Bounds on the maximum numbers of clear two-factor interactions for 2 (n 1+n 2 ) (k 1 +k 2 ) fractional factorial split-plot designs

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1 1816 Science in China: Series A Mathematics 2006 Vol. 49 No DOI: /s Bounds on the maximum numbers of clear two-factor interactions for 2 (n 1+n 2 ) (k 1 +k 2 ) fractional factorial split-plot designs ZI Xuemin ZHANG Runchu & LIU Minqian Department of Statistics School of Mathematical Sciences and LPMC Nankai University Tianjin China Correspondence should be addressed to Liu Minqian ( mqliu@nankai.edu.cn) Received July ; accepted July Fractional factorial split-plot (FFSP) designs have an important value of investigation for their special structures. There are two types of factors in an FFSP design: the whole-plot (WP) factors and sub-plot (SP) factors which can form three types of two-factor interactions: WP2fi WS2fi and SP2fi. This paper considers FFSP designs with resolution or under the clear effects criterion. It derives the upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis for FFSP designs and gives some methods for constructing the desired FFSP designs. It further examines the performance of the construction methods. Keywords: fractional factorial design split-plot clear effects criterion resolution. 1 Introduction Two-level fractional factorial (FF) designs are very important in factor screening experiments and many scientific investigations. A 2 n k design denotes a regular twolevel FF design with n factors and 2 n k runs which is completely determined by k generators. The names of the factors denoted by or a b... 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 the wordlength. Associated with every 2 n k design are asetofk words called the generators. The group formed by the k generators is called the defining contrast subgroup. Let A i denote the number of words of length i in the defining contrast subgroup of a 2 n k design then the vector W =(A 3...A n )is called the wordlength pattern of the design. The resolution of a 2 n k design is defined as the smallest i such that A i > 0 (ref. [1]). A 2 n k design with resolution r is usually denoted by 2 n k r. Various criteria have been recommended for selecting regular FF designs in different situations. Among them the minimum aberration is a common criterion for selecting appropriate FF designs under the hierarchical assumption (ref. [2]). But when there is no design with resolution V or higher the minimum aberration

2 Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs 1817 criterion does not always lead to the best designs. For this reason ref. [3] proposed the concepts of clear main effects and clear two-factor interactions (2fis) which are not aliased with any other main effect or two-factor interaction (2fi). The clear effects criterion is especially suitable for the case when some prior knowledge is known. If the three-factor or higher-order interactions are negligible the clear effects are estimable. In terms of the estimation capacity the designs containing the most clear main effects and the most clear 2fis are preferred. The existing results on this criterion include refs. [4 13]. In recent years there has been increasing interest in the study of fractional factorial split-plot (FFSP) designs see for example refs. [13 21] for details. If the levels of some of the factors are difficult or expensive to be changed or controlled it may be impractical or even impossible to perform the experimental runs of FF designs in a completely random order. This motivates us to use FFSP designs to meet the special demands. To perform an FFSP design with n factors we often first randomly choose one of the factorial level-settings of these say n 1 hard-to-change factors and then run all of the level-combinations of the remaining n 2 (= n n 1 ) factors in a random order with the n 1 factors fixed. This is repeated for each level-combination of the n 1 factors. If the design matrix for this experimental setup is identical to a 2 n k FF design then it is said to be a 2 (n1+n2) (k1+k2) FFSP design where k = k 1 + k 2.Then 1 and n 2 factors are called the whole-plot (WP) and sub-plot (SP) factors respectively. And there are k 1 and k 2 WP and SP fractional generators respectively. It is better if WP factors are included in the SP fractional generators. But any SP factor cannot be contained in the WP fractional generators (ref. [14]). If one regards an FFSP design as an FF design then the concepts of resolution and clear effects are applied to the former in the usual manner. For FFSP designs we divide the 2fis into three types: WP2fi SP2fi and WS2fi where a WP2fi or SP2fi means a 2fi in which both factors are WP or SP factors and a WS2fi means a 2fi in which one factor is a WP factor and the other is an SP factor. In practice especially in robust parameter experiments FFSP designs can be used in an important case where the WP and SP factors are regarded as the control and noise factors respectively (ref. [17]). As the control by control 2fi and the control by noise 2fi are very important in robust parameter experiments (ref. [22]) the WP2fis and WS2fis deserve particular attention in FFSP designs. In most situations it is reasonable to assume that interactions involving three or more factors are negligible. Then design with resolution V or higher permits the estimation of all the main effects and 2fis. In what follows we focus on the case where the experimenter cannot afford a design with resolution V or higher. A resolution design with the maximum number of clear 2fis allows the joint estimation of the whole main effects and the clear 2fis as many as possible in the presence of other 2fis. It is a desirable design when we are interested in estimating 2fis besides the main effects. For a resolution design we can assume that the magnitude of the main effects is much larger than that of the 2fis. Although the presence of 2fis which are not clear can bias the estimates of the main effects this bias

3 1818 Science in China Series A: Mathematics will not be substantial. Thus in this paper we mainly focus on the problem of clear WP2fis and WS2fis in FFSP designs. The upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis in 2 (n1+n2) (k1+k2) FFSP designs with resolution or will be given. The remainder of the article is organized as follows. We provide the preliminaries in the next section. In sec. 3 we mainly present the bounds on the maximum numbers of clear WP2fis and WS2fis for 2 (n1+n2) (k1+k2) FFSP designs with resolution and examine the performance of our construction methods. In sec. 4 the bounds on the maximum numbers of these two types of clear 2fis for 2 (n1+n2) (k1+k2) FFSP designs with resolution are derived. Sec. 5 contains the summary remarks. For simplicity in the rest parts of the paper we use 2 (n1+n2) (k1+k2) designs to indicate 2 (n1+n2) (k1+k2) FFSP designs. 2 Representation and notation for 2 (n1+n2) (k1+k2) designs Throughout this paper we let n = n 1 + n 2 k = k 1 + k 2 p 1 = n 1 k 1 p 2 = n 2 k 2 p = n k and the levels be labelled as +1 and 1 in each column. Let c 1...c p be p independent 2 p 1 columns. As we know a saturated design with 2 p runs and 2 p 1 columns can be obtained by taking all products of the p independent columns. Denoting the set of 2 p 1 columns in this saturated design by H(c 1...c p ) we can obtain a 2 n k design with resolution at least by selecting n columns from H(c 1...c p ) (ref. [6]). In order to get a 2 (n1+n2) (k1+k2) design we have to consider two designs H and H W whereh = H(c 1...c p ) and H W = H(c 1...c p1 ) is a closed subset of H generated by c 1...c p1. According to the structure of an FFSP design a 2 (n1+n2) (k1+k2) design can be generated as follows. First we choose n 1 columns form H W as WP factors with p 1 independent columns and then take n 2 columns from H\H W as SP factors with p 2 independent columns. Denote these n 1 and n 2 columns by B 1 and B 2 respectively. Then B 1 B 2 corresponds to a 2 (n1+n2) (k1+k2) design. Without loss of generality we assume that the p 1 independent columns in B 1 are c 1...c p1 and the p 2 independent columns in B 2 are c p1+1...c p1+p 2 = c p. Obviously B 1 B 2 