design selection and classification for hadamard matrices using generalized minimum aberration...

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Design Selection and Classification for HadamardMatrices Using Generalized Minimum AberrationCriteriaLih-Yuan Denga & Boxin Tanga

a Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152Published online: 01 Jan 2012.

To cite this article: Lih-Yuan Deng & Boxin Tang (2002) Design Selection and Classification for Hadamard Matrices UsingGeneralized Minimum Aberration Criteria, Technometrics, 44:2, 173-184, DOI: 10.1198/004017002317375127

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Design Selection and Classi’ cation forHadamard Matrices Using GeneralizedMinimum Aberration Criteria

Lih-Yuan Deng and Boxin Tang

Department of Mathematical SciencesUniversity of MemphisMemphis, TN 38152

( [email protected] )

Deng and Tang (1999) and Tang and Deng (1999) proposed and justi� ed two criteria of generalizedminimum aberration for general two-level fractional factorial designs. The criteria are de� ned using aset of values called J characteristics. In this article, we examine the practical use of the criteria in designselection. Speci� cally, we consider the problem of classifying and ranking designs that are based onHadamard matrices. A theoretical result on J characteristics is developed to facilitate the computation.The issue of design selection is further studied by linking generalized aberration with the criteria ofef� ciency and estimation capacity. Our studies reveal that generalized aberration performs quite wellunder these familiar criteria.

KEY WORDS: Confounding frequency vector; Isomorphic design; Nonregular design; Orthogonaldesign; Plackett–Burman design; Projection property; Regular design.

1. INTRODUCTION

Regular fractional factorials, often known as 2mƒp designs,are the most studied in the design literature, and widely usedin industrial experiments (Box, Hunter, and Hunter 1978).A regular design has a simple aliasing structure in that anytwo effects are either orthogonal or fully aliased. The runsize of a regular design is a power of 2, thus leaving largegaps in the available choices of run size. Minimum aberra-tion (Fries and Hunter 1980) is the most popular criterionfor selecting regular factorials. For a design D, let Ak4D5

be the number of words of length k in the de� ning rela-tion. Then W 4D5 D 6A34D51 : : : 1Am4D57 is called the wordlength pattern. The resolution of design D is then the small-est r such that Ar 4D5 ¶ 1. Minimum aberration is de� nedas a criterion that sequentially minimizes the components ofW4D5 D 6A34D51 : : : 1Am4D57. Two-level minimum aberrationdesigns are discussed in Chen (1992), Tang and Wu (1996),Chen and Hedayat (1996), Huang, Chen, and Voelkel (1998),Cheng, Steinberg, and Sun (1999), and many others.

This article considers two-level orthogonal fractional facto-rials, by which we mean that, for every two columns of thedesign matrix, the four level combinations 4ƒ1ƒ5, 4ƒ1 C5,4C1ƒ5, and 4C1C5 occur with the same frequency. A richand readily available class of orthogonal designs can be con-structed by choosing columns from Hadamard matrices, ofwhich Plackett–Burman designs (1946) as well as regular fac-torials are special cases. A square matrix of 1 is called aHadamard matrix if its columns are mutually orthogonal. Theorder n of a Hadamard matrix is a multiple of 4, except for thetrivial cases n D 112. For a Hadamard matrix of order n withits � rst column consisting of all C1s, we can remove this � rstcolumn to obtain a saturated design of n runs. Choosing mcolumns for m µ nƒ 1 from this saturated design gives rise toa design for m factors. Designs based on Hadamard matricesare useful screening designs, and � ll the gaps between the run

sizes available from regular designs. An orthogonal factorial iscalled nonregular if it does not belong to the class of regulardesigns. A nonregular design exhibits some complex alias-ing structure in that there exist two effects that are partiallyaliased. Partially aliased effects may be jointly estimable, asshown in Hamada and Wu (1992).

Deng and Tang (1999) recently proposed generalized mini-mum aberration, called minimum G aberration, as a systematiccriterion to compare and assess the “goodness” of general two-level factorials. In Tang and Deng (1999), minimum G2 aber-ration, a relaxed variant of minimum G aberration, is proposedand studied. They justi� ed the criteria by showing that thecriteria lead to designs that minimize the contamination oftwo-factor interactions on the estimation of main effects.

In this article, we examine the practical use of the gener-alized aberration criteria in design selection. Since Hadamardmatrices provide a rich and readily available class of orthog-onal designs, we restrict our attention to designs based onHadamard matrices. We show that the confounding frequencyvector associated with the generalized aberration criteria pro-vides a quick and convenient way for reducing a large numberof designs to a manageable number of nonisomorphic designs(to be de� ned in Section 2). A theoretical result is proved tofacilitate the computation. Once a collection of nonisomorphicdesigns is obtained, they can be ranked by either of the twoaberration criteria. As is true for any design criterion, oneshould also regard the rankings given by generalized minimumaberration as tentative. Through several examples, we furtherstudy the issue of design selection by connecting general-ized aberration with the criteria of ef� ciency and estimation

© 2002 American Statistical Association andthe American Society for Quality

TECHNOMETRICS, MAY 2002, VOL. 44, NO. 2

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174 LIH-YUAN DENG AND BOXIN TANG

capacity. Our studies show that generalized aberration per-forms quite well under these familiar design criteria.

The article is organized as follows. Section 2 is devotedto the method of classifying and ranking designs based onHadamard matrices using the generalized aberration criteria.Section 3 discusses the application of our method to designsof 12, 16, 20, and 24 runs. Section 4 explores the relationshipbetween the criteria of generalized aberration and those ofef� ciency and estimation capacity.

2. CLASSIFICATION AND RANKING WITHGENERALIZED MINIMUM ABERRATION

This section examines the problem of selecting designsfrom Hadamard matrices according to the generalized aber-ration criteria. Section 2.1 reviews the criteria of generalizedminimum aberration proposed in Deng and Tang (1999) andTang and Deng (1999). In Section 2.2, we present our methodfor classifying and ranking designs based on Hadamardmatrices.

2.1 Generalized Minimum Aberration Criteria

We use an n� m matrix of C and ƒ to represent an orthog-onal design D of n runs for m factors, where each row of thematrix corresponds to a run and each column to a factor. Fors D 8c11 : : : 1 ck9, a subset of k columns of D, de� ne

jk4s5 DnX

iD1

ci1 ¢ ¢ ¢ cik1 Jk4s5 D —jk4s5— (1)

where cij is the ith component of column cj . Since D isorthogonal, we have J14s5 D J24s5 D 0. When D is a regulardesign, Jk4s5 must equal either 0 or n, with 0 correspondingto orthogonality and n to full aliasing. If Jk4s5 D n, these k

columns in s form a word of length k in the de� ning rela-tion. The quantity jk4s5 has a simple interpretation. For exam-ple, for a set s of three columns, j34s5=n is the correlationbetween the main effect i and the two-factor interaction s ƒ i,where i 2 s. Similarly, for a set s of four columns, j44s5=nis the correlation between the two-factor interactions i1i2 ands ƒ 8i11 i29, where 8i11 i29 s.

Suppose that r is the smallest integer such that max—s—Dr

Jr 4s5 > 0, where the maximization is over all subsets of r

columns of D. The generalized resolution of D is de� ned as

R4D5 D r Cµ

1 ƒ max—s—Dr

Jr 4s5=n

¶0 (2)

When D is regular, its generalized resolution is the same as itsusual resolution. Generalized resolution enjoys some appealingprojection properties. For details, see Deng and Tang (1999).

For orthogonal designs, Jk4s5 must be a multiple of 4 (Dengand Tang 1999, prop. 3). Let n D 4t, and fkj be the frequencyof k column combinations that give Jk4s5 D 44t C 1 ƒ j5 forj D 11 : : : 1 t. The confounding frequency vector is de� ned as

F 4D5 D 6F34D53 : : : 3 Fm4D571

where Fk4D5 D 4fk11 : : : 1 fkt50 (3)

The minimum G-aberration criterion introduced in Deng andTang (1999) is de� ned as follows. For two designs D1 and D2,

let fi4D15 and fi4D25 be the ith entries of F 4D15 and F4D25,respectively, where i D 11 : : : 1 4m ƒ 25t. Let l be the smallestinteger such that fl4D15 6D fl4D25. If fl4D15 < fl4D25, thenD1 has less G aberration than D2. If no design has less Gaberration than D1, then D1 has minimum G aberration.

Tang and Deng (1999) proposed a relaxed version of mini-mum G aberration, called minimum G2 aberration. Let

Bk4D5 D nƒ2X

—s—Dk

6Jk4s5720 (4)

For two designs D1 and D2, let r be the smallest integer suchthat Br 4D15 6D Br 4D25. If Br 4D15 < Br 4D25, D1 has less G2

aberration than D2. If no design has less G2 aberration thanD1, then D1 has minimum G2 aberration. For regular designs,both G aberration and G2 aberration criteria reduce to mini-mum aberration.

