Metrika DOI 10.1007/s00184-014-0514-8

Blocked semifoldovers of two-level orthogonal designs Po Yang · Chang-Yun Lin · William Li

Received: 20 December 2013 © Springer-Verlag Berlin Heidelberg 2014

Abstract Follow-up experimentation is often necessary to the successful use of fractional factorial designs. When some effects are believed to be significant but cannot be estimated using an initial design, adding another fraction is often recommended. As the initial design and its foldover (or semifoldover) are usually conducted at different stages, it may be desirable to include a block factor. In this article, we study the blocking effect of such a factor on foldover and semifoldover designs. We consider two general cases for the initial designs, which can be either unblocked or blocked designs. In both cases, we explore the relationships between semifoldover of a design and its corresponding foldover design. More specifically, we obtain some theoretical results on when a semifoldover design can estimate the same two-factor interactions or main effects as the corresponding foldover. These results can be important for those who want to take advantage of the run size savings of a semifoldover without sacrificing the ability to estimate important effects. Keywords Foldover · Semifoldover · Block factor · Nonregular design · Indicator function · Generalized resolution

P. Yang (B) Department of Statistics, University of Manitoba, Winnipeg, MB R3T 2N2, Canada e-mail: [email protected] C.-Y. Lin Department of Applied Mathematics and Institute of Statistics, National Chung Hsing University, Taichung 40227, Taiwan e-mail: [email protected] W. Li Supply Chain and Operations Department, University of Minnesota, Minneapolis, MN 55455, USA e-mail: [email protected]

123

P. Yang et al.

1 Introduction Fractional factorial designs have been commonly used in practice due to their economic run size. A two-level fractional factorial design is called regular if any two effects are either orthogonal or fully aliased; otherwise, it is called a nonregular design. One problem for using a fractional factorial design is that some effects are aliased with others and cannot be estimated. In order to break the aliasing, a useful technique is to add new runs to initial designs. Foldover is a way to obtain the new runs by changing the signs of one or more columns of an initial design. The new runs form a new fraction, called a foldover of the initial design. The combination of the initial runs and the new runs is called a combined foldover design or foldover design in short. We call a set of factors whose signs are reversed in the new fraction a foldover plan. Many textbooks (see Box et al. 2005; Wu and Hamada 2009) have discussed foldover techniques. It is well known that for a resolution I I I design, if the new runs are obtained by changing the signs of all columns, then the foldover design is a resolution I V design. Montgomery and Runger (1996) showed that if the new runs are obtained by changing the signs of the column of an interested main effect, then the foldover resolution I V design can de-alias all the two-factor interactions that contain the main effect. Li and Lin (2003) studied foldover regular designs and Li et al. (2003) investigated foldover nonregular designs. They provided catalogues of optimal foldover designs. When adding only half of the new runs to the initial design, the resulting design is called a combined semifoldover design or semifoldover design in short. When an initial design is a resolution I V design, Barnett et al. (1997) used a practical example to explain that if we choose the half new runs in a special way (see Sect. 2), then the semifoldover design can estimate as many two-factor interactions as the corresponding foldover design. Mee and Peralta (2000) showed that this is true for any regular resolution I V design. Balakrishnan and Yang (2009) and Edwards (2011) extended this result to nonregular designs. Furthermore, Mee and Xiao (2008) also studied the properties of foldovers and semifoldovers of a special class of designs, called even designs (to be defined in Sect. 3). Blocking is an important technique used in experimental designs, which can effectively reduce systematic variations. As new runs in a foldover or semifoldover are usually conducted at a different time, it may be desirable to include a block factor that is usually defined to equal +1 for runs in the initial designs and −1 for runs in the foldover (or semifoldover). Following Ai et al. (2010), we call the block factor an implicit block factor. Ye and Li (2003) studied the theoretical properties of the blocking effects on the foldovers of regular designs. Ai et al. (2010) extended their work and studied the effect of the implicit block factor when the initial design is a blocked regular designs that contain block factors itself. Ou et al. (2011) further extended the work to considering the blocking effect when the initial design is a blocked nonregular design. For semifoldover designs, such blocking effects can be studied in a similar fashion. Due to the nature of semifoldovers, the results can be more complex than those for foldover designs. Among the existing work on the impact of the implicit block factor in a semifoldover, Yang and Li (2014) considered the semifoldover of (unblocked) regular designs. They showed that the main results in Mee and Peralta (2000) still

123

Blocked semifoldovers of two-level orthogonal designs

hold when the implicit block factor is included. Their work was further extended by Yang (2013), who studied the impact of the implicit block factor on the semifoldover designs when the initial design is a blocked regular design. In this article, we focused on the semifoldover designs when the initial designs are both unblocked and blocked designs. While the results in Yang and Li (2014) and Yang (2013) were obtained for regular designs, our results are applicable to both regular and nonregular designs. When the initial designs are unblocked even designs, we extend the main results of Mee and Xiao (2008) and Edwards (2011), which applied to semifoldovers of even design without the implicit block factor. The mathematical tool used in this article is the indicator function. Fontana et al. (2000) showed that any two-level fractional factorial unreplicated design can be uniquely represented by an indicator polynomial function, called indicator function. Ye (2003) proved that this is also true for a replicated fractional factorial design. Indicator function is a useful tool for studying two-level fractional factorial designs, especially for nonregular designs. The application of indicator function for studying nonregular designs can be found in many papers. For example, Cheng et al. (2004) investigated two-level blocked nonregular designs and defined optimal blocking criteria through indicator functions. For more information about indicator function and its applications, see Li et al. (2003), Balakrishnan and Yang (2006a, b), Ou et al. (2011). The remainder of the article is organized as follows. Section 2 introduces the indicator function, generalized resolution, foldover designs, and semifoldover designs. Section 3 discusses the blocking effect of the implicit block factor on semifoldover designs when the initial design is an unblocked design. In Sect. 4, we investigate the impact of the implicit block factor on semifoldover designs when the initial design is a blocked design. Concluding remarks are given in Sect. 5. 2 Indicator functions and follow-up designs Let F be a 2 p full factorial design with n runs. Each run (or point) is denoted by x = (x1 , x2 , . . . , x p ), where xi = ±1, i = 1, 2, . . . , p. Then any two-level fractional factorial design D with p factors can be seen as a collection of points in F. The indicator function, F(x), of D is a function defined on F such that  F(x) =

rx 0

if x ∈ D if x ∈ F \ D,

(1)

where rx is the number of replicates of the point x in D. Assume that P contains all the subsets of {1, 2, . . . , p}. Define X I (x) = i∈I xi , where I ∈ P. In particular, X ∅ (x) = 1. Fontana et al. (2000) and Ye (2003) showed that F(x) has a unique polynomial form, that is,  b I X I (x), (2) F(x) =  p