corresponds to a 2 (n1+n2) (k1+k2) design if and only if B 1 H W B 2 H \ H W (1) B 1 = n 1 B 2 = n 2 where S means the number of the elements in the set S and B i contains p i independent columns for i =1 2. In this paper we do not differentiate the factor from the column and also use the symbol {c i c j } to denote the 2fi c i c j. Let M(p) be the maximum number of n for which there exists a 2 n k design of resolution V for given p = n k. For a 2 (n1+n2) (k1+k2) design if the number of the WP factors n 1 >M(p 1 ) then not all WP2fis are clear in the design. And if the number of all factors n>m(p) then not all 2fis are clear. Therefore we only need to consider n 1 >M(p 1 )orn>m(p). Ref. [23] provided the values of M(p) whichare and 65 respectively for p = and 12. The last

4 Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs 1819 three entries in the list marked with are the best known bounds on M(p). Ref. [13] obtained the following results concerning the existence of 2 (n1+n2) (k1+k2) designs with different kinds of clear 2fis. Lemma 1. (i) There exist 2 (n1+n2) (k1+k2) designs if n 1 2 p1 1 and n 2 2 p 1 2 p1 1. (ii) There exist 2 (n1+n2) (k1+k2) designs containing clear WP2fis if and only if n 1 2 p1 2 +1andn 2 2 p 2 2 p1 2. (iii) There exist 2 (n1+n2) (k1+k2) designs containing clear WP2fis if and only if n 1 2 p1 1 and n 2 2 p 1 2 p1 1. (iv) There exist 2 (n1+n2) (k1+k2) designs containing clear WS2fis or SP2fis if and only if n 1 2 p1 1 and n 2 2 p 2 n (v) There exist 2 (n1+n2) (k1+k2) designs containing clear WS2fis or SP2fis if and only if n 1 2 p1 1andn 2 2 p 1 n 1. These conclusions are helpful for confirming the domains of n 1 and n 2 we are interested in for the two cases of resolutions and. When 2 p1 2 +1<n 1 2 p1 1 and 2 p 2 2 p1 2 <n 2 2 p 1 2 p1 1 resolution designs exist but they do not have any clear WP2fi thus if a design has clear WP2fis for such an n 1 and n 2 then it must be of resolution. When 2 p1 1 <n 1 2 p1 1and2 p 2 n 1 +1<n 2 2 p 1 n 1 a design containing clear WS2fis or SP2fis must be of resolution. 3 2 (n1+n2) (k1+k2) designs with clear WP2fis and WS2fis In this section we mainly consider the case of 2 (n1+n2) (k1+k2) designs with clear WP2fis and WS2fis. The upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis for 2 (n1+n2) (k1+k2) designs will be derived. A method for constructing the 2 (n1+n2) (k1+k2) designs with the numbers of clear WP2fis and WS2fis attaining the lower bounds will be provided and some instances will be given to illustrate the performance of our bounds. 3.1 Bounds on the maximum numbers of clear WP2fis and WS2fis Let α W (p 1 p 2 ; n 1 n 2 )andα WS (p 1 p 2 ; n 1 n 2 ) denote the maximum numbers of clear WP2fis and WS2fis in a 2 (n1+n2) (k1+k2) design respectively. The fact that the n 1 WP main effects and α W clear WP2fis are not mutually aliased with each other implies α W 2 p1 1 n 1. And the fact that the n 2 SP main effects and α WS clear WS2fis are not mutually aliased with each other implies α WS 2 p 1 (2 p1 1) n 2. Therefore the upper bounds on α W and α WS are established. Theorem 1. In a 2 (n1+n2) (k1+k2) design the maximum numbers α W of clear WP2fis and α WS of clear WS2fis are bounded above by α Wu =2 p1 1 n 1 and α WSu =2 p 2 p1 n 2 respectively. A method for constructing 2 (n1+n2) (k1+k2) designs containing clear WP2fis and WS2fis is given below. Let ñ j =2 p j +2 j 2 for j =1...JwhereJ = p/2 and x denotes the largest integer not exceeding x. It is obviously that ñ 1 > > ñ J.Whenn>ñ 1 =2 p 1 the

