Mixed-level fractional factorial split-plot designs ... - Springer Link

Metrika (2012) 75:953–962 DOI 10.1007/s00184-011-0361-9

Mixed-level fractional factorial split-plot designs containing clear effects Shengli Zhao · Xiangfei Chen

Received: 14 October 2010 / Published online: 10 June 2011 © Springer-Verlag 2011

Abstract Mixed-level designs are widely used in the practical experiments. When the levels of some factors are difficult to be changed or controlled, fractional factorial split-plot (FFSP) designs are often used. This paper investigates the sufficient and necessary conditions for a 2(n 1 +n 2 )−(k1 +k2 ) 4s1 FFSP design with resolution III or IV to have various clear factorial effects, including two types of main effects and three types of two-factor interaction components. The structures of such designs are shown and illustrated with examples. Keywords

Clear · Mixed-level design · Resolution · Split-plot

1 Introduction Fractional factorial (FF) designs are commonly used in factorial experiments. In factorial investigations, especially those involving physical experiments, there are often both two-level and four-level factors, and then mixed-level designs are used. Such designs can be constructed from two-level designs by the method of replacement, which was first formally introduced by Addelman (1962) and developed by Wu (1989). If the levels of some factors are difficult 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 fractional factorial split-plot (FFSP) designs to meet the special demands (see Box and Jones 1992 for a discussion on split-plot designs in industrial experiments). Clear effects (Wu and Chen 1992) is a popular optimality criterion for selecting designs. Under the weak assumption that interactions involving three or more factors S. Zhao (B) · X. Chen School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China e-mail: [email protected]

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can be ignored, a clear effect can be estimated unbiasedly. When some background knowledge suggests that certain two-factor interactions are potentially important, the designs with clear two-factor interactions are preferred. The corresponding factors can be assigned to the columns so that the potentially important two-factor interactions are estimable. We refer to Wu and Chen (1992) for some examples. In a factorial experiment with both two-level and four-level factors, if the levels of some factors are difficult to be changed or controlled, or the experiment is carried out in more than one steps, mixed-level split-plot designs are attractive to the experimenter. Wu and Chen (1992) mentioned an example in a circuit-pack assembly process. It is suitable to carry out the experiment using a mixed-level split-plot design with clear two-factor interactions if there are both two-level and four-level factors in the example (see Wu and Chen 1992 for details). So, it is desirable to know when a regular mixed-level split-plot design can have clear effects. This paper considers the regular split-plot designs with n two-level factors and a four-level factor and gives a complete classification of the existence of clear effects of such designs. Section 2 gives the notations and definitions. The sufficient and necessary conditions under which a regular FFSP design can have various clear effects are given in Sect. 3. Conclusions are given in Sect. 4. 2 Notations and definitions We first introduce a 2(n 1 +n 2 )−(k1 +k2 ) FFSP design (see Yang et al. 2006 for more details). Write p1 = n 1 − k1 , p2 = n 2 − k2 and p = p1 + p2 , given p independent 2 p × 1 columns, say a1 , . . . , a p1 and b1 , . . . , b p2 . Let H be the saturated design generated by a1 , . . . , a p1 , b1 , . . . , b p2 by taking all the possible products of these p columns. Let Ha be the closed subset of H including a1 , . . . , a p1 and their possible products. Then a 2(n 1 +n 2 )−(k1 +k2 ) design can be generated as follows. First, take n 1 columns from Ha as whole-plot (WP) factors with p1 independent columns, and take n 2 columns from H \Ha as sub-plot (SP) factors with p2 independent columns. Next, denote these n 1 and n 2 columns by B1 = {c1 , . . . , cn 1 } and B2 = {cn 1 +1 , . . . , cn 1 +n 2 }, respectively. And then (B1 , B2 ) corresponds to a 2(n 1 +n 2 )−(k1 +k2 ) design. We now turn to consider an FFSP design with both two-level and four-level factors. For simplicity, this paper considers only the designs with a four-level SP factor. Similar techniques work if the single four-level factor is a WP factor. For a two-level FFSP design (B1 , B2 ), we suppose these exists a triplet {d1 , d2 , d3 } of B2 satisfied d1 d2 d3 = I . Replacing the three two-level factors {d1 , d2 , d3 } with a four-level factor A, we get an FFSP design with a four-level SP factor. Denote such a mixed-level design as 2(n 1 +n 2 )−(k1 +k2 ) 4s1 , where n 1 and n 2 are the numbers of WP and SP two-level factors, respectively. If one regards a 2(n 1 +n 2 )−(k1 +k2 ) 4s1 design as a 2(n 1 +(n 2 +3))−(k1 +(k2 +1)) design, then we have p1 = n 1 − k1 and p2 = (n 2 + 3) − (k2 + 1). Clearly, we always have p2 ≥ 2. Hereafter, denote p1 = n 1 − k1 , p2 = (n 2 + 3) − (k2 + 1) and p = p1 + p2 . According to the discussion above, a 2(n 1 +n 2 )−(k1 +k2 ) 4s1 design D corresponds to a subset C = {c1 , . . . , cn 1 ; cn 1 +1 , . . . , cn 1 +n 2 ; d1 , d2 , d3 }