I ∈P

p where b I = 1/2 x∈D X I (x) for any I ∈ P. Clearly, the constant b∅ = n/2 . For a general two-level factorial design, a word of a design D is defined as a term in (2)

123

P. Yang et al.

which has non-zero coefficient. In particular, for a regular design, (2) contains only the words that are the products of two aliased effects. The coefficients of the words in an indicator function satisfy |b I /b∅ | ≤ 1 for any I ∈ P. For a regular design, |b I /b∅ | = 1 for any I ∈ P which is such that b I = 0. For a non-regular design, |b I /b∅ | can be fractional, which indicates the partial aliasing in the design. In general, |b I /b∅ | indicates the degree of aliasing associated with the word X I (x). The generalized word length L of the word X I (x) is defined as the number of the letters in X I (x) plus 1 − |b I /b∅ |. The generalized resolution, denoted by R, of a factorial design is defined as the smallest generalized word length of the words in F(x) (Li et al. 2003). When changing the signs of one or more factors of D, we get a foldover of the initial design. Denoted by γ a foldover plan which includes the factors whose signs are reversed in the foldover and Dγ the foldover after the foldover plan γ . If γ contains only one factor, that is, γ = {x j }, where j = 1, 2, . . . , or p, then we denote Dγ by Dx j . The design matrix of the combined foldover design has the form 

 D . Dγ

Let P1 be a subset of P such that, for any I ∈ P1 , b I = 0 and X I (x) includes an odd number of the factors in γ . Let P2 be a subset of P such that, for any I ∈ P2 , b I = 0 and X I (x) contains 0 or an even number of the factors in γ . In particular, ∅ ∈ P2 . Now, (2) can be written as 

F(x) =

b I X I (x) +

I ∈P1



b I X I (x).

(3)

b I X I (x).

(4)

I ∈P2

The indicator function of Dγ is F  (x) =



b I X I (x) −

I ∈P2

 I ∈P1

When adding the foldover to the initial design, we get a foldover design. Its indicator function, which can be easily obtained from (3) and (4), is Fo (x) = 2



b I X I (x).

(5)

I ∈P2

For any design A, denoted by Aα X J (x) the fraction which contains the runs in A satisfying α X J (x) = 1, where α = 1 or −1 and X J (x) is a main effect or an interaction. Then, for an initial design D, a semifoldover design is obtained by adding either DγX J (x) or Dγ−X J (x) to D. The design matrix of the semifoldover design has the form   D α X (x) . Dγ J

123

Blocked semifoldovers of two-level orthogonal designs α X (x)

We denote the design by D ∪ Dγ J . Note that the sign “∪” here means the combination of the runs in D and Dγα X J (x) , some runs may be repeated. The alias structure of the design D ∪ Dγα X J (x) is very complicated. This can be seen from its indicator function (see Balakrishnan and Yang 2009). In general, this design cannot estimate as many effects as the corresponding foldover design. Barnett et al. (1997) found that when the two combined designs, as shown below, are considered, a semifoldover design may estimate as many effects as the corresponding foldover design when interactions between three or more factors are negligible. They explained it using a practical example. Mee and Peralta (2000) applied this idea to regular designs. Balakrishnan and Yang (2009) extend this method to general factorial designs. The α X (x) combined design 1, denoted by D X J (x) ∪ Dγ J , consists of the original runs that satisfy X J (x) = 1 and the new runs that satisfy α X J (x) = 1. The combined design 2, denoted by D −X J (x) ∪ Dγα X J (x) , contains the rest of the original runs and the new runs that satisfy α X J (x) = 1. The design matrices of the two combined designs have the forms 

D X J (x) α X (x) Dγ J



 and

 D −X J (x) α X (x) , Dγ J α X (x)

respectively. The two combined designs use the same runs as D ∪ Dγ J . However, the new runs are used twice in the previous case, one time in each combined design. The alias structures for the two combined designs are much less complicated compared to that of D ∪ Dγα X J (x) . We can estimate some effects from combined design 1 and some effects from combined design 2. Together, we usually can estimate more effects in the two combined designs than in D ∪ Dγα X J (x) . In general, if an initial design is a regular design, then the two combined designs are regular designs (see Balakrishnan and Yang 2009). Balakrishnan and Yang (2009) showed that the resulting semifoldover design when X (x) −X (x) adding Dγ J to D has the same alias structure as the one when adding Dγ J to D. Thus, we only need to consider the semifoldover design obtained by adding X (x) Dγ J to D. In this case, following the notation of Mee and Peralta (2000), we say that the semifoldover design is obtained by folding over on γ and subsetting on X J (x), which is called a semifoldover plan and is denoted by f o = γ and ss = X J (x). Barnett et al. (1997) studied a case study of the semifoldover of a resolution I V design. They showed that the two combined designs, obtained by folding over on a main effect and subsetting on a main effect, together can estimate the same two-factor interactions as the corresponding foldover design. Mee and Peralta (2000) showed that this is true for any regular resolution I V design. Balakrishnan and Yang (2009) and Edwards (2011) obtained that the indicator X (x) function of the combined design 1 D X J (x) ∪ Dγ J is F1 (x) =

 I ∈P2

b I X I (x) + X J (x)



b I X (x)

(6)

I ∈P2

123

P. Yang et al. X (x)

and the indicator function of the combined design 2 D −X J (x) ∪ Dγ J F2 (x) =



b I X I (x) − X J (x)

I ∈P2



b I X I (x).

is (7)

I ∈P1

They showed that a semifoldover resolution 4 ≤ R < 5 design obtained by folding over on a main effect and subsetting on a main effect can estimate the same two-factor interactions as the corresponding foldover nonregular design. 3 Blocking on semifoldover of an unblocked design Since the new runs are usually performed at a different time, the initial runs and the new added runs can be seen as in two blocks. If the block effect is thought to be important, we need to include the block factor when selecting a design. Let x B be the implicit block factor and assume that x B equals 1 for the initial design and −1 for the foldover fraction. The design matrix of the blocked foldover design has the form   D 1n , Dγ −1n where 1n is the n-dimensional column vector of ones. We denote the blocked foldover design by D ∪x B Dγ , where “∪x B ” represents the combination of two fractions when x B is included. Let F(x) be the indicator function of D. Then the indicator function of (D, 1n ) defined on   F 12p F −12p is P(x, x B ) =

1 (1 + x B )F(x). 2

(8)

To show this, one can check that, by (1),  P(x, x B ) =

rx 0

if (x, 1) ∈ (D, 1n ) otherwise.

