Minimum Aberration and Related Criteria for Fractional Factorial Designs

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1 Minimum Aberration and Related Criteria for Fractional Factorial Designs Hegang Chen Division of Biostatistics and Bioinformatics 660 West Redwood Street University of Maryland School of Medicine Baltimore, MD and Ching-Shui Cheng Department of Statistics University of California Berkeley, CA March 30, Introduction Fractional factorial designs have a long history of successful use in scientific investigations and industrial experiments. This important subject was treated in Chapter 13 of Hinkelmann and Kempthorne (2005), Design and Analysis of Experiments, Volume 2, hereafter referred to as HK2. Several criteria for choosing fractional factorial designs, including the popular criterion of minimum aberration, were briefly presented in Section of HK2, and the issue of optimal blocking of fractional factorial designs was discussed in Section These criteria were proposed for choosing designs with better capability of estimating lower-order effects, albeit with different interpretations of such capability. The differences in the interpretations sometimes lead to inconsistences or even contradictions among the different criteria, and one should not expect any criterion to work in all circumstances. In this chapter, we give a 1

2 more comprehensive and in-depth discussion of these criteria, including clarifications of their relationship. We also extend optimal blocking to cover models with random block effects. For simplicity, we only consider two-level designs though many of the results can be extended easily to the case where the number of factor levels is a prime power. Suppose there are n two-level treatment factors. Then there are a total of 2 n factor-level combinations, called treatment combinations. A complete factorial requires 2 n runs. A 2 m th fraction, referred to as a 2 n m fractional factorial design, consists of 2 n m of the 2 n treatment combinations. We mainly focus on fractional factorial designs that can be constructed by using defining relations as discussed in Section 13.3 of HK2. Such designs are called regular fractional factorial designs. Nonregular designs will be briefly discussed near the end of this chapter. We first review some notations and basic concepts. Each treatment factor is represented by a letter such as A, B, C,..., and each of the 2 n 1 factorial effects (main effects and interactions) is represented by a string of letters, called a word, consisting of the letters associated with the factors that are involved. For example, the main effect of factor A is also denoted by A, and the interaction of factors A, B and D is denoted by ABD. The number of letters in a word is called its length. A regular 2 n m fractional factorial design is defined by a set of m independent interactions and is constructed by solving equations such as (13.6) on p.514 of HK2. The words that represent the m independent interactions are called independent defining words. Products of independent defining words, subject to the rule that even powers of any letter are deleted, are called defining words. There are a total of 2 m 1 defining words, and the corresponding factorial effects are called defining effects of the fraction. The defining effects constitute the defining relation, and cannot be estimated. The other 2 n 2 m factorial effects are partitioned into 2 n m 1 alias sets each of size 2 m. Each of these effects is estimable if all its aliases are assumed to be negligible. 2

3 Example 1. Consider the design d defined by two independent defining effects ABCE and BCDF. Then ADEF = (ABCE)(BCDF ) is also a defining effect. The defining relation is written as I = ABCE = BCDF = ADEF. The 60 factorial effects other than the three defining effects are partitioned into 15 alias sets each of size 4, where the aliases of each factorial effect can be obtained by multiplying all the defining words by the word representing that effect: A = BCE = ABCDF = DEF, B = ACE = CDF = ABDEF, C = ABE = BDF = ACDEF, D = ABCDE = BCF = AEF, E = ABC = BCDEF = ADF, F = ABCEF = BCD = ADE, AB = CE = ACDF = BDEF, AC = BE = ABDF = CDEF, AD = BCDE = ABCF = EF, AE = BC = ABCDEF = DF, AF = BCEF = ABCD = DE, BD = ACDE = CF = ABEF, BF = ACEF = CD = ABDE, ABD = CDE = ACF = BEF, ABF = CEF = ACD = BDE. Note that three alias sets are underlined. This is not relevant here and is to be discussed in Example 8 (Section 10) when we present blocked fractional factorial designs. An important property of a 2 n m fractional factorial design is its resolution. Box and Hunter (1961) defined the resolution of a regular fractional factorial design to be the length of the shortest defining word. We say that d is a 2 n m r design if it is a 2 n m fractional factorial design of resolution r. Under a design of resolution r, no s-factor interaction is aliased with any other effect involving less than r s factors. Under the hierarchical assumption that lower-order effects are more important than higher-order effects and that effects of the same 3

4 order are equally important, the experimenter may prefer a design that has the highest possible resolution. The design in Example 1 is a resolution IV design; all the main effects are aliased with some three-factor and higher-order interactions. However, not all 2 n m designs of maximum resolution are equally good. Fries and Hunter (1980) introduced the minimum aberration criterion for further discriminating 2 n m designs of the same resolution. For each regular 2 n m fractional factorial design d, let A i (d) be the number of defining words of length i and W (d) be the vector W (d) = (A 1 (d), A 2 (d),, A n (d)). Then W (d) is called the wordlength pattern of d. Given two 2 n m fractional factorial designs d 1 and d 2, d 1 is said to have less aberration than d 2 if A s (d 1 ) < A s (d 2 ) where s is the smallest integer such that A s (d 1 ) A s (d 2 ). A 2 n m design has minimum aberration if no other 2 n m design has less aberration. In other words, the criterion of minimum aberration sequentially minimizes A 1 (d), A 2 (d),..., etc. Chen and Hedayat (1996) proposed a weaker version of minimum aberration. A regular 2 n m design of maximum resolution r max is said to have weak minimum aberration if it minimizes the number of words of length r max among all the designs of resolution r max. Intuitively minimum and weakly minimum aberration designs are expected to produce less aliasing among lower-order effects, a desirable feature under the hierarchical assumption. The design in Example 1 has minimum aberration among all the designs. For any 2 n m design d, let d be the design obtained from d by switching the two levels. Then the treatment combinations in d and d together form a regular 2 n (m 1) design, called the foldover of d. If d is a resolution III design, then the foldover of d has resolution IV and all its defining words are of even lengths; see p.544 and p.545 of HK2. Designs that only have defining words of even lengths are called even designs. One can also add a factor at constant level to d. Then the foldover of the resulting 2 n+1 (m+1) design is a 2 (n+1) m 4

