Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2015

Acta Mathematica Sinica, English Series Jul., 2015, Vol. 31, No. 7, pp. 1163 1170 Published online: June 15, 2015 DOI: 10.1007/s10114-015-3616-y Http://www.ActaMath.com Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2015 Some Properties of β-wordlength Pattern for Four-level Designs Wei Wei SHENG Xia Ming LI 1) Yu TANG School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China E-mail : shengweiwei333@163.com xiamlee@suda.edu.cn ytang@suda.edu.cn Abstract Fractional factorial designs have played a prominent role in the theory and practice of experimental design. For designs with qualitative factors under an ANOVA model, the minimum aberration criterion has been frequently used; however, for designs with quantitative factors, a polynomial regression model is often established, thus the β-wordlength pattern can be employed to compare different fractional factorial designs. Although the β-wordlength pattern was introduced in 2004, its properties have not been investigated extensively. In this paper, we will present some properties of β-wordlength pattern for four-level designs. These properties can help find better designs with quantitative factors. Keywords β-wordlength pattern, four-level, fractional factorial design MR(2010) Subject Classification 62K15 1 Introduction Fractional factorial designs have been extensively used in industrial, agricultural and scientific experiments. If we use an appropriate design to conduct experiments, we cannot only reduce the cost, but also keep high efficiency. In this sense, it is of great importance to provide a good criterion to compare different designs. For regular designs with qualitative factors, the minimum aberration (MA) criterion proposed by Fries and Hunter [6] has been widely employed. Later on, Deng and Tang [3] and Tang and Deng [10] generalized the criterion for non-regular designs with two levels. Using coding theory, Xu and Wu [13] proposed the generalized minimum aberration and justified it for the designs with qualitative factors under the ANOVA model. In fact, as pointed in Wu and Hamada [12], the idea behind the (generalized) MA criterion is the hierarchical ordering principle, i.e., lower order effects are more likely to be important than higher order effects and effects with the same order are considered to have equal importance. Some other criteria are also proposed in literature, such as the minimum generalized aberration (see Ma and Fang [7]), the general minimum lower order confounding (see Zhang et al. [14]) and the minimum hybrid aberration (Pang and Liu [9]). More discussions about the criteria for designs with qualitative factors can be found in Mukerjee and Wu [8], Wu and Hamada [12] and references therein. Received November 30, 2013, revised August 21, 2014, accepted October 8, 2014 Supported by NSFC (Grant No. 11271279), NSF of Jiangsu Province (Grant No. BK2012612) and Qing Lan Project 1) Corresponding author

1164 Sheng W. W., et al. For designs with quantitative factors, things become more complicated. As Cheng and Wu [1] and Fang and Ma [5] pointed out, designs with the same wordlength patterns (thus cannot be distinguished under the MA criterion) may have different statistical inference abilities. When multiple-level quantitative factors are involved, normally a polynomial regression model (or a response surface model) will be established to analyze data collected by a fractional factorial design. In such cases, level permutation of factors can alter the geometrical structure of the design and ultimately change the efficiency when the coefficients of the regression model are estimated. In order to systematically compare designs with quantitative factors, Cheng and Ye [2] proposed the β-wordlength pattern and classified geometrically non-isomorphic designs. Under their framework, lower degree effects are more likely to be important than higher degree effects and effects with the same degree are considered to have equal importance, thus designs with better β-wordlength pattern are recommended. In [2], β-wordlength pattern was originally defined using the concept of indicator function. Although the relationship between them is quite clear, it is not easy to analyze more properties of β-wordlength pattern due to the complicated definition of indicator function itself. Recently, Tang and Xu [11] simplified the expression of β-wordlength pattern, and obtained some interesting results related to the properties of β-wordlength pattern for three-level regular designs. The current paper continues their approach and provides some results for four-level regular designs. The rest of the paper is organized as follows. In Section 2, some basic concepts and notations are introduced. Section 3 discusses a simple case for 4 n 1 regular designs and Section 4 generalizes the situation for 4 n k regular designs. Finally, the last section gives some conclusion and discussion. 