By the definition of the indicator function and the uniqueness of its polynomial form, (8) is the indicator function of (D, 1n ). Similarly, the indicator function of (Dγ , −1n ) is P  (x, x B ) =

123

1 (1 − x B )F  (x). 2

Blocked semifoldovers of two-level orthogonal designs

By the property 2 in Li et al. (2003), the indicator function of D ∪x B Dγ is Po (x, x B ) = P(x, x B ) + P  (x, x B ) 1 1 = (1 + x B )F(x) + (1 − x B )F  (x). 2 2

(9)

The indicator function (9) was obtained by Ye and Li (2003) in their original unpublished version of the article. By (3) and (4), Eq. (9) can be written as   b I X I (x) + x B b I X I (x). (10) Po (x, x B ) = I ∈P2

I ∈P1 X (x)

We now consider the indicator function of the semifoldover design. When Dγ J or −X (x) Dγ J is added to D and x B is included, we get the combined semifoldover design α X (x) D ∪x B Dγ J , where α = 1 or −1. Its design matrix has the form   D 1n , α X (x) Dγ J −1m where m is the number of runs that such that α X J (x) = 1 in Dγ . Its indicator function is given in Proposition 3.1. Proposition 3.1 Let D be a two-level fractional factorial design. Assume that x B is included and a blocked semifoldover design of D is obtained by the semifoldover plan f o = γ and ss = α X J (x). Then the indicator function of the blocked semifoldover design D ∪x B Dγα X J (x) is Ps (x, x B ) =

  1 1 (3 + x B ) b I X I (x) + (1 + 3x B ) b I X I (x) 4 4 I ∈P2 I ∈P1 ⎛ ⎞   1 b I X I (x) − b I X I (x)⎠. + α(1 − x B )X J (x) ⎝ 4 I ∈P2

I ∈P1

Proof Let (3) and (4) be the indicator functions of D and its foldover design Dγ , respectively. By the definition of the indicator function and the uniqueness of its polynomial form, one can check that the indicator function of (Dγα X J (x) , −1m ) is Ps (x, x B ) =

1 (1 − x B )(1 + α X J (x))F  (x). 4

(11)

By the property 2 in Li et al. (2003) and (8), one can get the indicator function of X (x) D ∪x B Dγ J , which is Ps (x, x B ) =

1 1 (1 + x B )F(x) + (1 − x B )(1 + α X J (x))F  (x). 2 4

By (3) and (4), we obtain the result desired.



123

P. Yang et al.

One can see that the alias structure of the blocked semifoldover design is very complicated in general. Thus, we will consider the two combined designs introduced in Sect. 2. When x B is included, the design matrices of the two combined designs are 

D X J (x) α X (x) Dγ J

1n1 −1m



 and

 1n2 , −1m

D −X J (x) α X (x) Dγ J

respectively. Here n 1 and n 2 are the numbers of runs that satisfy α X J (x) = 1 and −α X J (x) = 1 in D, respectively. Clearly, n 1 + n 2 = n. Lemma 3.2 below provides indicator functions of the two combined designs, D X J (x) ∪x B DγX J (x) and D −X J (x) ∪x B DγX J (x) , of the blocked semifoldover design X (x) when adding Dγ J to D with consideration of x B . Lemma 3.2 Let D be a two-level fractional factorial design. Assume that a blocked semifoldover design is obtained by the semifoldover plan f o = γ and ss = X J (x). When the block factor x B is included, the indicator function of the combined design 1 X (x) D X J (x) ∪x B Dγ J is ⎫ ⎧ ⎬  1⎨ P1 (x, x B ) = b I X I (x) F1 (x) + x B (1 + X J (x)) ⎭ 2⎩

(12)

I ∈P1

X (x)

and the indicator function of the combined design 2 D −X J (x) ∪x B Dγ J

is

⎧ ⎞⎫ ⎛ ⎬   1⎨ P2 (x, x B ) = b I X I (x) − X J (x) b I X I (x)⎠ , F2 (x) + x B ⎝ ⎭ 2⎩ I ∈P1

(13)

I ∈P2

where F1 (x) and F2 (x) refer to (6) and (7), respectively. Proof Let (3) and (4) be the indicator functions of D and its foldover design Dγ , respectively. By the definition of the indicator function and the uniqueness of its polynomial form, one can check that the indicator function of (D X J (x) , 1n1 ) is H (x, x B ) =

1 {(1 + x B )(1 + X J (x))F(x)}. 4

(14)

By the property 2 in Li et al. (2003), (14) and (11), we get the indicator function of the combined design 1, which is P1 (x, x B ) = H (x, x B ) + Fs (x, x B ) 1 = {(1 + x B )(1 + X J (x))F(x) + (1 − x B )(1 + X J (x))F  (x)}. 4

123

Blocked semifoldovers of two-level orthogonal designs

This function can be simplified to (12). Similarly, the indicator function of the combined design 2 is P2 (x, x B ) =

1 {(1 + x B )(1 − X J (x))F(x) + (1 − x B )(1 + X J (x))F  (x)}, 4 

which can be simplified to (13). −X (x)

Similarly, when a semifoldover design is obtained by adding Dγ J to D, we can −X (x) obtain that the indicator function of the combined design 1 D X J (x) ∪ Dγ J is P1 (x, x B ) =

 1 (1 + x B )(1 + X J (x))F(x) + (1 − x B )(1 − X J (x))F  (x) 4 −X J (x)

and the indicator function of the combined design 2 D −X J (x) ∪ Dγ P2 (x, x B ) =