5 even design of resolution IV. More generally, the foldover of a design of odd resolution r has resolution r + 1. Throughout this chapter, we also denote the run size 2 n m of a 2 n m fractional factorial design by N. To prevent aliasing among the main effects, we only consider designs of resolution III or higher. Such designs, called resolution III+ designs, must have n N 1. Resolution III designs with n = N 1 are unique up to isomorphism, and are called saturated designs. It can be shown that designs of resolution IV+ must have n N/2. Resolution IV designs with n = N/2 are also unique up to isomorphism. Such a design can be constructed by folding over a saturated design of run size N/2 that is supplemented by a factor at constant level, as described in the previous paragraph. Therefore, resolution IV designs with n = N/2 are even designs. It can be shown that every even design of size N is a foldover design and can be constructed by deleting factors from a resolution IV design with N/2 factors. Therefore we call resolution IV designs with n = N/2 maximal even designs. In Section 2 we discuss projections of regular fractional factorial designs onto subsets of factors, and show that minimum aberration designs have good projection properties. In Section 3 we provide a better understanding of minimum aberration by investigating alias structures of minimum aberrations designs. This is crucial for justifying minimum aberration as a good surrogate for the criterion of maximum estimation capacity under model uncertainty. Two other criteria, number of clear two-factor interactions and estimation index are presented in Sections 4 and 5, respectively. Relationship between aberration, estimation capacity and estimation index is addressed in Section 6. In Section 7, we present a method of constructing minimum aberration designs via their complementary designs. After a review of some properties of orthogonal arrays as well as extending the concepts of estimation capacity and clear two-factor interactions to nonregular designs in Section 8, a brief introduction to generalized minimum aberration, a natural extension of the minimum aberration criterion to nonregular designs, is presented in Section 9. The last section of this chapter is devoted 5

6 to optimal blocking. 2 Projections of Fractional Factorial Designs In factor screening experiments, among a large number of factors to be examined, typically only a few are expected to be active. Therefore information about the design when it is restricted to a small number of factors is valuable. Let d be a 2 n m r design. For any k of the n factors, the design obtained by dropping the other n k factor is called a k-dimensional projection of d. Let d be a 2 n m r design. Since the defining words of a lower-dimensional projection of d are also defining words of d, all lower-dimensional projections of d are of resolution r or higher. We first consider the r-dimensional projections. Box and Hunter (1961) pointed out that any k-dimensional projection with k < r is a replicated complete 2 k design. Projections onto r factors are replicated fractional factorial designs of resolution r if the r factors form a defining word, and are replicated full factorials otherwise. Specifically, each defining word of length r gives an r-dimensional projection that consists of 2 n m r+1 replicates of a 2 r 1 r design. There are A r (d) such r-dimensional projections. Each of the remaining ( ) n r Ar (d) projections consists of 2 n m r copies of a full 2 r factorial. Example 2. Let d be a V design with the following defining relation and wordlength pattern: I = ABCDG = ABEF H = CDEF GH, W (d) = (0, 0, 0, 0, 2, 1, 0, 0). Then all the k-dimensional projections of d with k < 5 are 2 6 k replicates of a full 2 k factorial. Projections onto the two subsets of five factors {A, B, C, D, G} and {A, B, E, F, H}, which form the two defining words of length five respectively, are replicated V designs, and each of the other five-dimensional projections consists of two replicates of a full 2 5 factorial. Chen (1998) studied the relationship between projections of a fractional factorial design 6

7 and its wordlength pattern, and completely characterized the projections of a 2 n m r onto r + 1 to r + [(r 1)/2] dimensions. design Theorem 1 (Chen, 1998) Let d be a 2 n m r design. For r + 1 k r + [(r 1)/2], each k-dimensional projection of d is a possibly replicated 2 k design, or a replicated 2 k 1 design. For j = r, r + 1,..., k, ( ) n j Aj (d) of the ( ) n k j k k-dimensional projections consist of 2 n m k+1 replicates of a 2 k 1 j design, and the other ( ) n ( ) k kj=r n j Aj (d) k-dimensional projections k j consist of 2 n m k copies of a 2 k design. In Example 2, we have discussed the k-dimensional projections of a V design for all k 5. Theorem 1 can be used to determine the 6- and 7-dimensional projections. For k = 6, six projections consist of two copies of a V replicates of a V I 7-dimensional projections, six are V design, one projection consists of two design, and the remaining projections are 2 6 designs. Among the eight designs, and two are V I designs. Theorem 1 shows that for r + 1 k r + [(r 1)/2], the number of k-dimensional projections that have resolution j, r j k, is proportional to A j (d). Therefore by sequentially minimizing the A j (d) s, minimum aberration designs have good projection properties in that they produce fewer projections of low resolutions. Example 3. There are three non-isomorphic IV designs: d 1 : I = ABCF = BCDG = ADF G, W (d 1 ) = (0, 0, 0, 3, 0, 0, 0), d 2 : I = ABCF = ADEG = BCDEF G, W (d 2 ) = (0, 0, 0, 2, 0, 1, 0), d 3 : I = DEF G = ABCDF = ABCEG, W (d 3 ) = (0, 0, 0, 1, 2, 0, 0). It can be shown that the maximum attainable resolution for a design is IV, and d 3 has minimum aberration. Design d 3 has only one replicated IV among its four-dimensional projections, while d 1 and d 2 have three and two such projections respectively. Among the twenty-one 5-dimensional projections of d 3, three are replicated IV 7 designs and two consist