To see the motivation behind the criteria of generalizedaberration, consider the following scenario. Suppose that theexperimenter is interested in estimating the main effects.Although she can safely assume the absence of interactionsinvolving three or more factors, she suspects that some two-factor interactions may be nonnegligible. Without furtherknowledge about these potentially important two-factor inter-actions, she has the option of using a large design that allowsestimation of all two-factor interactions in addition to themain effects. Facing a limited budget, she rules out thisoption. With the above considerations, she prefers a designthat not only allows her to estimate the main effects, but alsominimizes the contamination of two-factor interactions on theestimation of main effects.

Let d11 : : : 1 dm be the columns of design D of n runs for m

factors. The true model under the above scenario is given byyi

D ‚0 CPmjD1 ‚jdij

CPmk<l ‚kldikdil

C…i, where ‚j is the maineffect of factor j and ‚kl is the interaction between factorsk and l, and …i’s are independent random errors with zeromean and a constant variance. However, the main effect modelyi

D ‚0C Pm

jD1 ‚jdijC …i is � tted. Simple calculation shows

that the least squares estimator O‚j of ‚j from the � tted modelhas expectation (taken under the true model)

E4 O‚j5 D ‚jC nƒ1

mX

k<l

j34dj1 dk1dl5‚kl (5)

for j D 11 : : : 1 m, where j34dj 1dk1dl5 is de� ned in (1). Notethat Equation (5) provides an aliasing pattern for the design.Lin and Draper (1993) constructed alias tables for Plackett–Burman designs of up to 100 runs.

Note that —j34dj1 dk1dl5— D J34dj1 dk1dl5. To minimize thecontamination of two-factor interactions ‚kl’s on the estima-tion of main effects ‚j’s, we are led to choose a design forwhich J34dj1 dk1dl5’s are all small. There are many ways onecan achieve this. One possibility is to choose a design thatminimizes

PmjD1

Pmj<k<l6J34dj1 dk1dl5=n72 D 3B3, where B3 is

de� ned in (4). This method leads to minimum G2 aberra-tion. For a more general result, see Tang and Deng (1999). Aconservative approach is to minimize maxj<k<l J34dj1dk1 dl5,which is equivalent to maximizing the generalized resolutionas de� ned in (2). Sequentially minimizing f311 : : : 1 f3t , asrequired by minimum G aberration, is just an elaborate ver-sion of the idea of minimizing maxj<k<l J34dj1 dk1dl5, where


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DESIGN SELECTION AND CLASSIFICATION FOR HADAMARD MATRICES 175

f311 : : : 1 f3t are the components of the confounding frequencyvector associated with J3 values as given in (3). This estab-lishes a justi� cation for minimum G aberration. Yet anotherpossibility is to minimize the number of nonzero J3’s. Wereturn to this point when we discuss an example in Section 4.

2.2 Classi’ cation and Ranking for Designs Basedon Hadamard Matrices

Two designs are isomorphic if one can be obtained fromthe other by permuting rows, permuting columns, switchingC and ƒ, or a combination of the above. To choose a suit-able design from a collection of designs, the problem will begreatly simpli� ed if we can identify the set of nonisomor-phic designs before applying any design selection criterion.Although the total number of designs under consideration maybe quite large, the number of nonisomorphic designs is oftenvery small. Designs under consideration in this article arethose based on Hadamard matrices. Sun and Wu (1993) iden-ti� ed all of the nonisomorphic designs that can be constructedby choosing columns from Hadamard matrices of order 16.For regular designs, Chen, Sun, and Wu (1993) obtained allof the nonisomorphic designs of up to 64 runs.

Isomorphism checking is very computer intensive, and infea-sible for larger designs. We propose the use of minimumG aberration for this purpose. Speci� cally, we use the con-founding frequency vector (CFV) associated with minimumG aberration to differentiate designs. The idea is simple. Iftwo designs are isomorphic, they have the same CFV, whichcan easily be seen from (1) and (3). Two different CFV’stherefore correspond to two nonisomorphic designs. Insteadof directly working with isomorphism, we proceed to � nd theset of designs that have different CFV’s. The above argumentimplies that the resulting set of designs is nonisomorphic.

There are some obvious practical advantages of using theCFV to classify designs. Design classi� cation based on theCFV is less computationally expensive than that based onisomorphism, and therefore applicable to larger designs. Inaddition, design classi� cation based on the CFV is more infor-mative and statistically relevant. The concept of design iso-morphism is of combinatorial nature. Just knowing that twodesigns are nonisomorphic, we can hardly say much aboutpossible differences in their statistical properties. Design clas-si� cation based on the CFV, on the other hand, is informativeas the CFV of a design provides certain essential informationabout how the effects are confounded.

Two designs that have the same CFV are not necessarilyisomorphic. Therefore, the number of nonisomorphic designsidenti� ed by the CFV is generally smaller than the totalnumber of nonisomorphic designs. So it is possible that ourapproach may miss some designs. However, as shown laterin Section 3, the discrepancy is usually quite small. Our goalhere is not to identify the complete collection of noniso-morphic designs, but rather to obtain quickly a collection ofnonisomorphic designs, which can supply suf� cient � exibilityin meeting the needs of design situations often encounteredin practice.

Once a collection of nonisomorphic designs is identi� ed,they can be ranked by either of the two aberration criteria. Asis true for any design criterion, one should always view the

rankings given by the aberration criteria as tentative. Differentdesign situations may call for different designs. If more thanone design satis� es the user’s needs, the user can select thedesign that performs well under either or both of the two aber-ration criteria. This ensures that the selected design has therobust property as given in Section 2.1, in addition to meetingthe user’s needs. We elaborate on these points in Section 4.

The rest of the subsection discusses some computationalissues. The CFV de� ned in (3) has length 4m ƒ 25n=4. Thefollowing result is used to reduce the length of the CFV bya factor of 2. Recall that Jk4s5 must be a multiple of 4. Thisresult can be strengthened as follows.

Proposition 1. Let D be an orthogonal design of n runsfor m factors.

a. If n is a multiple of 8, then Jk4s5 is also a multiple of 8.b. If n is not a multiple of 8, then Jk4s5 is a multiple of

8 for k D 4w C 11 4w C 2, but is not a multiple of 8 for k D4w C 314w C 4, where w D 011121 : : : .

Proposition 1, proved in Appendix A, can be used to shortenthe CFV as follows. Let Lk be the list of possible valuesof Jk4s5 in decreasing order. When n is a multiple of 8, wehave Lk

D 6n1 n ƒ 81 : : : 1 07. When n is not a multiple of 8,we have Lk

D 6n ƒ 41 n ƒ 121 : : : 107 for k D 4w C 114w C 2,and Lk

D 6n1 n ƒ 81 : : : 147 for k D 4w C 31 4w C 4, wherew D 011121 : : : . The total number of entries in Lk is g C 1,where g is the largest integer not exceeding n=8. Let f ü

kj bethe frequency of Jk4s5 taking the value of the jth entry of Lk,for j D 11 : : : 1 g C 1. Since

PgC1jD1 f ü

kjD m

k

¢is a constant, we

only need to consider f ükj for j D 11 : : : 1 g. The CFV of design

D can now be rede� ned as

F 4D5 D 6F34D53 : : : 3 Fm4D571

where Fk4D5 D 4f ük11 : : : 1 f ü

kg50 (6)

Its length is 4m ƒ 25g µ 4m ƒ 25n=8. Note that the length ofthe original CFV is 4m ƒ 25n=4.

Even with the above simpli� cation, computation of theCFV is still time consuming for large m. The total num-ber of required calculations of Jk4s5 with 3 µ k µ m is2m ƒ m

1

¢ƒ m

2

¢ƒ 1, which increases exponentially with m.

Great savings in computing time can be achieved by usingtwo or three leading terms of the CFV to classify designs.We de� ne the MA-4 classi� er as a classi� cation and rankingcriterion based only on 6F34D53 F44D57, where F34D5 andF44D5 are de� ned in (6). The total number of requiredcalculations of Jk4s5 is reduced to m

3

¢C m

4

¢. Similarly, the

MA-5 classi� er is de� ned to be a criterion based only on6F34D53F44D53 F54D57.

3. APPLICATION TO DESIGNS OF12, 16, 20, AND 24 RUNS

Our method for design classi� cation and ranking inSection 2 is now applied to designs of 12, 16, 20, and 24runs, emphasizing designs of 16 runs as this case involvesboth regular and nonregular designs.