is

1 {(1 + x B )(1 − X J (x))F(x) + (1 − x B )(1 − X J (x))F  (x)}. 4

One can simplify the two functions and find that P1 (x, x B ) contains the same words as P2 (x, x B ) and P2 (x, x B ) includes the same words as P1 (x, x B ). Thus, we only consider the semifoldover design obtained by adding the new runs which satisfy X J (x) = 1. In the rest of this article, three usual assumptions for blocked factorial designs will be followed: (1) interactions involving three or more treatment factors are negligible; (2) interactions between block factors and treatment factors are negligible; (3) interactions between block factors are as important as the main effects of block factors. For more information about the assumptions, see Wu and Hamada (2009). Proposition 3.3 Let D be a two-level factorial design with generalized resolution 4 ≤ R < 5. For any foldover plan γ , if a two-factor interaction can be estimated in D ∪ Dγxi , where xi is a main effect, then it can also be estimated in the corresponding blocked semifoldover design D ∪x B Dγxi . Proof We consider the combined design 1 first. The indicator function of the combined design 1, D xi ∪ Dγxi , is (6) when X J (x) = xi . When x B is included, the indicator function of the corresponding combined design 1 D xi ∪x B Dγxi is (12) when X J (x) = of (12) is the indicator function of D xi ∪ Dγxi , and the second xi . Note that the first part part of (12) is x B (1+xi ) I ∈P1 b I X I (x), in which each word contains x B and three or more treatment factors. Since the interactions between a block factor and one or more treatment factors are assumed negligible, we obtain that if a two-factor interaction can be estimated in D xi ∪ Dγxi , then it is also estimable in D xi ∪x B Dγxi . Similarly, one can show that if a two-factor interaction can be estimated in D −xi ∪ xi  Dγ , then it can also be estimated in D xi ∪x B Dγxi . As a special case of Proposition 3.3, for a factorial design with resolution 4 ≤ R < 5, if a two-factor interaction is estimable in a semifoldover design D ∪ Dxxij , where x j is a main effect and j may not equal i, then it is also estimable in the

123

P. Yang et al.

corresponding blocked semifoldover design D ∪x B Dxxij . Since D ∪ Dxxij can estimate the same two-factor interactions as the corresponding foldover design D ∪ Dx j without consideration of x B (see Balakrishnan and Yang 2009; Edwards 2011), we have the following Theorem 3.4. Theorem 3.4 Let D be two-level factorial design with generalized resolution 4 ≤ R < 5. A blocked semifoldover design, D∪x B Dxxij , where xi and x j are main effects and i may not equal j, can estimate the same two-factor interactions as the corresponding unblocked foldover design D ∪ Dx j . Example 1 Consider a six-factor nonregular design D with 24 runs, as shown in Table 1 in “Appendix”. Its indicator function is f (x) =

1 (3 − x1 x2 x3 x4 + x1 x2 x3 x5 − x1 x2 x3 x6 − x1 x2 x4 x5 + x1 x2 x4 x6 8 + x1 x2 x5 x6 − x1 x3 x4 x5 − x1 x3 x4 x6 + x1 x3 x5 x6 + x1 x4 x5 x6 − x2 x3 x4 x5 − x2 x3 x4 x6 − x2 x3 x5 x6 − x2 x4 x5 x6 + x3 x4 x5 x6 ).

One can see that all the two-factor interactions are aliased with two-factor interactions in this design. If f o = {x6 }, then by (5), the indicator function of the unblocked foldover design is: f o (x) =

1 (3 − x1 x2 x3 x4 + x1 x2 x3 x5 − x1 x2 x4 x5 − x1 x3 x4 x5 − x2 x3 x4 x5 ). 4

It follows that all the two-factor interactions that include x6 can be estimated. If a semifoldover design is obtained by the semifoldover plan f o = {x6 } and ss = x5 and x B is not included, then by (6), the indicator function of the combined design 1 D x5 ∪ Dxx65 is f 1 (x) =

1 (3 − x1 x2 x3 x4 + x1 x2 x3 x5 − x1 x2 x4 x5 − x1 x3 x4 x5 − x2 x3 x4 x5 8 + 3x5 − x1 x2 x3 x4 x5 + x1 x2 x3 − x1 x2 x4 − x1 x3 x4 − x2 x3 x4 )

and, by (7), the indicator function of the combined design 2 D −x5 ∪ Dxx65 is f 2 (x) =

1 (3 − x1 x2 x3 x4 + x1 x2 x3 x5 − x1 x2 x4 x5 − x1 x3 x4 x5 − x2 x3 x4 x5 8 + x1 x2 x3 x5 x6 − x1 x2 x4 x5 x6 − x1 x2 x6 + x1 x3 x4 x5 x6 − x1 x3 x6 − x1 x4 x6 + x2 x3 x4 x5 x6 + x2 x3 x6 + x2 x4 x6 − x3 x4 x6 ).

One can check that x1 x6 , x2 x6 , x3 x6 and x4 x6 can be estimated in the combined design 1 and x5 x6 can be estimated in the combined design 2.

123

Blocked semifoldovers of two-level orthogonal designs

If f o = {x6 }, ss = x5 and x B is included, then by (12), the indicator function of the combined design 1 D x5 ∪x B Dxx65 is f 3 (x, x B ) =

1 ( f 1 (x) + x B (1 + x5 )(−x1 x2 x3 x6 + x1 x2 x4 x6 + x1 x2 x5 x6 16 − x1 x3 x4 x6 + x1 x3 x5 x6 + x1 x4 x5 x6 − x2 x3 x4 x6 − x2 x3 x5 x6 − x2 x4 x5 x6 + x3 x4 x5 x6 ))

and, by (13), the indicator function of the combined design 2 D −x5 ∪x B Dxx65 is f 4 (x, x B ) =

1 ( f 2 (x) + x B (−x1 x2 x3 x6 + x1 x2 x4 x6 + x1 x2 x5 x6 − x1 x3 x4 x6 16 + x1 x3 x5 x6 +x1 x4 x5 x6 −x2 x3 x4 x6 −x2 x3 x5 x6 −x2 x4 x5 x6 +x3 x4 x5 x6 − 3x5 + x1 x2 x3 x4 x5 − x1 x2 x3 + x1 x2 x4 + x1 x3 x4 + x2 x3 x4 )).

