Split-plot designs with general minimum lower-order confounding

SCIENCE CHINA Mathematics

. ARTICLES .

April 2010 Vol. 53 No. 4: 939–952 doi: 10.1007/s11425-010-0070-2

Split-plot designs with general minimum lower-order confounding WEI JiaLin1,4 , YANG JianFeng1 , LI Peng3 & ZHANG RunChu1,2,∗ 1School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China; and School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China; 3School of Mathematical Sciences, Capital Normal University, Beijing 100037, China; 4School of Sciences, Tianjin University of Commerce, Tianjin 300134, China Email: [email protected], [email protected], [email protected], [email protected]

2KLAS

Received June 1, 2009; accepted December 9, 2009

Abstract Split-plot designs have been widely used in industrial experiments. Up to now, most methods for choosing this kind of designs are based on the minimum aberration (MA) criterion. Recently, by introducing a new pattern, called aliased effect-number pattern (AENP), Zhang et al. proposed a general minimum lowerorder confounding (denoted by GMC for short) criterion and established a general minimum confounding (also denoted by GMC for saving notations) theory. It is proved that, the GMC criterion selects optimal designs in a more elaborate manner than the existing ones, and when an experimenter has a prior about the importance ordering of factors in experiments the GMC designs are better than other optimal designs. In this paper we extend the GMC criterion to the split-plot design case and give a GMC-FFSP criterion for ranking split-plot designs. Some comparisons of the new criterion with the MA-MSA-FFSP criterion are given, and the optimal 32-run split-plot designs up to 14 factors under the two criteria are tabulated for comparison and application. Keywords MSC(2000):

fractional factorial design, GMC criterion, minimum aberration, split-plot 62K05, 62K15

Citation: Wei J L, Yang J F, Li P, et al. Split-plot designs with general minimum lower-order confounding. Sci China Math, 2010, 53(4): 939–952, doi: 10.1007/s11425-010-0070-2

1

Introduction

When an experiment is performed, the experimental runs are often required to be completely randomized. However, sometimes it is impractical to actualize this requirement. For example, sometimes, in a practical experiment it is difficult or costly to change the levels of some factors. In this situation, fractional factorial split-plot (FFSP) designs which involve a two-phase randomization are a preferred option for experimenters. In FFSP designs, the factors with hard-to-change levels are called ones of the whole plot (WP), and the factors with relatively-easy-to-change levels are called ones of the subplot (SP). [5] gave an excellent discussion on this kind of designs. First, we recall some basic concepts and notations of FFSP designs. The design matrix of an FFSP design looks exactly like that of a usual fractional factorial (FF) design, and the only difference between them is in the randomization structure on the runs of design. The twophase randomization in an FFSP design inducing several novel features makes it different from a usual ∗ Corresponding

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FF design. The most notable of these, as summarized in [4], are the following two: (a) not all factors have the same status; (b) the inference is possible at two distinct levels of accuracy. To carry through an FFSP design with n factors, we often first randomly choose one of the factorial level-settings of the hard-to-change factors (with n1 denoting the number of the factors) and at the levelsetting run all the level-combinations of the remaining n2 (= n − n1 > 0) factors in a random order. We do so repeatedly for each of level-settings of the n1 factors. Actually, such an FFSP design can be treated as a usual 2n−m FF design but written as a 2(n1 +n2 )−(m1 +m2 ) design, where n1 is the number of WP factors with capital letters A, B, C, . . . denoting them and n2 is the number of SP factors with lowercase letters p, q, r, s, . . . denoting them. In such a design, there are m1 and m2 (m = m1 + m2 ) WP and SP factorial defining words, respectively. A so-called WP defining word means that in it there is no SP factor, and a SP defining word contains at least one SP factor. However, for a FFSP design, a necessary requirement is that, it is allowed that a SP defining word can contain any number of WP factors, but it is not allowed that a SP defining word contains only one SP factor, since if so the split-plot nature of this kind of experiments will be destroyed (see [12]). Accordingly, the 2fi’s in FFSP designs can be divided into three types: WP2fi, SP2fi and WS2fi, where a WP2fi is the interaction of two WP factors, a SP2fi is the interaction of two SP factors, and a WS2fi is the one of one WP factor and one SP factor. Obviously, when an FFSP design is considered as an FF design, the concepts of FF designs, such as resolution, WLP, MA and clear effects, also are correspondingly applicable to FFSP designs. Up to now, the most existing results of choosing optimal FFSP designs are of the MA type. A 2(n1 +n2 )−(m1 +m2 ) FFSP design, as a 2−m fraction of the full factorial 2n design, denotes a design with 2n−m runs, n factors and m independent defining words. It is determined by the m words and we denote the subgroup generated by the m independent words by G. As usual, the WP and SP factors in a word also are called letters and the number of letters in a word is called the length of the word. For a design d, let Ai (d) denote the number of words with length i in G(d). Then the vector (A1 , A2 , . . . , An ) is called the word length pattern (WLP) of the design d. For a set of parameters n1 , n2 , m1 and m2 , among all the 2(n1 +n2 )−(m1 +m2 ) FFSP designs the one sequentially minimizing the Ai ’s in the word length (A1 , A2 , . . . , An ) is called an MA-FFSP design, and the rule for selecting FFSP designs is called the MA-FFSP criterion. [12] adopted the MA-FFSP criterion to choose an optimal regular two-level FFSP design for a thinfilm coating experiment. [2] developed a new sequential construction method and compiled a catalog of MA two-level FFSP designs with 8 and 16 runs via primarily algorithmic approaches. Continuing with the two-level case, [3] listed MA-FFSP designs with up to 32 runs. [15–17] discussed split-plot designs with blocking. In addition, split-plot designs do not have the interchangeability between WP factors and SP factors so that there frequently exist several non-isomorphic FFSP designs which have MA. To overcome this problem, [18] explored a criterion of minimum secondary aberration (MSA), called the MAMSA-FFSP criterion, which significantly narrows the class of competing non-isomorphic MA designs and hence often yields the unique optimal one. [1] constructed MA-MSA-FFSP designs in terms of consulting designs. [22] constructed this kind of designs under weak MA. [21, 28] took further investigations at considering clear effects. Another way of the study of FFSP designs is to focus on the aspect of the D-optimal criterion. For the results on this we can see, say, [9–11, 13]. However, in many cases, an MA-MSA-FFSP design is not really optimal in the sense of effect estimation. Let us see the following example. Example 1.1.

Consider a 2(5+4)−(1+3) MA-MSA-FFSP design d1 : I = ABCDE = ABDpq = ACDpr = BCDps.

Its WLP is (0, 6, 8, 0, 0, 1, 0) and it has 9 clear main effects (ME’s) and only 8 clear 2fi’s (4 WP2fi’s, 4 WS2fi’s and 0 SP2fi). We have another 2(5+4)−(1+3) FFSP design d2 : I = ABCDE = ABpq = ACpr = BCps with WLP (0, 7, 7, 0, 0, 0, 1). Obviously, the design d2 is not an MA-MSA-FFSP design, but it has 9 clear ME’s and 15 clear 2fi’s (8 WP2fi’s, 7 WS2fi’s and 0 SP2fi). This indicates that, considering estimating

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lower-order effects, the non-MA-FFSP design d2 is much better than the MA-MSA-FFSP design d1 because of d2 having seven more clear 2fi’s than d1 . The above fact motivates us to seek some new criterion for selecting optimal FFSP designs, which will be better than the existing ones in the most of practical experiments. In this paper we extend the GMC criterion proposed by [24] to the case of FFSP designs. In Section 2 we establish a new criterion for choosing FFSP design, called the GMC-FFSP criterion. Then, make some comparisons of the new criterion with the MA-MSA-FFSP criterion in Section 3. A comparison with the clear effect criterion and some other related considerations is given in Section 4. In Section 5 we describe an algorithm for searching GMC-FFSP and MA-MSA-FFSP designs. Optimal 32-run split-plot designs under the two different criteria up to 14 factors are completely tabulated in the Appendix.