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3.1 Designs of 12 Runs

As is well known, there is exactly one nonisomorphicHadamard matrix of order 12, one version of which is givenby the 12 run Plackett and Burman design. Previously, Lin andDraper (1992) and Wang and Wu (1995) studied the projectiondesigns of this Plackett and Burman design, and found that,for m D 51 6, there are exactly two nonisomorphic designs,and that for any other m, there is exactly one nonisomorphicdesign. Our method reaches the same conclusion. For m D 5,the CFV’s, as de� ned in (6), of the two nonisomorphic designsare 603 0307 and 603 0317. The design with CFV 6030307 isbetter according to both G and G2 aberration. This conclusioncoincides with that of Wang and Wu (1995). For m D 6, theCFV’s of the two nonisomorphic designs are 6030303 17 and6030313 07. Clearly, in either case, the MA-5 classi� er is ableto differentiate the two nonisomorphic designs.


Appendix B contains the � ve (exactly) nonisomorphicHadamard matrices of order 16 (Hall 1961), labeled as H16- I,H16- II, H16- III, H16- IV, and H16-V. In particular, designH16- I is regular, and corresponds to a Plackett–Burmandesign. For 3 µ m µ 14, we perform a complete search of

15m

¢designs for each of the Hadamard designs, H16- I–H16-V;

Table 1 gives the number of nonisomorphic classes identi� edby the MA-4 classi� er, as well as the total number of classesfor each m. The exact number of classes, as found by Sunand Wu (1993) through direct checking of isomorphism, isincluded for comparison.

Clearly, selecting m columns from different Hadamarddesigns may yield the same projection. Therefore, the totalnumber of classes is usually less than the sum of the numbersof classes found in the � ve matrices. It is interesting toobserve that, for m D 31415, H16- III has all of the classes,and in fact, for any m, H16-III has more classes than anyother Hadamard design. When m D 13114, the � ve Hadamarddesigns produce disjoint classes of projections. We alsoobserve that MA-4 is able to differentiate all of the isomorphicprojections for m D 31 41 5113, and 14. In general, the MA-4classi� er performs quite well in differentiating nonisomorphicdesigns. Discrepancy is only moderate even for m D 819110.

For each m with 3 µ m µ 14, the designs found by theMA-4 classi� er are � rst ranked by the criterion of minimumG aberration. A partial list of these designs is given in Table 2.For m µ 5, we list all of the designs identi� ed by MA-4. Form ¶ 6, the top � ve nonregular designs, as ranked by minimumG aberration, are presented, and in addition, all of the regular

Table 1. Numbers of Classes Identi’ ed by MA-4 Classi’ er

Type\m 3 4 5 6 7 8 9 10 11 12 13 14

H16-I 2 3 4 5 6 6 5 4 3 2 1 1H16-II 3 5 10 18 25 29 26 19 13 8 4 2H16-III 3 5 11 25 45 57 53 40 26 13 7 3H16-IV 3 5 10 17 21 21 17 12 8 5 3 2H16-V 3 5 10 19 26 26 21 16 11 6 3 2

Total (MA-4) 3 5 11 26 50 69 74 71 52 31 18 10

Sun and Wu (1993) 3 5 11 27 55 80 87 78 58 36 18 10

designs are included in Table 2. In Table 2, label 160m0j isused to denote the jth best design of m factors according tominimum G aberration. For example, design 16.6.17 denotesthe 17th best design of 6 factors, according to minimum G

aberration, among the 26 designs identi� ed by the MA-4 clas-si� er. The column under “Type” indicates which Hadamarddesign a particular design comes from. The following schemeis used for obtaining “Type.” Since H16- I is the most popular,we � rst check whether or not a given design can be foundfrom H16- I. If not, we next check if it can be found fromH16- III. Otherwise, we proceed to check if it can be foundfrom the remaining Hadamard designs, in the following order:H16- II, H16-V, and H16- IV. Besides H16- I, our order roughlyre� ects the decreasing richness in the number of classes, sothat the user will be able to reconstruct most designs usingtwo (H16- I and H16- III) or three (H16- I, H16- III, and H16- II)Hadamard designs.

For each m, we also rank the designs using minimumG2 aberration based on 4B31B45. The ranks of these designs,along with their 4B31B45 vectors, are included in Table 2. Fora � xed m, the rank of a design D is de� ned to be r C1, wherer is the number of designs that have less G2 aberration thanD. For example, the rank of design 16.6.23 is 15 because thereare 14 designs among all of the 26 designs of 6 factors thatare better than 16.6.23 according to minimum G2 aberration.From Table 2, we observe that the rankings of designsgiven by the two aberration criteria are quite consistent witheach other, and that minimum G-aberration designs all haveminimum G2 aberration. Minimum G2-aberration designs areunique for 3 µ m µ 8, but not so for 9 µ m µ 14. MinimumG-aberration designs are regular for 3 µ m µ 8 and nonregularfor 9 µ m µ 14. On the other hand, one can always � nd aregular minimum G2-aberration design for any m.

To conclude our discussion on designs of 16 runs, weapply the MA-5 classi� er to the � ve Hadamard matricesH16- I–H16-V. These results, along with those given by thefull CFV (Li 2000), are presented in Table 3.

MA-5 gives one more design for m D 9, and produces thesame number of designs for other m values. Classi� cationusing the full CFV gives a few more designs for 7 µ m µ 12.Table 3 shows that, for designs of 16 runs, MA-4 has almostthe same differentiating power as the full CFV.


There are exactly three nonisomorphic Hadamard matri-ces of order 20 (Hall 1965). These are given in Appendix C


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Table 2. Partial Collection of Designs of 16 Runs Identi’ ed by MA-4 Classi’ er, Where 16.x .yDenotes the y th Best Design of x Factors, as Ranked by Minimum G-aberration; See Sections 2

and 4 for a More Detailed Discussion on the Issue of Design Selection

Design F3 : [1618] F4: [1618] G2 (B31B4) Type Columns selected

16.3.1 (010) — 1 (010) I {1 2 3}16.3.2 (011) — 2 (02510) III {7 11 12}16.3.3 (110) — 3 (110) I {1 2 13}

16.4.1 (010) (010) 1 (010) I {7 8 10 13}16.4.2 (010) (110) 2 (011) I {6 7 9 11}16.4.3 (011) (011) 3 (0251 025) III {2 6 11 14}16.4.4 (012) (010) 4 (0510) III {5 6 8 15}16.4.5 (110) (010) 5 (110) I {9 10 13 14}

16.5.1 (010) (010) 1 (010) I {8 9 13 14 15}16.5.2 (010) (110) 2 (011) I {3 4 8 9 12}16.5.3 (011) (012) 3 (0251 05) III {3 6 7 11 15}16.5.4 (012) (012) 4 (051 05) III {3 4 8 9 14}16.5.5 (013) (010) 5 (07510) III {2 3 4 9 13}16.5.6 (014) (010) 6 (110) III {3 4 8 12 13}16.5.7 (014) (012) 8 (11 05) III {3 4 8 12 14}16.5.8 (014) (110) 9 (111) III {7 9 10 13 14}16.5.9 (110) (010) 6 (110) I {3 9 10 11 13}16.5.10 (112) (012) 10 (1051 05) III {3 5 6 10 14}16.5.11 (210) (110) 11 (211) I {4 6 7 8 13}

16.6.1 (010) (310) 1 (013) I {2 3 4 7 9 10}16.6.2 (012) (114) 2 (0512) III {4 5 7 8 10 11}16.6.3 (014) (014) 3 (111) III {1 5 9 10 12 14}16.6.4 (014) (016) 6 (11105) III {1 5 6 8 10 14}16.6.5 (014) (110) 3 (111) III {3 9 10 11 12 13}16.6.6 (014) (114) 7 (112) III {4 5 7 8 12 13}16.6.17 (110) (110) 3 (111) I {1 3 6 8 12 13}16.6.23 (210) (010) 15 (210) I {2 3 4 6 13 14}16.6.24 (210) (110) 17 (211) I {1 3 6 8 14 15}16.6.26 (410) (310) 26 (413) I {1 3 6 9 10 11}

16.7.1 (010) (710) 1 (017) I {2 4 5 12 13 14 15}16.7.2 (014) (318) 2 (115) III {2 3 6 10 11 12 13}16.7.3 (016) (1112) 3 (10514) III {1 3 4 6 9 13 14}16.7.4 (018) (0112) 4 (213) III {1 5 7 9 11 12 15}16.7.5 (018) (0114) 9 (21305) IV {1 2 4 6 9 11 12}16.7.6 (018) (118) 4 (213) III {3 4 6 8 9 14 15}16.7.38 (210) (310) 4 (213) I {6 7 8 10 11 12 13}16.7.46 (310) (210) 25 (312) I {6 7 8 10 11 13 15}16.7.47 (310) (310) 35 (313) I {2 4 6 7 8 11 13}16.7.49 (410) (310) 45 (413) I {2 4 6 7 8 11 12}16.7.50 (710) (710) 50 (717) I {1 6 8 11 12 14 15}