It follows that, except for the words in f 1 (x) and f 2 (x), all other words in f 3 (x, x B ) and f 4 (x, x B ) contain x B and at least three treatment factors. Since the interactions between block factors and treatment factors are assumed negligible, it follows that x1 x6 , x2 x6 , x3 x6 and x4 x6 can be estimated in the combined design 1 D x5 ∪x B Dxx65 and x5 x6 can be estimated in the combined design 2 D −x5 ∪x B Dxx65 . Therefore, all the two-factor interactions that contain x6 can be estimated in the blocked semifoldover design. A general two-level factorial design is called even if all the words in its indicator function contain even number of letters (Edwards 2011). Many useful designs are even designs. For example, when the number of factors is larger than 11, all regular resolution IV designs with run sizes 8, 16, and 32 are even, and when the number of factors p is a multiple of 4, any orthogonal array with run size 2 p constructed by folding over a Hadamard matrix of order p is even. Mee and Xiao (2008) showed that for any even two-level regular resolution I V design and any foldover plan, the semifoldover design obtained by subsetting on a main effect can estimate the same two-factor interactions as the corresponding foldover design. Edwards (2011) extended this result to even orthogonal arrays. By Proposition 3.3, we can obtain Theorem 3.5 immediately. The result applies to both regular and nonregular initial design. Theorem 3.5 Let E be a two-level even orthogonal array with generalized resolution 4 ≤ R < 5. For any foldover plan γ , a blocked semifoldover design E ∪x B E γxi , where xi is a main effect, can estimate the same two-factor interactions as the corresponding foldover design E ∪ E γ . Theorems 3.4 and 3.5 indicate that, with consideration of the implicit block factor x B , the blocked semifoldover designs with generalized resolution 4 ≤ R < 5 can estimate the same two-factor interactions as the corresponding unblocked foldover designs. For a two-level factorial design with generalized resolution 3 ≤ R < 4, some main effects are aliased with two-factor interactions. Mee and Peralta (2000) mentioned that semifoldover resolution I I I designs usually do not work as well as the corresponding

123

P. Yang et al.

foldover designs. However, sometimes, we may be interested in estimating a main effect. In particular, if a main effect is found to be important in an initial screening experiment and it is expensive to run a foldover design, then a semifoldover design can be used. Balakrishnan and Yang (2009) and Edwards (2011) showed that if a semifoldover resolution 3 ≤ R < 4 design is obtained by folding over on all the factors and subsetting on a main effect xi , then xi can be de-aliased. But it is not true when x B is included, this can be easily seen from the indicator function (13) since xi is aliased with x B . However, we have the following Proposition 3.6. Proposition 3.6 Let D be a two-level factorial design with generalized resolution 3 ≤ R < 4. For any foldover plan γ and a main effect xi , if a main effect x j = xi can be de-aliased in D ∪ Dγxi , then it can also be de-aliased in the corresponding blocked semifoldover design D ∪x B Dγxi . Proof When subsetting on a main effect, the combined design 1 of the semifoldover design cannot estimate any main effect. Hence, we need to consider only combined design 2. Note that all the words in the second part of (13) contain three or more treatment factors except for one word xi x B . Similar to Proposition 3.3, if a main effect x j = xi can be de-aliased in D −xi ∪ Dγxi , then it can also be de-aliased in the corresponding blocked semifoldover design.  4 Blocking on semifoldover of a blocked design Consider a two-level factorial design D with n runs and p factors. We can construct a blocked design by selecting q columns from the p columns and assigning them to the block factors. We call a set of block factors a blocking plan. Assume that the q columns have n q different level combinations. Following Cheng et al. (2004), in order to assign the n q level combinations to n q blocks such that all the n q blocks contain the same number of runs, all n q level combinations must appear the same number of times. In this case, we say that the q columns are eligible. For example, any q columns in an orthogonal array of strength t, where q ≤ t, are eligible for constructing n q = 2q blocks. The remaining k = p − q columns can be assigned to treatment factors. Note that, according to Cheng et al. (2004), the selected q columns should also satisfy that main treatment effects are not aliased with block factors or interactions of block factors and no word in the indicator function of the blocked design contains only block factors. We denote the constructed blocked design by Db . Assume that the eligible q columns {xb1 , xb2 , . . . , xbq } are assigned to the q block factors and the remaining k columns {1, 2, . . . , s}\{xb1 , xb2 , . . . , xbq } are assigned to the treatment factors. Then the indicator function for a blocked nonregular design Db can be written as   b I X I (x) + b I X I (x), S(x) = I ∈P t

I ∈P b

where P t contains all the subsets of {1, 2, . . . , s}\{b1 , b2 , . . . , bq } and P b = P\P t . Note that ∅ ∈ P t . Therefore, for any I ∈ P t , if I = ∅ and b I = 0, then X I (x) consists of only treatment factors, called pure-type words; and for any I ∈ P b , if b I = 0, then

123

Blocked semifoldovers of two-level orthogonal designs

X I (x) consists of both block factors and treatment factors, called mixed-type words. From the discussion in the previous paragraph, we can see that each mixed-type word contains at least two treatment factors. For more discussion about nonregular blocked designs, see Cheng et al. (2004). Given a foldover plan γ , we can write the indicator function of Db as: S(x) =



b I X I (x) +

I ∈P1t



b I X I (x) +

I ∈P2t



b I X I (x) +

I ∈P1b



b I X I (x),

(15)

I ∈P2b

where P1t , P2t ∈ P t , P1b , P2b ∈ P b and are such that for any I ∈ P1t or I ∈ P1b , if b I = 0, then X I (x) contains an odd number of factors in γ ; and for any I ∈ P2t or I ∈ P2b , if b I = 0, then X I (x) contains 0 or an even number of factors in γ . Denote by Db,γ the foldover fraction of Db after a foldover plan γ . If γ contains only one factor, say γ = {xi }, where i = 1, 2, . . . , p, then we denote Db,γ by Db,xi . In this section, we call Db ∪ Db,γ a blocked foldover design and Db ∪x B Db,γ an implicit blocked foldover design. By (5), the indicator function of Db ∪ Db,γ is So (x) = 2





b I X I (x) + 2

I ∈P2t

b I X I (x).

(16)

I ∈P2b

By (10), the indicator function of Db ∪x B Db,γ is 

Q o (x, x B ) =

I ∈P2t X (x)

When Db,γJ



b I X I (x) +

b I X I (x) + x B

 I ∈P1t

I ∈P2b



b I X I (x) + x B

b I X I (x).

I ∈P1b X (x)

is added to the initial design Db , we call Db ∪ Db,γJ

(17) a blocked

X (x)

semifoldover design and Db ∪x B Db,γJ an implicit blocked semifoldover design. By (6), (7), and (15), one can obtain that the indicator function of the combined design 1 X (x) X (x) Db J ∪ Db,γJ of the blocked semifoldover design is ⎛

S1 (x) =



b I X I (x) +

I ∈P2t



⎞   ⎜ ⎟ b I X I (x) + X J (x) ⎝ b I X I (x) + b I X I (x)⎠ I ∈P2t

I ∈P2b

−X J (x)

and the indicator function of the combined design 2 Db

I ∈P2b X (x)

∪ Db,γJ

(18) is

⎛ S2 (x) =

 I ∈P2t

b I X I (x) +

 I ∈P2b

⎞

 ⎜ ⎟ b I X I (x) − X J (x) ⎝ b I X I (x) + b I X I (x)⎠. I ∈P1t

I ∈P1b

(19)