8 of two copies of a V design, whereas d 1 has nine projections that are replicated IV designs, and d 2 has six such projections. 3 Estimation Capacity In Section 2, we showed that minimum aberration designs have good projection properties. In this section, we introduce the criterion of estimation capacity, and provide another justification of minimum aberration by showing that it is a good surrogate for maximum estimation capacity. We first investigate the alias structures of minimum aberration designs. Let g = 2 n m 1. Then under a 2 n m design d of resolution III+, g n of the g alias sets do not contain main effects. Let f = g n and, without loss of generality, assume that the first f alias sets do not contain main effects. For 1 i g, let m i (d) be the number of two-factor interactions in the ith alias set. From each defining word of length three, say ABC, we can identify three two-factor interactions AB, AC and BC that are aliased with main effects. It follows that under any design d of resolution III+, the number of two-factor interactions that are not aliased with main effects is equal to f m i (d) = i=1 ( ) n 3A 3 (d). (1) 2 Cheng, Steinberg and Sun (1999) further showed that A 4 (d) = 1 g 6 { [m i (d)] 2 i=1 ( ) n }. (2) 2 It follows from (1) and (2) that a minimum aberration design of resolution III+ maximizes f i=1 m i (d), and minimizes g i=1[m i (d)] 2 among those which maximize f i=1 m i (d). The second step tends to make the m i (d) s as equal as possible. Then since f i=1 m i (d) is equal to the number of two-factor interactions that are not aliased with main effects, one can conclude that a minimum aberration design of resolution III+ maximizes the number of 8

9 two-factor interactions that are not aliased with main effects, and tend to distribute these two-factor interactions very uniformly over the alias sets that do not contain main effects. Likewise, a minimum aberration design of resolution V+ maximizes the number of threefactor interactions that are not aliased with two-factor interactions among the resolution V+ designs, and tend to distribute these three-factor interactions very uniformly over the alias sets that do not contain main effects and two-factor interactions, etc. This property is important for understanding the statistical meaning of minimum aberration and relating it to the criterion of estimation capacity, which we now define. Estimation capacity was introduced by Sun (1993) as a measure of the capability of a design d to handle and estimate different potential models involving interactions. For simplicity, assume that the main effects are of primary interest and their estimates are required. Furthermore, all the three-factor and higher-order interactions are assumed to be negligible. For any 1 k ( ) n 2, Let Ek (d) be the number of models containing all the main effects and k two-factor interactions such that all the effects in the model are estimable under d, where k can be thought of as the number of active two-factor interactions. It is desirable to have E k (d) as large as possible. A design d 1 is said to dominate another design d 2 if E k (d 1 ) E k (d 2 ) for all k, with strict inequality for at least one k. We say that d has maximum estimation capacity if it maximizes E k (d) for all k. It turns out that E k (d) is a function of m(d) = (m 1 (d),, m f (d)). It is easy to see that E k (d) = 0 for k > f, and E k (d) = k m ij (d), if k f. (3) 1 i 1 < <i k f j=1 In other words, E k (d) is the kth elementary symmetric function of the m i (d) s. Cheng, Steinberg and Sun (1999) argued that a design d has large estimation capacity if it (i) maximizes f i=1 m i (d), and (ii) the m i (d ) s are as equal as possible. This is because, by Proposition F.1 on p. 78 of Marshall and Olkin (1979), E k (d) as given in (3) is a Schur 9

10 concave function of m(d) = (m 1 (d),, m f (d)) and is nondecreasing in each component of m(d). By the discussion in the paragraph following (2), minimum aberration is a good surrogate for maximum estimation capacity. Table 1 shows the values of m i (d) s, 1 i f, for 32-run minimum aberration designs with 9 n 29. For 16 n 21 and 24 n 29, the m i (d) values of minimum aberration 2 n (n 5) designs differ from one another by at most one. This distribution is the most uniform possible. It follows from the Schur-concavity of E k (d) that these designs maximize E k (d) for all k; a little extra work shows that it is also true for n = 22 and 23. That is, for n 16, minimum aberration 32-run designs have maximum estimation capacity (Cheng, Steinberg and Sun, 1999). However, for n < 16, minimum aberration designs typically do not maximize E k (d) for all k, but they still maximize E k (d) unless k is too large. Note that for n < 16, the minimum aberration designs are of resolution IV. This points to an important difference between resolution III and IV designs. We shall return to this in Sections 5 and 6. Cheng and Mukerjee (1998) obtained some general results on the construction of designs with maximum estimation capacity, in particular, those with the m i (d) s differing from one another by at most 1. 10