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of H , where {c1 , . . . , cn 1 } ⊂ Ha are the WP factors, {cn 1 +1 , . . . , cn 1 +n 2 } ⊂ H \Ha are the two-level SP factors, and {d1 , d2 , d3 } ⊂ H \Ha with d1 d2 d3 = I are the three two-level factors which we replace with a four-level factor. As a subset of H, C corresponds to a 2(n 1 +n 2 +3)−(k1 +k2 +1) FF design D as well. The defining contrast subgroup G of D has two types of defining words (see Zhao and Zhang 2008 for details). The first one does not involve any of the d j ’s, which is called the type 0. The second one involves one of the d j ’s, which is called type 1. Let Ai0 and Ai1 be the numbers of type 0 and type 1 words of length i in G, respectively. The resolution of D is defined to be the smallest i such that Ai j (D) is positive for at least one j. In this article, we (n +n )−(k1 +k2 ) 1 use 2R 1 2 4s to denote a 2(n 1 +n 2 )−(k1 +k2 ) 4s1 design with resolution R. In the following, a 2(n 1 +n 2 )−(k1 +k2 ) 4s1 design D is said to be determined by (1) with d1 d2 d3 = I if the four-level factor A of D is obtained from {d1 , d2 , d3 } by replacement. We call d1 , d2 and d3 the main effect components of A. For convenience, we also call {d1 , d2 , d3 } the four-level factor of D. Both of the main effects of the two-level factors and the main effect components of the four-level factor are called the main effect components. For the same reason, both the two-factor interaction of two two-level factors and the two-factor interaction components (2FICs) of a two-level factor and a four-level factor are called 2FICs. We divide the 2FICs into three types: WP2FIC, SP2FIC and WS2FIC, where a WP2FIC (or SP2FIC) means a 2FIC in which both factors are WP (or SP) factors, and similarly a WS2FIC means a 2FIC in which one factor is a WP factor and the other is an SP factor. In this paper, we do not differentiate factor from column, and use the symbols {ci , c j } and {ci , dl } to denote the 2FICs ci c j and ci dl . A main effect component of a factor is said to be clear if it is not aliased with any main effect component of the other factors or any 2FIC. A 2FIC is said to be clear if it is not aliased with any main effect component or any other 2FIC. A main effect or two-factor interaction is said to be clear if all its components are clear. In the following section, we mainly focus on investigating the sufficient and neces(n +n )−(k1 +k2 ) 1 4s designs with R = III or IV have clear sary conditions under which 2R 1 2 main effects or 2FICs. 3 Main results First, we establish the sufficient and necessary conditions under which a (n +n )−(k1 +k2 ) 1 4s design has clear WP main effects or WP2FICs. 2III1 2 (n +n )−(k +k )

1 2 1 Theorem 1 There exist 2III1 2 4s designs containing clear WP main effects p −1 1 and n 2 ≤ 2 p−1 − 2 p1 −1 − 3, where p1 = or WP2FICs if and only if n 1 ≤ 2 n 1 − k1 , p2 = (n 2 + 3) − (k2 + 1) and p = p1 + p2 .

(n +n )−(k +k )

1 2 1 Proof Suppose that a 2III1 2 4s design D is determined by (1) with d1 d2 d3 = I and c1 is clear, we can get c1 ci ∈ Ha \{c1 , . . . , cn 1 } for i = 2, . . . , n 1 , which implies that n 1 − 1 ≤ 2 p1 − 1 − n 1 , i.e., n 1 ≤ 2 p1 −1 . Note that

c1 c j ∈ (H \Ha )\{cn 1 +1 , . . . , cn 1 +n 2 ; d1 , d2 , d3 }

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for j = n 1 + 1, . . . , n 1 + n 2 , and c1 dl ∈ (H \Ha )\{cn 1 +1 , . . . , cn 1 +n 2 ; d1 , d2 , d3 } for l = 1, 2, 3. Since c1 c j and c1 dl are different with each other for j = n 1 + 1, . . . , n 1 + n 2 , l = 1, 2, 3, we have n 2 + 3 ≤ |(H \Ha )\{cn 1 +1 , . . . , cn 1 +n 2 ; d1 , d2 , d3 }| = 2 p − 2 p1 − n 2 − 3, where | · | denotes the number of elements in a set. Thus we can get n 2 ≤ 2 p−1 − 2 p1 −1 −3. If D has a clear WP2FIC, similar arguments can lead to the same conditions. When n 1 = 2 p1 −1 and n 2 = 2 p−1 − 2 p1 −1 − 3, let B1 = {a1 } ∪ H (a2 , . . . , a p1 ), B2 = H (a2 , . . . , a p1 , b1 , . . . , b p2 )\H (a2 , . . . , a p1 ) (n 1 +n )−(k1 +k )