123

P. Yang et al.

If x B is included, then, from (12), (13), and (15), the indicator function of the X (x) X (x) combined design 1 Db J ∪x B Db,γJ is ⎧ ⎪ 1⎨

⎞⎫ ⎪ ⎬  ⎜ ⎟ b I X I (x) + b I X I (x)⎠ Q 1 (x, x B ) = S1 (x) + x B (1 + X J (x)) ⎝ ⎪ 2⎪ ⎩ ⎭ I ∈P t I ∈P b ⎛

1

and the indicator function of the combined design 2

Q 2 (x, x B ) =

⎧ ⎪ 1⎨ 2⎪ ⎩

S2 (x) + x B (

 I ∈P1t

1

−X (x) Db J

∪x B



b I X I (x) +

X (x) Db,γJ

2

is

b I X I (x)

I ∈P1b

⎤⎫ ⎪ ⎬  ⎢ ⎥ −X J (x) ⎣ b I X I (x) + b I X I (x)⎦ , ⎪ ⎭ I ∈P t I ∈P b ⎡

(20)

(21)

2

where S1 (x) and S2 (x) refer to (18) and (19), respectively. −X (x) Similarly, one can check that when adding Db,γ J to the initial design Db , the resulting two combined designs have the same alias structures as the ones above, respectively. Thus, we only consider the blocked semifoldover designs and the implicit X (x) blocked semifoldover designs obtained by adding Db,γJ to Db . Yang (2013) showed that if one foldover plan contains only treatment factors and the other foldover plan contains the same treatment factors and some block factors, then the corresponding two implicit blocked semifoldover designs, obtained by subsetting on the same main treatment effect or interaction of treatment factors, can de-alias the same effects. This result can be extended to nonregular designs, which is given below. Theorem 4.1 Assume that the initial design is a two-level blocked factorial design. Then, with consideration of x B , the two implicit blocked semifoldover designs, obtained by the following two semifoldover plans, respectively, can de-alias the same effects: (1) f o1 = δ and ss = X J (x); (2) f o2 = δ ∪ β and ss = X J (x), where δ contains only treatment factors, β contains only block factors, and X J (x) is a main treatment effect or an interaction of treatment factors. Proof Assume that (20) is the indicator function of the combined design 1, denoted by C1 , of the implicit blocked semifoldover design under the semifoldover plan (1). Clearly, under the semifoldover plan (2), P1t and P2t will not change. Hence, under the semifoldover plan (2), the indicator function of the combined design, denoted by C1 , can be written as

123

Blocked semifoldovers of two-level orthogonal designs

Q 1 (x, x B ) ⎫ ⎧ ⎪ ⎪ ⎬ ⎨   1 = b I X I (x)+x B (1 + X J (x)) b I X I (x) , L(x, x B )+(1+ X J (x)) ⎪ 2⎪ ⎭ ⎩ b b I ∈P2

I ∈P1

(22) where ⎫ ⎧ ⎬   1⎨ b I X I (x) + x B (1 + X J (x)) b I X I (x) , L(x, x B ) = (1 + X J (x)) ⎭ 2⎩ t t I ∈P2



I ∈P1



P1b is such that for any I ∈ P1b , X I (x) contains an odd number of factors in δ ∪ β,   and P2b is such that for any I ∈ P2b , X I (x) contains an even number of factors in   δ ∪ β. Note that P1b ∪ P2b = P1b ∪ P2b = P b . b ∪ P b , P b = P b ∪ P b , P b = P b ∪ P b , and P b = Now, write P1b = P11 12 2 21 22 1 11 12 2   b ∪ P b , where P b and P b , i, j = 1, 2, are such that for any I ∈ P b or P b , X (x) P21 I 22 ij i1 ij i1 

b or P b , X (x) includes an odd number of block factors in β and for any I ∈ Pi2 I i2  b b , P b = includes an even number of block factors in β. It is easy to see that P11 = P21 12 b , P b = P b , and P b = P b . P12 11 22 21 22 Let

G 1 (x, x B ) = (1 + X J (x))



b I X I (x) + x B (1 + X J (x))

b I ∈P22



b I X I (x)

b I ∈P12

and G 2 (x, x B ) = (1 + X J (x))

 b I ∈P21

b I X I (x) + x B (1 + X J (x))



b I X I (x).

b I ∈P11

Then, we can write (20) and (22) as Q 1 (x, x B ) =

1 {L(x, x B ) + G 1 (x, x B ) + G 2 (x, x B )} 2

(23)

and

1 {L(x, x B ) + G 1 (x, x B ) + x B G 2 (x, x B )}, (24) 2 respectively. Note that all the words in G 2 (x, x B ) include block factors. Let X K (x) be a treatment effect which can be a main effect or an interaction of treatment factors. Since interactions between block factors and treatment factors are assumed negligible, if X K (x) can be de-aliased in C1 , then each word in X K (x)G 2 (x, x B ) contains at least one treatment factor. It follows that each word in X K (x)x B G 2 (x, x B ) contains at least Q 1 (x, x B ) =

123

P. Yang et al.

one treatment factor. Thus, by (23) and (24), X K (x) can be de-aliased in C1 . Similarly, if X K (x) can be de-aliased in C1 , then it can also be de-aliased in C1 . Using the similar method, one can show that the combined design 2 under the two semifoldover plans can also de-alias the same treatment effects. We have proved the theorem.  Theorem 4.2 shows that a blocked semifoldover design and an implicit blocked semifoldover design can estimate as many two-factor interactions as the corresponding blocked foldover design and implicit blocked foldover design, respectively. Theorem 4.2 Assume that the length of the shortest pure-type word of a two-level blocked factorial design Db is 4 ≤ L < 5. Then the blocked semifoldover design xi , where xi is a main treatment effect, and the implicit blocked semifoldover Db ∪ Db,x i xi design Db ∪x B Db,x can estimate the same interactions of two treatment factors as i the corresponding blocked foldover design Db ∪ Db,xi and implicit blocked foldover design Db ∪x B Db,xi , respectively. xi Proof We will consider the implicit blocked semifoldover design Db ∪x B Db,x only i xi since the proof for the blocked semifoldover design Db ∪ Db,xi is similar. Let x j be a main treatment effect and xi x j can be de-aliased in the corresponding implicit blocked foldover design Db ∪x BDb,xi . Then, by (17), each mixed-type word in xi x j x B I ∈P b b I X I (x) and xi x j I ∈P b b I X I (x) contains at least two treatment fac1 2   tors, since each word in x B I ∈P b b I X I (x) and I ∈P b b I X I (x) contains at least two 1 2  treatment factors by assumption. Thus, each word in xi x j [xi x B I ∈P b b I X I (x)] and 1  xi x j [xi I ∈P b b I X I (x)] contains at least one treatment factor. Since each word in 2  at least four treatment factors, it follows that each word in I ∈P1t b I X I (x) contains  xi x j x B (1 + X J (x))( I ∈P t b I X I (x)) contains at least one treatment factor. Now, Let 1 X J (x) = xi in (20), we can see that, except for the word xi , each pure-type word contains either at least four letters without xi or at least five letters including xi . Thus except for xi x j and x j , all other words in xi x j Q 1 (x, x B ) are either interactions of at least three treatment factors or interactions of block factors and at least one treatment xi . Since each factor. Under the assumptions, xi x j + x j is estimable in Dbxi ∪x B Db,x i xi main effect is estimable in Db , it follows that xi x j is estimable in Db ∪x B Db,x .  i

