StaisticalPapers48,235-248(2007) Statistical Papers © Springer-Verlag 2007

2 m41 designs with minimum aberration or weak minimum aberration

Peng-Fe i Li ~ 2, M i n -Qi an Liu 1, R u n - C h u Zhang I

Department of Statistics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China (corresponding author, e-mail: mqliu@nankai.edu.cn)

2 Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Received: July 1, 2004; revised version: May 9, 2005

For measuring the goodness of 2m41 designs, Wu and Zhang (1993) proposed the minimum aberration (MA) criterion. MA 2"~41 designs have been constructed using the idea of complementary designs when the number of two-level factors, m, exceeds y,n where n is the total number of runs. In

5n this paper, the structures of MA 2m41 designs are obtained when m > 5-~' Based on these structures, some methods are developed for constructing MA 2"~41 designs for 5n n n < m < n. When m < 5n T~ < m < y as well as for y _ - 1W, there is no general method for constructing MA 2m41 designs. In this case, we obtain lower bounds for A3o and A31, where A30 and A31 are the numbers of type 0 and type 1 words with length three respectively. And a method for constructing weak minimum aberration (WMA) 2m41 designs (A3o and A31 achieving the lower bounds) is demonstrated. Some MA or WMA 2"~41 designs with 32 or 64 runs are tabulated for practical use, which supplement the tables 'in Wu and Zhang (1993), Zhang and Shao (2001) and Mukerjee and Wu (2001).

K e y w o r d s Minimum aberration; Resolution; Weak minimum aberra- tion; Wordlength pat tern.

2000 M a t h e m a t i c s S u b j e c t Class i f i ca t ions : 62K15, 62K05.

1 I n t r o d u c t i o n

Regular two-level fractional factorial designs are the most commonly used designs for factorial experiments. A 2 m-p design denotes a design with m two-level factors and can be constructed by using a defining relation. The

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numbers 1, 2 , . . . , m attached to the factors are called letters and a product (juxtaposition) of any subset of these letters is called a word. The number of letters in a word is called the length of the word. Associated with every 2 m-p design is a set of p words called generators. The set of distinct words formed by all possible products involving the p generators gives the defining relation of the design. Let Ai(D) denote the number of words of length i in the defining relation of a 2 m-p design D, then the vector W ( D ) = (A3(D), A 4 ( D ) , . . . , A,~(D)) is called the wordlength pattern of D.

An important characteristic of 2 m-p design is its resolution. The reso- lution of a 2 m-p design is defined to be the smallest r such that A r > 0 (Box and Hunter, 1961). A 2 m-p design with resolution r is usually denoted by 2r m-p. In such a design, no c-factor effect is confounded with any other effect containing less than r - c factors. Experimenters always prefer to use a design with the maximum resolution. However, not all 2 m-p designs with the same resolution are equally good. To further discriminate 2 m-p designs, Fries and Hunter (1980) proposed the minimum aberration (MA) criterion. That is,

D e f i n i t i o n 1 For two designs D1 and D2, suppose r is the smallest value such that Ar(D~) ~ Ar(D2). D1 is said to have less aberration than D2 if At (D1) < A~(D2). I f no design has less aberration than D1, then D1 is said to have min imum aberration (MA).

Let's illustrate these concepts through the following example.

E x a m p l e 1 Suppose 1, 2, 3, 4, 5 are five independent factors. Consider the following two 2 7-2 designs:

D 1 : 1 , 2 , 3 , 4 , 5 , 6 = 1 2 3 , 7 = 2 4 5 , D2 : 1, 2, 3, 4, 5, 6 = 1234, 7 = 2345.

The defining relations and wordlength patterns of these two designs are given by

D1 : I = 1236 = 2457 = 134567, and W ( D I ) = (0,2,0,1,0) , D2 : I = 12346 = 23457 = 1567, and W(D2) = (0, 1, 2, 0, 0).

Thus both D1 and D2 have resolution IV, but D2 has less aberration than D1.

There have been extensive discussions on MA designs in the litera- ture, such as Chen and Wu (1991), Chen (1992, 1998), Chen, Sun and Wu (1993), Chen and Hedayat (1996), Tang and Wu (1996), Cheng and Mukerjee (1998), Cheng, Steinberg and Sun (1999) and Butler (2003).