16.8.1 (010) (1410) 1 (0114) I {1 2 5 9 10 11 12 14}16.8.2 (018) (6116) 2 (2110) III {2 3 6 7 8 9 14 15}16.8.3 (0112) (1124) 3 (317) III {1 3 5 7 8 11 13 15}16.8.4 (0112) (2124) 8 (318) III {1 3 5 7 8 10 13 15}16.8.5 (0112) (3116) 3 (317) III {2 3 4 5 10 12 13 14}16.8.6 (0112) (710) 3 (317) III {5 8 9 11 12 13 14 15}16.8.56 (310) (710) 3 (317) I {1 2 3 4 6 9 12 15}16.8.64 (410) (510) 14 (415) I {1 2 3 4 6 9 13 15}16.8.65 (410) (610) 27 (416) I {1 2 3 4 6 9 14 15}16.8.68 (510) (510) 56 (515) I {1 2 5 7 11 12 13 15}16.8.69 (710) (710) 69 (717) I {1 2 3 7 8 10 13 14}

16.9.1 (0116) (1410) 1 (4114) III {2 8 9 10 11 12 13 14 15}16.9.2 (0120) (6124) 3 (5112) III {2 3 4 5 8 10 11 12 13}16.9.3 (0122) (2136) 5 (505111) V {4 5 6 7 8 9 10 14 15}16.9.4 (0124) (2132) 12 (6110) V {4 5 6 7 8 9 11 12 14}16.9.5 (0124) (3124) 7 (619) III {2 4 7 10 11 12 13 14 15}16.9.58 (410) (1410) 1 (4114) I {1 2 3 4 6 7 8 9 12}16.9.70 (610) (910) 7 (619) I {1 2 3 4 5 8 9 10 11}16.9.71 (610) (1010) 12 (6110) I {1 2 3 4 5 6 7 8 11}16.9.72 (710) (910) 55 (719) I {1 2 3 7 9 10 11 12 15}16.9.74 (810) (1010) 72 (8110) I {1 2 3 4 5 7 10 12 15}

(continued)


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Table 2 (continued)

Design F3: [1618] F4: [1618] G2 (B31B4) Type Columns selected

16.10.1 (0132) (10132) 1 (8118) III {2 3 6 7 8 9 10 11 12 13}16.10.2 (0132) (14116) 1 (8118) III {2 4 8 9 10 11 12 13 14 15}16.10.3 (0132) (1810) 1 (8118) III {2 3 8 9 10 11 12 13 14 15}16.10.4 (0134) (4152) 7 (805117) V {2 3 4 5 8 9 10 11 14 15}16.10.5 (0134) (6144) 7 (805117) IV {2 3 4 5 7 8 10 11 12 13}16.10.67 (810) (1810) 1 (8118) I {2 3 5 7 8 9 10 12 14 15}16.10.69 (910) (1610) 13 (9116) I {2 4 5 6 8 9 10 11 13 14}16.10.70 (1010) (1510) 61 (10115) I {2 4 5 6 8 9 10 11 12 13}16.10.71 (1010) (1610) 64 (10116) I {2 4 5 6 8 9 10 11 13 15}

16.11.1 (0148) (8172) 1 (12126) V {4 5 6 7 8 9 10 11 12 13 14}16.11.2 (0148) (10164) 1 (12126) IV {2 4 5 8 9 10 11 12 13 14 15}16.11.3 (0148) (14148) 1 (12126) III {2 3 4 5 8 9 10 12 13 14 15}16.11.4 (0148) (18132) 1 (12126) III {2 3 4 8 9 10 11 12 13 14 15}16.11.5 (0148) (2610) 1 (12126) II {4 5 6 7 8 9 10 11 12 14 15}16.11.50 (1210) (2610) 1 (12126) I {1 2 3 4 5 6 8 12 13 14 15}16.11.51 (1310) (2510) 28 (13125) I {1 2 3 4 5 6 8 11 12 13 14}16.11.52 (1310) (2610) 47 (13126) I {1 2 3 4 5 6 8 10 11 13 14}

16.12.1 (0164) (15196) 1 (16139) V {1 3 4 6 8 9 10 11 12 13 14 15}16.12.2 (0164) (23164) 1 (16139) III {2 3 4 5 8 9 10 11 12 13 14 15}16.12.3 (0164) (3910) 1 (16139) II {4 5 6 7 8 9 10 11 12 13 14 15}16.12.4 (0168) (101112) 10 (17138) IV {2 3 4 5 6 7 8 9 10 11 12 14}16.12.5 (1164) (14196) 10 (17138) V {1 3 4 7 8 9 10 11 12 13 14 15}16.12.30 (1610) (3910) 1 (16139) I {1 2 3 4 6 7 8 9 11 12 13 14}16.12.31 (1710) (3810) 10 (17138) I {1 2 3 4 6 7 8 11 12 13 14 15}

16.13.1 (0188) (151160) 1 (22155) IV {2 4 5 6 7 8 9 10 11 12 13 14 15}16.13.2 (2180) (151160) 1 (22155) V {3 4 5 6 7 8 9 10 11 12 13 14 15}16.13.3 (2180) (231128) 1 (22155) III {2 3 4 5 6 8 9 10 11 12 13 14 15}16.13.4 (4172) (101180) 1 (22155) V {1 2 3 4 5 6 8 9 10 11 12 13 14}16.13.5 (4172) (191144) 1 (22155) III {2 3 4 5 6 7 8 9 10 11 12 13 14}16.13.18 (2210) (5510) 1 (22155) I {2 3 4 6 7 8 9 10 11 12 13 14 15}

16.14.1 (01112) (211224) 1 (28177) IV {2 3 4 5 6 7 8 9 10 11 12 13 14 15}16.14.2 (4196) (171240) 1 (28177) V {1 2 3 4 5 6 8 9 10 11 12 13 14 15}16.14.3 (4196) (291192) 1 (28177) III {2 3 4 5 6 7 8 9 10 11 12 13 14 15}16.14.4 (6188) (151248) 1 (28177) IV {1 2 3 4 5 6 7 8 9 10 11 12 13 14}16.14.5 (7184) (141252) 1 (28177) V {1 2 3 4 5 6 7 8 9 10 11 12 13 14}16.14.10 (2810) (7710) 1 (28177) I {2 3 4 5 6 7 8 9 10 11 12 13 14 15}

under labels H20-Q, H20-P, and H20-N. Wang and Wu (1995)found that H20-Q is equivalent to the Plackett and Burmandesign. For m D 3141 : : : 1 18, the number of nonisomorphicclasses identi� ed by applying the MA-5 classi� er to all threeHadamard designs is 2, 3, 10, 34, 51, 80, 125, 125, 80, 51,34, 10, 3, 2, 1, and 1, respectively. We now rank our designsusing both aberration criteria. It turns out that the two criteriagive identical rankings. (We have carried out some theoreti-cal studies that con� rm the above empirical observation. Thedetails are omitted here.) We present in Table 4 the top threedesigns of 20 runs for 3 µ m µ 16. For m D 6, a few moredesigns are given, as they will be used in Section 4.

Table 3. Total Numbers of Classes Identi’ ed by MA-4, MA-5,and the Full CFV

Type\m 3 4 5 6 7 8 9 10 11 12 13 14

MA-4 3 5 11 26 50 69 74 71 52 31 18 10MA-5 3 5 11 26 50 69 75 71 52 31 18 10Full CFV 3 5 11 26 53 74 78 75 56 32 18 10Sun and Wu (1993) 3 5 11 27 55 80 87 78 58 36 18 10


According to Kimura (1989), there are exactly 60 noniso-morphic Hadamard matrices of order 24. We have performeda complete search for m µ 8 by applying MA-5 to all of the60 Hadamard matrices. The number of classes identi� ed byMA-5 for m D 314151617, and 8 are 4, 10, 49, 408, 3,805,and 29,196, respectively. We present in Table 5 the top threedesigns. These top designs can all be found from the 11thHadamard matrix of order 24, which is available from N.Sloane’s home page http://www.research.att.com/njas, andgiven in Appendix C of this article. Unlike the case n D 20,minimum G aberration is not equivalent to minimum G2

aberration for n D 24. However, we have veri� ed that the twocriteria both rank the three designs in Table 5 as the best, thesecond best, and the third best.

Finally, we refer to Deng, Li, and Tang (2000) for amore comprehensive listing of the designs of 16, 20, and24 runs. A complete listing is available from the authors uponrequest.