Example 2 Consider the design in Example 1. We could use the first column as a block factor to arrange the 24 runs to two blocks. B1 and B2 in the last column of Table 1 represent block 1 and block 2, respectively. Denote x1 by xb . Then the indicator function of the resulting blocked design Db is f b (x) =

1 (3 − xb x2 x3 x4 + xb x2 x3 x5 − xb x2 x3 x6 − xb x2 x4 x5 + xb x2 x4 x6 8 + xb x2 x5 x6 − xb x3 x4 x5 − xb x3 x4 x6 + xb x3 x5 x6 + xb x4 x5 x6 − x2 x3 x4 x5 − x2 x3 x4 x6 − x2 x3 x5 x6 − x2 x4 x5 x6 + x3 x4 x5 x6 ).

Since interactions of block factors and treatment factors are assumed negligible, one can see that all the interactions of two treatment factors are aliased with the interactions of two treatment factors.

123

Blocked semifoldovers of two-level orthogonal designs

If f o = {x6 } and x B is not included, by (16), the indicator function of the blocked foldover design Db ∪ Db,x6 is f ob (x) =

1 (3 − xb x2 x3 x4 + xb x2 x3 x5 − xb x2 x4 x5 − xb x3 x4 x5 − x2 x3 x4 x5 ). 4

Note that all the interactions of two treatment factors that include x6 can be estimated. If a semifoldover design is obtained by the semifoldover plan f o = {x6 } and ss = x5 and x B is not included, then by (18), the indicator function of the combined x5 is design 1 Dbx5 ∪ Db,x 6 f 1b (x) =

1 (3 − xb x2 x3 x4 + xb x2 x3 x5 − xb x2 x4 x5 − xb x3 x4 x5 − x2 x3 x4 x5 8 + 3x5 − xb x2 x3 x4 x5 + xb x2 x3 − xb x2 x4 − xb x3 x4 − x2 x3 x4 ).

x5 is and, by (19), the indicator function of the combined design 2 Db−x5 ∪ Db,x 6

f 2b (x) =

1 (3 − xb x2 x3 x4 + xb x2 x3 x5 − xb x2 x4 x5 − xb x3 x4 x5 − x2 x3 x4 x5 8 + xb x2 x3 x5 x6 − xb x2 x4 x5 x6 − xb x2 x6 + xb x3 x4 x5 x6 − xb x3 x6 − xb x4 x6 + x2 x3 x4 x5 x6 + x2 x3 x6 + x2 x4 x6 − x3 x4 x6 ).

One can check that x2 x6 , x3 x6 and x4 x6 can be estimated in the combined design 1 and x5 x6 can be estimated in the combined design 2. Thus, the blocked semifoldover x5 can estimate the same interactions of two treatment factors as the design Db ∪ Db,x 6 blocked foldover design Db ∪ Db,x6 . If x B is included, by (17), (20) and (21), we can obtain the indicator functions of the corresponding implicit blocked foldover design Db ∪x B Db,x6 and the two combined x5 x5 designs Dbx5 ∪x B Db,x and Db−x5 ∪x B Db,x , respectively. One can similarly check 6 6 that both the implicit blocked foldover design and the implicit blocked semifoldover design can estimate all the interactions of two treatment factors that include x6 . Note that for this initial blocked design, the implicit blocked foldover design Db ∪x B Db,x6 can de-alias the same interactions of two treatment factors as the blocked foldover design Db ∪ Db,x6 . But it is not always the case. For instance, if we use the first two columns as block factors, then one can check that when x B is included, the implicit blocked foldover design cannot de-alias the same number of interactions of treatment factors as the the corresponding blocked foldover design when x B is not included. When folding over on a main effect and subsetting on a different main effect, the resulting implicit blocked semifoldover design may not de-alias as many effects as the corresponding implicit blocked foldover design if we take into account the implicit block factor x B (see Yang 2013). However, when x B is not included, we have Proposition 4.3. Proposition 4.3 Assume that the length of the shortest pure-type word of a two-level blocked factorial design Db is 4 ≤ L < 5. Then a blocked semifoldover design

123

P. Yang et al. x

Db ∪ Db,xj i , where xi and x j are main treatment effects and i = j, can estimate as many two-factor treatment interactions as the corresponding blocked foldover design Db ∪ Db,xi . x

x

Proof Consider the combined fraction Db j ∪ Db,xj i . First, from (18), all the words in x Db j

x ∪ Db,xj i do not contain xi . Second, except for the word x j , each pure-type word x x in Db j ∪ Db,xj i contains at least three letters. Thus, xi + xi x j and xi xk , where k = j, x x are estimable in Db j ∪ Db,xj i . Since xi is estimable in Db , it follows that xi x j and xi xk x are estimable in Db ∪ Db,xj i . 

