, 0.43, 0.25) while they are (0.67, 0.60, 0.67, 0.57, 0.67) for Nguyen’s designs of size 12, 20, 24, 28 and 30. It has to be noted that Nguyen’s designs can accommodate two more factors for the same number of runs. The results for Lin’s designs are very similar to the ones for Nguyen’s designs. Due to the way the designs are constructed there will be a few two-factor interactions that are fully aliased. The percentage is very low. For the five designs mentioned above it is from 0.75% to 0.08%, decreasing with n. An even more striking difference is obtained when the number of different projections into three dimensions is considered. For the designs given above these are 3, 7, 6, 9 and 10 for our design while the number for Nguyen’s designs are 6, 18, 22, 27, 20. Again the numbers for Nguyen’s design and Lin’s designs are very similar. Finally if we look at the minimum number of distinct runs in a projection onto three factors these are 7, 7, 7, 7 and 8 for our designs and 6, 6, 7, 7 and 7 for Nguyen’s designs. Thereby, using the Paley design for n ¼ 32, our construction method provides us with a ð32; 60; 3Þ screen where the absolute value of the inner product between main effect columns and twofactor interaction columns does not exceed 0.25. In addition the Plackett–Burman designs for n=44, 48, 60, 68 72, 80 and 84 provide us with (44, 84, 3), (48, 92, 3), (60, 116, 3), (68, 132, 3), (72, 140, 3), (80, 156, 3) and (84, 164, 3) screens respectively. 5. Concluding remarks We have presented a simple way of constructing supersaturated designs from orthogonal non-regular two-level arrays. For an arbitrary column c in a n k orthogonal two-level design with balanced columns, Dn , let Dþ n=2 be the half fraction corresponding to þ1 values in c and D n=2 the one corresponding to 1 values in c. The corresponding supersaturated design is then given by " þ # Dn=2 Dþ n=2 2ðk1Þ : ¼ Dn D D n=2 n=2 The designs obtained are projectivity P ¼ 3 designs or near projectivity P ¼ 3 designs. They have also low maximum absolute value of the inner product between main effect columns and two-factor interaction columns. In addition their number of different projections on to a subset of three factor columns is much lower than for Nguyen’s and Lin’s designs with a similar number of runs. Hence they are expected to be well suitable screening designs when factor sparsity applies and only a few two-factor interactions are active.
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