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Table 4. Partial Collection of Designs of 20 Runs Identi’ ed MA-5 Classi’ er, Where 20.x.y Denotesthe y th Best Design of x Factors, as Ranked by Minimum G Aberration; See Sections 2 and 4 for

a More Detailed Discussion on the Issue of Design Selection; In the Table, {¢ ¢ ¢}c Denotesthe Complementary Set, Consisting of Columns That Are Not Selected

Design F3: [20112] F4: [20112] F5: [1618] Type Columns selected

20.3.1 (010) — — N { 1 8 14}20.3.2 (011) — — N { 1 9 18}

20.4.1 (010) (010) — N { 3 13 15 19}20.4.2 (010) (011) — N { 2 7 15 17}20.4.3 (011) (010) — N { 1 7 14 19}

20.5.1 (010) (010) (010) N { 3 4 8 18 19}20.5.2 (010) (010) (011) N { 3 4 8 17 18}20.5.3 (010) (011) (010) N { 2 3 6 8 9}

20.6.1 (010) (011) (012) N { 4 8 11 13 17 19}20.6.2 (010) (011) (013) N { 2 5 7 8 13 14}20.6.3 (010) (012) (011) N { 2 5 7 8 13 18}20.6.4 (010) (012) (013) N { 2 5 7 9 15 19}20.6.5 (010) (013) (010) N { 1 3 6 7 8 9}

20.6.30 (013) (011) (011) N { 1 3 6 10 12 17}20.6.31 (013) (011) (012) N { 1 3 5 11 16 19}20.6.32 (013) (012) (011) N { 1 3 6 10 13 14}20.6.33 (014) (013) (012) N { 5 6 7 8 9 15}20.6.34 (014) (013) (013) N { 5 6 7 8 10 15}

20.7.1 (010) (013) (017) N { 3 6 7 10 12 13 19}20.7.2 (010) (013) (019) N { 3 6 7 10 12 15 19}20.7.3 (010) (014) (015) N { 1 2 9 10 11 13 15}

20.8.1 (010) (016) (0124) N { 1 2 6 7 16 17 18 19}20.8.2 (010) (018) (0116) N { 5 6 8 13 14 15 17 19}20.8.3 (010) (0110) (0114) P { 1 3 4 8 9 13 14 18}

20.9.1 (010) (0118) (0134) P { 1 2 3 4 9 13 14 18 19}20.9.2 (010) (0118) (0154) P { 1 2 3 8 9 13 14 18 19}20.9.3 (011) (0114) (0134) P { 2 4 5 6 7 13 14 16 17}

20.10.1 (010) (0130) (0172) P { 4 5 6 7 10 11 12 13 16 17}20.10.2 (012) (0122) (0172) P { 1 2 3 4 8 11 12 14 16 18}20.10.3 (013) (0118) (0172) Q { 2 3 8 9 12 13 14 15 16 19}

20.11.1 (015) (0130) (01142) P { 1 2 3 4 8 9 13 14 15 18 19}20.11.2 (016) (0126) (01140) N { 1 3 4 6 7 8 9 16 17 18 19}20.11.3 (017) (0121) (01150) N { 1 2 4 6 10 11 12 13 15 17 18}

20.12.1 (018) (0139) (01240) N { 5 6 7 8 9 10 15}c

20.12.2 (0110) (0133) (01244) N { 2 3 6 9 15 16 17}c

20.12.3 (0110) (0135) (01242) P { 6 7 11 12 15 16 17}c

20.13.1 (0114) (0147) (01390) N { 1 3 4 5 10 15}c

20.13.2 (0114) (0147) (01391) N { 5 6 7 8 9 15}c

20.13.3 (0115) (0143) (01393) N { 1 8 11 16 18 19}c

20.14.1 (0120) (0160) (01601) N { 1 8 11 14 16}c

20.14.2 (0120) (0161) (01598) N { 15 16 17 18 19}c

20.14.3 (0120) (0161) (01599) N { 10 12 13 14 15}c

20.15.1 (0126) (0181) (01891) N { 6 9 15 16}c

20.15.2 (0127) (0180) (01881) N { 10 12 16 19}c

20.15.3 (0127) (0181) (01879) N { 5 15 17 19}c

20.16.1 (0132) (01108) (011296) N { 6 12 16}c

20.16.2 (0133) (01107) (011284) N { 6 12 15}c

4. GENERALIZED MINIMUM ABERRATION ANDOTHER CRITERIA

Suppose that we can safely assume negligible interactionsinvolving three or more factors, but that we suspect that sometwo-factor interactions (2� ’s) may be important. The goal hereis to estimate the main effects, and to obtain as much informa-tion as possible about nonnegligible 2� ’s. In this case, a gooddesign should be able to satisfactorily answer one or more ofthe following questions.

1. How does the presence of nonnegligible 2� ’s affect theestimation of main effects?

2. Does our design allow the estimation of nonnegligible2� ’s?

3. If our design allows estimation of these 2� ’s, does ithave high ef� ciency?

Question (1) has been discussed in Section 2.1. To address(2) and (3), we use the criteria of estimation capacity (Cheng,Steinberg, and Sun 1999) and average D ef� ciency. Consider


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Table 5. Partial Collection of Designs of 24 Runs Identi’ ed by MA-5 Classi’ er, Where 24.x .y Denotesthe y th Best Design of x Factors, as Ranked by Minimum G Aberration; See Sections 2 and 4 for

a More Detailed Discussion on the Issue of Design Selection

Design F3 : [2411618] F4 : [2411618] F5 : [2411618] Columns selected

24.3.1 (01010) — — {1 2 4}

24.3.2 (01011) — — {1 12 13}24.3.3 (01110) — — {1 12 23}24.3.4 (11010) — — {1 2 3}

24.4.1 (01010) (01011) — {1 2 4 7}24.4.2 (01010) (11010) — {2 3 4 5}24.4.3 (01011) (01010) — {1 2 4 6}

24.5.1 (01010) (01015) (01010) {1 2 4 7 13}24.5.2 (01011) (01013) (01011) {1 2 4 6 9}24.5.3 (01012) (01011) (01010) {1 2 4 12 20}

24.6.1 (01010) (010115) (01010) {1 3 5 7 9 11}24.6.2 (01012) (01019) (01014) {1 3 5 9 13 14}24.6.3 (01012) (010111) (01012) {1 2 4 7 8 13}

24.7.1 (01010) (010135) (01010) {1 3 5 9 13 14 20}24.7.2 (01014) (010121) (010112) {1 2 4 7 8 16 20}24.7.3 (01014) (010123) (01018) {1 3 4 8 14 17 19}

24.8.1 (01010) (010170) (01010) {1 3 5 9 13 14 20 23}24.8.2 (01016) (010152) (010118) {1 3 5 7 9 13 20 23}24.8.3 (01017) (010153) (010114) {1 3 4 7 11 14 17 19}

the set of models containing all of the main effects and f

2� ’s for f D 11 2131 : : : . Let Ef be the number of estimablemodels, and let Df be the average of D ef� ciencies. In the fol-lowing, we use three examples to illustrate how the criteria ofgeneralized aberration are related to the criteria of minimizingbias, estimation capacity Ef , and average ef� ciency Df .

4.1 Designs 16.9.1 and 16.9.58

Consider designs 16.9.1 and 16.9.58 in Table 2. Our curios-ity about them is natural. Traditional wisdom points to the useof 16.9.58 because it is a regular minimum aberration design.The criterion of G2 aberration ranks both designs as the best,while minimum G aberration ranks 16.9.1 as the best and16.9.58 as the 58th best among the 74 designs of 9 factors. Letus take a closer look at the two designs. For 16.9.58, Equation(5) becomes