For a factorial design with the length of the shortest pure-type word 3 ≤ L < 4, Proposition 4.4 below shows that the implicit block factor x B does not affect the estimation of a main effect if it can be estimated in the corresponding blocked semifoldover design without the consideration of x B . Proposition 4.4 Assume that the length of the shortest pure-type word of a two-level blocked factorial design Db is 3 ≤ L < 4. Then a main treatment effect x j that xi , where xi is a main can be de-aliased in the blocked semifoldover design Db ∪ Db,x i treatment effect and i = j, can also be de-aliased in the corresponding implicit xi . blocked semifoldover design Db ∪x B Db,x i Proof Note that a main treatment effect cannot be de-aliased in the combined design 1 of the blocked semifoldover designs when subsetting on a main treatment effect. xi . Since x j can be de-aliased in Db ∪ Consider the combined design 2 Db−xi ∪x B Db,x i xi Db,x and all the words in i ⎧ ⎪ ⎨

⎤⎫ ⎪ ⎬   ⎢ ⎥ xB b I X I (x) + b I X I (x) − xi ⎣ b I X I (x) + b I X I (x)⎦ ⎪ ⎪ ⎩ I ∈P t ⎭ I ∈P t I ∈P b I ∈P b 1

⎡

1

2

xi . contain xi , by (21), x j can be de-aliased in Db ∪x B Db,x i

2



5 Conclusions Both foldover and semifoldover designs have been studied extensively in the literature. However, none of the existing studies considered the impact of the implicit block factor on nonregular designs. This article investigates the properties of such designs. We first derived the indicator functions of the semifoldover designs in 3.1, and then obtained the two indictor functions of the two combined designs in (12) and (13). When the initial design is unblocked with the resolution of 4 ≤ L < 5, we found that with the addition of the implicit block factor, the resulting blocked semifoldover designs can estimate the same two-factor interactions as the corresponding unblocked semifoldover designs. If the foldover and subsetting are on main effects, then the blocked semifoldover design can estimate the same two-factor interactions as the corresponding unblocked foldover designs.

123

Blocked semifoldovers of two-level orthogonal designs

When the initial design is a blocked design, we showed that, for semifoldovers including the implicit block factor, any foldover plan containing block factors is equivalent to a foldover plan containing treatment factors only. If the length of the shortest pure-type word of the design is 4 ≤ L < 5, then a semifoldover design estimate the same two-factor interactions as the corresponding foldover designs. This is true both with and without the consideration of the implicit block factor, as long as the foldover and subsetting are conducted on the same main effect. However, if foldover and subsetting are conducted on different main effects, Proposition 4.3 shows a semifoldover design can estimate the same two-factor interactions only when the implicit block factor is not included. In general, as a semifoldover requires only half as many runs as the corresponding foldover, the results we obtained in this article can be useful for those who want to explore the use of semifoldover designs without sacrificing the ability to estimate important effects. The obtained results may also be useful when searching for optimal designs since we do not need to consider all the foldover and semifoldover plans. Acknowledgments The authors would like to express their sincere thanks to the anonymous referee and the editor for their valuable comments and suggestions which led to significant improvement in the presentation and clarity of the paper. Lin’s research was partially supported by the National Science Council of Taiwan (Grant No. NSC 102-2118-M-005-002). Li’s research was supported by the Research and Teaching Supplements system in Carlson School of Management at the University of Minnesota.

6 Appendix See Table 1. Table 1 A resolution 4 23 non-regular design with 6 factors and 24 runs x1

x2

x3

x4

x5

x6

−1

−1

1

−1

−1

−1

−1

−1

−1

1

−1

1

B1

−1

−1

1

−1

1

−1

B1

−1

−1

1

1

1

1

B1

−1

−1

−1

−1

1

1

B1

−1

−1

−1

1

−1

−1

B1

−1

1

−1

−1

−1

−1

B1

−1

1

−1

1

1

−1

B1

−1

1

−1

−1

1

1

B1

−1

1

1

1

−1

1

B1

−1

Block B1

1

1

−1

−1

1

B1

1

−1

1

−1

−1

1

B2

1

−1

1

1

−1

−1

B2

1

1

1

1

−1

B1

−1

−1

−1

−1

1

B2

−1 1

123

P. Yang et al. Table 1 continued x1

x2

x3

x4

x5

x6

Block

1

−1

−1

1

1

−1

B2

1

−1

1

1

1

1

B2

1

−1

−1

−1

1

−1

B2

1

1

1

−1

1

1

B2

1

1

1

1

−1

−1

B2

1

1

−1

−1

−1

−1

B2

1

1

1

−1

1

−1

B2

1

1

−1

1

−1

1

B2

1

1

−1

1

1

1

B2

References Ai M, Xu X, Wu J (2010) Optimal blocking and foldover plans for regular two-level designs. Stat Sin 20:183–207 Balakrishnan N, Yang P (2006a) Classification of three-word indicator functions of two-level factorial designs. Ann Inst Stat Math 58:595–608 Balakrishnan N, Yang P (2006b) Connections between the resolutions of general two-level factorial designs. Ann Inst Stat Math 58:609–618 Balakrishnan N, Yang P (2009) De-aliasing effects using semifoldover techniques. J Stat Plan Inference 139(9):3102–3111 Barnett J, Czitrom V, John P, Lión L (1997) Using fewer wafers to resolve confounding in screening experiments. In: Czitrom V, Spagon PD (eds) Statistical case studies for industrial process improvement. SIAM, Philadelphia, pp 235–250 Box G, Hunter S, Hunter W (2005) Statistics for experimenters: design, innovation, and discovery. Wiley, New York Cheng S, Li W, Ye K (2004) Blocked nonregular two-level factorial designs. Technometrics 46(3):269–279 Edwards D (2011) Optimal semifoldover plans for two-level orthogonal designs. Technometrics 53(3):274– 284 Fontana R, Pistone G, Rogantin M (2000) Classification of two-level factorial fractions. J Stat Plan Inference 87(1):149–172 Li W, Lin D (2003) Optimal foldover plans for two-level fractional factorial designs. Technometrics 45(2):142–149 Li W, Lin D, Ye K (2003) Optimal foldover plans for two-level nonregular orthogonal designs. Technometrics 45(4):347–351 Mee R, Peralta M (2000) Semifolding 2k− p designs. Technometrics 42(2):122–134 Mee R, Xiao J (2008) Optimal foldovers and semifolding for minimum aberration even fractional factorial designs. J Qual Tech 40(4):448–460 Montgomery D, Runger G (1996) Foldovers of 2k− p resolution I V experimental designs. J Qual Tech 28(4):446–450 Ou Z, Qin H, Li H (2011) Optimal blocking and foldover plans for nonregular two-level designs. J Stat Plan Inference 141(5):1635–1645 Wu J, Hamada M (2009) Experiments: planning analysis and parameter design optimization. Wiley, New York Yang P (2013) Optimal blocking and semifoldover plans for 2n− p designs. Stat Probab Lett 83(2):624–630 Yang P, Li W (2014) Blocked two-level semifoldover designs. J Stat Plan Inference 144(1):133–140 Ye K (2003) Indicator function and its application in two-level factorial designs. Ann Stat 31:984–994 Ye K, Li W (2003) Some properties of blocked and unblocked foldovers of 2k− p designs. Stat Sin 13: 403–408

123