Science in China: Series A M athematics 2006 Vol. 49 No. 12 1816–1829
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DOI: 10.1007/s11425-006-2032-2
Bounds on the maximum numbers of clear two-factor interactions for 2(n1+n2)−(k1+k2) fractional factorial split-plot designs ZI Xuemin, ZHANG Runchu & LIU Minqian Department of Statistics, School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China Correspondence should be addressed to Liu Minqian (email:
[email protected]) Received July 14, 2005; accepted July 12, 2006
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 III or IV 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:
1
fractional factorial design, split-plot, clear effects criterion, resolution.
Introduction
Two-level fractional factorial (FF) designs are very important in factor screening experiments and many scientific investigations. A 2n−k design denotes a regular twolevel FF design with n factors and 2n−k runs which is completely determined by k generators. The names of the factors denoted by 1, 2, . . . 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 2n−k design are a set of k words called the generators. The group formed by the k generators is called the defining contrast subgroup. Let Ai denote the number of words of length i in the defining contrast subgroup of a 2n−k design, then the vector W = (A3 , . . . , An ) is called the wordlength pattern of the design. The resolution of a 2n−k design is defined as the smallest i such that Ai > 0 (ref. [1]). A 2n−k design with resolution r is usually denoted by 2n−k . Various criteria have been recommended for selecting regular FF r 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 www.scichina.com
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Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs
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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 n1 , hard-to-change factors and then run all of the level-combinations of the remaining n2 (= n − n1 ) factors in a random order with the n1 factors fixed. This is repeated for each level-combination of the n1 factors. If the design matrix for this experimental setup is identical to a 2n−k FF design, then it is said to be a 2(n1 +n2 )−(k1 +k2 ) FFSP design, where k = k1 + k2 . The n1 and n2 factors are called the whole-plot (WP) and sub-plot (SP) factors, respectively. And there are k1 and k2 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 IV 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 III 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
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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 III or IV 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 III, 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 IV 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 = n1 + n2 , k = k1 + k2 , p1 = n1 − k1 , p2 = n2 − k2 , p = n − k and the levels be labelled as +1 and −1 in each column. Let c1 , . . . , cp be p independent 2p × 1 columns. As we know, a saturated design with 2p runs and 2p − 1 columns can be obtained by taking all products of the p independent columns. Denoting the set of 2p − 1 columns in this saturated design by H(c1 , . . . , cp ), we can obtain a 2n−k design with resolution at least III by selecting n columns from H(c1 , . . . , cp ) (ref. [6]). In order to get a 2(n1 +n2 )−(k1 +k2 ) design, we have to consider two designs H and HW , where H = H(c1 , . . . , cp ), and HW = H(c1 , . . . , cp1 ) is a closed subset of H generated by c1 , . . . , cp1 . According to the structure of an FFSP design, a 2(n1 +n2 )−(k1 +k2 ) design can be generated as follows. First, we choose n1 columns form HW as WP factors with p1 independent columns and then take n2 columns from H\HW as SP factors with p2 independent columns. Denote these n1 and n2 columns by B1 and B2 respectively. Then B1 ∪ B2 corresponds to a 2(n1 +n2 )−(k1 +k2 ) design. Without loss of generality, we assume that the p1 independent columns in B1 are c1 , . . . , cp1 and the p2 independent columns in B2 are cp1 +1 , . . . , cp1 +p2 = cp . Obviously, B1 ∪ B2 corresponds to a 2(n1 +n2 )−(k1 +k2 ) design if and only if ⎧ ⎨ B ⊆H , B ⊆H \H , 1 W 2 W (1) ⎩ |B1 | = n1 , |B2 | = n2 , where |S| means the number of the elements in the set S and Bi contains pi independent columns for i = 1, 2. In this paper, we do not differentiate the factor from the column, and also use the symbol {ci , cj } to denote the 2fi ci cj . Let M (p) be the maximum number of n for which there exists a 2n−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 n1 > M (p1 ), 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 n1 > M (p1 ) or n > M (p). Ref. [23] provided the values of M (p) which are 5, 6, 8, 11, 17, 23, 32∗ , 41∗ and 65∗ , respectively for p = 4, 5, 6, 7, 8, 9, 10, 11 and 12. The last
Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs
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three entries in the list marked with ∗ are the best known bounds on M (p). (n +n )−(k1 +k2 ) Ref. [13] obtained the following results concerning the existence of 2IV1 2 designs with different kinds of clear 2fis. Lemma 1. (n +n )−(k1 +k2 ) (i) There exist 2IV1 2 designs if n1 2p1 −1 and n2 2p−1 − 2p1 −1 . (n +n )−(k1 +k2 ) (ii) There exist 2IV1 2 designs containing clear WP2fis if and only if p1 −2 p−2 p1 −2 n1 2 + 1 and n2 2 −2 . (n +n )−(k1 +k2 ) (iii) There exist 2III1 2 designs containing clear WP2fis if and only if p1 −1 p−1 p1 −1 n1 2 and n2 2 −2 . (n +n )−(k1 +k2 ) (iv) There exist 2IV1 2 designs containing clear WS2fis or SP2fis if and p1 −1 p−2 only if n1 2 and n2 2 − n1 + 1. (n1 +n2 )−(k1 +k2 ) (v) There exist 2III designs containing clear WS2fis or SP2fis if and only if n1 2p1 − 1 and n2 2p−1 − n1 . These conclusions are helpful for confirming the domains of n1 and n2 we are interested in for the two cases of resolutions III and IV. When 2p1 −2 + 1 < n1 2p1 −1 and 2p−2 − 2p1 −2 < n2 2p−1 − 2p1 −1 , resolution IV designs exist but they do not have any clear WP2fi, thus if a design has clear WP2fis for such an n1 and n2 , then it must be of resolution III. When 2p1 −1 < n1 2p1 − 1 and 2p−2 − n1 + 1 < n2 2p−1 − n1 , a design containing clear WS2fis or SP2fis must be of resolution III. 3
(n +n2 )−(k1 +k2 )
2III1
designs with clear WP2fis and WS2fis (n +n )−(k +k )