2

Establishing of the GMC-FFSP criterion

In order to establish a new better criterion for choosing FFSP designs, we first recall some concepts of the existing criteria on selecting optimal regular FF designs. It is well-known that in FF designs the effect hierarchy principle is the most commonly basic one to be followed for selecting good designs (see [20]), which states that, (i) lower-order effects are more likely to be important than higher order effects, and (ii) effects of the same order are equally likely to be important. So, a good design should have as many as possible lower-order effects which are least confounded with each other. Clearly, the minimum aberration (MA) criterion proposed by [8] follows the effect hierarchy principle. The criterion and MA designs have been studied for three decades so that they become very popular now. [6] proved that MA designs have some robust property on estimating lower-order effects. This property makes MA designs to be a good choice when the experimenter does not have any information about the importance order of factors in his/her experiment. However, in the most of practical cases, an experimenter often has such a prior before his/her experiment. In this situation, is the MA design the best choice? Perhaps people shall give a negative answer to this question and think that there might be some designs to be better. Recently, by introducing an aliased effect number pattern (AENP) and based on the effect hierarchy principle, [24] proposed a new criterion, called the general minimum lower-order confounding (GMC) criterion for choosing optimal 2n−m designs. The AENP, as a set, consists of the numbers   n # (k) C , i, j = 1, . . . , n, k = 0, 1, . . . , K with K = , j j i j j (k)

where # denotes the number of ith-order effects each of which is exactly aliased with k jth-order i Cj effects and k is called the severe degree of an ith-order effect aliased with jth-order effects. Then the # (0) # (1) # (Kj ) vector # ) presents the dispersion of the total number of ith-order effects i Cj = ( i Cj , i Cj , . . . , i Cj aliased with jth-order effects at different severe degrees from the least to the most. Thus, the set AENP provides a complete information for the confounding between effects of a design in a more elaborate and explicit manner. Under the effect hierarchy principle, they rank the numbers in the AENP into the following sequence # # # # # # # # C = (# 1C2 , 2C1 , 2C2 , 0C3 , 1C3 , 2C3 , 3C1 , 3C2 , 3C3 , . . .)

#

(1)

and called the design that sequentially maximizes the components in the sequence (1) a GMC design. # For saving notations we also use AENP to denote the sequence (1). By noting that # j C1 (j  2) and 0Cj (j  3) in (1) can be determined by their previous terms, [25] gave the simplified version of (1): # # # # # C = (# 1C2 , 2C2 , 1C3 , 2C3 , 3C2 , 3C3 , . . .)

#

(2)

for designs with resolution three or higher. Hence, a GMC design is also the one sequentially maximizing (k) the components # i Cj ’s in (2). In addition, we call the theory based on the pattern AENP the general

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minimum confounding (also denoted by GMC for saving notations) theory because of the wide applications of the AENP (see [7, 14, 23, 25, 26], say). Since the ideas of the AENP and GMC criterion are relatively new, before passing on to the main results of the paper, we first recall some points from [23] in order to highlight their significance and the connection with other existing criteria. First, each of the MA, CE, and MEC criteria sequentially minimizes or maximizes terms which are determined by the AENP. In other words, all the major existing criteria are determined by the AENP, though none of them completely exploits the AENP. Since the AENP comprehensively describes the confounding pattern among the factorial effects, this underscores the need of utilizing the information included in the AENP in its entirety. The GMC criterion aims precisely at doing so and hence holds the promise of the development of a unified theory encapsulating the existing criteria. In addition to being satisfying from a theoretical viewpoint, such a unified approach can help the practitioner as well. Second, by using the result in [27] and (3.3) in [24], they indicated, in particular, the following more explicit connection between the GMC and MA criteria: K

2 1 (k) k# A3 = 1C2 , 3

k=0

K

2 1 (k) A4 = k# 2C2 , 6

k=0

1 A5 = 10

 K3

(k) k# 2C3

− (n − 3)

k=1

K2 

(k) k# 1C2

 ,

(3)

k=0

and so on. The above not only helps us to understand GMC in the light of MA, but also indicates a feature where these two criteria differ. From (2) and (3) we note that, the MA criterion considers the averages of the terms in the AENP and (3), and hence an MA design has the property of minimum lowerorder confounding on an average. On the other hand, the GMC criterion considers the terms individually, without averaging, and handles these individual terms sequentially, which means that a GMC design has the property of individually minimum lower-order confounding. In this sense, the GMC criterion is more elaborate and informative than the MA criterion. Because of this, the two criteria differ with regard to the situations where they are relevant. To illustrate this point, let us see the following examples. Consider the three pairs of the GMC and MA designs of 213−7 , 214−8 and 216−10 given in Table 4 of [24]. All the ME’s are clear in these designs. But they have different numbers of clear 2fi’s. With letters denoting factors, the structures of their clear ME’s and 2fi’s, respectively denoted by the bold dots and the lines, are shown in Figures 1–3. a br cr d er f rP r r ` X X   P PP    ` X X   H  H  HH   H ` X X   P    P ` H  H    H X X ` @ @ @ @ @ P  P   PP X X   ` @  P  P    H @  H  X H ` @ X   @H @ ` X  X  r  P  P  P ` H   H   H H X X `  P  P P X X `H @  H  P H  X  P  @ X P H @ ` @ @ m `  X X    P P P ` H     X  H X H H `  @ @ P@ XH X `H  H Xr P X r  r  r  r P    @   H P P P @ `r g

h i j k (a) The GMC design with 36 clear 2fi’s Figure 1

g m d e f h i j k l (a) The GMC design with 25 clear 2fi’s

g m e f h i j k l (a) The GMC design with 29 clear 2fi’s Figure 3

r r

h i j k (b) The MA design with 20 clear 2fi’s

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g m e f h i j k l (b) The MA design with 8 clear 2fi’s

A comparison of the GMC and MA 214−8 designs (both have 14 clear ME’s)

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A comparison of the GMC and MA 213−7 designs (both have 13 clear ME’s)

ar br h X P ! a h XX h Hh Hh  P Q   ! a  Xh P
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Figure 2

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f

a br c d rP r re X P P rH  X H H @ XP P H H H   X@ @ @ P  X P P HH@ @  X H  @ HH XH P P   X@ P  P X@ XP   H @  P H @ XH P X@ H   H PH XH Xr @  r  @ r @r PH H @ P  Pr P

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g m e f h i j k l (b) The MA design with 0 clear 2fi’s

A comparison of the GMC and MA 216−10 designs (both have 16 clear ME’s)

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From the figures we can see that, when an experimenter has a prior about the ordering of the factors according to their importance in his/her experiment, a GMC design should be a more suitable choice, since they offer more choices to clearly estimate the different important 2fi’s in addition to ME’s via an orderly identification of the important factors with the columns labeled a, b, c, d, . . . In contrast, by the property of minimum lower-order confounding, on an average, as enjoyed by MA, if an experimenter does not have any such a prior, perhaps an MA design is more suitable, since they treat all 2fi’s equally in so far as their estimation is concerned. Although an MA design may sacrifice some 2fi’s to be clearly estimated, it may be attractive from the consideration of model robustness in [6]. We will see below that, without exception, the similar facts also happen for the case of FFSP designs discussed in this paper (see Section 3). Split-plot designs have their own characteristic. Two-phase randomization gives rise to two kinds of random errors, the WP error term and SP error term. Different error terms have different precisions. This implies that the power to detect significant effects in data analysis is not the same for the two kinds of effects, WP and SP, and inferences to them may have two distinct levels of accuracy. See [4] for details of the explanation. In split-plot designs, effects involving only WP factors are called WP-type effects, and effects involving at least one SP factor are called SP-type effects. Correspondingly, an alias set is said to be of WP-type if it contains at least one WP-type effect, or of SP-type otherwise. The significance of an effect in an alias set of WP-type is assessed by using the WP error term, and that of SP-type is assessed by using the SP error term. These rules were originally developed by [4] and were subsequently summarized by [3]. # (1) # (0) C(w) denote the number of ith-order SP-type effects in WP-type alias sets and i(s) C(w) the Let i(s) number of ith-order SP-type effects not in any WP-type alias set. Clearly,  i    n2 n1 # (0) # (1) C + C = . i(s) (w) i(s) (w) l i−l l=1