5 1820 Science in China Series A: Mathematics 2 (n1+n2) (k1+k2) design has no clear 2fi (ref. [5]). Suppose that n = ñ j for some j. Let H j = H(c 1...c j ) be the subset of H generated by c 1...c j and H p j = H(c j+1...c p ) be the subset of H generated by c j+1...c p. Let H i = H(c 1...c i ) where i =0...min(Ij) withi = p 1 /2 H p1 i = H(c j+1...c j+p1 i) andh 0 =. Let B 1 = H i H p1 i and B 2 = C 1 C 2 where C 1 = H j \H i and C 2 = H p j \H p1 i. Note that here we only consider the situation of H i H j and H p1 i H p j. Then the design D ij = B 1 B 2 contains n =2 p j +2 j 2 columns where there exist n 1i =2 p1 i +2 i 2 WP columns. Since no column belongs to both H i and H p1 i for any a H i and any b H p1 i it is easily seen that ab is a clear WP2fi. And there is no clear WP2fi within H i or H p1 i. Thus the number of clear WP2fis in D ij is (2 i 1)(2 p1 i 1) = 2 p1 1 n 1i which achieves the upper bound given by Theorem 1. At the same time the WS2fi ab is clear for any a H i and b C 2 or any a H p1 i and b C 1. Hence the number of clear WS2fis in D ij is (2 i 1)(2 p j 2 p1 i )+(2 p1 i 1)(2 j 2 i ). Now consider the case ñ j n>ñ j+1 for some j =1...Jwhereñ J+1 is defined as ñ J+1 =2(2 J 1) + 1. When p is even ñ J < ñ J+1 andthecaseñ J >n>ñ J+1 in fact does not exist and can be ignored. For some n 1i n 1 >n 1i+1 i =0...min(Ij)and (ñ j n) (n 1i n 1 ) 0 where n 1I+1 =2(2 I 1) + 1 suppose D is obtained from D ij by deleting any n 1i n 1 columns from H p1 i and any (ñ j n) (n 1i n 1 ) columns from C 2. Note that when (ñ j n) (n 1i n 1 ) < 0 we cannot obtain a corresponding FFSP design. It can be easily verified that n 1 (2 i 1) > 2 p1 i 1 and n (2 j 1) > 2 p j 1. Therefore there is no clear WP2fi among the n 1 (2 i 1) columns selected from H p1 i and no clear 2fi among the n (2 j 1) columns selected from H p1 i C 2. This implies the numbers of clear WP2fis and WS2fis are simply (2 i 1)(n 1 2 i +1) and (2 i 1)(n 2 2 j +2 i )+(n 1 2 i + 1)(2 j 2 i ) respectively. Consider the case of n 1 min(n 1I n 1I+1 ). Note that min(n 1I n 1I+1 )isn 1I for even p 1 and n 1I+1 for odd p 1. We can obtain a design D from D Ij by taking any n 1 /2 columns from H I any n 1 n 1 /2 columns from H p1 I and deleting any ñ j n (n 1I n 1 ) columns from C 2. Thus the numbers of clear WP2fis and WS2fis are at least n 1 /2 (n 1 n 1 /2 ) and n 1 /2 (n 2 2 j +2 I )+(n 1 n 1 /2 )(2 j 2 I ) respectively. Next we look at the case n min(ñ J ñ J+1 ). Note that min(ñ J ñ J+1 )isñ J for even p and ñ J+1 for odd p. For n 1 = n 1i the design D ij can be constructed as follows. Let B 1 = H i H p1 i for some integer i satisfying max(0 p/2 p 2 ) i p 1 /2 where H i and H p1 i are taken from H J and H p J respectively. Let B 2 = C 1 C 2 wherec 1 is a set containing n/2 (2 i 1) columns from H J \H i and C 2 is a set containing n n/2 (2 p1 i 1) columns from H p J \H p1 i. Then for the design D ij = B 1 B 2 the numbers of clear WP2fis and WS2fis are at least (2 i 1)(2 p1 i 1) and (2 i 1)(n n/2 2 p1 i +1)+(2 p1 i 1)( n/2 2 i +1) respectively. For the case n 1i >n 1 >n 1i+1 a respective design D can be obtained from D ij by deleting some columns from H p1 i and C 2 similarly as what we have done. Then the numbers of clear WP2fis and WS2fis are at least (2 i 1)(n 1 2 i +1) and (2 i 1)(n 2 n/2 +2 i 1) + (n 1 2 i +1)( n/2 2 i + 1) respectively. For

6 Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs 1821 the case n 1 min(n 1I n 1I+1 ) a respective design D = B 1 B 2 can be constructed by taking n 1 /2 columns from H I and n 1 n 1 /2 columns from H p1 I to form B 1 and taking n/2 n 1 /2 columns from H J \H I and n 2 n/2 + n 1 /2 columns from H p J \H p1 I to form B 2. It is obvious that there exist n 1 /2 (n n 1 /2 ) clear WP2fis and n 1 /2 (n 2 n/2 + n 1 /2 )+(n n 1 /2 )( n/2 n 1 /2 ) clear WS2fis in the design D. Summarizing the above results we have Theorem 2. For i =0...min(Ij) j =1...J where J = p/2 and I = p 1 /2 the lower bounds α Wl and α WSl on the maximum numbers of clear WP2fis and WS2fis are respectively represented as follows. For simplicity denote s i =2 i 1 s ij =2 j 2 i and t = n/2 n 1 /2. s i (n 1 s i ) if n 1i n 1 >n 1i+1 α Wl = n 1 /2 (n 1 n 1 /2 ) if n 1 min(n 1I n 1I+1 ); and α WSl = s i (n 2 s ij )+(n 1 s i )s ij if ñ j n>ñ j+1 n 1i n 1 >n 1i+1 n 1 /2 (n 2 s Ij )+(n 1 n 1 /2 )s Ij if ñ j >n>ñ j+1 n 1 min(n 1I n 1I+1 ) s i (n 2 n/2 + s i )+(n 1 s i )( n/2 s i ) if n min(ñ J ñ J+1 ) n 1i n 1 >n 1i+1 n 1 /2 (n 2 t)+(n n 1 /2 )t if n min(ñ J ñ J+1 ) n 1 min(n 1I n 1I+1 ). 