2 and B = (B1 , B2 ), then B is a 2III 2 design with n 1 = 2 p1 −1 and n 2 = p−1 p −1 1 −2 . Replacing {b1 , b2 , b1 b2 } with a four-level factor A in B2 , we obtain a 2 (n +n )−(k1 +k2 ) 1 2III1 2 4s design with n 1 = 2 p1 −1 and n 2 = 2 p−1 − 2 p1 −1 − 3. It is easy to check that both the WP main effect a1 and WP2FICs a1 c are clear for any c ∈ B1 \{a1 }. When n 1 < 2 p1 −1 or (and) n 2 < 2 p−1 − 2 p1 −1 − 3, we can obtain the design containing clear WP main effects or WP2FICs by deleting some columns from B1 \{a1 } or (and) B2 \{b1 , b2 , b1 b2 }. This completes the proof.

By the construction result in the proof of Theorem 1, one can find that when n 1 ≤ (n +n )−(k1 +k2 ) 1 2 p1 −1 and n 2 ≤ 2 p−1 − 2 p1 −1 − 3, there exist 2III1 2 4s designs containing both clear WP main effects and WP2FICs. The following Theorem 2 shows us the (n +n )−(k1 +k2 ) 1 conditions of a 2III1 2 4s design containing clear four-level SP main effect. (n +n )−(k +k )

1 2 1 4s designs containing clear four-level SP Theorem 2 There exist 2III1 2 p main effect if and only if n 1 ≤ 2 1 − 1 and n 2 ≤ 2 p−2 − n 1 − 1, where p1 = n 1 − k1 , p2 = (n 2 + 3) − (k2 + 1) and p = p1 + p2 .

(n +n )−(k +k )

1 2 1 Proof We suppose that a 2III1 2 4s design D is determined by (1) with d1 d2 d3 = I . Since {c1 , . . . , cn 1 } ⊂ Ha , we must have n 1 ≤ 2 p1 − 1. If the fourlevel factor {d1 , d2 , d3 } is clear, then di c j ∈ H \C, i = 1, 2, 3, j = 1, 2, . . . , n 1 + n 2 . Note that the columns di c j , i = 1, 2, 3, j = 1, 2, . . . , n 1 + n 2 , are different with each other, we have 3(n 1 + n 2 ) ≤ |H \C| = 2 p − 1 − n 1 − n 2 − 3, i.e., n 2 ≤ 2 p−2 − n 1 − 1. When n 1 ≤ 2 p1 − 1 and n 2 = 2 p−2 − n 1 − 1, let B1 be any n 1 -subset of Ha with p1 independent columns and

B2 = {b1 , b2 , b1 b2 } ∪ ({b2 } ⊗ (H (a1 , . . . , a p1 , b3 , . . . , b p2 )\B1 )), hereafter, S1 ⊗ S2 = {s1 s2 : s1 ∈ S1 , s2 ∈ S2 } for two sets S1 and S2 , then |B2 | = 2 p−2 −1−n 1 +3. Replacing {b1 , b2 , b1 b2 } with a four-level factor A in B = (B1 , B2 ), (n +n )−(k1 +k2 ) 1 we get a 2III1 2 4s design D with n 1 ≤ 2 p1 − 1 and n 2 = 2 p−2 − n 1 − 1. From the structure of D, we can easily check that A is clear. For n 2 < 2 p−2 − n 1 − 1,

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1 2 1 2III1 2 4s designs containing clear four-level factor can be obtained by deleting some columns from B2 \{b1 , b2 , b1 b2 }. This completes the proof of Theorem 2.

(n +n )−(k +k )

1 2 1 4s design can have clear fourThe above theorem tells us when a 2III1 2 level factor. Its proof shows the structure of such designs and is useful for the purpose (n +n )−(k1 +k2 ) 1 4s design can of construction. Now one may want to know when a 2III1 2 have clear SP two-level main effects, SP2FICs or WS2FICs. The following Theorem 3 provides an answer.

(n 1 +n 2 )−(k1 +k2 ) 1 4s designs containing clear Theorem 3 (a) For p2 ≥ 2, there exist 2III p WS2FICs if and only if n 1 ≤ 2 1 − 1 and n 2 ≤ 2 p−1 − n 1 − 3; (n +n )−(k1 +k2 ) 1 4s designs containing clear SP two-level (b) For p2 ≥ 3, there exist 2III1 2 main effects or SP2FICs if and only if n 1 ≤ 2 p1 − 1 and n 2 ≤ 2 p−1 − n 1 − 3; (n +n )−(k1 +k2 ) 1 4s designs containing clear SP two-level (c) For p2 = 2, there exist 2III1 2 main effects or SP2FICs if and only if n 1 ≤ 2 p1 − 2 and n 2 ≤ 2 p−1 − n 1 − 3, where p1 = n 1 − k1 , p2 = (n 2 + 3) − (k2 + 1) and p = p1 + p2 .