E4‚15 D ‚1 C ‚28 C ‚36 C ‚45 C ‚79

E4‚25 D ‚2C ‚181 E4‚35 D ‚3

C ‚161

E4‚45 D ‚4 C ‚151 E4‚55 D ‚5 C ‚14

E4‚65 D ‚6C ‚131 E4‚75 D ‚7

C ‚191

E4‚85 D ‚8C ‚121 E4‚95 D ‚9

C ‚17

while for design 16.9.1, Equation (5) becomes

E4‚15 D ‚1C 0054‚24

C ‚25C ‚26

C ‚34C ‚35

C ‚37C ‚48

C‚59 C ‚68 C ‚69 C ‚78 C ‚79 ƒ ‚27

ƒ‚36ƒ ‚49

ƒ ‚585

E4‚25 D ‚2C 0054‚14

C ‚15C ‚16

ƒ ‚1751

E4‚35 D ‚3C 0054‚14

C ‚15C ‚17

ƒ ‚165

E4‚45 D ‚4 C 0054‚12 C ‚13 C ‚18 ƒ ‚1951

E4‚55 D ‚5 C 0054‚12 C ‚13 C ‚19 ƒ ‚185

E4‚65 D ‚6C 0054‚12

C ‚18C ‚19

ƒ ‚1351

E4‚75 D ‚7 C 0054‚13 C ‚18 C ‚19 ƒ ‚125

E4‚85 D ‚8 C 0054‚14 C ‚16 C ‚17 ƒ ‚1551

E4‚95 D ‚9C 0054‚15

C ‚16C ‚17

ƒ ‚145

where, for convenience, the most confounded factor is des-ignated as factor 1 in both designs. In both cases, 2� ’s biasthe estimates of the main effects, but do so in different ways.Without detailed knowledge about nonnegligible 2� ’s, we haveto work on the coef� cients of the 2� terms in the aboveequations. A conservative approach in minimizing the coef-� cients, as is done by minimum G aberration, is to mini-mize their maximum value, which equals 05 for 160901 and 1for 1609058. This explains why minimum G aberration favors160901 over 1609058. A more aggressive approach is to mini-mize the sum of the squared coef� cients, which is 3B3 D 12for both designs. Therefore, 160901 and 1609058 are the samein terms of minimum G2 aberration. A simple, and somewhataggressive, method in minimizing the coef� cients is to mini-mize the number of nonzero coef� cients, which represents thenumber of 2� ’s that bias the main effects. Since the numberof nonzero coef� cients is 12 for 1609058 and 48 for 160901,this method favors 1609058 over 160901. Many other possiblemethods exist for minimizing the bias terms, and for example,one may be interested in using a Ge-aberration criterion asde� ned in Tang and Deng (1999). Which of the two designs


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Table 6. Designs of 16 Runs for 6 Factors: An Evaluation of K D 15f

¢Models With f 2’ ’s

f D 1 (K D 15) f D 2 (K D 105) f D 3 (K D 455) f D 4 (K D 1365)

Design Nf Df Nf Df Nf Df Nf Df

16.6.1 0 100000 9 09143 115 07473 645 0527516.6.2 0 09000 3 07500 43 05706 273 0389016.6.3 0 08000 0 06048 10 04275 115 0278416.6.4 0 08000 0 05940 3 04041 43 0247416.6.5 1 08000 17 06000 127 04176 561 0265916.6.6 0 08000 5 05810 69 03758 424 0211016.6.17 3 08000 42 06000 265 04176 1002 0265916.6.23 6 06000 69 03429 371 01846 1239 0092316.6.24 6 06000 71 03238 385 01538 1280 0062316.6.26 12 02000 105 00000 455 00000 1365 00000

is better depends on the problem at hand, and perhaps moreheavily on the experimenter’s personal preference.

The above discussion centers on the contamination of 2� ’son the estimation of the main effects. The following consider-ation gives design 160901 the edge on design 1605058. Supposethat, after the signi� cant main effects are identi� ed, we areinterested in estimating some 2� ’s among the factors hav-ing signi� cant main effects. Design 160901 makes it possible.Note that design 160901 has a generalized resolution of 305,implying that its projection onto any three factors contains acopy of a complete 23 factorial. Therefore, when projectedonto any three factors, design 160901 allows the estimationof all of the main effects and 2� ’s, a property not shared bydesign 1606058.

4.2 Designs of 16 Runs for 6 Factors

Consider the 10 designs of 16 runs for 6 factors givenin Table 2. Here, we evaluate the performance of minimumG aberration using the criteria of design ef� ciency Df andestimation capacity Ef , given earlier in this section. Theresults are given in Table 6.

Note that 6 factors produce 15 2� ’s. For convenience,instead of Ef , we present Nf

D K ƒ Ef , the number ofnonestimable models, where K is the total number of modelswith f two-factor interactions. From Table 6, we observe thatthe ranking of these designs given by the Df criterion is veryconsistent with that given by minimum G aberration. The Ef

ranking is also quite consistent with the G-aberration rankingin that designs ranked higher by G aberration generally have

Table 7. Designs of 20 Runs for 6 Factors: An Evaluation of K D 15f

¢Models With f 2’ ’s

Design D1 D2 D3 D4 D5 D6 D7 N4 N5 N6 N7

20.6.1 08400 06771 05215 03818 02642 01714 01031 0 0 2 1820.6.2 08400 06778 05231 03840 02665 01735 01048 0 0 1 920.6.3 08400 06700 05045 03560 02334 01405 00764 0 0 1 920.6.4 08400 06711 05066 03584 02351 01411 00759 0 0 1 920.6.5 08400 06608 04826 03238 01971 01074 00514 0 0 4 3620.6.30 06480 03950 02243 01173 00556 00233 00083 4 44 220 66420.6.31 06480 03936 02218 01143 00529 00213 00072 4 44 220 66420.6.32 06480 03883 02127 01051 00460 00174 00055 5 55 275 81720.6.33 05840 03033 01366 00514 00152 00031 00003 30 330 11630 4141020.6.34 05840 03065 01409 00543 00166 00035 00004 27 297 11485 41131

better estimation capacity. Design 160604 is the best accordingto the Ef criterion.

Design 160601 is of resolution IV, and therefore two-factorinteractions do not bias the main effects. If we are interestedin estimating only the main effects, design 160601 is the best.Even if there is one signi� cant 2� , design 160601 can estimateall of the models containing the main effects and one 2� . How-ever, when there is more than one signi� cant 2� , design 160604performs better than design 160601 in terms of Ef . Design160604 allows all of the models containing the main effectsand two 2� ’s to be estimated. Only 3 out of 455 models withf D 3 2� ’s are not estimable, and the percentage of estimablemodels is 452=455 D 99%. Even for f D 4, the percentage ofestimable models is still very high, at 1322=1365 D 97%.

4.3 Designs of 20 Runs for 6 Factors

In Section 3, the MA-5 classi� er identi� es 34 designs of20 runs for 6 factors. The best � ve and worst � ve designs,in terms of G aberration, are examined here. The Df and Ef

values for f D 1121 : : : 17 are calculated and given in Table 7.As before, here we give Nf

D K ƒEf , the number of nones-timable models, where K D 15

f

¢is the total number of models

with f 2� ’s. For f D 11 213, NfD 0 for all of the 10 designs.

Again, we see that minimum G aberration does select designsthat are also good in terms of Df and Ef criteria. Each of thetop � ve designs allows estimation of all of the models withf D 5 2� ’s. For f D 617, the percentage of estimable modelsis at least 99%. If we compare these designs with those of16 runs, we see that, with just 4 additional runs, much morecan be achieved in terms of the Ef criterion.


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ACKNOWLEDGMENTS

The authors thank two referees, an associate editor, andthe editor for their constructive comments on this article. Theresearch of Boxin Tang is partially supported by NSF grantDMS-9971212.

APPENDIX A: PROOF OF PROPOSITION 1

The (Hadamard) product of two vectors u D 4u11 u21 : : : 1 un50

and v D 4v11 v21 : : : 1 vn50 is de� ned as uv D 4u1v11 u2v21 : : : 1

unvn50. For a set of vectors s D 8v11 v21 : : : 1vp9, de� ne4s5 D v1v2 ¢ ¢ ¢ vp to be their product. Our proof is done byinduction. Since Jk4s5 D —jk4s5—, it is suf� cient to proveProposition 1 for jk4s5. Since D is orthogonal, we � rst havej14s5 D j24s5 D 0. Now, assume that Proposition 1 is true forany set s, with —s— µ k ƒ 1. Consider an s D with —s— D k.Let u1 v be any two columns in s. Let sƒu, sƒv be the subsetsof s with u, v removed, respectively. Similarly, let sƒuƒv bethe subset of s with both u and v removed. Consider the n� 2matrix given by u and v. Since D is orthogonal, each of thefour pairs 6C1C71 6C1 ƒ71 6ƒ1C7, and 6ƒ1 ƒ7 occurs exactlyt times, where n D 4t. Consider the four groups of rows thathave the four pairs above. Let x11 x21 x31 x4 be the numbersof positive entries in 4s5 corresponding to the four groupsof rows that have the four pairs 6C1C71 6C1 ƒ71 6ƒ1C7, and6ƒ1ƒ7, respectively. Then the numbers of negative entries in4s5 for the four groups of rows are t ƒ x11 t ƒ x21 t ƒ x31 andt ƒx4, respectively. Now, the n rows of s are divided into eightsubgroups according to the signs of u1 v and 4s5. Relevant j

values, and their counterparts within each subgroup, are givenin the following table.