3.2 Performance of our construction method We now examine the performance of those lower and upper bounds just obtained. For illustration consider the cases of p =5andp 1 = 4. The ranges of M(p) <n 2 p 1 and M(p 1 ) <n 1 2 p1 1 are 6 <n 16 and 5 <n 1 8 respectively. From Theorems 1 and 2 the bounds for designs with some n and n 1 arecalculatedandshownintable 1. Table 1 also presents some outcomes for p =6andp 1 = 4. It can be seen from Table 1 that most entries give α Wu = α Wl. And for some cases α WSu = α WSl or they only differ a little. In this and the following tables the entries of the upper bounds marked with the asterisk ( ) are the total numbers of the respective 2fis rather than the upper bounds given by our theorems. Now let us illustrate the construction method given above in the following example. Example 1. For p =5andp 1 = 4 we consider the cases of n = ñ 2 =10 and n 1 =6 8. For n 1 = n 12 = 6 suppose H = H 1 H 2 whereh 1 = H(1 2) and H 2 = H(3 4 5). Let B 1 = H 1 H(3 4) and B 2 = H 2 \H(3 4) then B 1 B 2 corresponds to a 2 (6+4) (2+3) design which contains 9 clear WP2fis and 12 clear WS2fis according

7 1822 Science in China Series A: Mathematics Table 1 The lower and upper bounds on α W and α WS p p 1 n n 1 n 2 α Wu α Wl α WSu α WSl to Theorem 2. For n 1 = n 11 =8letB 1 = {1} H(3 4 5) and B 2 = H 1 \{1} = {2 12} then B 1 B 2 corresponds to a 2(8+2) (4+1) design which contains 7 clear WP2fis and 14 clear WS2fis. For n =10andn 1 =6 8 comparing the results with the upper bounds derived from Theorem 1 one can check that the upper and lower bounds are identical for WP2fis and WS2fis respectively. And in each of these two designs the sum of the respective numbers of clear WP2fis and WS2fis is the maximum number of clear 2fis for designs (see ref. [6]). For ñ 2 >n=9> ñ 3 and n 1 = n 12 =6let B 2 = B 2 \{5} thenb 1 B 2 is a 2 (6+3) (2+2) design containing 9 clear WP2fis and 9 clear WS2fis. 4 2 (n1+n2) (k1+k2) designs with clear WP2fis and WS2fis In this section FFSP designs with resolution will be considered. The upper and lower bounds on the respective maximum numbers of clear WP2fis and WS2fis will be

8 Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs 1823 derived. And an example will be given to illustrate the performance of our construction method. 4.1 Bounds on the maximum numbers of clear WP2fis and WS2fis For a 2 (n1+n2) (k1+k2) design denote the maximum numbers of clear WP2fis and WS2fis by β W (p 1 p 2 ; n 1 n 2 )andβ WS (p 1 p 2 ; n 1 n 2 ) respectively. To get the upper bounds some notations from refs. [7 24] are used here. For a 2 n k design D let f =2 p 1 n andm l (D) be the number of 2fis in the lth alias set not containing the main effects where l =1...f. And let N u =#{1 l f : m l (D) =u} for u 0 be the number of alias sets that contain u 2fis. Then the number of clear 2fis is C(D) =N 1. From Lemma 4.5 of ref. [7] and Theorem 8 of ref. [11] we can easily obtain that if D contains the maximum number of clear 2fis then N U =0andthe following two lemmas hold where U = n/2. Lemma 2. (i) For N U =0 C 1o =2n 3+4e + 8e/(n 5) if n>5 is odd C(D) C 1e =2n 2+4e + (8e +2)/(n 4) if n>4iseven; (ii) for N U =0andN U 1 > 0 C 2o =3(n 2) if n is odd C(D) C 2e =2n 3 if n is even; (iii) for N U =0andN U 1 =0 C 3o =2n 5+4e + (8e 10)/(n 7) C(D) C 3e =2n 4+4e + (8e 4)/(n 6) if n>7 is odd if n>6iseven where e =2 p 2 +1 n. Lemma 3. If n 8 the maximum number β(p n) of clear 2fis in a 2 n k design is bounded above by min{c 1o max{c 2o C 3o }} if n is odd β u (p n) = min{c 1e max{c 2e C 3e }} if n is even. From Lemmas 2 and 3 the upper bound on the maximum number of clear WP2fis can be derived easily. Similar to the above for the WP section of a 2 (n1+n2) (k1+k2) design D we can regard it as a respective 2 n1 k1 design. Suppose that f =2 p1 1 n 1 and let N v be the number of alias sets that contain v WP2fis. Then the number of clear WP2fis is C (D) =N 1 and we have Theorem 3. (i) For N V =0whereV = n 1/2 C C 1o (D) =2n 1 3+4e + 8e /(n 1 5) if n 1 > 5 is odd C 1e =2n 1 2+4e + (8e +2)/(n 1 4) if n 1 > 4iseven;