Particularly,

# (0) 1(s)C(w)

is the number of SP-type ME’s not aliased with any WP-type effects. Therefore,

considering the split-plot structure, the number n2 . Hence we take #sp

(0)

# # C = ( 1(s) C(w) = n2 , # 1C2 , 2C2 ,

# (0) 1(s)C(w)

for an FFSP 2(n1 +n2 )−(m1 +m2 ) design must be

# (0) # # # # # (0) 2(s)C(w) , 1C3 , 2C3 , 3C2 , 3C3 , 3(s)C(w) , . . .)

(4)

(0)

# as the AENP for FFSP designs. In this pattern, the reason that 2(s) C(w) is put behind # 2C2 is as follows. After considering the confounding of 2fi’s, it follows to concern the estimation accuracy of 2fi’s. Note that, for an FFSP design, all WP-type effects are tested against the WP level error, while SP-type effects are tested against the WP or SP level error. As the WP level error is typically larger than the SP level error (see [4]), a good FFSP design should have lower-order SP-type effects not aliased with WP effects # (0) C(w) just indicates the number of such SP 2fi’s and hence it should follow as many as possible. The 2(s) # 2C2 .

(0)

# Similarly, 3(s) C(w) should be put behind # 3C3 , and so on. The definition of GMC-FFSP designs is now given below. sp

Definition 2.1. Suppose that d and d are two 2(n1 +n2 )−(m1 +m2 ) designs. Let # Cl , which may sp sp # (k) # (0) C(w) , is the l-th component of # C. Suppose # Ct is the first component of has the form i Cj or i(s) sp sp sp sp #sp C such that # Ct (d) = # Ct (d ). If # Ct (d) > # Ct (d ), then d is said to have less lower-order confounding than d . A 2(n1 +n2 )−(m1 +m2 ) design d is said to have GMC if no other 2(n1 +n2 )−(m1 +m2 ) design has less lower-order confounding than d, and also we call such design a GMC-FFSP design. From the definition, we can see that a GMC-FFSP design sequentially maximizes the components of C in (4).

#sp

3

Comparison of GMC-FFSP and MA-MSA-FFSP criteria

For comparison, let us recall the MA-MSA-FFSP criterion first.

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For a set of given parameters n1 , n2 , m1 and m2 , it is possible to have many MA-FFSP designs. [18] explored a finer criterion, named as the minimum secondary aberration (MSA), to narrow the competing MA-FFSP designs. Let Bi (d) be the number of distinct ith-order SP-type factorial effects that appear in the WP-type alias sets. Considering the structure of split-plot designs, we have B1 (d) = 0. A good FFSP design should assign the SP-type factorial effects with order as low as possible to WP-type alias sets (see [18]). This implies that we should sequentially minimize B2 (d), B3 (d), . . . [18] called W ∗ (d) = (B2 (d), . . . , Bn (d)) as the secondary wordlength pattern (SWLP) and used it for further ranking these MA-FFSP designs as follows. For two nonisomorphic MA 2(n1 +n2 )−(m1 +m2 ) designs, d1 and d2 , d1 is said to have less secondary aberration than d2 if there exists a positive integer r such that Br (d1 ) < Br (d2 ) and Bi (d1 ) = Bi (d2 ) for i < r. A MA-FFSP design has MSA, called a MA-MSA-FFSP design, if no other MA design has less secondary aberration. We call the criterion combining the two steps of MA and MSA an MA-MSA-FFSP criterion. Since the MA-MSA-FFSP design is the best among the MA-FFSP designs, we only need to compare the GMC-FFSP criterion with the MA-MSA-FFSP criterion. # (k) C(w) in (4), we can get the following proposition immediComparing the definitions of Bi here and i(s) ately.    n1  # (0) − i(s)C(w) (d) Proposition 3.1. For a 2(n1 +n2 )−(m1 +m2 ) FFSP design d, we have Bi (d) = il=1 nl2 i−l  1 +n2 and ni=2 Bi (d) = 2n1 (2m2 − 1). Proof. The first part of the proposition is obvious, so we only need to consider the second part. (n1 +n2 )−(m1 +m2 ) In a 2 FFSP design, there are 2n1 −m1 WP-type alias sets. Each of such sets involves m1 +m2 m1 − 2 SP-type effects. Totally, there are 2n1 −m1 (2m1 +m2 − 2m1 ) = 2n1 (2m2 − 1) SP-type effects 2 n1 +n2 in such WP-type alias sets. On the other hand, it is obvious that i=2 B (d) means the total number n1 +ni 2 of SP-type effects that are in WP-type alias sets. Therefore, we have i=2 Bi (d) = 2n1 (2m2 − 1). As usual, from the definitions of the WLP of a FFSP design and clear effects and Lemma 1 in [24], (1) (0) (0) # (1) and the numbers of clear ME’s and 2fi’s are # and # we also have that, Ai = # 1C2 2C2 − 1C2 , i C0 respectively. Thus, the above facts tell us that, the pattern of MA-FFSP, WLP, and the pattern of MSA, SWLP, are all the functions of the pattern of GMC-FFSP, AENP-FFSP, and hence the GMC-FFSP criterion is the best one for choosing optimal designs with least lower-order confounding in a more elaborate and explicit manner. It means that, in the sense of minimum lower-order confounding, the GMC-FFSP criterion is better than the MA-MSA-FFSP criterion. In the following, we give some examples to illustrate the comparison. Example 3.1.

Consider the following two 2(5+4)−(1+3) FFSP designs.

d1 : I = ABCDE = ABDpq = ACDpr = BCDps,

d2 : I = ABCDE = ABpq = ACpr = BCps.

Here, design d1 is an MA-MSA-FFSP design and design d2 is a GMC-FFSP design. In fact, we have (0) # # # (0) =# and # 1C2 (d2 ) = (9), 2C2 (d1 ) = (8, 24, 0, 4) and 2C2 (d2 ) = (15, 0, 21). Note that 1C2 2C2 are the numbers of clear ME’s and clear 2fi’s respectively from [24], because both designs have resolution IV. It is obvious that both designs have 9 clear ME’s, but d2 has 7 more clear 2fi’s than d1 . So we can say that d2 is better than d1 in the sense of less confounding between the lower-order effects. For more comparisons between them, refer to Table 1. # 1C2 (d1 )

Table 1

Comparisons between 2(5+4)−(1+3) MA-MSA-FFSP design d1 and GMC-FFSP design d2

# # # (0) 1 C2 ; 2 C2 ; 2(s)C(w)

WLP number of the clear ME’s number of the total clear 2fi’s numbers of the clear WP, WS and SP 2fi’s respectively

MA-MSA-FFSP design d1

GMC-FFSP design d2

9; 8, 24, 0, 4; 20 0, 6, 8, 0, 0, 1, 0 9 8 4, 4, 0

9; 15, 0, 21; 20 0, 7, 7, 0, 0, 0, 1 9 15 7, 8, 0

An intuitive expression of the example in Table 1 is shown in Figure 4. In the figure, as before we use letters to denote factors, and the bold dots, red and blue lines respectively to denote the clear ME’s,

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clear WP and WS 2fi’s of the designs. Thus, the structures of clear ME’s and different kinds of clear 2fi’s of the 2(5+4)−(1+3) MA-MSA-FFSP design d1 and GMC-FFSP design d2 are displayed. From the figure we can clearly see that, d2 is a better choice than d1 if an experimenter has a prior about the importance ordering of the factors in experiments, which is just like the case without block and split-plot.