Proof First, we give the proofs of “only if ” parts. For (a) and (b). Because the WP factors belong to Ha , we have n 1 ≤ 2 p1 − 1. We (n 1 +n 2 )−(k1 +k2 ) 1 suppose that a 2III 4s design D has clear WS2FICs (or SP2FICs). Regard(n 1 +n 2 +3)−(k1 +k2 +1) FF design D , the clear WS2FICs (or SP2FICs) of D ing D as a 2III are clear two-factor interactions of D . Thus, we have n 1 + n 2 + 3 ≤ 2 p−1 (Chen and Hedayat 1998). Arguments similar to those in Theorem 2 can be used to show that (n +n )−(k1 +k2 ) 1 4s design has clear SP two-level main effects. n 1 + n 2 + 3 ≤ 2 p−1 if a 2III1 2 For (c). Suppose p2 = 2. Arguments similar to those in parts (a) and (b) can show that n 2 ≤ 2 p−1 − n 1 − 3, so we need only to prove n 1 ≤ 2 p1 − 2. If n 1 = 2 p1 − 1, then every column in Ha is a WP factor. Without loss of generality, we can assume the fourlevel factor is {b1 , b2 , b1 b2 }. Note that H = Ha ∪{b1 , b2 , b1 b2 }∪(Ha ⊗{b1 , b2 , b1 b2 }) and the two-level SP factors must belong to Ha ⊗ {b1 , b2 , b1 b2 }. Thus each two-level SP main effect or SP2FIC is the interaction of a WP factor and a component of the four-level factor and it is not clear. Therefore we have n 1 ≤ 2 p1 − 2. Now we turn to the proof of “if ” parts. It is sufficient to give the construction meth(n +n )−(k1 +k2 ) 1 ods of the 2III1 2 4s designs containing SP two-level main effects, SP2FICs or WS2FICs. For (a). Let B1 = Ha = {c1 , . . . , c2 p1 −1 } and {d1 , d2 , d3 } = {b1 , b2 , b1 b2 }, then d1 d2 d3 = I and c1 d1 ∈ H \Ha . There are altogether 2 p−1 −2 disjoint pairs of columns in H which join {c1 , d1 } to form 2 p−1 − 2 distinct words of length four (see Chen and Hedayat 1998, Corollary 1). Among them, there are 2 p1 − 2 pairs being of the form {ci , c1 d1 ci }, where ci ∈ Ha \{c1 } and c1 d1 ci ∈ H \Ha . The remaining 2 p−1 −2 p1 pairs with both columns in H \Ha are denoted by {esi , eti }, i = 1, . . . , 2 p−1 − 2 p1 . Since both {d2 , c1 d3 } and {d3 , c1 d2 } belong to the latter 2 p−1 − 2 p1 pairs, we can select one of each pair {esi , eti } (i = 1, . . . , 2 p−1 − 2 p1 ) as the elements of B2 such that d2 ∈ B2 and d3 ∈ B2 . By selecting d1 into B2 , we have |B2 | = 2 p−1 − 2 p1 + 1. Replacing (n +n )−(k1 +k2 ) 1 {d1 , d2 , d3 } in (B1 , B2 ) with a four-level factor A, we get a 2III1 2 4s design D with n 1 = 2 p1 − 1 WP factors and n 2 = 2 p−1 − n 1 − 3 two-level SP factors. By the