9 1824 Science in China Series A: Mathematics (ii) for N V =0andN V 1 > 0 C C 2o (D) =3(n 1 2) if n 1 is odd C 2e =2n 1 3 if n 1 is even; (iii) for N V =0andN V 1 =0 C C 3o (D) =2n 1 5+4e + (8e 10)/(n 1 7) if n 1 > 7 is odd C 3e =2n 1 4+4e + (8e 4)/(n 1 6) if n 1 > 6iseven where e =2 p n 1. Theorem 4. If n 1 8 the maximum number β W of clear WP2fis in a 2 (n1+n2) (k1+k2) design is bounded above by min{c 1o max{c 2oC 3o}} if n 1 is odd β Wu = min{c 1e max{c 2e C 3e }} if n 1 is even. As for the upper bound on the maximum number of clear WS2fis we have Theorem 5. The maximum number β WS of clear WS2fis in a 2 (n1+n2) (k1+k2) design is bounded above by β WSu = [ˆn(2 p 2 p1 n 2 ) n 1 n 2 ]/(ˆn 1) whereˆn = min(n 1 n 2 ). Proof. Let D = {d 1...d n1 d n1+1...d n1+n 2 } be a 2 (n1+n2) (k1+k2) design having β WS clear WS2fis and E = {e 1...e βws } be the set of clear WS2fis. Because e i is clear e i B 2 in (1). Now consider all the n 1 n 2 WS2fis d i d j for 1 i n 1 and n 1 +1 j n 1 + n 2.Itisobviousthatd i d j B 2 because D is of resolution. If any d i d j E thend i d j F wheref = H\(H W B 2 E). There are 2 p 2 p1 n 2 β WS columns in F. For any two pairs (d i1 d j1 )and(d i2 d j2 ) where d i1 d i2 B 1 and d j1 d j2 B 2 in (1) if d i1 d j1 = d i2 d j2 thend h1 is distinct from d h2 forh = i j. Thus for any column in F atmostˆn interactions d i d j can equal it where ˆn =min(n 1 n 2 ). Therefore β WS +ˆn(2 p 2 p1 n 2 β WS ) >n 1 n 2 which completes the proof. After deriving the upper bounds on the maximum numbers of clear WP2fis and WS2fis in FFSP designs with resolution we continue to discuss the lower bounds on β W and β WS. By constructing some designs we obtain these lower bounds which are summarized as follows. The detailed construction method is given in Appendix. Theorem 6. For j =2...J i =2...min(Ij) where J = p/2 and I = p 1 /2 the lower bounds β Wl and β WSl on the maximum numbers of clear WP2fis and WS2fis are respectively given as follows. For simplicity denote s i =2i 2 s i = 2 p1 i 2 s ij =2 j 2 i s Ij =2 j 2 I s ij =2p j 2 p1 i and t = n/2 n 1 /2. 2(n 1 2) + 1 if ñ 2 n>ñ 3 n 1 2 n 1 >n 1 3 β Wl = s i s i if n ñ 3 n = n 1 i fori =2...min(Ij) s i (n 1 s i ) if ñ j >n>ñ j+1 n 1 i >n 1 >n 1 i+1 n 1 /2 (n 1 n 1 /2 ) if ñ j >n>ñ j+1 n 1 n 1 I+1 s i (n 1 s i ) if n ñ J+1 n 1 i >n 1 >n 1 i+1 n 1 /2 (n 1 n 1 /2 ) if n ñ J+1 n 1 n 1 I+1 ;

10 Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs 1825 and 2n 2 if ñ 2 n>ñ 3 n 1 2 n 1 >n 1 3 β WSl = s i s ij + s i s ij if n =ñ j n = n 1 i s i (n 2 s ij )+s i s ij if ñ j >n>ñ j+1 n = n 1 i s i (n 2 s ij )+(n 1 s i )s ij if ñ j >n>ñ j+1 n 1 i >n 1 >n 1 i+1 n 1 /2 (n 2 s Ij )+(n 1 n 1 /2 )s Ij if ñ j >n>ñ j+1 n 1 n 1 I+1 s i ( n/2 s i )+(n 2 n/2 + s i )s i if n ñ J+1 n = n 1 i (n 1 s i )( n/2 s i )+(n 2 n/2 +s i )s i if n ñ J+1 n 1 i >n 1 >n 1 i+1 n 1 /2 (n 2 t)+(n 1 n 1 /2 )t if n ñ J+1 n 1 n 1 I+1 where ñ j =2p j +2 j 3 n 1 i =2 p1 i +2 i 3 ñ J+1 =2(2J 2) + 1 and n 1I+1 = 2(2 I 2) Performance of our construction method Now let us examine the performance of our construction method for 2 (n1+n2) (k1+k2) designs. Table 2 tabulates the lower and upper bounds for some designs with p = 7and p 1 =6p =8andp 1 =5p =8andp 1 = 7 respectively. From this table we can see that most of the lower bounds differ not so much from the respective upper bounds except for the case of WS2fis with p =8andp 1 = 5. Since the precise maximum numbers of clear WP2fis and WS2fis (though they are still not known) are between the respective upper and lower bounds these comparisons reveal that our construction Table 2 The lower and upper bounds on β W and β WS p p 1 n n 1 n 2 β Wu β Wl β WSu β WSl

11 1826 Science in China Series A: Mathematics method performs well for constructing 2 (n1+n2) (k1+k2) designs with clear WP2fis and WS2fis as many as possible. Let us see an example which illustrates the procedure of obtaining these lower bounds. Example 2. For p =7p 1 =6n =ñ 3 =21andn 1 = n 1 3 = 13 let be the 7 independent columns. Let Oâ = { } Eâ = { } Oˆb = { } Eˆb = { } ando a = Oâ E a = Eâ O b = { } E b = { }. Suppose D = P Q wherep = Oâ (45Eâ\{1245}) andq = Oˆb (12Eˆb). Then the WP section is B 1 = P 1 Q 1 = { } { }whereP 1 = O a (45E a \{1245}) and Q 1 = O b (12E b ). And the SP section is B 2 =(P \P 1 ) (Q\Q 1 )={ } { }. It can be proved that D has resolution and the numbers of clear WP2fis and WS2fis are 36 and 48 respectively which have the minor discrepancies with the upper bounds derived from Theorems 4 and 5. 