However, in some experiments there are some factors that have four lev- els. Addelman (1962) constructed this kind of design from two-level designs by the method of replacement. Wu (1989) improved Addelman's construc- tion method by introducing the method of grouping. Wu, Zhang and Wang (1992) extended Wu's grouping scheme to cover more general designs. Given

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the many 2m4 k designs (designs with m two-level factors and k four-level fac- tors) constructed from the above methods, Wu and Zhang (1993) proposed the minimum aberration (also denoted by MA) criterion for measuring their goodness. For practical use, we are mainly concerned with the construction of MA 2"~41 designs here, but the methods can be easily extended to the construction of MA 2m42 or 2rn81 designs.

Now we briefly describe the construction of 2m41 designs by the method of replacement and the MA criterion proposed by Wu and Zhang (1993). Throughout this paper let q = m - p be the number of independent factors, rt : 2 q be the total number of runs, and let the term two-factor interactions mean the interactions between the two-level factors only. Note that a design with n runs and rn factors can be denoted by an n x m matrix, and the rows and columns are identified with the runs and factors, respectively. Thus in what follows in this paper, we will not differentiate between the factor and column. We can represent the n - 1 columns in a saturated two-level design (denoted by Ha) with n runs by the q independent columns denoted by 1, 2 , . . . , q and their interactions of order 2 to q, that is 12, 13 , . . . , 12.. • q (Wu and Zhang, 1993). Any three columns of the form (a,b, ab), where ab is the interaction column between two-level columns a and b, can be replaced by a four-level column without affecting the orthogonality (Addel- man, 1962). The replacement is done according to the rule shown in Table 1.

Table 1 Rule for replacing any three columns of the form (a, b, ab) by a four-level column

a b ab four-level column 0 0 0 0 0 1 1 ~ 1 1 0 1 2 1 1 0 3

Note that when the two levels 0 and 1 are replaced by 1 and - 1 for the two- level columns, the rule shown in Table 1 is still valid for the replacement. To discriminate 2m41 designs, we now introduce the MA criterion. Let D be a 2m41 design, where the two-level column is represented by c1 , . . . , c,~, the four-level column is represented by A = (al, a2, a3 = ala2), and the cj 's and ai 's are chosen from H,~. In this paper, we assume any two columns of cj 's and ai 's are distinct. There are two types of words in the defining relation of this design. The first involves only the cj's, which is called type 0. The second involves one of ai's and some of the cj's, which is called type 1. Because ala2aa = I, any two ai's that appear in a word can be represented by the third ai. Therefore these two types exhaust the possibilities. For a 2m41 design D, let A~o(D) and Ai l (D) respectively be the number of type 0 and type 1 words of length i in the defining relation of D. The vector W ( D ) = ( A a ( D ) , A 4 ( D ) , . . . ) is called the wordlength pattern of D, where Ai(D) = {A io (D) ,A i l (D)} for i >_ 3. The MA criterion is widely used for

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measuring 2"~41 designs. In practice, A30(D) and A31(D) are of main inter- est. If MA designs are hard to be constructed, then a modified version of the MA criterion can be used, that is the weak minimum aberration (WMA) criterion.

D e f i n i t i o n 2 Let D1 and D2 be two 2m41 designs and r be the smallest value such that At(D1) ~ At(D2). I f Aio(D1) < Ai0(D2), or Ai0(D1) = Ai0(D2) but Ai l (D1) < Ail(D2), then D1 has less aberration than D2. I f no design has less aberration than D1, then D1 is said to have minimum aberration (MA). A 2m41 design is said to have weak minimum aberration (WMA) if its A30 and A31 are minimized sequentially.

To illustrate the concepts for 2m41 designs, let 's see the following exam- ple.

E x a m p l e 2 Based on D1 and D2 in Example 1, let A = (1, 2, 12) be the four-level column, the following two 2541 designs with 32 runs can be ob- tained:

D 3 : A , 3, 4, 5, 6 =123 , 7 = 2 4 5 , Da : A, 3, 4, 5, 6 = 1234, 7 = 2345,

where 3, 4, 5, 6, 7 represent the five two-level columns. Denoting 1, 2, 12 by al , a2 and a 3 respectively, we have the defining relations of D3 and D4, that is

D3 : I = a 3 3 6 = a2457 =a134567, D4 : I = a3346 = a23457 = a1567.