From the above table, it is straightforward to see thatjk4s5 D 24x1 C x2 C x3 C x45 ƒ 4t, jkƒ14sƒu5 D 24x1 C x2 ƒx3

ƒ x45, jkƒ14sƒv5 D 24x1ƒ x2

C x3ƒ x45, and jkƒ24sƒuƒv5 D

24x1ƒ x2

ƒ x3C x45. Summing up the above equations, we

obtain

jk4s5 D 8x1ƒ 4t ƒ jkƒ14sƒu5 ƒ jkƒ14sƒv5 ƒ jkƒ24sƒuƒv50 (A.1)

Proposition 1 follows by combining (A.1) with the fact thatj1 D j2 D 0. &

(u) (v) (s) (c) D Count (s)(c) (s)(u)(c) (s)(v)(c) (s)(u)(v)(c)

1 C C C x1 x1 x1 x1 x1

2 C C ƒ t ƒx1 x1 ƒ t x1 ƒ t x1 ƒ t x1 ƒ t

3 C ƒ C x2 x2 x2 ƒx2 ƒx2

4 C ƒ ƒ t ƒx2 x2 ƒ t x2 ƒ t t ƒ x2 t ƒ x2

5 ƒ C C x3 x3 ƒx3 x3 ƒx3

6 ƒ C ƒ t ƒx3 x3 ƒ t t ƒx3 x3 ƒ t t ƒ x3

7 ƒ ƒ C x4 x4 ƒx4 ƒx4 x4

8 ƒ ƒ ƒ t ƒx4 x4 ƒ t t ƒx4 t ƒ x4 x4 ƒ t

Total 4t j k (s) jkƒ1(sƒu ) j kƒ1(sƒv ) jkƒ2(sƒuƒv )

APPENDIX B: DESIGNS H16-I, H16-II,H16-III, H16-IV, AND H16-V

Design H16- I is a regular design which is also a Plackett–Burman design. It can be generated by successive cyclicpermuting of the row 4C1 C1C1C1 ƒ1C1ƒ1 C1 C1ƒ1ƒ1 C,ƒ1ƒ1 ƒ5 and adding a vector of minus signs as thelast row.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15C C C C C C C C C C C C C C C CC C C C C C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ C C C C ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ C C C CC C ƒ ƒ C C ƒ ƒ C C ƒ ƒ C C ƒ ƒC C ƒ ƒ C C ƒ ƒ ƒ ƒ C C ƒ ƒ C CC C ƒ ƒ ƒ ƒ C C C C ƒ ƒ ƒ ƒ C CC C ƒ ƒ ƒ ƒ C C ƒ ƒ C C C C ƒ ƒC ƒ C ƒ C ƒ C ƒ C ƒ C ƒ C ƒ C ƒC ƒ C ƒ C ƒ C ƒ ƒ C ƒ C ƒ C ƒ CC ƒ C ƒ ƒ C ƒ C C ƒ C ƒ ƒ C ƒ CC ƒ C ƒ ƒ C ƒ C ƒ C ƒ C C ƒ C ƒC ƒ ƒ C C ƒ ƒ C C ƒ ƒ C ƒ C C ƒC ƒ ƒ C C ƒ ƒ C ƒ C C ƒ C ƒ ƒ CC ƒ ƒ C ƒ C C ƒ C ƒ ƒ C C ƒ ƒ CC ƒ ƒ C ƒ C C ƒ ƒ C C ƒ ƒ C C ƒ

(H16-II)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15C C C C C C C C C C C C C C C CC C C C C C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ C C C C ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ C C C CC C ƒ ƒ C C ƒ ƒ C C ƒ ƒ C C ƒ ƒC C ƒ ƒ C C ƒ ƒ ƒ ƒ C C ƒ ƒ C CC C ƒ ƒ ƒ ƒ C C C C ƒ ƒ ƒ ƒ C CC C ƒ ƒ ƒ ƒ C C ƒ ƒ C C C C ƒ ƒC ƒ C ƒ C ƒ C ƒ C ƒ C ƒ C ƒ C ƒC ƒ C ƒ C ƒ C ƒ ƒ C ƒ C ƒ C ƒ CC ƒ C ƒ ƒ C ƒ C C ƒ ƒ C C ƒ ƒ CC ƒ C ƒ ƒ C ƒ C ƒ C C ƒ ƒ C C ƒC ƒ ƒ C C ƒ ƒ C C ƒ ƒ C ƒ C C ƒC ƒ ƒ C C ƒ ƒ C ƒ C C ƒ C ƒ ƒ CC ƒ ƒ C ƒ C C ƒ C ƒ C ƒ ƒ C ƒ CC ƒ ƒ C ƒ C C ƒ ƒ C ƒ C C ƒ C ƒ

(H16-III)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15C C C C C C C C C C C C C C C CC C C C C C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ C C C C ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ C C C CC C ƒ ƒ C C ƒ ƒ C C ƒ ƒ C C ƒ ƒC C ƒ ƒ C C ƒ ƒ ƒ ƒ C C ƒ ƒ C CC C ƒ ƒ ƒ ƒ C C C C ƒ ƒ ƒ ƒ C CC C ƒ ƒ ƒ ƒ C C ƒ ƒ C C C C ƒ ƒC ƒ C ƒ C ƒ C ƒ C ƒ C ƒ C ƒ C ƒC ƒ C ƒ C ƒ ƒ C C ƒ ƒ C ƒ C ƒ CC ƒ C ƒ ƒ C C ƒ ƒ C ƒ C C ƒ ƒ CC ƒ C ƒ ƒ C ƒ C ƒ C C ƒ ƒ C C ƒC ƒ ƒ C C ƒ C ƒ ƒ C ƒ C ƒ C C ƒC ƒ ƒ C C ƒ ƒ C ƒ C C ƒ C ƒ ƒ CC ƒ ƒ C ƒ C C ƒ C ƒ C ƒ ƒ C ƒ CC ƒ ƒ C ƒ C ƒ C C ƒ ƒ C C ƒ C ƒ

(H16-IV)


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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15C C C C C C C C C C C C C C C CC C C C C C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ C C C C ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ C C C CC C ƒ ƒ C C ƒ ƒ C C ƒ ƒ C C ƒ ƒC C ƒ ƒ C C ƒ ƒ ƒ ƒ C C ƒ ƒ C CC C ƒ ƒ ƒ ƒ C C C ƒ C ƒ C ƒ C ƒC C ƒ ƒ ƒ ƒ C C ƒ C ƒ C ƒ C ƒ CC ƒ C ƒ C ƒ C ƒ C C ƒ ƒ ƒ ƒ C CC ƒ C ƒ C ƒ C ƒ ƒ ƒ C C C C ƒ ƒC ƒ C ƒ ƒ C ƒ C C ƒ ƒ C ƒ C C ƒC ƒ C ƒ ƒ C ƒ C ƒ C C ƒ C ƒ ƒ CC ƒ ƒ C C ƒ ƒ C C ƒ C ƒ ƒ C ƒ CC ƒ ƒ C C ƒ ƒ C ƒ C ƒ C C ƒ C ƒC ƒ ƒ C ƒ C C ƒ C ƒ ƒ C C ƒ ƒ CC ƒ ƒ C ƒ C C ƒ ƒ C C ƒ ƒ C C ƒ

(H16-V)

APPENDIX C: DESIGNS H20-N, H20-P, AND H20-Q,AND THE 11TH HADAMARD MATRIX OF ORDER 24

C C C C C C C C C C C C C C C C C C CC C C C C C C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ ƒ C C C C C ƒ ƒ ƒ ƒ ƒƒ ƒ ƒ ƒ ƒ C C C C ƒ C C C C C ƒ ƒ ƒ ƒC ƒ ƒ ƒ C C C ƒ ƒ C C C ƒ ƒ ƒ C C ƒ ƒƒ C ƒ ƒ C C ƒ C ƒ C C ƒ C ƒ ƒ ƒ ƒ C Cƒ ƒ C ƒ C ƒ ƒ C C C ƒ C ƒ C ƒ C ƒ C ƒƒ ƒ ƒ C C ƒ C ƒ C C ƒ ƒ C C ƒ ƒ C ƒ Cƒ ƒ C C ƒ ƒ C C ƒ ƒ C C ƒ ƒ ƒ ƒ C C CC C ƒ ƒ ƒ ƒ C ƒ C ƒ ƒ C C ƒ ƒ C ƒ C Cƒ C C ƒ ƒ C ƒ ƒ C ƒ C ƒ ƒ C ƒ C C ƒ CC ƒ ƒ C ƒ C ƒ C ƒ ƒ ƒ ƒ C C ƒ C C C ƒC ƒ C ƒ ƒ C C ƒ ƒ C ƒ ƒ ƒ C C ƒ ƒ C Cƒ C C ƒ ƒ ƒ C C ƒ C ƒ ƒ C ƒ C C C ƒ ƒƒ C ƒ C ƒ C ƒ ƒ C C ƒ C ƒ ƒ C ƒ C C ƒC ƒ ƒ C ƒ ƒ ƒ C C C C ƒ ƒ ƒ C C ƒ ƒ CC ƒ C ƒ C ƒ ƒ ƒ C ƒ C ƒ C ƒ C ƒ C C ƒC C ƒ ƒ C ƒ ƒ C ƒ ƒ ƒ C ƒ C C ƒ C ƒ Cƒ C ƒ C C ƒ C ƒ ƒ ƒ C ƒ ƒ C C C ƒ C ƒƒ ƒ C C C C ƒ ƒ ƒ ƒ ƒ C C ƒ C C ƒ ƒ C