2 Concepts and Notations AdesignD, denoted by (N,s n ), with N runs and n factors, each with s levels, is an N n matrix, whose entries take values from the set of residue classes modulo p, i.e., Z p = {0, 1,...,s 1}. Let p 0 (x) 1andp j (x) beaj-th polynomial on Z s,where1 j s 1, such that s 1 0, if i j; p i (x)p j (x) = s, if i = j. x=0 The set {p 0 (x),p 1 (x),...,p s 1 (x)} is called an orthogonal polynomial basis (see Draper and Smith [4, Chapter 22]). For a design D =(d il ) N n,letf 1,...,F n be its n factors. When j 1 + + j n = j, F j 1 1 Fj n n is called an interaction with degree j. The orthogonal polynomial contrast coefficient of F j 1 1 Fj n n is an N 1 vector, whose i-th element is p j1 (d i1 ) p jn (d in ). Following [13], for a design D, denoted as (N,s n ), consider the ANOVA model, Y = X 0 α 0 + X 1 α 1 + + X n α n + ɛ, where Y is the N 1 response vector, α 0 is the intercept, X 0 is an N 1 all-one vector, α j is the vector of all interactions of j-th order, X j is the orthogonal contrast coefficient matrix of α j and ɛ represents the random error. Let n j =(s 1) j ( n j ), X j =(x (j) ik ) N n j,

Some Properties of β-wordlength Pattern for Four-level Designs 1165 and n j N A j (D) =N 2 k=1 x (j) 2 ik i=1 for j =0, 1,...,n. (2.1) Then the vector (A 1 (d),...,a n (d)) is called the (generalized) wordlength pattern of design D. For two designs D (1) and D (2),wecallD (1) haslessaberrationthand (2) if there exists a positive integer r, 1 r n, such that A r (D (1) ) <A r (D (2) ), and A i (D (1) )=A i (D (2) ), i =1,...,r 1. If there does not exist any other design, which has less aberration than D (1), then D (1) is said to be a (generalized) minimum aberration design. Following [2], for a design D, denoted as (N,s n ), consider the polynomial model, Y = Z 0 θ 0 + Z 1 θ 1 + + Z K θ K + ɛ, where Y is the N 1 response vector, θ j is the vector of all interactions of degree j, Z j is the orthogonal contrast coefficient matrix of θ j and ɛ represents the random error. Let Z j =(z (j) ak ) N n, j where n j represents the number of all interactions with degree j and n j N β j (D) =N 2 z (j) 2 ak for j =0, 1,...,K. (2.2) k=1 a=1 Then the vector (β 1 (d),...,β K (d)) is called the β-wordlength pattern of design D. HereK = n(s 1) is the highest degree of all interactions. Cheng and Ye [2] suggested that a good design with quantitative factors should sequentially minimize (β 1,...,β K ). Noticing that, for j 1 + + j n = j, z (j) ak = p j 1 (d a1 ) p jn (d an ), we have N n 2 β j (D) =N 2 p jl (d al ) for j =0, 1,...,K. (2.3) 0 j 1,...,jn s 1 j 1 + +jn=j a=1 l=1 When a design with qualitative factors is considered, levels of the factors may not be restricted on the set of residue classes Z p. For example, regular fractional factorial designs listed in textbooks are often constructed on finite fields. Let q beaprimeoraprimepowerandgf(q) be the finite field of order q. For a positive integer s, denote V (s, q) as the s-dimensional linear space, formed by all s-th row vectors on GF(q). Obviously, the cardinality of V (s, q) isq s. A design D is called to be a full factorial design if it contains all elements of V (s, q) as its runs. AdesignD is called to be a regular design if all its runs form a subspace of V (s, q). A regular deign D, denoted by q s k,representsa1/q k -fraction of the full factorial design, and all its q s k runs form a group. When we consider a design with quantitative factors and calculate its β-wordlength pattern, theentriesofthedesignmustbevaluesinz p. If the level q is a prime, it is well known that Z p and GF(q) are isomorphic, thus we can map all entries of a regular design to Z p and then calculate its β-wordlength pattern directly. If the level q is a prime power, things become complicated, because Z p and GF(q) are no longer isomorphic. In order to calculate the β- wordlength pattern of a regular design with a prime power level, we must firstly find a suitable map to transform all the entries of the design from GF(q) toz q. Obviously, different maps will lead to different β-wordlength patterns.