So, A31(D3) = 1, A41(D3) = 1, A61(D3) = 1, A41(D4) = 2, A51(D4) = 1 and also D4 has less aberrat ion than D3. From Wu and Zhang (1993), we can also know D4 has MA.

There are only two papers in the literature that consider MA 2m41 designs. Mukerjee and Wu (2001) considered the situation when m > ~ and Zhang and Shao (2001) constructed MA 2m41 designs for m = q, q + 1. In this paper, alternative methods are developed which allow MA designs with n runs and m > Tg5~ two-level factors to be constructed. When m <_ 5nl__g, lower bounds of A30 and A31 are obtained and a method for constructing WMA designs is demonstrated. In Section 2, some existing results and useful lemmas are given. Theorems on constructing MA and WMA 2m41 designs are presented in Sections 3 and 4 respectively. Examples are provided for illustration after the theorems. Section 5 contains some conclusions and discussions. Some newly obtained designs are presented in the Appendix.

2 E x i s t i n g r e s u l t s a n d s o m e usefu l l e m m a s

For the convenience of presenting the existing and new results, in this section and the subsequent sections, we assume the two levels of any two-level factor are labelled as 1 and - 1 . First we consider the construction of saturated resolution IV two-level designs with n runs and ~ factors (denoted by Fn).

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It follows from Margolin (1969) that a saturated resolution IV design must

design of H~. That is to say, Fn ( Ha 1~ ) be the blocked combined 2 2 •

- H ~ - 1 9 ( H ~ )

Define Gn = H~_ , then H~ -- (F~, G~). The following four properties of 2

F~ and G~ are very useful for constructing MA and WMA 2~41 designs.

L e m m a 1 (i) I f f l E Fn and f2 E Fn, then f l f 2 c G~, where f l f 2 is the interaction column between f l and f2.

(ii) I f f l E F~ and f2 E G~, then f l f 2 E F~, where f l f 2 is the interaction column between f l and f 2 .

(iii) I f D is a 2~v -(m-q) design with n runs, then D should be selected from 5 n n Fn and A i (D) = 0 for odd i, when ~ + 1 < m < -~.

(iv) I f D is a 2~i -('~-q) design with n runs, m > ~, and A3(D) is minimized, then D can be written in the form D = (F~, do), where do comes from Gn.

Properties (i) and (ii) are obvious, so we omit their proofs. Property (iii) can be found in Butler (2003) and Property (iv) is obtained by Chen and Hedayat (1996).

Butler (2003) developed a new method for constructing MA 2 r e - ( m - q )

designs. Some of the results are very useful for constructing MA 2m41 de- signs, which are summarized in the following lamina.

L e m m a 2 Let D be a 2 re-(m-q) design with r >_ III. Define T = (tij) = DD' and Mk(D) = n -2 E~,j=I tkj • Then the following equations hold.

k-2 (i) Mk(D) = k!Ak(D) + ~i=3 ~k i (m)Ai (D) + constant for k > 3, where s k i ( m ) ' s are constants only depending on i, k, m and n. Especially, M4(D) = 24A4(D) + constant and M6(D) = 720A6(D) + (360m - 960)A4 (D) + constant.

(ii) I f D c Fn, then Mk(D) = M k ( F n \ D ) + constant for even k and Mk(D) = 0 for odd k.

(iii) I f D = (F~,d0) and do c Gn, then Mk(D) = Mk(do) + constant for k>_3.

The constants in the above lemma may depend on k, m and n, but not on the particular choice of D, and the summation in (i) equals zero when the superscript is less than the subscript. In the following of this paper, constants and the summations have the same properties respectively. From (i) and (ii) of the above lemma and some simple calculations, we have

L e m m a 3 Let D be a 2 m-(m-q) design and D C Fn, then

A4(D) = Aa(Fn \D) + constant, and

n _ m ) A 4 ( F n \ D ) + constant. A6(D) = A6(Fn \D) + (-~

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Proof From (i) of Lemma 2, we get M4(D) = 24Aa(D) + constant and Ma ( F,~ \ D ) = 24A4 ( F~ \ D ) + constant. Following the relationship between M4(D) and Ma(F~\D) in (ii) of Lemma 2, we can easily obtain the first equation. The second equation follows similarly. []

In fact, following the equations in Lemma 2, we could express Ai(D) for i _> 8 and even i. However such details are rarely needed in practical use.