(H20-Q)

C C C C C C C C C C C C C C C C C C CC C C C C C C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ ƒ C C C C C ƒ ƒ ƒ ƒ ƒƒ ƒ ƒ ƒ ƒ C C C C ƒ C C C C C ƒ ƒ ƒ ƒC ƒ ƒ ƒ C C C ƒ ƒ C C C ƒ ƒ ƒ C C ƒ ƒƒ C ƒ ƒ C C C ƒ ƒ C ƒ ƒ C C ƒ ƒ ƒ C Cƒ ƒ C ƒ C ƒ ƒ C C C C C ƒ ƒ ƒ ƒ ƒ C Cƒ ƒ ƒ C C ƒ ƒ C C C ƒ ƒ C C ƒ C C ƒ ƒC ƒ ƒ C ƒ ƒ C ƒ C ƒ ƒ C ƒ C ƒ ƒ C C CC ƒ ƒ C ƒ C ƒ C ƒ ƒ C ƒ C ƒ ƒ C ƒ C Cƒ C C ƒ ƒ ƒ C C ƒ ƒ ƒ C C ƒ ƒ C C ƒ Cƒ C C ƒ ƒ C ƒ ƒ C ƒ C ƒ ƒ C ƒ C C C ƒC ƒ C ƒ ƒ ƒ C C ƒ C ƒ ƒ ƒ C C C ƒ C ƒC ƒ C ƒ ƒ C ƒ ƒ C C ƒ ƒ C ƒ C ƒ C ƒ Cƒ C ƒ C ƒ C ƒ C ƒ C ƒ C ƒ ƒ C ƒ C C ƒƒ C ƒ C ƒ ƒ C ƒ C C C ƒ ƒ ƒ C C ƒ ƒ CC C ƒ ƒ C ƒ ƒ ƒ C ƒ ƒ C C ƒ C C ƒ C ƒC C ƒ ƒ C ƒ ƒ C ƒ ƒ C ƒ ƒ C C ƒ C ƒ Cƒ ƒ C C C ƒ C ƒ ƒ ƒ C ƒ C ƒ C ƒ C C ƒƒ ƒ C C C C ƒ ƒ ƒ ƒ ƒ C ƒ C C C ƒ ƒ C

(H20-P)

C C C C C C C C C C C C C C C C C C CC C C C C C C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒC C C C ƒ ƒ ƒ ƒ ƒ C C C C C ƒ ƒ ƒ ƒ ƒƒ ƒ ƒ ƒ ƒ C C C C ƒ C C C C C ƒ ƒ ƒ ƒC ƒ ƒ ƒ C C C ƒ ƒ C C C ƒ ƒ ƒ C C ƒ ƒƒ C ƒ ƒ C C C ƒ ƒ C ƒ ƒ C C ƒ ƒ ƒ C Cƒ ƒ C ƒ C ƒ ƒ C C C C C ƒ ƒ ƒ ƒ ƒ C Cƒ ƒ ƒ C C ƒ ƒ C C C ƒ ƒ C C ƒ C C ƒ ƒC ƒ ƒ C ƒ ƒ C ƒ C ƒ ƒ C ƒ C ƒ ƒ C C CC ƒ ƒ C ƒ C ƒ C ƒ ƒ C ƒ C ƒ ƒ C ƒ C Cƒ C C ƒ ƒ ƒ C ƒ C ƒ C ƒ C ƒ ƒ C C ƒ Cƒ C C ƒ ƒ C ƒ C ƒ ƒ ƒ C ƒ C ƒ C C C ƒC ƒ C ƒ ƒ ƒ C C ƒ C ƒ ƒ ƒ C C C ƒ ƒ CC ƒ C ƒ ƒ C ƒ ƒ C C ƒ ƒ C ƒ C ƒ C C ƒƒ C ƒ C ƒ C ƒ ƒ C C ƒ C ƒ ƒ C C ƒ ƒ Cƒ C ƒ C ƒ ƒ C C ƒ C C ƒ ƒ ƒ C ƒ C C ƒC C ƒ ƒ C ƒ ƒ ƒ C ƒ C ƒ ƒ C C C ƒ C ƒC C ƒ ƒ C ƒ ƒ C ƒ ƒ ƒ C C ƒ C ƒ C ƒ Cƒ ƒ C C C ƒ C ƒ ƒ ƒ ƒ C C ƒ C C ƒ C ƒƒ ƒ C C C C ƒ ƒ ƒ ƒ C ƒ ƒ C C ƒ C ƒ C

(H16-N)

C C C C C C C C C C C C C C C C C C C C C C CC C C C C ƒ ƒ ƒ ƒ ƒ ƒ C C C C C C ƒ ƒ ƒ ƒ ƒ ƒC C C C C C C C C C C ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒC C C ƒ ƒ C C ƒ ƒ ƒ ƒ C C ƒ ƒ ƒ ƒ C C C C ƒ ƒC C C ƒ ƒ ƒ ƒ C C ƒ ƒ ƒ ƒ C C ƒ ƒ C C ƒ ƒ C CC C C ƒ ƒ ƒ ƒ ƒ ƒ C C ƒ ƒ ƒ ƒ C C ƒ ƒ C C C CC ƒ ƒ C C C C ƒ ƒ ƒ ƒ ƒ ƒ C ƒ C ƒ C ƒ C ƒ C CC ƒ ƒ C C ƒ ƒ C C ƒ ƒ C ƒ ƒ ƒ ƒ C ƒ C C C C ƒC ƒ ƒ C C ƒ ƒ ƒ ƒ C C ƒ C ƒ C ƒ ƒ C C ƒ C ƒ CC ƒ ƒ ƒ ƒ C C C C ƒ ƒ ƒ C ƒ C C C ƒ ƒ ƒ C ƒ CC ƒ ƒ ƒ ƒ C C ƒ ƒ C C C ƒ C C ƒ C ƒ C ƒ ƒ C ƒC ƒ ƒ ƒ ƒ ƒ ƒ C C C C C C C ƒ C ƒ C ƒ C ƒ ƒ ƒƒ C ƒ C ƒ C ƒ C ƒ C ƒ C ƒ ƒ C C ƒ C ƒ ƒ C C ƒƒ C ƒ C ƒ C ƒ C ƒ C ƒ ƒ C C ƒ ƒ C ƒ C C ƒ ƒ Cƒ C ƒ C ƒ ƒ C ƒ C ƒ C C ƒ ƒ C ƒ C C ƒ C ƒ ƒ Cƒ C ƒ ƒ C C ƒ ƒ C ƒ C ƒ C C ƒ ƒ C C ƒ ƒ C C ƒƒ C ƒ ƒ C ƒ C C ƒ ƒ C ƒ C ƒ C C ƒ ƒ C C ƒ C ƒƒ C ƒ ƒ C ƒ C ƒ C C ƒ C ƒ C ƒ C ƒ ƒ C ƒ C ƒ Cƒ ƒ C C ƒ C ƒ ƒ C ƒ C ƒ ƒ C C C ƒ ƒ C C C ƒ ƒƒ ƒ C C ƒ ƒ C C ƒ ƒ C C C C ƒ ƒ ƒ ƒ ƒ ƒ C C Cƒ ƒ C C ƒ ƒ C ƒ C C ƒ ƒ C ƒ ƒ C C C C ƒ ƒ C ƒƒ ƒ C ƒ C ƒ C C ƒ C ƒ ƒ ƒ C C ƒ C C ƒ C C ƒ ƒƒ ƒ C ƒ C C ƒ ƒ C C ƒ C C ƒ C ƒ ƒ ƒ ƒ C ƒ C Cƒ ƒ C ƒ C C ƒ C ƒ ƒ C C ƒ ƒ ƒ C C C C ƒ ƒ ƒ C

(H24-11)

[Received January 1998. Revised April 2001.]

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design selection and classification for hadamard matrices using generalized minimum aberration...

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