11 Table 1. m 1 (d),, m f (d) for minimum aberration 2 n (n 5) designs with 9 n 29. n r f m 1 (d),, m f (d) ,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,4, ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, ,3,3,3,3,4,4,4,4,4,4,4,4,4,4,0,0,0,0, ,4,4,4,4,4,4,4,4,5,5,5,5,5,5,0,0,0, ,5,5,5,5,5,5,5,5,5,5,5,6,6,6,0,0, ,6,6,6,6,6,6,6,6,6,6,6,6,6,7,0, ,7,7,7,7,7,7,7,7,7,7,7,7,7,7, ,8,8,8,8,8,8,8,8,8,8,8,8,8, ,8,8,8,8,8,8,8,8,8,8,8,8, ,8,8,8,8,8,8,8,8,8,8,8, ,8,8,8,8,8,8,8,8,9,9, ,8,8,8,8,9,9,9,9,9, ,9,9,9,9,9,9,9,9, ,8,10,10,10,10,10,10, ,11,11,11,11,11,11, ,12,12,12,12,12, ,12,12,12,12, ,12,12,12, ,13,13, ,14, ,14 r denotes resolution. 4 Clear Two-factor interactions It is prudent to include the main effects and a set of judiciously selected two-factor interactions in a first approximation to the response model. Therefore resolution III and IV designs are important for factor screening experiments. Wu and Chen (1992) classified two-factor interactions into three categories ineligible, eligible, and clear. A two-factor interaction is called ineligible if it is aliased with at least one main effect, eligible if it is not aliased with any main effect, and clear if it is neither aliased with main effects nor aliased with other two-factor interactions. Clearly a two- factor interaction is ineligible if it is part of a defining word of length three, eligible if it is not part of a defining word of length three, and clear if 11

12 it is not part of a defining word of length three or four. It is obvious that if estimates of all the main effects are required, then ineligible two-factor interactions are not estimable. On the other hand, if we can assume that the two-factor and higher order interactions aliased with an eligible two-factor interaction are negligible, then that eligible two-factor interaction can be estimated. Under the assumption of negligible three-factor and higher order interactions, all clear two-factor interactions are estimable. We use two example to illustrate these concepts. Example 4. There are five non-isomorphic III designs (see Chen, Sun and Wu (1993)) with the following independent defining words and wordlength patterns: d 1 : d 2 : d 3 : d 4 : d 5 : I = ABCDE = ABF = BCG = CDH = ABCJ W (d 1 ) = (0, 0, 7, 9, 6, 6, 3, 0, 0) I = ABCDE = ABF = ACG = BCH = ABCJ W (d 2 ) = (0, 0, 8, 10, 4, 4, 4, 1, 0) I = ABCDE = ABF = ACG = ADH = ABCJ W (d 3 ) = (0, 0, 6, 10, 8, 4, 2, 1, 0) I = ABCDE = ABF = ADG = BCH = CDJ W (d 4 ) = (0, 0, 6, 9, 9, 6, 0, 0, 1) I = ABCDE = ABF = ACG = ADH = BCDJ W (d 5 ) = (0, 0, 4, 14, 8, 0, 4, 1, 0). In this case, the maximum resolution is three, and there are no clear two-factor interactions under any of the five resolution III designs. The minimum aberration design d 5 has 4 defining words of length three and 14 defining words of length four: I = ABF = ACG = ADH = AEJ = BCF G = BDF H = BEF J = CDGH = CEGJ = DEHJ = BCDJ = BCEJ = BDEG = BGHJ = CDEF = CF HJ = DF GJ = EF GH =. The twelve two-factor interactions AB, AC, AD, AE, AF, AG, AH, AJ, BF, CG, DH, EJ are ineligible, and the other 24 two-factor interactions are eligible. The following example shows that when both resolution III and IV designs exist, designs of higher resolution do not necessarily have more clear two-factor interactions. Example 5. There are three III resolution III designs and one IV 12 resolution IV design

13 (Chen, Sun and Wu (1993)). Consider the following two designs, where d 1 is of resolution III and d 2 is of resolution IV: d 1 : I = ABC = ADEF = BCDEF W (d 1 ) = (0, 0, 1, 1, 1, 0) d 2 : I = ABCE = BCDF = ADEF W (d 2 ) = (0, 0, 0, 3, 0, 0). Note that d 2 is the design in Example 1. From the alias sets given in Example 1, it can be seen that under d 2 there is no clear two-factor interaction. However, there are six clear two-factor interactions, {BD, BE, BF, CD, CE, CF }, under the resolution III design d 1. Not only designs of higher resolution may have fewer clear two-factor interactions, Chen, Sun and Wu (1993) listed several examples of minimum aberration designs that are not the best in terms of the number of clear two-factor interactions. This seems surprising at the first look, since intuitively, minimum aberration designs should be the best designs with respect to the estimation of 2-factor interactions (HK2, p.522). This, however, is not surprising once one realizes that minimum aberration designs tend to distribute the twofactor interactions that are not aliased with main effects nearly uniformly over the alias sets, as demonstrated in the previous section. A two-factor interaction is clear if and only if it is the only two-factor interaction in an alias set that does not contain any main effect. Thus the number of clear two-factor interactions is equal to the number of m i (d) s, 1 i f, that are equal to 1. For a given run size, unless the number of factors is small, a minimum aberration design, due to nearly uniform distribution of the two-factor interactions over the alias sets not containing main effects, will have more than one two-factor interaction in each of such alias sets, resulting in no clear two-factor interaction. This shows that minimum aberration typically runs counter to large numbers of clear two-factor interactions. We have demonstrated in the previous section that minimum aberration is a good surrogate for maximum estimation capacity, a criterion under model uncertainty. Indeed as Fries and Hunter (1980) indicated, minimum aberration was for situations in which prior knowledge 13