1166 Sheng W. W., et al. 3 Properties of β-wordlength Pattern for a 4 n 1 Design Table 1 A regular 4 3 1 design with β 3 (D)=0 0 0 0 0 1 2 0 1 1 0 ξ ξ 0 ξ 2 ξ 2 1 0 1 1 1 0 1 ξ ξ 2 1 ξ 2 ξ ξ 0 ξ ξ 1 ξ 2 ξ ξ 0 ξ ξ 2 1 ξ 2 0 ξ 2 ξ 2 1 ξ ξ 2 ξ 1 ξ 2 ξ 2 0 0 0 1 0 2 0 0 3 3 1 1 1 1 0 2 1 2 3 1 3 0 2 1 0 2 0 3 2 2 2 2 3 1 3 1 3 3 0 0 3 2 1 3 3 2 Typically, to construct a four-level regular design 4 n k, we first define a primitive element of GF(4), ξ, satisfying ξ 2 + ξ +1 = 0. A regular 4 n k design is formed by taking n k primitive columns and other k columns of their linear combinations. For example, the left side of Table 1 is aregular4 3 1 design, which takes the last column as the sum of the first two primitive columns. Notice here all operations are performed over GF(4). Now comes the problem: different ways of assigning levels to be 0, 1, 2, 3 lead to different β-wordlength patterns. For a regular design, we always have β 0 =1andβ 1 = β 2 = 0; but in general, β 3 of the design is not zero if its levels are assigned improperly. In fact, for regular 4 n 1 minimum aberration designs, we have the following lemma. Lemma 3.1 Let D be a regular 4 n 1 minimum aberration design on GF(4). Let D 0 be a design by conducting linear permutation of the dependent column of D. Letφ(x) be a bijection map from GF(4) to Z 4. Denote by φ(d 0 ) the design obtained by mapping all elements of D 0 using φ. Thenβ n (φ(d 0 )) 0. Proof Without loss of generality, assume the sum of each row of D 0 is b 0,whereb 0 GF(4). When n is odd, if b 0 {φ 1 (0),φ 1 (1)}, thenineachrowofd 0, there must be odd number of x, where x {φ 1 (0),φ 1 (1)}. Otherwise, if there exists a row of D 0 containing even number of elements denoted by x in {φ 1 (0),φ 1 (1)} and odd number of elements denoted by y in {φ 1 (2),φ 1 (3)}, then the sum of these x is either 0 or φ 1 (0)+φ 1 (1) (= φ 1 (2)+φ 1 (3), as φ 1 (0) + φ 1 (1) + φ 1 (2) + φ 1 (3) = 0); the sum of these y s is either φ 1 (2) or φ 1 (3). So the total sum of the row is either φ 1 (2) or φ 1 (3), which is a contradiction. So in each row of φ(d 0 ), there must be odd number of x,wherex {0, 1}. Noticing that D 0 is a 4 n 1 minimum

Some Properties of β-wordlength Pattern for Four-level Designs 1167 aberration design, we have A j (φ(d 0 )) = 0 for j =1, 2,...,n 1, which makes β j (φ(d 0 )) = 0 for j =1, 2,...,n 1. According to the definition of β-wordlength pattern, we have N n 2 β n (φ(d 0 )) = N 2 p 1 (d al ). (3.1) a=1 l=1 Since p 1 (0) and p 1 (1) are both less than 0, N n a=1 l=1 p 1(d al ) < 0, which makes β n (φ(d 0 )) 0. Similarly, if b 0 {φ 1 (2),φ 1 (3)}, thenineachrowofd 0, there must be even number of x, where x {φ 1 (0),φ 1 (1)}, which will make N