(a) The GMC-FFSP design with 15(7,8,0) clear 2fi’s Figure 4

(b) The MA-MSA-FFSP deign with 8(4,4,0) clear 2fi’s

A comparison of the 2(5+4)−(1+3) GMC-FFSP and MA-MSA-FFSP designs

To compare the GMC-FFSP designs and MA-MSA-FFSP designs more clearly, we take one more example, which is the 2(3+6)−(0+4) designs shown in Table 3 of the Appendix. For the two optimal designs, their structures of clear ME’s and clear 2fi’s (WP, WS and SP) are intuitively shown in Figure 5 (with the same notations as above except for using green lines to denote the clear SP2fi). From the figure, again we can note the difference between the two designs and find that, the 2(3+6)−(0+4) GMCFFSP design is also much better than the MA-MSA-FFSP design on the aspect of estimating lower-order effects, when an experimenter has a prior of the importance ordering of factors in his/her experiments.

(a) The GMC-FFSP design with 15(2,8,5) clear 2fi’s Figure 5

(b) The MA-MSA-FFSP deign with 8(2,6,0) clear 2fi’s

A comparison of the 2(3+6)−(0+4) GMC-FFSP and MA-MSA-FFSP designs

In the following, we compare the two kinds of optimal designs from other aspects, say, how easy to de-alias some 2fi’s. Example 3.2.

Consider the following two 2(4+6)−(0+5) designs: d3 : I = BDpq = ABpr = CDps = ABCDpt = ACpu, d4 : I = BDpq = BCpr = ADps = CDpt = ABpu. (0)

# # Design d3 is an MA-MSA-FFSP design and d4 is a GMC-FFSP design. (# 1C2 ; 2C2 ; 2(s)C(w) )’s of d3 and 2 d4 are (10; 0 , 45; 24) and (10; 0, 6, 27, 12; 24) respectively. We can see that, although both designs have the same number of clear ME’s and clear 2fi’s, their confounding degrees between lower-order effects are different. In d3 45 2fi’s are confounded with two other 2fi’s, while in d4 6 2fi’s are confounded with only one other 2fi and 27 2fi’s are confounded with two other 2fi’s. This indicates that the number of 2fi’s with less confounding in the design d4 is much more than that in the design d3 . If we need to de-alias a few 2fi’s, say, less than 15, for estimating, perhaps the run number of follow-up experiments needed by design d4 will be much less than that by d3 . From the above two examples, we can see that FFSP designs with GMC tend to have more clear effects and less confounding between lower-order effects. The following example illustrates the usage of GMC-FFSP designs if some prior is available for the factorial effects.

Example 3.3. Consider the GMC-FFSP 2(5+4)−(1+3) design d2 in Example 3.1 again. In this design, all the ME’s are clear. The 15 clear 2fi’s in the design are AD, AE, BD, BE, CD, CE, DE, Dp, Dq, Dr, Ds, Ep, Eq, Er and Es. Suppose we want to assign a 2(5+4)−(1+3) design with five WP factors denoted by 1, 2, 3, 4, 5 and four SP factors denoted by 6, 7, 8, 9. If we have a prior information that all 2fi’s involving factors 1 or/and 2 are of special interest and need to be estimated clearly, then we can put factor 1 to the column D and factor 2 to the column E in the design matrix. In this sense, we can easily see that, the GMC-FFSP design d2 is better than the MA-MSA-FFSP design d1 by using the prior, since the former has more clear 2fi’s than the latter. The same thing also holds for the two designs d3 and d4

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in Example 3.2, in which the confoundings between lower-order effects of the two designs are different although they have the same number of clear effects. So we also can properly assign the factors to the columns of the GMC design so as to make the important 2fi’s with the least confounding. Remark 3.1.

In the table of the appendix, the

# (0) 2(s)C(w)

values are the same for most GMC and MA (0)

# designs with the same parameters. For the cases in which they are different, the 2(s) C(w) value of the GMC design is found, in the table, to be slightly smaller than that of the MA design. In fact, the slight # difference is not so important compared with the former terms # 1C2 and 2C2 . Particularly, for the GMC # (0) # design with smaller 2(s) C(w) value, the leading two terms # 1C2 and 2C2 in the AENP are much better than those of MA designs with the same parameters.

4

Comparison with the clear effects criterion and some other related discussions

Next, we consider a comparison with the clear effects criterion which sequentially maximizes the numbers of clear ME’s and clear 2fi’s. Since the definition of clear effects in the case of FFSP designs is the same as that of the usual FF designs, all the properties of clear effects of the GMC-FFSP criterion are the same as that of the GMC criterion of usual FF designs. As a result, we still have the following conclusion. Proposition 4.1. For given n1 n2 , m1 and m2 , if optimal designs under the clear effects criterion exist, then the GMC-FFSP design must be the best one among all optimal designs under the clear effects criterion, where the meaning of “best” is under the comparison in Definition 2.1 of the GMC-FFSP criterion. And the GMC-FFSP criterion can be used for designs without any clear effect. The above proposition means that, the GMC-FFSP criterion is better than the clear effects criterion. The same with the case of usual FF designs, the AENP-FFSP in (4) seems to be a little complicated. But, actually, in practice the interactions of three and higher orders are negligible, and then the AENPFFSP for choosing optimal designs can be simplified as #sp

(0)

# # C = ( 1(s) C(w) = n2 , # 1C2 , 2C2 ,

# (0) 2(s)C(w) ).

A natural question is if a GMC-FFSP design can be obtained from a GMC FF design by assigning some factors as WP factors and the others as SP factors. The answer to this question is negative. Let us note the following example. Example 4.1.

Consider the following 210−5 design: I = ABCpq = ABCDr = ABDps = ACDpt = BCDpu.

As an FF design, it is a GMC design. Suppose we are interested in running a 2(4+6)−(0+5) FFSP # (0) experiment. The 210−5 GMC FF design, however, have no split-plot structure since 1(s) C(w) (d) < n2 whenever any four factors are regarded as WP factors. So GMC-FFSP designs can not be obtained from GMC FF designs by randomly assigning some factors as WP factors and others as SP factors. The above example shows that, for a set of given parameters n1 n2 , m1 and m2 , probably there is no GMC FF design satisfying the condition of the FFSP design. The other possibility is that, although for a set of given parameters n1 n2 , m1 and m2 , there exists a GMC FF design, it is not an optimal one according to the GMC-FFSP criterion. Let us see the example below. Example 4.2.

We have the following two FFSP 2(4+7)−(1+5) designs: d5 : I = ABCD = Bpqr = ABqs = BCpt = ACqu = Cpqv, d6 : I = ABCD = ACpr = ABps = BCpt = ABqu = BCqv.