14 is diffuse concerning the possible greater importance of certain effects. On the other hand, to take advantage of clear two- factor interactions, one needs to have some knowledge about which two-factor interactions might be important. The two criteria are really designed for different objectives. Example 4 shows that there is no clear two-factor interaction under any III design. Chen and Hedayat (1998) provided a complete characterization of the existence of clear two-factor interactions under 2 n m designs of resolution III or IV, and revealed their structures. For a fixed number of runs N = 2 n m, the maximum resolution of any 2 n m fractional factorial design is equal to III when N/2 < n N 1. It is natural to ask whether in this case a 2 n m resolution III design contains any clear two-factor interaction. The following theorem gives a negative answer to this question. Theorem 2 When N/2 < n N 1, there is no clear two-factor interaction under any 2 n m design of resolution III. Although when the maximum resolution is III, no resolution III design can have clear two-factor interactions, there may be eligible two-factor interactions. As shown in the previous section, for any design d, the number of eligible two-factor interactions is equal to ( ) n 3A 2 3 (d). By minimizing A 3 (d), resolution III designs with (weak) minimum aberration maximize the number of eligible two-factor interactions. In Example 4, we observed that there is no clear two-factor interaction under any resolution III design. However, the minimum aberration design d 5 yields the maximum number (24) of eligible two-factor interactions. The following two theorems from Chen and Hedayat (1998) characterize the existence of two-factor interactions under resolution IV designs. Theorem 3 When N/4 + 1 < n N/2, there is no clear two-factor interaction under any 2 n m design of resolution IV, but there exist Resolution III 2 n m designs with clear two-factor 14

15 interactions. Let n(k) be the maximum number of factors n that can be accommodated in a 2 n (n k) design of resolution V+. Then for n(k) < n 2 k 1, the maximum resolution of a 2 n (n k) design is IV. Theorem 4 For n(k) < n N/4 + 1, where N = 2 k, there exist 2 n (n k) resolution IV designs that have clear two-factor interactions. It follows from Theorems 2, 3 and 4 that for 32 runs, no regular design with 16 < n 31 can have clear two-factor interactions; for 10 n 16, no resolution IV design can have clear two-factor interactions, but there are resolution III designs with clear two-factor interactions; for 6 < n 9, there are resolution III and resolution IV designs that have clear two-factor interactions. Some lower and upper bounds for the number of clear two-factor interactions were derived in Tang, Ma, Ingram and Wang (2002). Wu and Wu (2002) developed an approach to show whether a given design has the maximum number of clear two-factor interactions. Zhang, Li, Zhao and Ai (2008) defined a minimum lower order confounding (GMC) criterion, which can be viewed as a refined version of the criterion of maximizing the number of clear two-factor interactions. 5 Estimation Index In a factorial experiment, there may be a large number of interactions that are potentially important. Then we should choose a design that allows the estimation of as many such interactions as possible. For example, in a robust design experiment to study the effects of control and noise factors on certain responses of a product or process, we prefer a design that can be used to entertain models that contains all the main effects of control and noise factors and as many of their interactions as possible. 15

16 Example 6. Suppose we wish to perform an experiment with sixteen runs and six twolevel factors, labeled A, B, C, D, E and F, where A, B and C are control factors and the others are noise factors. The estimability of main effects and control-noise interactions is the primary concern. There are four non-isomorphic designs, among which the design in Example 1 (d 2 in Example 5) has minimum aberration. Again from the alias sets given in Example 1 we can see that under d 2, we are able to estimate at most seven two-factor interactions, and therefore it is not possible to entertain all nine control-noise interactions. This is because two-factor interactions appear in only seven of the nine alias sets that do not contain main effects. On the other hand, under the more aberration resolution III design d 1 in Example 5, none of the main effects and control-noise two-factor interactions are aliased among themselves. Therefore all the main effects and control-noise two-factor interactions are estimable if the other interactions are negligible. Although as a design of lower resolution, d 1 causes aliasing of some main effects and two-factor interactions, in many circumstances an experimenter may have prior knowledge that certain interactions are negligible. If the goal of the experiment is to explore as many interaction effects as possible, then selection decisions based on the wordlength pattern alone may not be sufficient. One should also examine the alias structure of the design. One can see from Table 1 that for all the 32-run minimum aberration designs with n 16, all the m i (d) s, 1 i f, are positive. That is, there is at least one two-factor interaction in every alias set that does not contain main effects. Under such designs, one can entertain models that contain all the main effects and up to f = 2 n m n 1 twofactor interactions, using up all the available degrees of freedom. On the contrary, most of the minimum aberration resolution IV designs with n 15 have some zero m i (d) s. For example, under the minimum aberration design, 15 of the m i (d) s are nonzero and 5 are zero. Such a design can be used to entertain only up to 15 two-factor interactions, even though there are 20 degrees of freedom that are not aliased with main effects. In Example 16

17 6, the minimum aberration design d 2 has 7 nonzero m i (d 2 ) s with 1 i 9, but for d 1, all the 9 m i (d 1 ) s with 1 i 9 are not equal to zero. We define the length of an alias set to be the length of the shortest word in the set. The estimation index of a design d, denoted by ρ(d), is defined as the largest length of all the alias sets. For a resolution III+ design, clearly a necessary and sufficient condition for m i (d) > 0 for all 1 i f, (i.e., there is at least one two-factor interaction in each alias set that does not contain main effects) is that ρ(d) = 2. So resolution III+ designs with estimation index two can be used to entertain some models containing all the main effects and up to 2 n m n 1 two-factor interactions, the largest number possible, if all the other interactions are negligible. Such designs are called second-order saturated designs by Block and Mee (2003). If the estimation index is greater than 2, then fewer than 2 n m n 1 two-factor interactions can be entertained. As Example 6 shows, for resolution IV designs, index two may not be achievable. Example 6 (revisited). In Example 6, the maximum attainable resolution is IV. The minimum aberration resolution IV design d 2 has estimation index three since it has six alias sets of length one, seven alias sets of length two and two alias sets of length three. The lower resolution design d 1 has estimation index two since it has six alias sets of length one and nine alias sets of length two. From the definitions of resolution and alias sets, it is not difficult to see the following relationship between resolution and estimation index. Proposition 1 Let d be a 2 n m design of resolution r and estimation index ρ. Then ρ [(r 1)/2], (4) where [x] is the largest integer less than or equal to x. 17