3 M A 2m41 des igns

In this section and the subsequent section, let D be a 2m41 design, Do be the 2 m-(m-q) design and A = (al,a2,a3 = ala2) be the four-level factor. The technique for constructing MA 2"~41 designs with y6 + 1 < m < ~ 5 ~ ~ is different from the case when rn _> ~,n so we partition this section into two subsections.

3.1 2m41 designs with rn >

Firstly, let's see the worldlength pattern of D, the first component is A30 (D), which is equal to A3(Do). So we must minimize A3(Do) in the first step. When m > ~ and A30(D) is minimized, from Lemma 1, we can write Do in the following form: Do = (F~, do), where do C Gn. When m = ~, there

exist designs with resolution IV (Bose, 1947) and Fn is the unique 2~v - (~ -z ) design up to isomorphism. Therefore al , a2 and a3 must come from Gn and D = (Fn,do, A) when m _> ~. Let d = (d0,A), then d is a 2"~-~41 design with n runs. Following the above argument and Lemma 2, we obtain the following theorem for constructing MA 2m41 designs.

T h e o r e m 1 Let D be a 2m41 design. If D satisfies the following two con- ditions:

(i) There exists a 2"~-~41 design d with n runs satisfying D = ( F n , d), (ii) d is an MA design,

then D is an MA design.

P r o o f Let Dr = (Do,at) for r = 1,2,3, then Aio(D) = Ai(Do) and 3 Ail(D) = ~ r = l Ai(Dr) - 3Ai(D0). From (i) of Lemma 2, we have

i - 2

Mi(Do) = i!Ai(Do) + E aiz(m)Al(Do) + constant, and l = 3

3 3 i - - 2 3

E Mi(Dr) = i! E Ai(Dr) + E ail(rn + 1) E At(Dr) + constant. r = l r = l / = 3 r = l

Then minimizing (A30 (D), A~I (D), A40 (D), Aal (D) , . . . ) sequentially is equiv- alent to minimizing (M3(D0) Y~'.3r= 1 M3(Dr), M4(D0), 3 , E r = l M4(Dr ) , . . . ) se- quentially. Let do be the collection of the two-level columns in d, A =

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(ax, a2, a3 = ala2) be the four-level factor in d and dr = (do, a t) for r = 1, 2, 3. Then minimizing (A30 (d), A31(d), A40 (d), A41 (d ) , . . . ) sequentially is equivalent to minimizing (M3(do), }-:~= 1 M3(dr), M4 (do), }-~3= 1 M4(d r ) , . . . ) . From (iii) of L e m m a 2, we obtain

Mi(do) = Mi(Do) + constant, and 3 3

E Mi(dr) = E Mi(Dr) + constant. r=l r~l

Then from the above argument, we can easily have the assertion. []

Now, lets's show how to construct an MA 22041 design with 32 runs by the above method.

Example 3 Let i, 2, 3,4 and 5 be the five independent columns of F32, and F32 = {1,2,3,4,5,6 = 123,7 = 124,8 = 125,9 = 134, t0 = 135, tl = 145, t2 = 234, t3 = 235, t4 = 245, t5 = 345, t6 = 12345}, where ti represents the (I0 + i)th factor, for example t2 means the 12th factor. Obviously, 12, 13, 14, 15 are four independent columns of G32. Let A = (12, 13, 23) be a four-level factor, do = (14, 15, 1245, 34) be the collection of 4 two-level factors and d -- (do, A). From Wu and Zhang (1993), d is an MA 2441 design. So D -- (F32, d) is an MA 22041 design.

Remark 1 When we use the above method, one thing we should note is that there are only q - 1 independent columns in Gn, which can be verified from the structure of Gn. So d is an MA design among all 2m-~41 designs with n runs that contain at most q - 1 independent columns. Actually, we can construct d in this way: first construct an MA 2m-~41 design d* with

n g runs, then let d = d* , d is what we want. Here, we can easily see tha t

d and d* have the same defining relation.