Design d5 is a GMC-FFSP design. As an FF design, d6 is a GMC design. Although the numbers # 1 C2 # (0) and # C of them are the same, the numbers of their C are different: d possesses 4 less SP-type 2fi’s 2 5 2 2(s) (w)

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tested against the WP error than d6 . So, the GMC-FFSP one d5 is better than d6 from the estimation accuracy point. The detailed comparison between the two designs is shown in Table 2. Table 2 # # # (0) 1 C2 ; 2 C2 ; 2(s)C(w)

5

Comparisons between GMC-FFSP design d5 and GMC FF design d6 d5

d6

11; 0, 0, 24, 16, 15; 44

11; 0, 0, 24, 16, 15; 40

Algorithm for the construction of GMC-FFSP designs

In this section, we give an algorithm to construct GMC-FFSP designs from FF designs. Given a 2n−m FF design, we choose any n1 factors as WP factors and the rest n2 (= n − n1 ) factors as SP factors. If # (0) (n1 +n2 )−(m1 +m2 ) design, and the following question is how 1(s)C(w) (d) = n2 , we have indeed obtained a 2 to determine the values of m1 and m2 . From Examples 4.1 and 4.2, it is clear that we need to find the GMC-FFSP designs from all the # (0) (n1 +n2 )−(m1 +m2 ) designs having the split-plot structure, i.e., 1(s) C(w) (d) = n2 . According to (8.1.2) 2 in [19], a 2(n1 +n2 )−(m1 +m2 ) FFSP design can be given, in [19]’s manner, by d(B) = {x : Bx = 0}, 

where B=

B11

0

B21

B22



is a matrix over GF (2), with B11 , B21 , B22 of orders m1 × n1 , m2 × n1 , m2 × n2 , respectively, such that rk(B11 ) = m1 , rk(B22 ) = m2 . It can be easily seen that, the number of words containing only WP factors in the split-plot design must be 2m1 − 1, which can be used to determine the value of parameter m1 . Thus the algorithm for searching optimal 2(n1 +n2 )−(m1 +m2 ) designs is as follows: Step 1. Given a 2n−m design and n1 , we assign n1 factors as WP factors and the remaining n2 = n− n1 # (0) factors as SP factors. If the design has split-plot structure ( 1(s) C(w) (d) = n2 ), go to Step 2; otherwise, go to the beginning of Step 1 and assign another n1 factors as WP factors. Step 2. Obtain the value of m1 for the design in Step 1. Then we can determine all the parameters n1 , n2 , m1 and m2 = m − m1 of the FFSP design found in Step 1. Step 3. Put designs which have the same parameters in one set and give a competitor for each of the GMC-FFSP criterion and MA-MSA-FFSP criterion. In the appendix, GMC-FFSP designs and MA-MSA-FFSP designs for n1 + n2  14 are completely listed for comparison. GMC-FFSP designs with 8 and 16 runs are omitted here since they are the same as MA-FFSP designs listed by [2]. Note that, for most parameters in the appendix, MA-MSA-FFSP designs are listed for the first time. Furthermore, the detailed information on the listed designs is also provided in the appendix. Especially, the numbers of clear WP2fi’s, clear WS2fi’s and clear SP2fi’s are first given, respectively, for each GMC-FFSP design and MA-MSA-FFSP design.

Acknowledgements The authors cordially thank the referees for their careful reading and helpful comments and suggestions for improving the paper. This work was supported by National Natural Science Foundation of China (Grant No. 10871104, 10971107) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20050055038).

References 1 Ai M Y, Zhang R C. Minimum secondary abberation fractional factorial split-plot designs in terms of consulting designs. Sci China Ser A, 2006, 49: 494–512

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2 Bingham D, Sitter R R. Minimum aberration two-level fractional factorial split-plot designs. Technometrics, 1999, 41: 62–70 3 Bingham D, Sitter R R. Design issues in fractional factorial split-plot experiments. J Quality Technol, 2001, 33: 2–15 4 Bisgaard S. The design and analysis of 2k−p × 2q−r split-plot experiments. J Quality Technol, 2000, 32: 39–56 5 Box G E P, Jones S. Split-plot designs for robust product experimentation. Appl Statist, 1992, 19: 3–26 6 Cheng C S, Steinberg D M, Sun D X. Minimum aberration and model robustness for two-level factorial designs. J Roy Statist Soc Ser B, 1999, 61: 85–93 7 Cheng Y, Zhang R C. On construction of general minimum lower order confounding 2n−m designs with N/4 + 1  n  9N/32. J Statist Plan Inference, in press, 2010, doi: 10.1016/j.jspi.2010.02.006 8 Fries A, Hunter W G. Minimum aberration 2k−p designs. Technometrics, 1980, 22: 601–608 9 Goos P. Optimal versus orthogonal and equivalent-estimation design of blocked and split-plot experiments. Statist Neerlandica, 2006, 60: 361–378 10 Goos P, Vandebroek M. Optimal split-plot Designs. J Quality Technol, 2001, 33: 435–450 11 Goos P, Vandebroek M. D-optimal split-plot designs with given numbers and sizes of whole plots. Technometrics, 2003, 45: 235–245 12 Huang P, Chen D, Voelkel J. Minimum aberration two-level split-plot designs. Technometrics, 1998, 40: 314–326 13 Jones B, Goos P. D-optimal design of split-split-plot experiments. Biometrika, 2009, 96: 67–82 14 Li P F, Zhao S L, Zhang R C. A theory on constructing 2m−p designs with general minimum lower order confounding. Submitted to Statist Sinica, 2009 15 Mcleod R G. Optimal block sequences for blocked fractional factorial split-plot designs. J Statist Plan Inference, 2008, 138: 2563–2573 16 Mcleod R G, Brewster J F. The design of blocked fractional factorial split-plot experiments. Technometrics, 2004, 46: 135–146 17 Mcleod R G, Brewster J F. Blocked fractional factorial split-plot experiments for robust parameter design. J Quality Technol, 2006, 38: 267–279 18 Mukerjee R, Fang K T. Fractional factorial split-plot designs with minimum aberration and maximum estimation capacity. Statist Sinica, 2002, 12: 885–903 19 Mukerjee R, Wu C F J. A Modern Theory of Factorial Designs. Springer Series in Statistics. New York: Springer Science+Business Media Inc, 2006 20 Wu C F J, Hamada M S. Experiments: Planning, Analysis and Parameter Design Optimization. New York: Wiley, 2000 21 Yang J F, Li P F, Liu M Q, et al. 2(n1 +n2 )−(k1 +k2 ) fractional factorial split-plot designs with clear effects. J Statist Plan Inference, 2006, 136: 4450–4458 22 Yang J F, Zhang R C, Liu M Q. Construction of fractional factorial split-plot designs with weak minimum aberration. Statist Prob Letters, 2007, 77: 1567–1573 23 Zhang R C, Cheng Y. General minimum lower order confounding designs: an overview and a construction theory. J Statist Plan Inference, 2010, 140: 1719–1730 24 Zhang R C, Li P, Zhao S L, et al. A general minimum lower-order confounding criterion for two-level regular designs. Statist Sinica, 2008, 18: 1689–1705 25 Zhang R C, Mukerjee R. Characterization of general minimum lower order confounding via complementary. Statist Sinica, 2009, 19: 363–375 26 Zhang R C, Mukerjee R. General minimum lower order confounding in block designs using complementary sets. Statist Sinica, 2009, 19: 1787–1802 27 Zhang R C, Park D K. Optimal blocking of two-level fractional factorial designs. J Statist Plan Inference, 2000, 91: 107–121 28 Zi X M, Zhang R C, Liu M Q. Bounds on the maximum numbers of clear two-factor interactions for 2(n1 +n2 )−(k1 +k2 ) fractional factorial split-plot designs. Sci China Ser A, 2006, 49: 1816–1829

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Appendix Table 3

32-run GMC-FFSP designs and MA-MSA-FFSP designs

Designs

n1 n2 m1 m2 4 3 2 1 5 4 4 3 3 2 1 6 5 5 4 4 3 3 2 1 7 6 6 5 5 5 4 4 4 3 3 3 3 2 2 1 1

2 3 4 5 2 3 3 4 4 5 6 2 3 3 4 4 5 5 6 7 2 3 3 4 4 4 5 5 5 6 6 6 6 7 7 8 8

0 0 0 0 1 1 0 1 0 0 0 2 2 1 1 0 1 0 0 0 3 3 2 2 1 1 1 1 0 1 1 0 0 0 0 0 0

1 1 1 1 1 1 2 1 2 2 2 1 1 2 2 3 2 3 3 3 1 1 2 2 3 (GMC) 3 (MA) 3 (GMC) 3 (MA) 4 3 (GMC) 3 (MA) 4 (GMC) 4 (MA) 4 (GMC) 4 (MA) 4 (GMC) 4 (MA)

8241 7341 7332 6432 6423 5 5 2 3 (GMC) 5 5 2 3 (MA) 5514 4614 4 6 0 5 (GMC) 4 6 0 5 (MA) 3 7 1 4 (GMC) 3 7 1 4 (MA) 3705 2805