18 Under a design with ρ = [(r 1)/2], one can estimate all the effects involving at most (r 1)/2 factors if the higher-order interactions are negligible. Except for a few cases, equality in (4) usually does not hold. However a 2 n m design with ρ = [(r 1)/2] + 1 should be fairly good. For r = 3 or 4, [(r 1)/2] + 1 = 2. The estimation index of a saturated design (n = 2 n m 1) is equal to 1; this is because under such a design every alias set contains one main effect. All the other resolution III+ designs have ρ(d) 2. We have seen that ρ(d) = 2 if and only if m i (d) > 0 for all 1 i f. An interesting fact is that this lower bound 2 is always achieved as long as N/2 < n < N 1. Note that this is when the maximum possible resolution is three. Theorem 5 (Chen and Cheng, 2004) If N/2 < n < N 1, then all resolution III 2 n m designs achieves the minimum possible estimation index 2. In Table 1, for all the 32-run minimum aberration designs of resolution III (those with n > 16), we do have m i (d) > 0 for all 1 i f. Resolution IV designs exist when n N/2. In this case, the following result holds. Theorem 6 (Chen and Cheng, 2004) If N/4 + 1 n N/2, then any resolution IV 2 n m design has estimation index at most 3. The resolution IV designs covered by Theorem 6 can have estimation indices 2 or 3. Unlike resolution III designs, in general most of these resolution IV designs have estimation index equal to 3. For 32-run designs, resolution IV designs with estimation index two exist only when n = 9, 10 and 16; see Table 1. This is a consequence of some results from coding theory and finite projective geometry, which have important implications in the structure and construction of resolution IV designs. A brief account is given below. The readers are referred to Chen and Cheng (2004, 2006) for more detailed discussions. A design of resolution IV+ is called maximal if and only if its resolution reduces to three whenever a factor is added. Maximal designs are important since all non-maximal designs 18

19 of resolution IV can be obtained from maximal ones by deleting some factors, i.e., they are projections of maximal designs. We state two theorems, which are translations of results from coding theory and finite projective geometry into design language (Davydov and Tombak, 1990; Bruen, Haddad and Wehlau, 1998; Bruen and Wehlau, 1999): Theorem 7 For any regular design d of resolution IV+, the following conditions are equivalent : (a) d has estimation index 2; (b) d is maximal; (c) d is second-order saturated. Theorem 7 shows that non-maximal resolution IV designs are not second-order saturated and have estimation index greater than 2. One intuitive explanation is as follows. The non-maximal designs must be constructed by deleting factors from maximal ones. When some factors are dropped, it does not change the alias sets where interactions of the other factors are located. Thus the number of nonzero m i (d) s does not increase even though extra degrees of freedom become available due to elimination of some main effects. For example, a maximal design of resolution IV has all the two-factor interactions distributed over 15 alias sets. A design of resolution IV can be constructed by deleting 5 factors from the maximal design of resolution IV. It still has 15 nonzero m i (d) s, but after deleting five factors from a design, we have 5 more, i.e., 20 alias sets that do not contain main effects. Suppose the two levels of each factor are denoted by 1 and 1, and each fractional factorial design d is represented by an N n matrix X(d), with each row corresponding to a run and each column corresponding to a factor. Then the matrix [ ] X(d) X(d) X(d) X(d) 19

20 represents a design with 2N runs and 2n factors. We call this design the double of d. Theorem 8 For N/4 + 1 n N/2, where N = 2 k, k 4, maximal designs of resolution IV+ exist if and only if n = N/2 or n = (2 i + 1)N/2 i+2 for some integer i such that 2 i k 2. A maximal design with n = (2 i + 1)N/2 i+2 can be obtained by repeatedly doubling a maximal regular resolution IV+ design with 2 i+2 runs and 2 i + 1 factors k i 2 times. By Theorems 7 and 8, for N = 32, maximal designs of resolution IV+ exist only for n = 9, 10 and 16, as noted before. For all the other n values, no resolution IV 2 n m design is maximal, and hence all the resolution IV 2 n m designs must have estimation indices greater than two. By Theorem 6, they are equal to three. Also, the maximal resolution IV+ designs in Theorem 8 can be constructed by repeatedly doubling maximal designs of smaller run sizes. For example, all maximal designs with n = 5N/16 can be constructed by repeatedly doubling the design defined by I = ABCDE. Remark 1. Besides folding over a saturated design, the maximal even design can also be constructed by repeatedly doubling the 2 2 complete factorial Saturated designs can be constructed by repeatedly doubling [ ] 1 1, 1 1 followed by deletion of the first column. Some important applications of Theorems 7 and 8 will be given in Section 7. Theorem 4 shows that N/4+1 is the maximum number of factors for resolution IV designs to have clear two-factor interactions. One class of resolution IV designs with n = N/4 + 1 was shown by Wu and Wu (2002) to have the maximum number of clear two-factor interactions. It can be seen that they are maximal designs. 20