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structure of D, the WS2FIC c1 d1 of D is clear. For n 1 < 2 p1 −1 and n 2 = 2 p−1 −n 1 −3, we only need to delete some ci ’s (i = 1) from B1 and add the corresponding c1 d1 ci ’s into B2 to get the required designs. Furthermore, for n 2 < 2 p−1 − n 1 − 3, deleting some columns from B2 \{d1 , d2 , d3 } results the demanded designs. For (b). Let B1 be any n 1 -subset of Ha and B2 = {b1 } ∪ ({b1 } ⊗ (Ha \B1 )) ∪ H (b2 , . . . , b p2 ) ∪ (Ha ⊗ H (b2 , . . . , b p2 )), then |B1 | = n 1 ≤ 2 p1 − 1 and |B2 | = 2 p−1 − n 1 . Replacing {b2 , b3 , b2 b3 } in (n +n )−(k1 +k2 ) 1 (B1 , B2 ) with a four-level factor A, we get a 2III1 2 4s design D with n 1 p−1 WP factors and n 2 = 2 − n 1 − 3 two-level SP factors. From the structure of (B1 , B2 ), we can easily see that the two-level SP main effect b1 is clear. For n 2 < (n +n )−(k1 +k2 ) 1 2 p−1 − n 1 − 3, 2III1 2 4s designs containing clear SP main effect b1 can be obtained by deleting some columns from B2 \{b1 , b2 , b3 , b2 b3 }. (n +n )−(k1 +k2 ) 1 In the following, we construct 2III1 2 4s designs containing clear SP2FICs. Let B1 = Ha = {c1 , . . . , c2 p1 −1 }. Since p2 ≥ 3, we can select {d1 , d2 , d3 } = {b1 , b2 , b1 b2 } and cn 1 +1 = b3 , then d1 d2 d3 = I and cn 1 +1 d1 ∈ H \Ha . Also, there are altogether 2 p−1 − 2 disjoint pairs of columns in H which join {cn 1 +1 , d1 } to form 2 p−1 − 2 distinct words of length four. Among them, there are 2 p1 − 1 pairs being of the form {ci , ci cn 1 +1 d1 }, where ci ∈ Ha and ci cn 1 +1 d1 ∈ H \Ha , i = 1, . . . , 2 p1 − 1. The remaining 2 p−1 − 2 p1 − 1 pairs with both columns in H \Ha are denoted by {esi , eti }, i = 1, . . . , 2 p−1 − 2 p1 − 1. Note that both {d2 , d3 cn 1 +1 } and {d3 , d2 cn 1 +1 } belong to the latter 2 p−1 − 2 p1 − 1 pairs. We can select one of each pair {esi , eti } (i = 1, . . . , 2 p−1 − 2 p1 − 1) as the elements of B2 such that {d2 , d3 } ⊂ B2 . We also select cn 1 +1 and d1 as the elements of B2 , then |B2 | = 2 p−1 − 2 p1 + 1. Replacing (n +n )−(k1 +k2 ) 1 4s design {d1 , d2 , d3 } in (B1 , B2 ) with a four-level factor A, we get a 2III1 2 p p−1 1 − n 1 − 3 two-level SP factors. D with n 1 = 2 − 1 WP factors and n 2 = 2 Clearly, the SP2FIC cn 1 +1 d1 is clear in D. For n 1 < 2 p1 − 1 and n 2 = 2 p−1 − n 1 − 3, we only need to delete some ci ’s from B1 and add the corresponding ci cn 1 +1 d1 ’s into B2 to get the required designs. Furthermore, for n 2 < 2 p−1 − n 1 − 3, deleting some columns from B2 \{d1 , d2 , d3 , cn 1 +1 } results the demanded designs. (n 1 +n 2 )−(k1 +k2 ) 1 For (c). Suppose p2 = 2. First, we construct 2III 4s designs containing clear SP2FICs. Let B1 = Ha \{a1 } = {c1 , . . . , c2 p1 −2 }, {d1 , d2 , d3 } = {b1 , b2 , b1 b2 } and cn 1 +1 = a1 b2 , then d1 d2 d3 = I and cn 1 +1 d1 = a1 b1 b2 ∈ H \Ha . Also, there are altogether 2 p−1 − 2 disjoint pairs of columns in H which join {cn 1 +1 , d1 } to form 2 p−1 − 2 distinct words of length four. Among them, there are 2 p1 − 1 pairs being of the form { f i , cn 1 +1 d1 f i }, where f i ∈ Ha and cn 1 +1 d1 f i ∈ H \Ha . The remaining 2 p−1 − 2 p1 − 1 pairs with both columns in H \Ha are denoted by {esi , eti }, i = 1, . . . , 2 p−1 − 2 p1 − 1. Note that {d3 , d2 cn 1 +1 } is among the 2 p1 − 1 pairs because of d2 cn 1 +1 = a1 , and {d2 , d3 cn 1 +1 } is among the 2 p−1 − 2 p1 − 1 pairs. We can select one of each pair {esi , eti } (i = 1, . . . , 2 p−1 − 2 p1 − 1) as the elements of B2 such that d2 ∈ B2 . Also selecting cn 1 +1 , d1 and d3 as the elements of B2 , we have |B2 | = 2 p−1 − 2 p1 + 2. Replacing {d1 , d2 , d3 } in (B1 , B2 ) with a four-level (n +n )−(k1 +k2 ) 1 factor A, we get a 2III1 2 4s design D with n 1 = 2 p1 − 2 WP factors and p−1 − n 1 − 3 two-level SP factors. The SP2FIC cn 1 +1 d1 is clear in D. For n2 = 2

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n 1 < 2 p1 − 2 and n 2 = 2 p−1 − n 1 − 3, we only need to delete some ci ’s from B1 and add the corresponding ci cn 1 +1 d1 ’s into B2 to get the required designs. Furthermore, for n 2 < 2 p−1 − n 1 − 3, deleting some columns from B2 \{d1 , d2 , d3 , cn 1 +1 } results (n +n )−(k1 +k2 ) 1 the 2III1 2 4s designs containing clear SP2FIC cn 1 +1 d1 . Deleting cn 1 +1 from the designs constructed in the last paragraph and adding (n 1 +n 2 )−(k1 +k2 ) 1 cn 1 +1 d1 into it, we get 2III 4s designs with n 1 ≤ 2 p1 − 2 and n 2 ≤ p−1 − n 1 − 3. The SP main effect cn 1 +1 d1 is clear in the designs. The proof is 2 completed. The following example illustrates the construction method in Theorem 3(a). (7+6)−(4+6)