1 2 designs with clear In this section, we mainly consider the case of 2III1 2 WP2fis and WS2fis. The upper and lower bounds on the maximum numbers of clear (n +n )−(k1 +k2 ) WP2fis and WS2fis for 2III1 2 designs will be derived. A method for con(n +n )−(k1 +k2 ) structing the 2III1 2 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 αW (p1 , p2 ; n1 , n2 ) and αW S (p1 , p2 ; n1 , n2 ) denote the maximum numbers of clear (n +n )−(k1 +k2 ) WP2fis and WS2fis in a 2III1 2 design respectively. The fact that the n1 WP main effects and αW clear WP2fis are not mutually aliased with each other implies αW 2p1 − 1 − n1. And the fact that the n2 SP main effects and αW S clear WS2fis are not mutually aliased with each other implies αW S 2p − 1 − (2p1 − 1) − n2 . Therefore the upper bounds on αW and αW S are established. (n +n )−(k1 +k2 ) Theorem 1. In a 2III1 2 design, the maximum numbers αW of clear WP2fis and αW S of clear WS2fis are bounded above by αW u = 2p1 − 1 − n1 and αW Su = 2p − 2p1 − n2 respectively. (n +n )−(k1 +k2 ) A method for constructing 2III1 2 designs containing clear WP2fis and WS2fis is given below. Let n ˜ j = 2p−j + 2j − 2, for j = 1, . . . , J, where J = p/2 and x denotes the largest integer not exceeding x. It is obviously that n ˜1 > · · · > n ˜ J . When n > n ˜ 1 = 2p−1 , the
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1 2 2III1 2 design has no clear 2fi (ref. [5]). Suppose that n = n ˜ j for some j. Let Hj = H(c1 , . . . , cj ) be the subset of H, generated by c1 , . . . , cj and Hp−j = H(cj+1 , . . . , cp ) be the subset of H, generated by cj+1 , . . . , cp . Let Hi = H(c1 , . . . , ci ), where i = 0, . . . , min(I, j) with I = p1 /2, Hp1 −i = H(cj+1 , . . . , cj+p1 −i ) and H0 = ∅. Let B1 = Hi ∪ Hp1 −i and B2 = C1 ∪ C2 , where C1 = Hj \Hi and C2 = Hp−j \Hp1 −i . Note that here we only consider the situation of Hi ⊆ Hj and Hp1 −i ⊆ Hp−j . Then the design Dij = B1 ∪ B2 contains n = 2p−j + 2j − 2 columns where there exist n1i = 2p1 −i + 2i − 2 WP columns. Since no column belongs to both Hi and Hp1 −i , for any a ∈ Hi and any b ∈ Hp1 −i , it is easily seen that ab is a clear WP2fi. And there is no clear WP2fi within Hi or Hp1 −i . Thus the number of clear WP2fis in Dij is (2i − 1)(2p1 −i − 1) = 2p1 − 1 − n1i , which achieves the upper bound given by Theorem 1. At the same time, the WS2fi ab is clear for any a ∈ Hi and b ∈ C2 or any a ∈ Hp1 −i and b ∈ C1 . Hence the number of clear WS2fis in Dij is (2i − 1)(2p−j − 2p1 −i ) + (2p1 −i − 1)(2j − 2i ). Now consider the case n ˜j n > n ˜ j+1 for some j = 1, . . . , J, where n ˜ J+1 is defined as n ˜ J+1 = 2(2J − 1) + 1. When p is even, n ˜J < n ˜ J+1 , and the case n ˜J > n > n ˜ J+1 in fact does not exist and can be ignored. For some n1i n1 > n1i+1 , i = 0, . . . , min(I, j) and (˜ nj − n) − (n1i − n1 ) 0, where n1I+1 = 2(2I − 1) + 1, suppose D is obtained from Dij by deleting any n1i −n1 columns from Hp1 −i and any (˜ nj −n)−(n1i −n1 ) columns from C2 . Note that when (˜ nj − n) − (n1i − n1 ) < 0, we cannot obtain a corresponding FFSP design. It can be easily verified that n1 − (2i − 1) > 2p1 −i−1 and n − (2j − 1) > 2p−j−1 . Therefore, there is no clear WP2fi among the n1 − (2i − 1) columns selected from Hp1 −i and no clear 2fi among the n − (2j − 1) columns selected from Hp1 −i ∪ C2 . This implies the numbers of clear WP2fis and WS2fis are simply (2i − 1)(n1 − 2i + 1) and (2i − 1)(n2 − 2j + 2i ) + (n1 − 2i + 1)(2j − 2i ), respectively. Consider the case of n1 min(n1I , n1I+1 ). Note that min(n1I , n1I+1 ) is n1I for even p1 and n1I+1 for odd p1 . We can obtain a design D from DIj by taking any n1 /2 columns from HI , any n1 − n1 /2 columns from Hp1 −I and deleting any n ˜ j − n − (n1I − n1 ) columns from C2 . Thus the numbers of clear WP2fis and WS2fis are at least n1 /2(n1 − n1 /2) and n1 /2(n2 − 2j + 2I ) + (n1 − n1 /2)(2j − 2I ), respectively. Next we look at the case n min(˜ nJ , n ˜ J+1 ). Note that min(˜ nJ , n ˜ J+1 ) is n ˜ J for even p and n ˜ J+1 for odd p. For n1 = n1i , the design DiJ can be constructed as follows. Let B1 = Hi ∪ Hp1 −i , for some integer i satisfying max(0, p/2 − p2 ) i p1 /2, where Hi and Hp1 −i are taken from HJ and Hp−J respectively. Let B2 = C1 ∪ C2 , where C1 is a set containing n/2 − (2i − 1) columns from HJ \Hi and C2 is a set containing n − n/2 − (2p1 −i − 1) columns from Hp−J \Hp1 −i . Then for the design DiJ = B1 ∪ B2 , the numbers of clear WP2fis and WS2fis are at least (2i − 1)(2p1 −i − 1) and (2i − 1)(n − n/2 − 2p1 −i + 1) + (2p1 −i − 1)(n/2 − 2i + 1), respectively. For the case n1i > n1 > n1i+1 , a respective design D can be obtained from DiJ by deleting some columns from Hp1 −i and C2 similarly as what we have done. Then the numbers of clear WP2fis and WS2fis are at least (2i − 1)(n1 − 2i + 1) and (2i − 1)(n2 − n/2 + 2i − 1) + (n1 − 2i + 1)(n/2 − 2i + 1), respectively. For
Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs
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the case n1 min(n1I , n1I+1 ), a respective design D = B1 ∪ B2 can be constructed by taking n1 /2 columns from HI and n1 − n1 /2 columns from Hp1 −I to form B1 and taking n/2 − n1 /2 columns from HJ \HI and n2 − n/2 + n1 /2 columns from Hp−J \Hp1 −I to form B2 . It is obvious that there exist n1 /2(n − n1 /2) clear WP2fis and n1 /2(n2 − n/2 + n1 /2) + (n − n1 /2)(n/2 − n1 /2) clear WS2fis in the design D . Summarizing the above results, we have Theorem 2. For i = 0, . . . , min(I, j), j = 1, . . . , J where J = p/2 and I = p1 /2, the lower bounds αWl and αW Sl on the maximum numbers of clear WP2fis and WS2fis are respectively represented as follows. For simplicity, denote si = 2i − 1, sij = 2j − 2i and t = n/2 − n1 /2. ⎧ ⎨ s (n − s ) if n1i n1 > n1i+1 , i 1 i αWl = ⎩ n1 /2(n1 − n1 /2) if n1 min(n1 , n1 ); I
and
αW Sl
I+1
⎧ ⎪ si (n2 − sij ) + (n1 − si )sij ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n1 /2(n2 − sIj ) + (n1 − n1 /2)sIj ⎪ ⎪ ⎪ ⎪ ⎨ =
⎪ ⎪ si (n2 − n/2 + si ) + (n1 − si )(n/2 − si ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n1 /2(n2 − t) + (n − n1 /2)t ⎪ ⎪ ⎪ ⎪ ⎩
if n ˜j n > n ˜ j+1 , n1i n1 > n1i+1 , if n ˜j > n > n ˜ j+1 , n1 min(n1I , n1I+1 ), if n min(˜ nJ , n ˜ J+1 ), n1i n1 > n1i+1 , if n min(˜ nJ , n ˜ J+1 ), n1 min(n1I , n1I+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 = 5 and p1 = 4. The ranges of M (p) < n 2p−1 and M (p1 ) < n1 2p1 −1 are 6 < n 16 and 5 < n1 8, respectively. From Theorems 1 and 2, the bounds for designs with some n and n1 are calculated and shown in Table 1. Table 1 also presents some outcomes for p = 6 and p1 = 4. It can be seen from Table 1 that most entries give αWu = αWl . And for some cases, αW Su = αW Sl 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 = 5 and p1 = 4, we consider the cases of n = n ˜ 2 = 10 and n1 = 6, 8. For n1 = n12 = 6, suppose H = H1 ∪ H2 , where H1 = H(1, 2) and H2 = H(3, 4, 5). Let B1 = H1 ∪H(3, 4) and B2 = H2 \H(3, 4), then B1 ∪B2 corresponds (6+4)−(2+3) to a 2III design, which contains 9 clear WP2fis and 12 clear WS2fis according
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p
p1
5
4
6
4
Table 1
The lower and upper bounds on αW and αW S
n
n1
n2
αWu
αWl
αW Su
αW Sl
8
6
2
9
9
12
6
9
6
3
9
9
13
9
10
6
4
9
9
12
12
10
8
2
7
7
14
14
11
7
4
8
6
12
4
11
8