5 Summary remarks This paper explores the clear 2fis problem for FFSP designs and classifies the 2fis in an FFSP design into WP2fis SP2fis and WS2fis. It provides the upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis for 2 (n1+n2) (k1+k2) designs with resolution or. And more importantly some construction methods are developed and the structures of these designs are revealed. This paper extends ref. [6] s results to the FFSP designs and it can be used to find desired FFSP designs under the clear effects criterion. Since the maximum numbers of clear WP2fis and WS2fis for FFSP designs are still not known at present our bounds could be useful for searching these values to some extent. Motivated by a practical need in robust parameter experiments the results here are mainly concentrated on the problem of clear WP2fis and WS2fis. The FFSP designs with as many as possible clear SP2fis may also be required in some situations. For this case the upper and lower bounds on the maximum number of clear SP2fis can also be obtained similarly. We note that it is difficult to obtain good bounds on the maximum number of clear SP2fis following our construction methods. How to derive more accurate bounds on the maximum number of clear SP2fis and what is the relationship between the clear effects and minimum aberration criteria for FFSP designs? These are still open problems for a further study. Appendix Proof of Theorem 6. Consider the case of n =2 p 2 +1 and n 1 =2 p Let c 1...c p be the independent columns. Suppose O c = {c t1 c th where h 1isoddand3 t 1 < <t h p 1} E c = {c t1 c th where h 2isevenand3 t 1 < <t h p 1} Oĉ = {c t1 c th where h 1isoddand3 t 1 < <t h p} Eĉ = {c t1 c th where h 2isevenand3 t 1 < <t h p}.

12 Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs 1827 It is obvious that O c =2 p1 3 and E c =2 p Let c 1c 2E c = {c 1c 2d d E c}. Obviously we have c 1c 2E c =2 p Suppose D 1 = B 1 B 2 and B 1 = A C where A = {c 1c 2} and C = O c {c 1c 2E c}. (2) Clearly c 1c 2 is clear. It is also easy to verify that B 1 =2 p and for any a A and c C thewp2fiac is clear. Therefore the number of clear WP2fis in the design D 1 is β Wl =2 p1 1 1 the subscript l is used to indicate that β Wl provides a lower bound on β W and it has the same meaning for other similar symbols below. Let B 2 = {Oĉ\O c} {c 1c 2Eĉ\c 1c 2E c}. (3) Then B 2 =2 p 2 2 p1 2. For any a A and c B 2theWS2fiac is clear. Hence the number of clear WS2fis is 2 p 1 2 p1 1. Consider n =ñ j =2 p j +2 j 3 for j =3...J = p/2 and n 1 = n 1 i =2 p1 i +2 i 3 for i =2...min(j I) where I = p 1/2. Let a 1...a ia i+1...a jb 1...b p1 ib p1 i+1... b p j be the p independent columns. Without loss of generality suppose a 1...a ib 1...b p1 i are the WP factors. Let Oâ = {a t1 a th where h 1isoddand1 t 1 < <t h j} Eâ = {a t1 a th where h 2isevenand1 t 1 < <t h j} Oˆb = {b t1 b th where h 1isoddand1 t 1 < <t h p j} Eˆb = {b t1 b th where h 2isevenand1 t 1 < <t h p j}. Similarly we denote O a = {a t1 a th where h 1isoddand1 t 1 < <t h i} E a = {a t1 a th where h 2isevenand1 t 1 < <t h i} O b = {b t1 b th where h 1isoddand1 t 1 < <t h p 1 i} E b = {b t1 b th where h 2isevenand1 t 1 < <t h p 1 i}. Suppose design D ij is given by D ij = P Q where P = Oâ (b 1b 2Eâ\a 1a 2b 1b 2)andQ = Oˆb (a 1a 2Eˆb). (4) Note that D ij =2 p j +2 j 3=n. Let the WP section be B 1 = P 1 Q 1 (5) where P 1 = O a (b 1b 2E a\a 1a 2b 1b 2)andQ 1 = O b (a 1a 2E b ). Then the SP section is B 2 = P 2 Q 2 (6) where P 2 = P \P 1 and Q 2 = Q\Q 1. First we consider the number of clear WP2fis in the design D ij. It can be verified that D ij has resolution and pq is clear for any p P 1 and any q Q 1whereq a 1a 2b 1b 2. The WP2fi a 1a 2 is not clear since Oâ is a saturated design of resolution so the number of the clear WP2fis in D ij is β Wl =(2 i 2)(2 p1 i 2). Consequently the WS2fi pq is clear if p P 1 and q Q 2orp P 2 and q Q 1whereq a 1a 2b 1b 2. Hence the number of clear WS2fis is (2 i 2)(2 p j 2 p1 i )+(2 p1 i 2)(2 j 2 i ).