4s1 design with clear WS2FICs. Example 1 Consider the construction of a 2III Then we get n 1 = 7, n 2 = 6, k1 = 4, k2 = 6, p1 = 3 and p2 = 2. Let B1 = Ha = {1, 2, 12, 3, 13, 23, 123}, d1 = 4, d2 = 5, d3 = 45 and c1 = 1. Then c1 d1 = 14 ∈ H \Ha . For the pairs which join {1, 4} to form length four words, the 6 ones containing a column of Ha are {2, 124}, {12, 24}, {3, 134}, {13, 34}, {23, 1234} and {123, 234}, and the other 8 pairs are {5, 145}, {15, 45}, {25, 1245}, {125, 245}, {35, 1345}, {135, 345}, {235, 12345} and {1235, 2345}. Let B2 = {4, 5, 45, 25, 125, 35, 135, 235, 1235}. By replacing {4, 5, 45} in (B1 , B2 ) with a four-level factor, we get a (7+6)−(4+6) 1 2III 4s design with clear WS2FIC {1, 4}. For any FFSP design with resolution IV, all the main effects are clear. The remaining part of this section is devoted to establish the conditions under which (n +n )−(k1 +k2 ) 1 4s designs may contain clear 2FICs. 2IV1 2 (n 1 +n 2 )−(k1 +k2 ) 1 4s designs containing clear WS2FICs or Theorem 4 There exist 2IV SP2FICs if and only if n 1 ≤ 2 p1 −1 and n 2 ≤ 2 p−2 −n 1 −1, where p1 = n 1 −k1 , p2 = (n 2 + 3) − (k2 + 1) and p = p1 + p2 . (n +n )−(k +k )

1 2 1 Proof First, we prove that no 2IV1 2 4s design can have any clear WS2FIC p −1 p−2 − n 1 − 1. Note that the WP section of a or SP2FIC if n 1 > 2 1 or n 2 > 2 (n 1 +n 2 )−(k1 +k2 ) 1 n 1 −k1 n 1 −k1 4s design is a 2IV design and there does not exist 2IV design 2IV p −1 p−2 p−2 − n 1 − 1, we have n 1 + n 2 > 2 − 1. We if n 1 > 2 1 . When n 2 > 2 (n +n )−(k1 +k2 ) 1 4s design D is determined by (1) with d1 d2 d3 = suppose that a 2IV1 2 I and D has a clear SP2FIC cn 1 +1 cn 1 +2 . Since D has resolution IV, we have cn 1 +1 cl1 ∈ H \C, l1 = 1, 2, . . . , n 1 + n 2 , l1 = n 1 + 1, cn 1 +2 cl2 ∈ H \C, l2 = 1, 2, . . . , n 1 + n 2 , l2 = n 1 + 1, n 1 + 2, cn 1 +i dl3 ∈ H \C, i = 1, 2, l3 = 1, 2, 3. Because cn 1 +1 cn 1 +2 is clear, we have cn 1 +1 cn 1 +2 cl4 ∈ H \C, l4 = 1, 2, . . . , n 1 + n 2 , l4 = n 1 + 1, n 1 + 2 and cn 1 +1 cn 1 +2 dl5 ∈ H \C, l5 = 1, 2, 3. The columns cn 1 +1 cl1 , cn 1 +2 cl2 , cn 1 +i dl3 , cn 1 +1 cn 1 +2 cl4 and cn 1 +1 cn 1 +2 dl5 are distinct with each other, therefore, H \C has at least 3(n 1 + n 2 ) + 4 columns. Thus we have 3(n 1 + n 2 ) + 4 ≤ 2 p − 1 − 3 − (n 1 + n 2 ) and n 1 + n 2 ≤ 2 p−2 − 2, which contradicts n 1 + n 2 > 2 p−2 − 1. For D having a clear WS2FIC, the proof is similar. Let E a = {a1 } ∪ ({a1 } ⊗ H (a2 , . . . , a p1 )), E ab = E a ⊗ H (b2 , . . . , b p2 ), B1 = E a and B2 = E ab ∪ {b1 }. Then B = (B1 , B2 ) is a resolution IV 2(n 1 +n 2 )−(k1 +k2 ) design with n 1 = |B1 | = 2 p1 −1 WP factors and n 2 = |B2 | = 2 p−2 − 2 p1 −1 + 1 SP factors.