Generators ABCDpq ABCpqr ABpqrs Apqrst ABCDE BCpq ABCD BCpqr CDpq ABDpr ABC Bpqrs ABCpr ABpqs ABpqs ABqrt Aqrst pqsu ABDE ABCF ACDpq BCD ACE ABpqr BDEpr BCpq ABCD ABCD ACpqr ABqs BDpq ACDpr BCps ABC ABqrs Aprt ABCpr ACpqs BCqt Bqrs Apqrt ABpqu Apqrt Apqsu Aqrsv CDEpq BDF pq ADGpq ABCD ABD ACE ABCF BCpqr ABDE ABCF ABpq BCDpr ACDqs Cprs ABDE ABC ABCDE ABpq ACpr BCps ABCDE ABDpq ACDpr BCDps ABCD ABpqr ACqs BCqt ABCD ABqr ABps BCpqt ADpq BCDpr ACps ABpt ABC ABpqs ABprt pqru ABC Aqrs ABpqrt Bpqu Cpqr BCqs ABpqt BCpu Apqr BCpqs ABCpt ABqu Bpqs Bprt pqru ABqrv Aqrs Aprt ABpqu Bpqrv Aprst Apqsu Apqrv qrsw Apqrt Apqsu Aprsv Aqrsw ACDE ABDF ABCG BCDH ADpq ABCD ABE ACF BCG Cpqr ABCE ABDF BCDG ABCDpq ABpr ABD ACE ABCF ACpqr BCqs ABDE ABCF BCDpq ABpr ACDps ACD ABE BCqr BCps ACpqt ACDpt ABDE ABCpqrt ABpqs ABC ABCDE CDpq BDpr ADps ABCpt ABCD ABpr BCpqs ABqt ACpqu BDpq BCpr ADps CDpt ABpu BDpq ABpr CDps ABCDpt ACpu ABC pqrs ABpqt ABpru ABqrv ABC ABqrs ABprt Bpqu Apqrv BCpqr ABCqs Apqt ACpu ABpv Apqs ABprt Bpqru Aqrv ABqw

WLP

SWLP

A3 , . . . , A 6

B1 , . . . , B4

ME 2fi(WP, WS, SP)

6; 15; 8 6; 15; 12 6; 15; 14 6; 15; 15 7; 15,6; 10 7; 15,6; 15 7; 15,6; 12 4,3; 21; 18 7; 15,6; 17 7; 15,6; 20 7; 15,6; 21 8; 13,12,3; 12 3,4,1; 22,6; 18 8; 13,12,3; 15 8; 13,12,3; 21 8; 13,12,3; 16 5,3; 16,12; 25 8; 13,12,3; 23 8; 13,12,3; 27 8; 13,12,3; 28 9; 15,0,21; 14 3,0,6; 21,12,3; 21 9; 8,24,0,4; 18 4,4,1; 15,18,3; 25 9; 15,0,21; 20 9; 8,24,0,4; 20 9; 15,0,21; 27 9; 8,24,0,4; 28 9; 8,24,0,4; 20 6,3; 15,0,21; 33 6,3; 12,18,6; 33 9; 15,0,21; 30 9; 8,24,0,4; 30 9; 15,0,21; 35 9; 8,24,0,4; 35 9; 15,0,21; 36 9; 8,24,0,4; 36

0,0,0,1 0,0,0,1 0,0,0,1 0,0,0,1 0,1,2,0 0,1,2,0 0,1,2,0 1,0,1,1 0,1,2,0 0,1,2,0 0,1,2,0 0,3,4,0 2,1,2,2 0,3,4,0 0,3,4,0 0,3,4,0 1,2,3,1 0,3,4,0 0,3,4,0 0,3,4,0 0,7,7,0 4,3,3,4 0,6,8,0 2,4,6,2 0,7 7,0 0,6,8,0 0,7,7,0 0,6,8,0 0,6,8,0 1,7,4,0 1,5,6,2 0,7,7,0 0,6,8,0 0,7,7,0 0,6,8,0 0,7,7,0 0,6,8,0

0,1,4,6 0,0,1,3 0,0,0,1 0,0,0,0 0,1,5,10 0,0,1,4 0,3,12,18 0,0,0,1 0,1,5,9 0,0,2,5 0,0,0,3 0,1,6,15 0,0,1,5 0,3,15,30 0,1,6,14 0,6,24,37 0,0,2,7 0,2,10,19 0,0,4,11 0,0,0,7 0,1,7,21 0,0,1,6 0,3,18,45 0,1,7,20 0,6,30,61 0,6,30,61 0,3,15,30 0,2,12,29 0,10,40,65 0,0,4,15 0,0,4,15 0,3,17,36 0,3,17,36 0,0,7,21 0,0,7,21 0,0,0,14 0,0,0,14

6 6 6 6 7 7 7 4 7 7 7 8 3 8 8 8 5 8 8 8 9 3 9 4 9 9 9 9 9 6 6 9 9 9 9 9 9

15 ( 6,8,1 ) 15 ( 3,9,3 ) 15 ( 1,8,6 ) 15 ( 0,5,10 ) 15 ( 9,6,0 ) 15 ( 0,12,3 ) 15 ( 5,8,2 ) 15 ( 0,12,6 ) 15 ( 3,9,3 ) 15 ( 1,10,4 ) 15 ( 0,6,9 ) 13 ( 0,12,1 ) 18 ( 0,15,3 ) 13 ( 4,7,2 ) 13 ( 0,8,5 ) 13 ( 3,7,3 ) 13 ( 0,9,4 ) 13 ( 2,7,4 ) 13 ( 1,7,5 ) 13 ( 0,7,6 ) 15 ( 0,14,1 ) 21 ( 0,18,3 ) 8 ( 0,6,2 ) 11 ( 0,9,2 ) 15 ( 7,8,0 ) 8 ( 4,4,0 ) 15 ( 0,8,7 ) 8 ( 0,4,4 ) 8 ( 0,4,4 ) 12 ( 0,12,0 ) 9 ( 0,8,1 ) 15 ( 2,8,5 ) 8 ( 2,6,0 ) 15 ( 1,8,6 ) 8 ( 1,7,0 ) 15 ( 0,8,7 ) 8 ( 0,8,0 )

10; 0,16,0,24,5; 16

0,18,0,8

0,1,8,28

10

0(0,0,0)

3,0 ,7; 18,6,21; 24

7,8,3,4

0,0,1,7

3

18(0,18,0)

10; 0,6,27,12; 21

0,16,0,12

0,3,21,63

10

0(0,0,0)

4,0,6; 8,30,3,4; 29

4,8,8,4

0,1,8,27

4

8(0,8,0)

10; 0,40,02,5; 24

0,10,16,0

0,6,36,91

10

0(0,0,0)

2,10,8,4

0,2,14,41

5

7(0,7,0)

(0) # # # 1C2 ; 2C2 ; 2(s)C(w)

2

5,4,1; 11,12,18,4; 33

Clear effects

5,4,1; 8,30,3,4;33

2,8,12,4

0,2,14,41

5

4(0,4,0)

10; 0,40,02,5; 25

0,10,16,0

0,10,50,105

10

0(0,0,0)

10; 0,40,0 ,5; 36

0,10,16,0

0,3,20,53

10

0(0,0,0)

10; 0,6,27,12; 24

0,16,0,12

0,15,60,105

10

0(0,0,0)

10; 0 ,45; 24

0,15,0,15

0,15,60,105

10

0(0,0,0)

7,3; 17,02,28; 42

1,14,7,0

0,0,7,28

7

14(0,14,0)

7,3; 11,12,18,4; 42

1,10,11,4

0,0,7,28

7

8(0,8,0)

10; 0,40,02,5; 37

0,10,16,0

0,5,27,58

10

0(0,0,0)

10; 0,40,02,5; 43

0,10,16,0

0,1,12,32

10

0(0,0,0)