3
7
7
13
3
12
7
5
8
6
11
5
12
8
4
7
7
12
4
13
7
6
8
6
10
6
13
8
5
7
7
11
5
14
7
7
8
6
9
7
14
8
6
7
7
10
6
15
7
8
8
6
8
8
16
8
8
7
7
8
8
10
6
4
9
9
24∗
12
11
6
5
9
9
30∗
15
11
7
4
8
6
28∗
24 18
12
6
6
9
9
36∗
12
7
5
8
6
35∗
30
13
6
7
9
9
41
21 31
13
7
6
8
6
42
13
8
5
7
7
40∗
35
14
6
8
9
9
40
24
14
8
6
7
7
42
42
15
6
9
9
9
39
27
15
7
8
8
6
40
18
15
8
7
7
7
41
19
16
6
10
9
9
38
30
16
7
9
8
6
39
19
16
8
8
7
7
40
20
to Theorem 2. For n1 = n11 = 8, let B1 = {1} ∪ H(3, 4, 5) and B2 = H1 \{1} = {2, 12}, (8+2)−(4+1) then B1 ∪ B2 corresponds to a 2III design, which contains 7 clear WP2fis and 14 clear WS2fis. For n = 10 and n1 = 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 210−5 designs (see ref. [6]). For n ˜2 > n = 9 > n ˜ 3 and n1 = n12 = 6, let III (6+3)−(2+2) B2 = B2 \{5}, then B1 ∪ B2 is a 2III design containing 9 clear WP2fis and 9 clear WS2fis. 4
(n +n2 )−(k1 +k2 )
2IV1
designs with clear WP2fis and WS2fis
In this section, FFSP designs with resolution IV will be considered. The upper and lower bounds on the respective maximum numbers of clear WP2fis and WS2fis will be
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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 (n +n )−(k +k )
1 2 For a 2IV1 2 design, denote the maximum numbers of clear WP2fis and WS2fis by βW (p1 , p2 ; n1 , n2 ) and βW S (p1 , p2 ; n1 , n2 ) respectively. To get the upper bounds, some notations from refs. [7, 24] are used here. For a 2n−k design D, let IV f = 2p − 1 − n, and ml (D) be the number of 2fis in the lth alias set not containing the main effects, where l = 1, . . . , f . And let Nu = #{1 l f : ml (D) = u} for u 0 be the number of alias sets that contain u 2fis. Then the number of clear 2fis is C(D) = N1 . 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 NU = 0 and the following two lemmas hold, where U = n/2. Lemma 2. (i) For NU = 0, ⎧ ⎨ C = 2n − 3 + 4e + 8e/(n − 5) if n > 5 is odd, 1o C(D) ⎩ C1e = 2n − 2 + 4e + (8e + 2)/(n − 4) if n > 4 is even;
(ii) for NU = 0 and NU−1 > 0, ⎧ ⎨ C = 3(n − 2) if n is odd, 2o C(D) ⎩ C2e = 2n − 3 if n is even; (iii) for NU = 0 and NU−1 = 0, ⎧ ⎨ C = 2n − 5 + 4e + (8e − 10)/(n − 7) 3o C(D) ⎩ C3e = 2n − 4 + 4e + (8e − 4)/(n − 6)
if n > 7 is odd, if n > 6 is even,
where e = 2p−2 + 1 − n. Lemma 3. If n 8, the maximum number β(p, n) of clear 2fis in a 2n−k design IV is bounded above by ⎧ ⎨ min{C , max{C , C }} if n is odd, 1o 2o 3o βu (p, n) = ⎩ min{C1e , max{C2e , C3e }} if n is even. From Lemmas 2 and 3, the upper bound on the maximum number of clear WP2fis (n +n )−(k1 +k2 ) can be derived easily. Similar to the above, for the WP section of a 2IV1 2 n1 −k1 design D, we can regard it as a respective 2IV design. Suppose that f = 2p1 −1−n1 and let Nv be the number of alias sets that contain v WP2fis. Then the number of clear WP2fis is C (D) = N1 , and we have Theorem 3. (i) For NV = 0, where V = n1 /2, ⎧ ⎨ C = 2n − 3 + 4e + 8e /(n − 5) if n1 > 5 is odd, 1 1 1o C (D) ⎩ C = 2n1 − 2 + 4e + (8e + 2)/(n1 − 4) if n1 > 4 is even; 1e
Science in China Series A: Mathematics
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(ii) for NV = 0 and NV −1 > 0, ⎧ ⎨ C = 3(n − 2) if n is odd, 1 1 2o C (D) ⎩ C = 2n1 − 3 if n1 is even; 2e (iii) for NV = 0 and NV −1 = 0, ⎧ ⎨ C = 2n − 5 + 4e + (8e − 10)/(n − 7) if n > 7 is odd, 1 1 1 3o C (D) ⎩ C = 2n1 − 4 + 4e + (8e − 4)/(n1 − 6) if n1 > 6 is even, 3e where e = 2p1 −2 + 1 − n1 . (n +n )−(k1 +k2 ) Theorem 4. If n1 8, the maximum number βW of clear WP2fis in a 2IV1 2 design is bounded above by ⎧ ⎨ min{C , max{C , C }} if n is odd, 1 1o 2o 3o βWu = ⎩ min{C , max{C , C }} if n1 is even. 1e