Remark 2 Mukerjee and Wu (2001) also studied the case when m _> ~, especially w h e n n - 4 - m = 1 , 2 , . . . , 1 2 a n d n - m = 2 r - w f o r r < q a n d w = 0, 1, 2, 3. Then when n = 64, their method will be difficult to const ruct MA 2m41 designs for m = 36, 37,..., 47. However, from the discussion in Remark 1, if we can construct MA 2m41 designs for m = 4, 5 , . . . , 15 with 32 runs, the cons t ruc t ion becomes easier following our method. 32-run MA 2~41 designs with m = 4, 5 , . . . , 9 can be found in Wu and Zhang (1993). And 32-run MA 2~41 designs for m = 10, 1 1 , . . . , 15 can be cons t ruc ted by our following methods and will be tabula ted in the Appendix.

3.2 2'~41 designs with m <

Before presenting the construction method, let's see the structure of D first,

that is the following theorem.

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8 P.-F. Li et al.

T h e o r e m 2 Let D be a 2m41 design with $-~ < m < ~ - 2. I f D is an MA design, then D satisfies the two conditions: Do C Fn and two of al, a2 and a3 are in In .

Proof When ~ < m < ~ - 2, the maximum resolution of Do is IV, which can be inferred from Corollary 3 of Chen and Hedayat (1998). If D is an MA design, then Do has resolution IV and Do C Fn (Lemma 1). From Lemma 1, we know tha t two or none of al , a2 and a3 are in Fn. Now we only need to prove that if all ai ~ F,~, then we can find a design tha t has less aberrat ion than D. Since m _< ~ - 2, there exist two columns a and b satisfying a, b ~ Fn and a, b ~ D. Let A* - (a, b, ab) be the four-level factor and D* = (Do, A*), next we show D* has less aberrat ion than D.

Let Do = ( f l , f 2 , . . . , fro) and Di = (ai f l , aif2 . . . . , aif,~) for i = 1, 2, 3. From Lemma 1, obviously Di C F~. Since ID01 = ]Di I = m, then IDoNDil = IDol + IDil - IDo U Dil >_ 2m - ~ for i = 1,2,3, where IDil means the cardinality of Di for i = 0, 1, 2, 3. Since if ai f j = fk for some i, j and k, then aifk = f j . Therefore, there are at least m - ~ two-factor interactions that are aliased with each ai for i = 1, 2, 3. Tha t is to say, A3o(D) = 0 and A31 (D) _> 3 ( m - ~). In D*, there is no two-factor interaction tha t is aliased

"~ two-factor interactions tha t are aliased with with a and b and at most ab. Hence, A30(D*) = 0 and A3z(D*) <_~-. "~ Because T ~ < m < y 5 ~ ~ - 2, then A31(D*) < A31(D). Thus, the proof is completed.

Without loss of generality, we assume al ~ Fn and a2 E Fn. Let F = Fn\(Do, al, a2) and DR = (F, A). The following lemma studies the relation- ship between the wordlength pat terns of D and D R.

L e m m a 4 Let D be a 2rn41 design which satisfies the two conditions in Theorem 2, then the wordlength patterns of D and DR satisfy the following equations.

(i) A31 (D) = A31 (DR) + constant, (ii) Aao(D) = A4o(DR) + A41(DR) + A31(DR) + constant,

(iii) A41 (D) = - A a l ( D R ) - 2An1 (On) + constant, (iv) A51(D) = A51(DR) + (~ - rn)A31(DR) + A41(DR) + constant, (v) A60(D) = A6o(DR)+A61(DR)+A51(DR)+(~-m) (A4o(DR)+A41(DR)+

Aaz (DR)) + constant, (vi) A61 (D) -- -A61 (DR) - 2A51 (DR) - (~ - m + 1)A41 (DR) - 2A4o(DR) --

(~ -- 2m)A31 (DR) + constant.

Proof Let Di = (Do, ai), Fi = (F, a¢), for i = 1, 2, D+ = (Do, al , a2), and F+ = (F, al , a2).