2

2

to be continued

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continued Designs

n1 n2 m1 m2 1905 9251 8342 7442 7 4 3 3 (GMC) 7 4 3 3 (MA) 6533 6 5 2 4 (GMC) 6 5 2 4 (MA) 5 6 2 4 (GMC) 5 6 2 4 (MA) 5 6 1 5 (GMC) 5 6 1 5 (MA) 4 7 1 5 (GMC) 4 7 1 5 (MA) 4706 3 8 1 5 (GMC) 3 8 1 5 (MA) 3 8 0 6 (GMC) 3 8 0 6 (MA) 2 9 0 6 (GMC) 2 9 0 6 (MA) 1 10 0 6 (GMC) 1 10 0 6 (MA) 10 2 6 1 9352 8 4 4 3 (GMC) 8 4 4 3 (MA) 7 5 4 3 (GMC) 7 5 4 3 (MA) 7534 6 6 2 5 (GMC) 6 6 2 5 (MA) 6634 5 7 2 5 (GMC) 5 7 2 5 (MA)

Generators Aqrt Apru pqrsv Apqsw Arsx BCDE ACF ACDG ADH BDJ BCpq ACDE ABDF ABCG BCDH ACpq CDpr CDpqs ABqr BDF ADG ABDE ABC ABCD ACEF ABEG CDqs ABpr ACDEps ABCE ABDF BCDG BCpq BDpr ADps ABCD ACE ABF BCqr BCps Bpqt BCDE ABDF ABpq BCpr ADps CDpt BCDE ABDF ADpq BDpr ACps CDpt BCD ACE ABpr BCpqs ABqt ACpqu ABC ABDE BCDpqu ABpqs Cptu Cqru ABDE ACpq BCpr CDps ABCDpt ABpu ABDE ADpq BCpr CDps ACpt BDpu ABCD Bpqr ABqs BCpt ACqu Cpqv ABCD ABCpqr BCps Cpqt ABqu BCqv ADpq ACpr BDps BCpt ABpu CDpv ABC Aprs Bpqrt Bru Aqrv ABpqw ABC Bprs Bqrt ABpqru Bpqv Apqrw BCpr Apqs ABqt BCqu ABpv Cpqw Apqr ACps ABCpqu ABpv ACqw Bpqt Bpqs Bprt pqru Aqrv ABqw ABrx Aqrs Apqt Bpru pqrv Bqrw ABqx Aqst Apru Apsv prsw Aqrx qrsy Aqst Apqu Arsv qrsw Apsx Apry BDE ABF ABCG ACH BCDJ CDK ADpq ABCDEpr ACDqr ABCEG ACH CEJ ABEF BCD ABDE BCDF BDpr BCps CDpq ACDG ABCH ACDE ABDF ACpr BDps ABpq ABCG BCDH ABDF BDE ABC CDpqt ADG ABqr Aqs ADF ABDE BCDpqt BDG ABC Bqr ABps ABDE ABCF BCpt ACpq BDpr ABCDps ACDG BCDE ABDF ADpq CDpr ABps BCpt BDpu BCDE ABDF BDpq ADpr BCps ABCDpt ABpu ABD ACE ABCF BCpr ABpqs BCqt ACpqu BCD ACE Apqr Cqs ABpt ABCqu Bpqv BCD ACE Apqr ABCqs Bpqt ABqu ABCpv

(0) # # # 1C2 ; 2C2 ; 2(s)C(w)

WLP

SWLP

Clear effects

A3 , . . . , A6 B1 , . . . , B4

ME 2fi(WP, WS, SP)

10; 0,40,02,5; 45

0,10,16,0

0,0,4,18

10

0(0,0,0)

2,2,5,2; 12,18,21,4; 18

6,12,16,12

0,1,9,36

2

10(0,10,0)

11; 02,24,16,15; 24

0,26,0,24

0,3,24,84

11

0(0,0,0)

2,2,0,6,1; 12,18,21,4; 33

8,12,10,12

0,1,9,35

2

10(0,10,0)

11; 0 ,24,16,15; 28

0,26,0,24

0,6,42,127

11

0(0,0,0)

11; 02,15,40; 28

0,25, 0,27

0,6,42,127

11

0(0,0,0)

5,0,6; 4,28,18,0,5; 38

4,14,16,8

0,2,16,55

5

4(0,4,0)

11; 0 ,24,16,15; 30

0,26,0,24

0,10,60,155

11

0(0,0,0)

11; 02,15,40; 30

0,25,0,27

0,10,60,155

11

0(0,0,0)

6,4,1; 10,16,0,24,5; 42

2,18,14,8

0,3,23,73

6

6(0,6,0)

6,4,1; 4,28,18,0,5; 42

2,14,22,8

0,3,23,73

6

0(0,0,0)

11; 02,24,16,15; 30

0,26,0,24

0,15,75,165

11

0(0,0,0)

11; 0 ,15,40; 30

0,25,0,27

0,15,75,165

11

0(0,0,0)

11; 02,24,16,15; 44

0,26,0,24

0,5,32,85

11

0(0,0,0)

11; 02,15,40; 44

0,25,0,27

0,5,32,85

11

0(0,0,0)

11; 0 ,15,40; 28

0,25,0,27

0,21,84,161

11

0(0,0,0)

6,4,1; 10,16,0,24,5; 51

2,18,14,8

0,1,13,44

6

6(0,6,0)

6,4,1; 4,28,18,0,5; 51

2,14,22,8

0,1,13,44

6

0 ( 0,0,0 )

11; 0 ,24,16,15; 45

0,26,0,24

0,7,39,90

11

0(0,0,0)

11; 02,15,40; 45

0,25,0,27

0,7,39,90

11

0(0,0,0)

11; 02,24,16,15; 52

0,26,0,24

0,2,18,48

11

0(0,0,0)

11; 0 ,15,40; 52

0,25,0,27

0,2,18,48

11

0(0,0,0)

11; 02,24,16,15; 55

0,26,0,24

0,0,8,26

11

0(0,0,0)

11; 02,15,40; 55

0,25,0,27

0,0,9,25

11

0(0,0,0)

2,0,8,0,2; 4,32,0,20,10; 20

8,22,24,20

0,1,10,45

2

4(0,4,0)

3,8,02,1; 11,0,24,16,15; 27

4,26,20,24

0,3,27,108

3

3 ( 0,3,0 )

12; 0 ,48,0,18; 32

0,39,0,48

0,6,48,169

12

0(0,0,0)

12; 03,36,30; 32

0,38,0,52

0,6,48,169

12

0(0,0,0)

2,0,3,4,3; 8,18,24,16; 42

10,19,21,25

0,3,24,84

2

8(0,8,0)

1,4,0,5,2; 7,24,27,8; 43

9,17,21,27

0,2,18,71

1

3(0,3,0)

12; 03,36,30; 35

0,38,0,52

0,10,70,215

12

0(0,0,0)

12; 03,48,0,18; 36

0,39,0,48

0,15,90,240

12

0(0,0,0)

12; 0 ,36,30; 36

0,38,0,52

0,15,90,240

12

0(0,0,0)

6,0,6; 0,36,0,24,0,6; 48

4,23,28,16

0,3,26,96

6

0(0,0,0)

5,6,0,1; 11,0,24,16,15; 51

3,26,22,24

0,5,37,117

5

5(0,5,0)

5,6,0,1; 11,0,15,40; 51

3,25,23,27

0,5,37,117

5

5(0,5,0)

2

2

2

2

2

2

3

3

to be continued

WEI JiaLin et al.