2e
3e
As for the upper bound on the maximum number of clear WS2fis, we have (n +n )−(k1 +k2 ) Theorem 5. The maximum number βW S of clear WS2fis in a 2IV1 2 design is bounded above by βW Su = [ˆ n(2p − 2p1 − n2 ) − n1 n2 ]/(ˆ n − 1), where n ˆ= min(n1 , n2 ). (n +n )−(k1 +k2 ) Proof. Let D = {d1 , . . . , dn1 , dn1 +1 , . . . , dn1 +n2 } be a 2IV1 2 design having βW S clear WS2fis, and E = {e1 , . . . , eβW S } be the set of clear WS2fis. Because ei is clear, ei ∈ B2 in (1). Now consider all the n1 n2 WS2fis di dj for 1 i n1 and n1 + 1 j n1 + n2 . It is obvious that di dj ∈ B2 because D is of resolution IV. If any di dj ∈ E, then di dj ∈ F , where F = H\(HW ∪ B2 ∪ E). There are 2p − 2p1 − n2 − βW S columns in F . For any two pairs (di1 , dj1 ) and (di2 , dj2 ), where di1 , di2 ∈ B1 and dj1 , dj2 ∈ B2 in (1), if di1 dj1 = di2 dj2 , then dh1 is distinct from dh2 , for h = i, j. Thus for any column in F , at most n ˆ interactions di dj can equal it, where n ˆ = min(n1 , n2 ). Therefore βW S + n ˆ (2p − 2p1 − n2 − βW S ) > n1 n2 , which completes the proof. After deriving the upper bounds on the maximum numbers of clear WP2fis and WS2fis in FFSP designs with resolution IV, we continue to discuss the lower bounds on βW and βW S . 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(I, j), where J = p/2 and I = p1 /2, the lower bounds βWl and βW Sl on the maximum numbers of clear WP2fis and WS2fis are respectively given as follows. For simplicity, denote si = 2i − 2, si = 2p1 −i − 2, sij = 2j − 2i , sIj = 2j − 2I , sij = 2p−j − 2p1 −i and t = n/2 − n1 /2. ⎧ ⎪ if n ˜ 2 n > n ˜ 3 , n12 n1 > n13 , ⎪ 2(n1 − 2) + 1 ⎪ ⎪ ⎪ ⎪ ⎪ si si if n n ˜ 3 , n = n1i , for i = 2, . . . , min(I, j), ⎪ ⎪ ⎪ ⎪ ⎨ s (n − s ) if n ˜ j > n > n ˜ j+1 , n1i > n1 > n1i+1 , i 1 i βWl = ⎪ ⎪ n1 /2(n1 − n1 /2) if n ˜ j > n > n ˜ j+1 , n1 n1I+1 , ⎪ ⎪ ⎪ ⎪ ⎪ if n n ˜ J+1 , n1i > n1 > n1i+1 , ⎪ ⎪ si (n1 − si ) ⎪ ⎪ ⎩ n /2(n − n /2) if n n ˜ , n n ; 1
1
1
J+1
1
1I+1
Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs
and
βW S l =
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1825
2n2
if n ˜ 2 n > n ˜ 3 , n12 n1 > n13 ,
si sij + si sij
if n = n ˜ j , n = n1i ,
si (n2 −sij )+si sij
if n ˜ j > n > n ˜ j+1 , n = n1i ,
si (n2 −sij )+(n1 −si )sij
if n ˜ j > n > n ˜ j+1 , n1i > n1 > n1i+1 ,
n1 /2(n2 − sIj ) + (n1 − n1 /2)sIj
if n ˜ j > n > n ˜ j+1 , n1 n1I+1 ,
si (n/2 − si ) + (n2 − n/2 + si )si
if n n ˜ J+1 , n = n1i ,
(n1 −si )(n/2−si)+(n2 −n/2+si)si
if n n ˜ J+1 , n1i > n1 > n1i+1 ,
n1 /2(n2 − t) + (n1 − n1 /2)t
if n n ˜ J+1 , n1 n1I+1 ,
˜ J+1 = 2(2J − 2) + 1 and n1I+1 = where n ˜ j = 2p−j + 2j − 3, n1i = 2p1 −i + 2i − 3, n I 2(2 − 2) + 1. 4.2 Performance of our construction method (n +n )−(k +k )
1 2 Now let us examine the performance of our construction method for 2IV1 2 designs. Table 2 tabulates the lower and upper bounds for some designs with p = 7 and p1 = 6, p = 8 and p1 = 5, p = 8 and p1 = 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 = 8 and p1 = 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 p
7
8
8
p1
6
5
7
The lower and upper bounds on βW and βW S
n
n1
n2
βWu
βWl
β W Su
β W Sl
20
12
8
46
36
50
48
20
13
7
40
36
51
42
21
13
8
40
36
49
48
22
14
8
38
24
48
16
22
15
7
36
26
49
14 88
28