(i) Because there are only words of even-length in F , , Aal(D) is equal to the number of length 4 words in the defining relation of D+ that contain

ala2. So

Aal(D) = A4(D+) - An(D1) - A4(D2) + A4(Do).

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From this equation and Lemma 3, we get

A31(D) = A4(F) - A4(F2) - A4(F1) + A4(F+) ~- constant

= A31(DR) + constant.

(ii) From Lemma 3, we have

A40(D) -- A4(Do) = A4(F+) + constant.

Note tha t there are three kinds (containing 0, 1, 2 factors of al and a2) of length 4 words in the defining relation of F+ and the number is A4o(DR), A41 (DR) and A31 (DR), respectively. Therefore A40(D) = A40(DR)+A41 (DR) +Aal (DR) + constant.

(iii) Since Do C Fn and al , a2 C Fn, then A41(D) is equal to the number of length 4 words in the defining relation of D+ that contain al or a2. Tha t is to say, A41(D) = A4(D1) + A4(D2) - 2A4(Do). Then from Lemma 3 and the equation in (ii), we have

A41(D) = A4(F2) + A4(F1) - 2A4(F+) + constant

= An(F2) + A4(F1) - 2(A40(DR) + A41(DR) + A31(DR)) -~- constant

= -A41 (DR) - 2A31 (DR) + constant.

The last three equations follow similarly. []

From this lemma and Theorem 2, we can easily obtain the following theorem, which will be helpful for constructing MA 2m41 designs.

T h e o r e m 3 Let D be a 2m41 design which satisfies the two conditions in 5 n n Theorem 2, where -f~ < m ~_ 7 - 2. I f D is the unique design (up to iso-

morphism) to minimize (A31(DR), A40(DR) +A41 (DR), A40(DR), A51(DR), A60(DR) + A61(DR), A60(DR) , " ") sequentially, then D is an MA design.

n In the above theorems, we don' t consider the case m = 7 - 1. In this case, if D is an MA design, then Do C F~ and ai ~t F~ for i =1, 2, 3.

When n > 8, all 2~v 1- (~-1-q) designs are isomorphic (Chen and Cheng,

2 0 0 0 ) , t h u s w e c a n s e l e c t D o = ( _ H ~ ) . T h e n d i f f e r e n t c h o i c e s o f a l , a2

and a3 = ala2 in G~ will result in 29-141 designs with the same wordlength pa t te rn when n > 8.

(H~) T h e o r e m 4 Let Do = - H and A = (al,a2,a3 = ala2), where n > 8

and ai E G~ for i = 1,2. Then D = (D0,A) is an MA 2~-141 design.

Based on Theorems 3 and 4, we can construct MA 2m41 designs for 5~ < m < ~ Now, let 's construct some MA designs with 32 runs for 1-~ 7" illustration. Some MA 2"~41 designs with 32 and 64 runs are shown in the Appendix.

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E x a m p l e 4 Suppose 1, 2, 3, 4 and 5 are the five independent columns of F32, where F32 is the same as tha t in Example 3.

(i) For m = 14, let A = (1,2, 12) and DR = {A}, which means DR is a design only containing the four-level factor. Then Aij(DR) = 0 for i > 3, j = 0, 1, and DR satisfies the condit ion in Theorem 3. Let Do = F32\{1, 2} = {3, 4, 5 ,123 ,124,125,134,135,145,234, 235,245,345, 12345} and D = (Do, A), then D is an MA 21441 design.

(ii) For m = 13, let A = (1,2, 12) and DR = {A, 12345}, then DR is a 2141 design with the four-level factor A and two-level factor 12345. Obviously, DR also satisfies the condit ion in Theorem 3. Let Do = F32\{1, 2, 12345} = {3, 4, 5,123,124, 125,134, 135,145,234, 235,245,345} and D = (Do ,A) , then D is an MA 21341 design.

(iii) For m = 15, f rom Theorem 4, let A = (12, 13, 23) be the four-level factor and Do = F32\{12345} = {1, 2, 3, 4, 5 ,123,124,125,134, 135,145,234, 235,245,345}, then D = (Do, A) is an MA 21541 design.