13 1828 Science in China Series A: Mathematics For ñ 2 >n>ñ 3 n 1 2 n 1 >n 1 3 and (ñ 2 n) (n 1 2 n 1) 0 the design is constructed by deleting any n 1 2 n 1 columns from C given in (2) and any (ñ 2 n) (n 1 2 n 1) columns from B 2 given in (3). Therefore the numbers of clear WP2fis and WS2fis in the design D ij are β Wl =2(n 1 2) + 1 and β WSl =2n 2 respectively. For ñ j >n>ñ j+1 with j =3...J where ñ J+1 = 2(2 J 2) + 1 and n 1 = n 1 i with i = 2...min(j I) the design can be derived from deleting any ñ j n columns from Q 2 given in (6). Similar to the above we have β Wl =(2 i 2)(2 p1 i 2) and β WSl =(2 i 2)(n 2 2 j +2 i )+(2 p1 i 2)(2 j 2 i ). For ñ j >n>ñ j+1 n 1 i >n 1 >n 1 i+1 and ñ j n (n 1 i n 1) 0 with j =3...J and i =2...min(j I) where n 1I+1 = 2(2 I 2) + 1 our design can be constructed by deleting the column a 1a 2b 1b 2 and the additional n 1i n 1 columns from Q 1 in (5) and ñ j n (n 1 i n 1) columns from Q 2 in (6). Then the numbers of clear WP2fis and WS2fis are β Wl =(2 i 2)(n 1 2 i +2) and β WSl =(2 i 2)(n 2 2 j +2 i )+(n 1 2 i + 2)(2 j 2 i ). For the case n 1 n 1I+1 and n j n (n 1 I+1 n 1) 0 we construct a design by selecting n 1/2 columns from P 1 and n 1 n 1/2 columns from Q 1 and deleting ñ j n (n 1 I+1 n 1) columns from Q 2. Then the numbers of clear WP2fis and WS2fis are β Wl = n 1/2 (n 1 n 1/2 ) and β WSl = n 1/2 (n 2 2 j +2 I )+(n 1 n 1/2 )(2 j 2 I ). For n ñ J+1 and n 1 = n 1i with i =2...I we can construct the design by selecting n/2 columns from P which include 2 i 2 columns in P 1 and n n/2 columns from Q in (4) which include 2 p1 i 1 columns in Q 1. Thus β Wl = (2 i 2)(2 p1 i 2) and β WSl =(2 i 2)(n 2 n/2 +2 i 2) + (2 p1 i 2)( n/2 2 i +2). For n ñ J+1 and n 1i >n 1 >n 1i+1 we obtain our design by selecting n 1 2 i + 2 columns excluding a 1a 2b 1b 2 from Q 1 n 2 n/2 +2 i 2 columns from Q 2 n/2 columns from P in (4) which include 2 i 2 columns in P 1. Hence β Wl =(2 i 2)(n 1 2 i +2) andβ WSl =(2 i 2)(n 2 n/2 + 2 i 2) + (n 1 2 i +2)( n/2 2 i +2). For n 1 ñ 1I+1 the design can be constructed by selecting n 1/2 columns from P 1 n/2 n 1/2 columns from P 2 n 1 n 1/2 columns from Q 1andn n/2 n 1 + n 1/2 columns from Q 2. Then β Wl = n 1/2 (n 1 n 1/2 ) and β WSl = n 1/2 (n 2 n/2 + n 1/2 ) +(n 1 n 1/2 )( n/2 n 1/2 ). This completes the proof. Acknowledgements The authors cordially thank two anonymous referees for their valuable comments which lead to the improvement of this paper. This work was partially supported by the National Natural Science Foundation of China (Grant Nos and ) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No ). Liu s research was also supported by the Science and Technology Innovation Fund of Nankai University and the Visiting Scholar Program at Chern Institute of Mathematics. References 1 Box G E P Hunter J S. The 2 k p fractional factorial designs. Technometrics : ; Fries A Hunter W G. Minimum aberration 2 k p designs. Technometrics : Wu C F J Chen Y. A graph-aided method for planning two-level experiments when certain interactions are important. Technometrics : Chen J Sun D X Wu C F J. A catalogue of two-level and three-level fractional factorial designs with small runs. Internat Statist Rev : Chen H Hedayat A S. 2 n m designs with resolution or containing clear two-factor interactions. J Statist Plann Inference : Tang B Ma F Ingram D et al. Bounds on the maximum numbers of clear two factor interactions for 2 m p designs of resolution and. Canad J Statist : Wu H Wu C F J. Clear two-factor interactions and minimum aberration. Ann Statist :

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