21 6 Estimation index, minimum aberration and maximum estimation capacity Minimum aberration was shown in Section 3 to be a good surrogate for maximum estimation capacity, a criterion for choosing designs that can be used to entertain the largest number of models containing a fixed number of two-factor interactions. However, minimum aberration and maximum estimation capacity do not always coincide. We have already pointed out in Section 3 that for 32 runs, all minimum aberration designs of resolution III (those with n > 16) have maximum estimation capacity, but minimum aberration designs of resolution IV (those with n 16), except for n = 16, maximize E k (d) unless k is too large, but may not maximize E k (d) for larger k s. This is intimately related to the phenomenon observed in the previous section that for N/2 < n < N 1, all the resolution III designs have estimation index two, but for n < N/2, most of the resolution IV designs have estimation index greater than two. When N/2 < n < N 1, since none of the m i (d) s is constrained to be equal to zero, we can say that the m i (d) s for a minimum aberration resolution III design are truly nearly equal. In this case, we do expect high consistency between minimum aberration and maximum estimation capacity. We wonder whether all minimum aberration 2 n m designs with N/2 < n < N 1 maximize E k (d) s for all k. This is known to be true for 16- and 32-run designs (Cheng, Steinberg and Sun, 1999). Under a resolution IV design with n = N/2, the maximal even design, the ( ) n 2 two-factor interactions are distributed uniformly over the 2 n m 1 1 alias sets that do not contain main effects, i.e., each of these alias sets contains the same number of two-factor interactions. This design has minimum aberration, estimation index two, and also has maximum estimation capacity over all designs. For n < N/2, in most cases, the minimum aberration designs have estimation indices 21

22 greater than two. For example, we noted earlier that under the minimum aberration design, 15 of the m i (d) s are nonzero and 5 are zero. Such a design can be used to entertain only up to 15 two-factor interactions. As a result, E k (d) = 0 for all k > 15. On the other hand, there exist resolution III designs with estimation index two. Such designs have positive E k (d) s for 16 k 20, and due to the continuity of E k (d) as a function of k, they also have larger E k (d) s than the minimum aberration design for the k s that are not much smaller than 16. Although the nonzero m i (d) s for a minimum aberration design are nearly equal, the constraint that some of them must be zero prevents the design from maximizing E k (d) for larger k s. For 32 runs, by the discussion in the paragraph preceding the previous one, the maximal design with n = 16 has minimum aberration and maximum estimation capacity. We note that the maximal design with n = 10 also has minimum aberration. It can be shown that it maximize E k (d) for all k as well. The fact that it has estimation index two plays an important role for it to have maximum estimation capacity. The readers are referred to Chen and Cheng (2004) for details. The third maximal design, with n = 9, however, does not have minimum aberration. In this case, the minimum aberration design can be obtained by deleting one factor from the maximal design with n = 10, and therefore has estimation index 3. In general, if a minimum aberration design of resolution III+ has estimation index 2, then it is expected to have large, if not maximum, E k (d) for all k s. On the other hand, if a minimum aberration design has estimation index 3, but another design has estimation index 2, then the minimum aberration design tends to be optimal for smaller k s, but not for larger k s. In this case, if the number of active two-factor interactions is expected to be large, then one may want to use a design that has minimum aberration among those with estimation index 2. 22

23 7 Complementary Design Theory for Minimum Aberration Designs Construction of minimum aberration designs has been studied by many authors. In this section we review a useful technique of constructing minimum aberration designs via complementary designs. Let d be a regular 2 n m design of resolution III+. Then D can be constructed by deleting f = 2 n m 1 n factors from the 2 n m 1 factors of a saturated design. The deleted factors form another regular design, denoted by d. We call it the complementary design of d. Tang and Wu (1996) derived identities that relate the wordlength pattern of d to that of d. Chen and Hedayat (1996) independently derived such identities for defining words of lengths three and four: A k (d) = C 0 + k 1 j=3 C j A j (d) + ( 1) k A k (d), 3 k n,. (5) where the C j s are constants not depending on d. It follows from (5) that sequentially minimizing A k (d) is equivalent to sequentially minimizing ( 1) k A k (d), i.e., maximizing A 3 (d), followed by minimizing A 4 (d), and then maximizing A 5 (d), etc. Therefore the determination of a minimum aberration design can be done via the selection of its complementary design. This is particularly useful when d is nearly saturated, in which case d has only a few factors and it is much easier to determine the wordlength pattern of d than that of d. This result was extended to regular s n m designs, where s is a prime power, by Suen, Chen and Wu (1997). Chen and Hedayat (1996) showed that the complementary designs of minimum aberration designs in the saturated designs have the same structure as long as they have the same number of factors, and determined all minimum aberration 2 n m designs with f 16. For n = N/2, we already know that the maximal even designs have minimum aberration, and that they can be constructed by applying the method of foldover to saturated designs 23

24 of size N/2 or by repeatedly doubling the complete 2 2 factorial. When n < N/2, the complementary design has more factors than the original design. Therefore the method of complementary designs described above is not useful. In this case, the minimum aberration designs must have resolution IV+, and can be constructed by deleting factors from certain maximal designs of resolution IV+, instead of the saturated designs. Can a similar complementary design theory be developed, with maximal resolution IV+ designs as the universes? By Theorem 8, there is no maximal design with 5N/16 < n < N/2. The following is an important consequence of this observation: Theorem 9 All the regular resolution IV designs with 5N/16 < n < N/2 are projections of the maximal even design of size N. It follows from Theorem 9 that all regular resolution IV designs with 5N/16 < n < N/2 are foldover and even designs. For example, to construct a minimum aberration 32-run design with 14 factors, we only have to determine which two factors to drop from the 16 factors of the 32-run maximal even design. This substantially simplifies the problem. Such a complementary design theory was provided by Butler (2003a); see also Chen and Cheng (2009). The following theorem was due to Butler (2003a). Theorem 10 For 5N/16 < n < N/2, a regular 2 n m design has minimum aberration if and only if it is an n-dimensional projection of the maximal even design of size N, and its complement in the maximal even design has minimum aberration among all (N/2 n)- dimensional projections of the maximal even design. Chen and Cheng (2009) obtained explicit identities that relate the wordlength pattern of an even design to that of its complement in the maximal even design. These identities were 24