4 W M A 2m41 d e s i g n s

In the above section, we proposed the methods for cons t ruc t ing MA 2m41 5n And some examples are provided for be t te r under- designs when m > Tg"

s tanding the methods . However, there is no general me thod for cons t ruc t ing MA 2m41 designs when m < 5,~ In this case, we will consider W M A 2m41 designs, which minimize A30 and A31 sequentially. Firstly, we will derive lower bounds of A3o and A31, which will serve as benchmarks for searching W M A designs. Then , we will propose a method to const ruct W M A 2m41 designs.

T h e o r e m 5 I f D is an MA 2m41 design, then A30(D) = 0 and Aal (D) = n 1) f o r n l < m < n 2. ( m - ~ + ~ - - 7

n _ 1 < m < n - 2, there exist 2~v -(m-q) designs (Bose, 1947). Proof W h e n ~ _ _ Then if D is an MA 2m41 design, we must have A30(D) = 0. To prove the

1) for ~ _ l < m < n _ 2 . second equality, we first prove A31 (D) > (m - ~ + ~ - - 7 Let D+ = (Do, al, a2), it is a 2 m+2-(m+2-q) design. Next we will only prove

n = n a n d A a l ( D ) > 2 f o r m = n + l . W h e n m > ~ + 1 , A31(D) _> 1 for m ~ _ the results follow similarly.

W h e n m = ~, if Aa l (D) = 0, then D+ has resolution at least IV and ala2 is clear (not aliased with any main effect or two-factor interaction), which contradicts Corol lary 3 of Chen and Hedayat (1998). Thus, A31(D) _> 1 for

Let D - i be the 2"~-141 design obtained by deleting the i th two-level m = ~ . column in D for i = 1 , . . . , m . W h e n m = ~ + 1, if Aa l (D) = 1, we have Aal(D- i ) <_ A a l ( D ) = 1. Since D - i is a 2~41 design, then A31(D- i ) _> 1. Hence, A31(D-i) = 1, which means the i th factor doesn ' t appear in the length three words of type 1 for i = 1 , . . . , m , thus A31(D) -- 0, which

n contradicts A31(D) = 1. So Aa l (D) _> 2 for m = ~ + 1.

2 4 5

n 1) for n - - 1 < m < ~ - - 2 . Now let us prove A31(D) _< ( m - ~ + ~ _ _ Let F , = {b l ,b2 , . . . b~}. From Chen and Hedayat (1998), we know tha t there are ~ - 1 disjoint two-factor interactions in Fn, say b3b4,. • •, b } - l b ~

without loss of generality, tha t are aliased with bib2. Let A* = (bl, b2, bib2) be a four-level factor, D~ = ( b 3 , b s , . . . , b ~ - l ) and D* = (D~,A*). Then A30(D*) = 0 and Aal (D*) = 0. By adding b4, b6,. . b- to D* sequentially,

' ' i -

we obta in 2"~41 designs with A30(D*) = 0 and A31(D*) = (m - ~ + 1) fo r th _ 1 _< m _< y " - 2. Since D is an MA design, then A30(D) = 0 and

n n 1) for n - 1 < m < y - 2. Combining the above A31(D) _< ( m - ~ + ~ _ _ arguments , the proof is completed. []

R e m a r k 3 The above theorem considers the lower bounds of A30 and A31. Actually, in the proof of this theorem, D* is a W M A 2"~41 design, since A30(D*) and A31(D*) a t ta in the lower bounds. From the proof, W M A 2m41 desigps can also be const ructed by deleting columns from Fn. Deleting b4, b6, . . •, bn/2 from Fn sequentially will result in such W M A designs. This me thod can be easily employed, especially when m is large. In the following, we will i l lustrate the me thod by construct ing W M A 21°41 design with 32 runs. Some 64-run W M A designs are tabula ted in the Appendix.

E x a m p l e 5 Suppose 1, 2, 3, 4 and 5 are the five independent columns of F32, where F32 is the same as tha t in Example 3. Let A = (1,2, 12) be the four-level factor. For const ruct ing W M A 21°41 designs, we consider the two-factor interact ions tha t are aliased with 12. T h a t is,

12 = 36 = 47 = 58 = 9t2 = tot3 = t i t 4 = t s t6 .