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For any c ∈ E a ∪ E ab , b1 c is clear in B. Now we can obtain a design B by adding the column a1 b1 b2 into B2 . Because {b1 , a1 b2 } is clear in design B, the new design B has only one length three word b1 (a1 b2 )(a1 b1 b2 ). Thus, by replacing {b1 , a1 b2 , a1 b1 b2 } (n +n )−(k1 +k2 ) 1 with a four-level factor, we can obtain a 2IV1 2 4s design D with n 1 = 2 p1 −1 p−2 and n 2 = 2 − n 1 − 1. Clearly, for any c1 ∈ B1 and c2 ∈ B2 \{b1 , a1 b2 }, b1 c1 is a clear WS2FIC, and b1 c2 is a clear SP2FIC. For convenience, we denote E a = {e1 , . . . , e2 p1 −1 }. When n 1 < 2 p1 −1 and n 2 = p−2 −n 1 −1, let B1∗ = {e1 , . . . , en 1 } and B2∗ = B2 ∪{b1 en 1 +1 , . . . , b1 e2 p1 −1 }, we can 2 get a resolution IV 2(n 1 +n 2 )−(k1 +k2 ) design B ∗ = (B1∗ , B2∗ ) with n 1 = |B1∗ | WP factors and n 2 = |B2∗ | = 2 p−2 − n 1 + 1 SP factors. By adding a1 b1 b2 into B2∗ and replacing (n 1 +n 2 )−(k1 +k2 ) 1 4s design {b1 , a1 b2 , a1 b1 b2 } with a four-level factor, we can obtain a 2IV p−2 − n 1 − 1 two-level SP factors. For any c1 ∈ B1∗ D with n 1 WP factors and n 2 = 2 and c2 ∈ B2∗ \{b1 , a1 b2 }, b1 c1 is a clear WS2FIC, and b1 c2 is a clear SP2FIC. When n 1 < 2 p1 −1 or (and) n 2 < 2 p−2 − n 1 − 1, deleting some suitable columns from B1∗ or (and) B2∗ can result the demanded designs. This completes the proof of Theorem 4. The construction method in the proof of Theorem 4 shows us that when n 1 ≤ 2 p1 −1 (n +n )−(k1 +k2 ) 1 and n 2 ≤ 2 p−2 − n 1 − 1, there exist 2IV1 2 4s designs containing both clear WS2FICs and SP2FICs. The following example illustrates the construction method. (8+7)−(4+7)

4s1 design with clear WS2FICs and Example 2 Suppose we want a 2IV SP2FICs. Then we have n 1 = 8, n 2 = 7, k1 = 4, k2 = 7, p1 = 4 and p2 = 2. Let E a = {1} ∪ ({1} ⊗ H (2, 3, 4)) = {1, 12, 13, 123, 14, 124, 134, 1234}, E ab = E a ⊗ H (6) = {16, 126, 136, 1236, 146, 1246, 1346, 12346}, (8+9)−(4+7)

design. By B1 = E a and B2 = E ab ∪ {5}. Then B = (B1 , B2 ) is a 2IV adding 156 into B2 and replacing {5, 16, 156} with a four-level factor, we obtain a (8+7)−(4+7) 1 2IV 4s design D. It is easy to check that the WS2FIC {5, c1 } and SP2FIC {5, c2 } are clear in D for any c1 ∈ B1 and c2 ∈ B2 \{5, 16}. (6+9)−(2+9)

4s1 design with clear WS2FICs and SP2FICs. Now suppose we need a 2IV We have n 1 = 6, n 2 = 9, k1 = 2, k2 = 9, p1 = 4 and p2 = 2. Let B1∗ = {1, 12, 13, 123, 14, 124} and B2∗ = B2 ∪ {1345, 12345}. Then B ∗ = (B1∗ , B2∗ ) is (6+11)−(2+9) design. Again by adding 156 into B2 and replacing {5, 16, 156} with a 2IV (6+9)−(2+9) 1 4s design D ∗ . And the WS2FIC {5, c1 } a four-level factor, we obtain a 2IV ∗ and SP2FIC {5, c2 } are still clear in D for any c1 ∈ B1∗ and c2 ∈ B2∗ \{5, 16}. (n +n )−(k1 +k2 ) 1 4s design can have clear The remaining question is when a 2IV1 2 WP2FICs. Theorem 5 below gives an answer to it. (n +n )−(k +k )

1 2 1 4s designs containing clear WP2FICs if and Theorem 5 There exist 2IV1 2 p −2 p−2 1 + 1 and n 2 ≤ 2 − 2 p1 −2 − 3, where p1 = n 1 − k1 , p2 = only if n 1 ≤ 2 (n 2 + 3) − (k2 + 1) and p = p1 + p2 .

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1 2 1 Proof First, we will show that no 2IV1 2 4s design can have any clear p −2 p−2 p −2 1 1 + 1 or n 2 > 2 −2 − 3. We suppose that D is a WP2FIC if n 1 > 2 (n +n )−(k1 +k2 ) 1 2IV1 2 4s design having clear WP2FICs and D is determined by (1) with d1 d2 d3 = I . Without loss of generality, we can suppose c1 = a1 , c2 = a2 and the interaction of c1 and c2 is clear, i.e., a1 a2 is clear. Let

H (a3 , . . . , a p1 ) = {α1 , . . . , α2 p1 −2 −1 }, H (a3 , . . . , a p1 , b1 , . . . , b p2 )\H (a3 , . . . , a p1 ) = {α2 p1 −2 , . . . , α2 p−2 −1 } and Si = {αi } ∪ ({αi } ⊗ H (a1 , a2 )) for i = 1, . . . , 2 p−2 − 1. Then we have H = H (a1 , a2 ) ∪