n, ifb 0 {0,φ 1 (0) + φ 1 (1)}, then N a=1 n a=1 l=1 p 1(d al ) > 0. In the same vein, for even n l=1 p 1(d al ) > 0; otherwise, N n a=1 l=1 p 1(d al ) < 0. Lemma 3.1 tells us that if we use the same function to map levels of each column to 0, 1, 2, 3, then β n (D 0 ) 0. However, if we assign levels as the right design of Table 1, its β 3 is zero. The assignment method used in Table 1 can be generalized. In fact, we can obtain the following result related to regular minimum aberration 4 n 1 designs. Theorem 3.2 By assigning levels of a regular minimum aberration 4 n 1 design, there always exists a design D, satisfying β n (D) =0. Proof Let x i, i =1, 2,...,n 1 ben 1 primitive columns. The n-th column is formed by taking the sum of the primitive columns. Now we assign the levels, (0, 1,ξ,ξ 2 ), of the first n 2primitive columns as (0, 1, 2, 3); of the (n 1)-th primitive column as (1, 0, 2, 3) and of the last n-th column as (2, 1, 0, 3). When n is odd, for any row (η 1,...,η n 2,η n 1,η n ) in the original design, where η i GF(4), there exists a distinct row (ξ 2 +η 1,...,ξ 2 +η n 2,ξ+η n 1, 1+η n ), as they share the same sum. Let (z 1,...,z n 2,z n 1,z n ) be the corresponding row of (η 1,...,η n 2,η n 1,η n ) after conducting the above level assignment strategy. Then row (ξ 2 + η 1,...,ξ 2 + η n 2,ξ + η n 1, 1+η n ) will correspond to (3 z 1,...,3 z n 2, 3 z n 1, 3 z n ), which is the resultant row when conducting the reflection operator to (z 1,...,z n 2,z n 1,z n ). Thus, n i=1 p 1(3 z i )= n i=1 p 1(z i ). All rows of D can then be partitioned into reflection pairs, which leads to β n (D) = 0. Similarly, when n is even, for any row (η 1,...,η n 3,η n 2,η n 1,η n ) in the original design, there exists a distinct row (η 1,...,η n 3,ξ 2 + η n 2,ξ+ η n 1, 1+η n ) with the same sum. Let (z 1,...,z n 2,z n 1,z n ) be the corresponding row of (η 1,...,η n 3,η n 2,η n 1,η n ). Then (η 1,...,η n 3,ξ 2 +η n 2,ξ+η n 1, 1+η n ) will correspond to (z 1,...,z n 3, 3 η n 2, 3 z n 1, 3 z n ). Since n i=n 2 p 1(3 z i )= n i=n 2 p 1(z i ), the contribution of the above paired rows vanishes when β n (D) iscalculated. 4 Properties of β-wordlength Pattern for a 4 n k Design To generalize the idea in Section 3 to find a good map for 4 n k designs, we need more concepts. Let D be an s-level design on Z p. Mapping each run of D from {z 1,z 2,...,z n } to {s 1 z 1,s 1 z 2,...,s 1 z n } forms a new design, which is called the mirror-image of D. Definition 4.1 When the mirror-image of a design D is itself, the design D is said to be mirror-symmetric. For a mirror-symmetric design, it is easy to show the following property. Theorem 4.2 If D is mirror-symmetric, then for any odd j, β j (D) =0.