25 used to further investigate the structures of (weak) minimum aberration designs of resolution IV. For 9N/32 < n 5N/16, a minimum aberration 2 n m design is either a projection of the maximal even design or the maximal design with 5N/16 factors. We need to compare the best projections of these two maximal designs with respect to the minimum aberration criterion. Chen and Cheng (2006) showed that the best projection of the maximal even design has more aberration than the best projection of the maximal design with 5N/16 factors. It follows that minimum aberration designs with 9N/32 < n 5N/16 are n-dimensional projections of the maximal design with 5N/16 factors; therefore they are not even (foldover) designs. In particular, the maximal design with 5N/16 factors is a minimum aberration design. They also showed that the maximal design with n = 9N/32 does not have minimum aberration. Instead the minimum aberration design for n = 9N/32 is also a projection of the maximal design with 5N/16 factors. By Theorem 8, for 17N/64 < n < 9N/32, minimum aberration designs can be projections of maximal designs with 9N/32, 5N/16 or N/2 factors. Again one needs to compare the best n-dimensional projections of these three maximal designs. In general, for n N/4 + 1, to determine a minimum aberration 2 n m design, one needs to compare the best n-dimensional projections of the maximal designs with at least n factors. A complementary design theory for the maximal designs would be useful for determining their best projections. By Theorem 8, all the maximal designs with N/4 + 1 n N/2 can be obtained by repeatedly doubling smaller maximal designs. Xu and Cheng (2008) developed a general complementary design theory for repeated doubles. It can be applied to saturated designs as well since the saturated designs can also be constructed by the method of doubling; see Remark 1. By applying the general result to saturated designs and maximal even designs, one obtains the results of Chen and Hedayat (1996), Tang and Wu (1996) and Butler (2003) as special cases. As an application, Xu and Cheng (2008) obtained the following improvement of Chen and Cheng s 25

26 (2006) result mentioned in the previous paragraph. Theorem 11 For N = 32 2 t, t 0, and 17N/64 n 5N/16, a minimum aberration design with N runs and n factors must be a projection of the maximal regular design of resolution IV with 5N/16 factors. As a numerical illustration, for 256 runs, Xu and Cheng (2008) determined the best n- factor projections of the maximal design with 80 factors for 69 n 79. This corresponds to complementary designs with one to eleven factors. Block (2003) considered the construction of minimum aberration designs from this maximal design by deleting one column at a time. The theoretical result in Xu and Cheng (2008) confirmed that the designs with 69 n 79 obtained by Block (2003) indeed have minimum aberration except for n = Nonregular Designs and Orthogonal Arrays Two-level regular designs are easy to construct and analyze, but the run sizes must be powers of two. On the other hand, nonregular designs are more flexible in terms of run sizes. To pave the way for extending the minimum aberration criterion to nonregular designs in the next section, we first review some basic properties of orthogonal arrays. We also briefly discuss extension of the concepts of estimation capacity and clear two-factor interactions to nonregular designs. Suppose the two levels of each factor are denoted by 1 and 1. Then as in the paragraph preceding Theorem 8 and elsewhere in this chapter, each N-run design d for n two-level factors can be represented by an N n matrix X(d), where each column corresponds to one factor and each row represents a factor-level combination. Let the ith column of X(d) be x i (d). Then x i (d) is the column of the model matrix that corresponds to the main effect of the ith factor. For each 1 k n and any subset S = {i 1,, i k } of {1,, n}, let x S (d) = x i1 (d) x ik (d), where for any two vectors x and y, x y is the componentwise 26

27 product of x and y. Then x S (d) is the column of the model matrix that corresponds to the interaction of factors i 1,, i k. Tang (2001) defined J S (d), called a J-characteristic, as the sum of all the entries of x S (d): J S (d) = N l=1 x li1 (d) x lik (d), where x li (d) is the (l, i)th entry of X(d), i.e., the level of the ith factor at the lth run. Definition. An orthogonal array OA(N, s n, t) is an N n matrix with s distinct symbols in the ith column, 1 i n, such that in each N t submatrix, all combinations of the symbols appear equally often as row vectors. The positive integer t is called the strength of the orthogonal array. For any subset S of {1,, n}, let S be the cardinality (size) of S. If X(d) is an orthogonal array of strength t, then it is easy to see that for any S such that S t, x S (d) contains the same number of 1 s and 1 s. It follows that J S (d) = 0 for all S such that S t. It can be shown that the converse is also true: Theorem 12 A fractional factorial design d is an orthogonal array of strength t if and only if J S (d) = 0 for all S such that S t. Under a regular design, if the k-factor interaction involving factors i 1,, i k appears in the defining relation, then N l=1 x li1 (d) x lik (d) is equal to either N or N; otherwise, J S (d) = 0; in particular J S (d) = 0 for all S such that S < r. This and Theorem 12 together imply the following result: Theorem 13 Each regular fractional factorial design of resolution r is an orthogonal array of strength r 1 For any two subsets S and T of the factors, we say that their corresponding factorial effects are orthogonal if x S (d) T x T (d) = 0. We have 27