8
20
18
12
160∗
28
9
19
15
12
171∗
86
29
9
20
15
12
99
88
30
8
22
18
12
176∗
60
30
9
21
15
12
189∗
58
37
9
28
15
12
189
72
28
20
8
95
84
114
112
29
21
8
91
84
113
112
30
20
10
95
84
108
60
30
21
9
91
84
110
54
30
22
8
89
40
112
88
36
20
16
95
84
98
96
36
21
15
91
84
98
90
37
21
16
91
90
97
96
Science in China Series A: Mathematics
1826
(n +n )−(k +k )
1 2 method performs well for constructing 2IV1 2 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 = 7, p1 = 6, n = n ˜ 3 = 21 and n1 = n13 = 13, let 1, . . . , 7 be the 7 independent columns. Let Oaˆ = {1, 2, 3, 123}, Eaˆ = {12, 13, 23}, Oˆb = {4, 5, 6, 7, 456, 457, 467, 567}, Eˆb = {45, 46, 47, 56, 57, 67, 4567}, and Oa = Oaˆ , Ea = Eaˆ , Ob = {4, 5, 6, 456}, Eb = {45, 46, 56}. Suppose D = P ∪ Q, where P = Oaˆ ∪ (45Eaˆ \{1245}) and Q = Oˆb ∪ (12Eˆb ). Then the WP section is B1 = P1 ∪ Q1 = {1, 2, 3, 123, 1345, 2345}∪{4, 5, 6, 456, 1245, 1246, 1256}, where P1 = Oa ∪(45Ea \{1245}) and Q1 = Ob ∪(12Eb ). And the SP section is B2 = (P \P1 )∪(Q\Q1 ) = {7, 457, 467, 567} ∪{1247, 1257, 1267, 124567}. It can be proved that D has resolution IV 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 III or IV. 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 = 2p−2 + 1 and n1 = 2p1 −2 + 1. Let c1 , . . . , cp be the independent columns. Suppose Oc
=
{ct1 · · · cth | where h 1 is odd and 3 t1 < · · · < th p1 },
Ec
=
{ct1 · · · cth | where h 2 is even and 3 t1 < · · · < th p1 },
Ocˆ
=
{ct1 · · · cth | where h 1 is odd and 3 t1 < · · · < th p},
Ecˆ
=
{ct1 · · · cth | where h 2 is even and 3 t1 < · · · < th p}.
Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs
1827
It is obvious that |Oc | = 2p1 −3 and |Ec | = 2p1 −3 −1. Let c1 c2 Ec = {c1 c2 d| d ∈ Ec }. Obviously, we have |c1 c2 Ec | = 2p1 −3 − 1. Suppose D1 = B1 ∪ B2 and B1 = A ∪ C, where A = {c1 , c2 } and C = Oc ∪ {c1 c2 Ec }.
(2)
Clearly, c1 c2 is clear. It is also easy to verify that |B1 | = 2p1 −2 + 1 and for any a ∈ A and c ∈ C, the WP2fi ac is clear. Therefore, the number of clear WP2fis in the design D1 is βWl = 2p1 −1 − 1, the subscript l is used to indicate that βWl provides a lower bound on βW and it has the same meaning for other similar symbols below. Let B2 = {Ocˆ\Oc } ∪ {c1 c2 Ecˆ\c1 c2 Ec }.
(3)
Then |B2 | = 2p−2 − 2p1 −2 . For any a ∈ A and c ∈ B2 , the WS2fi ac is clear. Hence the number of clear WS2fis is 2p−1 − 2p1 −1 . Consider n = n ˜ j = 2p−j + 2j − 3, for j = 3, . . . , J = p/2 and n1 = n1i = 2p1 −i + 2i − 3, for i = 2, . . . , min(j, I), where I = p1 /2. Let a1 , . . . , ai , ai+1 , . . . , aj , b1 , . . . , bp1 −i , bp1 −i+1 , . . . , bp−j be the p independent columns. Without loss of generality, suppose a1 , . . . , ai , b1 , . . . , bp1 −i are the WP factors. Let Oaˆ
=
{at1 · · · ath | where h 1 is odd and 1 t1 < · · · < th j},
Eaˆ
=
{at1 · · · ath | where h 2 is even and 1 t1 < · · · < th j},
Oˆb
=
{bt1 · · · bth | where h 1 is odd and 1 t1 < · · · < th p − j},
Eˆb
=
{bt1 · · · bth | where h 2 is even and 1 t1 < · · · < th p − j}.
Similarly, we denote Oa
=
{at1 · · · ath | where h 1 is odd and 1 t1 < · · · < th i},
Ea
=
{at1 · · · ath | where h 2 is even and 1 t1 < · · · < th i},
Ob
=
{bt1 · · · bth | where h 1 is odd and 1 t1 < · · · < th p1 − i},
Eb
=
{bt1 · · · bth | where h 2 is even and 1 t1 < · · · < th p1 − i}.
Suppose design Dij is given by Dij = P ∪ Q, where P = Oaˆ ∪ (b1 b2 Eaˆ \a1 a2 b1 b2 ) and Q = Oˆb ∪ (a1 a2 Eˆb ).