Then if we delete {6 = 123, 9 = 134, t3 = 235, t6 = 12345} from F32, we will get a W M A 21°41 design D = (Do, A), where Do = {3, 4, 5,124, 125,135,145, 234, 245,345} is the collection of 10 two-level factors. From Sitter, Chen and Feder (1997), we can know tha t this design is also an MA design. Similarly, we can const ruct W M A 2m41 designs for m = 7, 8, 9. Compared with the MA designs in Wu and Zhang (1993), all designs we obta in are also MA designs except for m = 8. Thus we will not tabula te these W M A designs in the Appendix.

5 C o n c l u s i o n s a n d d i s c u s s i o n s

This paper considers the construct ion of MA and W M A 2rn41 designs. Some examples are provided for il lustration and some newly cons t ruc ted designs are tabula ted for practical use. If the four-level factor is a blocking factor, then a slightly different MA criterion can be found in Zhang and Park (2000) or Cheng and Wu (2002). In this case, some modifications should be made to Theorems 1, 3 and 4. Then the results here also supplement the work of Li, Liu and Zhang (2005), which considers the cons t ruc t ion of MA 2 "~-(m-q) designs with two blocks. The methods can also be easily extended to the const ruct ion of MA and W M A 2"~42 designs. However, can the methods be

246

generalized to construct MA sm(s 2) or sm(s2) 2 designs for genral s? These is an interesting and open problem for further study.

A c k n o w l e d g e m e n t s

The authors would like to thank the Coordinating Editor and two anony- mous referees for their valuable comments and suggestions. This work was partially supported by the National Natural Science Foundation of China grants 10171051, 10301015, and the Science and Technology Innovation Fund of Nankai University.

A p p e n d i x

In the following three tables, A always represents the four-level factor and F64 = {1, 2, 3, 4, 5, 6,123,124,125,126,134,135,136,145,146,156,234,235, 236,245,246,256,345,346,356,456, 12345, 12346, 12356, 12456, 13456, 23456}.

Table 2 32-run MA 2m41 designs for 10 ~ m < 15

m Two-level factors Four-level factor A 10 11 12 13 14 15

3,4,5,124,125,135,145,234,245,345 3,4,5,123,124,125,135,145,234,245,345

3,4,5,123,124,125,134,135,145,234,245,345 3,4,5,123,124,125,134,135,145,234,235,245,345

3,4,5,123,124,125,134,135,145,234,235,245,345,12345 1,2,3,4,5,123,124,125,134,135,145,234,235,245,345

(1,2,12) (1,2,12) (1,2,12) (1,2,12) (1,2,12)

(12,13,23)

Table 3 64-run MA 2m41 designs for 21 < m < 31

m Two-level factors Four-level factor A 21 22 23 24 25 26 27 28 29 3O 31

F64\{ 1, 2, 3, 4, 5, 6,134,246, 12345, 23456, 12456} F64\{1, 2, 3, 4, 5, 6,134, 156,235,246}

F64\{ 1, 2, 3, 4, 5, 6,134, 156,235} F64\{ 1, 2, 3, 4, 5, 6, 12345, 23456}

F64\{ 1, 2, 3, 4, 5, 6, 23456} F64\{1, 2, 3, 4, 5, 6} F64\{1, 2, 3, 4, 5} F64\{ 1, 2, 3, 4} F64\{1, 2, 3} F~4\{1, 2} F64\{1}

(1,2,12) (1,2,12) (1,2,12) (1,2,12) (1,2,12) (1,2,12) (1,2,12) (1,2,12) (1,2,12) (1,2,12)

(12,13,23)

247

Table 4 64-run WMA 2m41 designs for 16 < m < 20, A = (1, 2, 12)

m

16 17 18 19 20

Two-level factors 3,4,5,6,134,135,136,145,146,156,345,346,356,456,13456,123

3,4,5,6,134,135,136,145,146,156,345,346,356,456,13456,123,124 3,4,5,6,134,135,136,145,146,156,345,346,356,456,13456,123,124,125

3,4,5,6,134,135,136,145,146,156,345,346,356,456,13456,123,124,125,126 3,4,5,6,134,135,136,145,146,156,345,346,356,456,13456,123,124,125,126,234

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