2 p−2 −1

∪ Si

i=1

2 −1 and the columns of C except for c1 and c2 must come from ∪i=1 Si . Note that for p−2 − 1), there is at most one column of Si belonging to C. any given i (1 ≤ i ≤ 2 Otherwise, D would have resolution at most III or a1 a2 could not be clear, which contradict the assumptions that D has resolution IV and a1 a2 is clear. Note that the WP factors {c3 , . . . , cn 1 } must belong to Si for i = 1, . . . , 2 p1 −2 −1 and the SP factors {cn 1 +1 , . . . , cn 1 +n 2 , d1 , d2 , d3 } must belong to Si for i = 2 p1 −2 , . . . , 2 p−2 − 1. Thus we have n 1 ≤ 2 p1 −2 + 1 and n 2 ≤ 2 p−2 − 2 p1 −2 − 3. Let p−2

E ab

B1 = {a1 , a2 } ∪ ({a2 } ⊗ H (a3 , . . . , a p1 )), = {a2 } ⊗ (H (a3 , . . . , a p1 , b1 , . . . , b p2 )\H (a3 , . . . , a p1 )), (n 1 +n )−(k1 +k )

2 and B2 = (E ab \{a2 b1 , a2 b1 b2 }) ∪ {a1 b1 }. Then B = (B1 , B2 ) is a 2IV 2 p −2 p−2 p −2 1 1 design with n 1 = |B1 | = 2 + 1 and n 2 = |B2 | = 2 −2 − 1. Clearly, (a1 b1 )(a2 b2 ) is clear in B. By adding a1 a2 b1 b2 into B, we get a new design B ∗ and (a1 b1 )(a2 b2 )(a1 a2 b1 b2 ) is the only length three word in B ∗ . By replacing (n +n )−(k1 +k2 ) 1 {a1 b1 , a2 b2 , a1 a2 b1 b2 } with a four-level factor in B ∗ , we obtain a 2IV1 2 4s p −2 p−2 p −2 design D with n 1 = 2 1 + 1 and n 2 = 2 − 2 1 − 3. Clearly, a1 a2 is a clear WP2FIC in D. When n 1 < 2 p1 −2 + 1 or (and) n 2 < 2 p−2 − 2 p1 −2 − 3, we only need to delete some suitable columns from B1 or (and) B2 to get the demanded designs. The proof is completed.

(5+9)−(1+9) 1 Example 3 We consider the construction of a 2IV 4s design with clear WP2FICs. We have n 1 = 5, n 2 = 9, k1 = 1, k2 = 9, p1 = 4 and p2 = 2. Let B1 = {1, 2, 23, 24, 234},

E ab = {25, 235, 245, 2345, 26, 236, 246, 2346, 256, 2356, 2456, 23456}, and B2 = {15, 235, 245, 2345, 26, 236, 246, 2346, 2356, 2456, 23456}.

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Then B = (B1 , B2 ) is a 2IV design. By adding 1256 into B2 and replacing (5+9)−(1+9) 1 {15, 26, 1256} with a four-level factor, we get a 2IV 4s design D with B1 and B2 \{15, 26} as its WP factors and two-level SP factors, respectively. The WP2FIC {1, 2} is clear in D. (5+9)−(1+9) 1 4s design in an experiment, if the background knowlWhen we need a 2IV edge suggests that the interaction of two of the WP factors is potentially important, the (5+9)−(1+9) 1 4s design D is a good candidate. We can first allocate the two WP above 2IV factors to the columns 1 and 2, and then the other factors to the remaining columns of D. 4 Conclusions We have discussed the mixed-level FFSP designs with clear main effects or 2FICs. Usually, the levels of a four-level factor are easy to be changed or controlled, so we (n 1 +n 2 )−(k1 +k2 ) 1 can put it in SP section. The sufficient and necessary conditions for 2R 4s designs with R = III or IV to have clear main effects or 2FICs are obtained. However, we may meet mixed-level FFSP designs with the four-level factors in WP section or both WP and SP sections in practice. If we need a 2(n 1 +n 2 )−(k1 +k2 ) 41 design D with the four-level factor in WP section, then D corresponds to a subset C of H similar to that in (1) just with a difference that {d1 , d2 , d3 } ⊂ Ha . The conditions for such designs with resolution III or IV to have clear main effects or 2FICs can also be obtained by methods similar to those used in the paper. Acknowledgments The authors would like to thank the editor and the two referees for the constructive suggestions and comments that lead to a significant improvement over the article. This work was partially supported by the NNSF of China Grants 10826059 and 10901092, the NSF of Shandong Province of China Grant Q2007A05, the China Postdoctoral Science Foundation Grant 20090451292 and the Scientific Research Start-up Foundation of QFNU Grant bsqd07028.

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