(4)
Note that |Dij | = 2p−j + 2j − 3 = n. Let the WP section be B1 = P1 ∪ Q1 ,
(5)
where P1 = Oa ∪ (b1 b2 Ea \a1 a2 b1 b2 ) and Q1 = Ob ∪ (a1 a2 Eb ). Then the SP section is B2 = P2 ∪ Q2 ,
(6)
where P2 = P \P1 and Q2 = Q\Q1 . First, we consider the number of clear WP2fis in the design Dij . It can be verified that Dij has resolution IV and pq is clear for any p ∈ P1 and any q ∈ Q1 , where q = a1 a2 b1 b2 . The WP2fi a1 a2 is not clear since Oaˆ is a saturated design of resolution IV, so the number of the clear WP2fis in Dij is βWl = (2i − 2)(2p1 −i − 2). Consequently, the WS2fi pq is clear, if p ∈ P1 and q ∈ Q2 , or p ∈ P2 and q ∈ Q1 , where q = a1 a2 b1 b2 . Hence the number of clear WS2fis is (2i − 2)(2p−j − 2p1 −i ) + (2p1 −i − 2)(2j − 2i ).
1828
Science in China Series A: Mathematics
For n ˜ 2 > n > n ˜ 3 , n12 n1 > n13 and (˜ n2 − n) − (n12 − n1 ) 0, the design is constructed by deleting any n12 − n1 columns from C given in (2) and any (˜ n2 − n) − (n12 − n1 ) columns from B2 given in (3). Therefore, the numbers of clear WP2fis and WS2fis in the design Dij are βWl = 2(n1 − 2) + 1 and βW Sl = 2n2 respectively. For n ˜ j > n > n ˜ j+1 with j = 3, . . . , J J where n ˜ J +1 = 2(2 − 2) + 1 and n1 = n1i with i = 2, . . . , min(j, I), the design can be
derived from deleting any n ˜ j − n columns from Q2 given in (6). Similar to the above, we i p1 −i have βWl = (2 − 2)(2 − 2) and βW Sl = (2i − 2)(n2 − 2j + 2i ) + (2p1 −i − 2)(2j − 2i ). For n ˜ j > n > n ˜ j+1 , n1i > n1 > n1i+1 and n ˜ j − n − (n1i − n1 ) 0, with j = 3, . . . , J I and i = 2, . . . , min(j, I), where n1I+1 = 2(2 − 2) + 1, our design can be constructed by deleting the column a1 a2 b1 b2 and the additional n1i − n − 1 columns from Q1 in (5) and n ˜ j − n − (n1i − n1 ) columns from Q2 in (6). Then the numbers of clear WP2fis and WS2fis are βWl = (2i − 2)(n1 − 2i + 2) and βW Sl = (2i − 2)(n2 − 2j + 2i ) + (n1 − 2i + 2)(2j − 2i ). For the case n1 n1I+1 and nj − n − (n1I+1 − n1 ) 0, we construct a design by selecting n1 /2 columns from P1 and n1 − n1 /2 columns from Q1 and deleting n ˜ j − n − (n1I+1 − n1 ) columns from Q2 . Then the numbers of clear WP2fis and WS2fis are βWl = n1 /2(n1 − n1 /2) and βW Sl = n1 /2(n2 − 2j + 2I ) + (n1 − n1 /2)(2j − 2I ). For n n ˜ J +1 and n1 = n1i with i = 2, . . . , I, we can construct the design by selecting n/2 columns from P , which include 2i − 2 columns in P1 and n − n/2 columns from Q in (4), which include 2p1 −i − 1 columns in Q1 . Thus βWl = (2i − 2)(2p1 −i − 2) and ˜ J +1 and βW Sl = (2i − 2)(n2 − n/2 + 2i − 2) + (2p1 −i − 2)(n/2 − 2i + 2). For n n n1i > n1 > n1i+1 , we obtain our design by selecting n1 − 2i + 2 columns excluding a1 a2 b1 b2 from Q1 , n2 − n/2 + 2i − 2 columns from Q2 , n/2 columns from P in (4) which include 2i − 2 columns in P1 . Hence βWl = (2i − 2)(n1 − 2i + 2) and βW Sl = (2i − 2)(n2 − n/2 + ˜ 1I+1 , the design can be constructed by 2i − 2) + (n1 − 2i + 2)(n/2 − 2i + 2). For n1 n selecting n1 /2 columns from P1 , n/2 − n1 /2 columns from P2 , n1 − n1 /2 columns from Q1 , and n − n/2 − n1 + n1 /2 columns from Q2 . Then βWl = n1 /2(n1 − n1 /2) and βW Sl = n1 /2(n2 − n/2 + n1 /2) + (n1 − n1 /2)(n/2 − n1 /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. 10301015 and 10571093) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20050055038). 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.
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Bounds on the maximum numbers of clear 2fis for 2-level FFSP designs
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