1168 Sheng W. W., et al. Proof The result simply follows from the fact that the orthogonal polynomial p m (x) isanodd function for any odd m, andp m (x) isanevenfunctionforanyevenm. For m =1,ξ and ξ 2, we define three maps from GF(4) to Z 4, such that if φ m (x) =z, then φ m (x + m) =3 z, wherex GF(4) and z Z 4. Now we consider 4 n k designs. Without loss of generality, assume the first n k columns are independent and the last k columns are linear combinations of the independent ones. Firstly, we have the following theorem. Theorem 4.3 Let D be a 4 n k design. Denote its j-th column by C j,where1 j n and assume the first n k columns are independent. For 1 j k, if C n k+j = C i1 + C i2 + + C iaj + ξ(c t1 + C t2 + + C tbj )+ξ 2 (C q1 + C q2 + + C qcj ), where i 1,...,i aj,t 1,...,t bj,q 1,...,q cj are all distinct positive integers in [1,n k], anda j,b j,c j are not all odd or not all even, then there exists a map ϕ from GF(4) to Z 4, such that β m (ϕ(d)) = 0, wherem is odd. Proof Letusfirstassumea j,b j,c j are odd, odd and even, respectively. To define the map of the design from GF(4) to Z 4, we consider two cases. For the first n k independent columns C j,where1 j n k, weusethemapφ ξ 2 defined in Section 3, while for the last k dependent columns C n k+j,where1 j k, weusethemapφ ξ. For any row (η 1,...,η n k,η n k+1,...,η n ) in the original design, where η i GF(4), there exists another row (ξ 2 + η 1,...,ξ 2 + η n k,η n k+1,...,η n), where η n k+j and η n k+j satisfy and a j η n k+j = η n k+j = (η is + ξ 2 )+ξ a j η is + ξ b j η ts + ξ 2 c j η qs b j (η ts + ξ 2 )+ξ 2 c j a j = η is + ξ b j η ts + ξ 2 c j = η n k+j + ξ 2 + ξ 3 +0 = η n k+j + ξ 2 +1 = η n k+j + ξ. (η qs + ξ 2 ) η qs + a j ξ 2 + ξ b j ξ 2 + ξ 2 c j ξ 2 Considering the maps φ ξ 2 for the first n k columns and φ ξ for the last k columns we use here, we know that if denote by (z 1,...,z n k,z n k+1,...,z n ) the corresponding row of (η 1,...,η n k,η n k+1,...,η n ) after conducting the above level assignment strategy, then there exists another row (3 z 1,...,3 z n k, 3 z n k+1,...,3 z n ) as the corresponding row of (ξ 2 + η 1,...,ξ 2 + η n k,ξ + η n k+1,...,ξ + η n ). That is to say, the transformed design is mirror-symmetric. So the result follows from Theorem 4.2. For other cases of a j,b j and c j, we can prove the result similarly, except for the different choices of the maps for the last k dependent columns C n k+j,where1 j k. More strictly, if (a j,b j,c j ) is (odd, even, odd) or (even, odd, even), then we use the map φ 1 ;if(a j,b j,c j )is

Some Properties of β-wordlength Pattern for Four-level Designs 1169 (even, odd, odd) or (odd, even, even), then we use the map φ ξ ;if(a j,b j,c j ) is (even, even, odd) or (odd, odd, even), then we use the map φ ξ 2. Notice that in Theorem 4.3, if a j,b j,c j are all odd or all even, then we cannot assign the same map, i.e., φ ξ 2 in the proof of Theorem 4.3, to each independent columns. Whether we can find a suitable map to make the transformed design mirror-symmetric depends on the structure of the original design. For example, the left design 4 3 1 in Table 1 is defined by C 3 = C 1 + C 2. Here a 1 =2andb 1 = c 1 = 0 are all even, but we can assign φ ξ 2, φ ξ and φ 1 to the three columns respectively to make the transformed design mirror-symmetric. However, we cannot find such a map for the following saturated design 4 5 3 in Table 2, which is defined by C 3 = C 1 + C 2 ; C 4 = C 1 + ξ C 2 ; C 5 = C 1 + ξ 2 C 2. Table 2 A regular 4 5 3 design 0 0 0 0 0 0 1 1 ξ ξ 2 0 ξ ξ ξ 2 1 0 ξ 2 ξ 2 1 ξ 1 0 1 1 1 1 1 0 ξ 2 ξ 1 ξ ξ 2 ξ 0 1 ξ 2 ξ 0 ξ 2 ξ 0 ξ ξ ξ ξ 1 ξ 2 0 1 ξ ξ 0 1 ξ 2 ξ ξ 2 1 ξ 2 0 ξ 2 0 ξ 2 ξ 2 ξ 2 ξ 2 1 ξ 1 0 ξ 2 ξ 1 0 ξ ξ 2 ξ 2 0 ξ 1 5 Conclusion and Discussion In this paper, we show that for a regular 4 n 1 minimum aberration design on GF(4), if we map all the elements of the designs to Z 4 using a unified transformation, we cannot get a design with β n = 0. However, if we map different columns with different transformations, we can choose an appropriate map to make many transformed 4 n k designs mirror-symmetric, which means β j s of the transformed designs are all zeros for odd j. Such results can help find designs with better β-wordlength patterns. Generalizing these properties to all 4 n k designs seems challenging, but is well worth further research.

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