Sci China Math

April 2010

Vol. 53

951

No. 4

continued Designs

n1 n2 m1 m2

Generators

BCDE ACpq BDpr BCps CDpt ABCDpu ABpv ABCD Bpqr ABqs Apqt 4 8 1 6 (GMC) ABCpqu Cpqv ABpw ABCD Bpqr ABqs Apqt 4 8 1 6 (MA) ABpu BCpv ACqw BDpq ADpr BCps ABpt 4807 ACpu CDpv ABCDpw ABC Aprs Brt Bqu Apqv 3 9 1 6 (GMC) ABqrw pqrx ABC Aqrs Apqt pqru 3 9 1 6 (MA) ABprv ABpqw Bpqrx BCqr BCps ACqt Bpqu 3 9 0 7 (GMC) Apqv ACpw Cpqx ABCpqr ACps Cpqt Apqu 3 9 0 7 (MA) ABpv ABqw BCqx Bpqs Bprt pqru Apqv 2 10 0 7 (GMC) Bqrw Aqrx Apry Aprs Apqt ABpu Bprv 2 10 0 7 (MA) Bqrw ABpqrx ABqy pqst Aqru Aqsv qrsw 1 11 0 7 (GMC) prsx Arsy pqrz Aprt Aqsu Aqrv pqrw 1 11 0 7 (MA) Apqrsx Apsy pqsz ACE BCDF CDG ACDH 11 2 7 1 ABCJ BCK ABCDL ABDpq ABDE BCF pr DF H 10 3 6 2 ADJ BDF K BDpq BF G ABC BDE ADF ABCDG ABps 9 4 5 3 (GMC) CDJ ACpq BCpr ABCH ABCDG ABEF BDEps 9 4 5 3 (MA) ABCH BCEJ ACDqs ACpr BCD ACDE ABDF ABCG 8544 ABpq BDpr ABCDpt BCDH BCps BCDpqu BDE ABqs ADG 7 6 4 4 (GMC) ABDF ABC Bqt Aqr ABCDpqu BDE Apr ABC 7 6 4 4 (MA) ADG ACsu ABqt ABDF ABCE ABDF BCDG 7635 BCpr ABCDps ABpt ACpu BDpq ABD ABCF Bqs ACE 6735 ABCpqr BCpt Cqu Apqv ABDEpv BCEF CDqv 6726 ACpr ABps BDtv ADuv ABCD ACD BCE Bpqr ABCqt 5 8 2 6 (GMC) ABqu ABpv Apqw Cqs BCDqw ABDE Apqr 5 8 2 6 (MA) ACptw Cpquw ABC Csw ABpv ABCE CDpq BCpr ABCDpt 5817 BDps ADpu ACpv ABpw ABCD BCqr Bpqs ACpt 4917 ACqu BCpv Apqw ABCpqx Apq Bpr ACDps Dpt ABDpx 4908 BCDpu ABCpv Cpw ABC Apqs Brt Bqu ABqrv 3 10 1 7 (GMC) Aqrw Aprx pqry ABC ABprs ABpqt Bpu 3 10 1 7 (MA) Aprv Aqrw Bpqrx Bqy Apqr BCqs ABpt BCpu 3 10 0 8 Cpqv ABqw ABCpqx ACqy ABqs ABpt ABpqru pqrv 2 11 0 8 Apqw ABrx Aqry Aprz 5716

(0) # # # 1C2 ; 2C2 ; 2(s)C(w)

WLP

SWLP

A3 , . . . , A6 B1 , . . . , B4

Clear effects

ME 2fi(WP, WS, SP)

12; 03,36,30; 35

0,38,0,52

0,21,105,245 12

0(0,0,0)

12; 03,48,0,18; 52

0,39,0,48

0,8,48,126

12

0(0,0,0)

12; 03,36,30; 53

0,38,0,52

0,7,46,129

12

0(0,0,0)

12; 03,36,30; 32

0,38,0,52

0,28,112,238 12

0(0,0,0)

5,6,0,1; 11,0,24,16,15; 61

3,26,22,24

0,2,20,66

5

5(0,5,0)

5,6,0,1; 11,0,15,40; 61

3,25,23,27

0,2,20,66

5

5(0,5,0)

12; 03,48,0,18; 54

0,39,0,48

0,9,54,135

12

0(0,0,0)

12; 0 ,36,30; 54

0,38,0,52

0,9,54,135

12

0(0,0,0)

12; 03,48,0,18; 62

0,39,0,48

0,3,26,71

12

0(0,0,0)

12; 0 ,36,30; 62

0,38,0,52

0,3,25,70

12

0(0,0,0)

12; 03 ,48,0,18; 66

0,39,0,48

0,0,13,39

12

0(0,0,0)

12; 0 ,36,30; 66

0,38,0,52

0,0,12,38

12

0(0,0,0)

2,02,8,3; 6,16,24,12,20; 22

12,30,41,44

0,1,11,55

2

6(0,6,0)

3,0,8,0,2; 2,20,24,16,10,6; 30 8,31,40,40

0,3,30,135

3

2(0,2,0)

4,8,02,1; 12,02,48,0,18; 36

4,39,32,48

0,6,54,217

4

4(0,4,0)

4,8,02,1; 12,02,36,30; 36

4,38,33,52

0,6,54,217

4

4(0,4,0)

13; 04,60,18; 40

0,55,0,96

0,10,80,285

13

0(0,0,0)

2,02,8,0,3; 6,16,24,12,20; 51

13,30,36,44

0,6,46,155

2

6(0,6,0)

0,6,0,4,3; 6,12,48,12; 54

10,24,39,54

0,3,29,122

0

0(0,0,0)

13; 04,60,18; 42

0,55,0,96

0,15,105,330 13,

0(0,0,0)

4,2,5,2; 2,20,24,16,10,6; 58

6,31,44,40

0,5,42,154

0(0,0,0)

13; 04,60,18; 42

0,55,0,96

0,21,126,350 13

0(0,0,0)

4,8,02 , 1; 12, 02 ,48,0,18; 60

4,39,32,48

0,8,56,174

4

4 ( 0,4,0 )

4,8,02,1; 12,02,36,30; 61

4,38,32,52

0,7,53,175

4

4 ( 0,4,0 )

13; 04,60,18; 40

0,55,0,96

0,28,140,350 13

0(0,0,0)

13; 04,60,18; 63

0,55,0,96

0,9,63,189

0(0,0,0)

4,8,02,1; 12,02,36,30; 36

4,38,32,52

0,36,144,342 4

4(0,0,4)

4,8,02 , 1; 12, 02 ,48,0,18; 72

4,39,32,48

0,3,29,97

4

4(0,4,0)

4,8,0 ,1; 12,0 ,36,30; 72

4,38,32,52

0,3,28,95

4

4(0,4,0)

13; 04,60,18; 63

0,55,0,96

0,12,72,190

13

0(0,0,0)

0,55,0,96

0,4,33,100

13

0(0,0,0)

3

3

3

2

4

2

13; 0 ,60,18; 73

4

13

to be continued

952

WEI JiaLin et al.

continued Designs

n1 n2 m1 m2

Generators

qst psu pqrsv pqrw 1 12 0 8 (GMC) Apqx rsy Aqrz Aprp Aqrst Aqru Apqsv Aprsw 1 12 0 8 (MA) psx Apqy pqrsz qsp ACDE ADF ABDG ACH 12 2 8 1 ABCDJ ABCK ABL CDpq BCDM ABCDF BCDG ABDH 11 3 7 2 ABCJ BDK ABL ACpq BCE CDpr BDF ps ABF G ABCF H 10 4 6 3 CF K CDqs ABpr ABCD ACE BCJ ACDEpt CEF ABEG 9 5 5 4 (GMC) ABCJ CDqt ABCpr Cps ACH ABD ADEpt ABEF ACEG 9 5 5 4 (MA) BCEJ ABDqt Drt BCD ABCH ACDst ACDE ABDF ABCG 8645 BDpq ABpr ABCDps BCpt ADpu BCDH ABDpq BDE ABDF ADG 7 7 4 5 (GMC) ACqrv Aps ABpt Bpu ABC ABCD ACE ABF BCG Aqv 7 7 4 5 (MA) BCpqr Cps ABCpqt Bpu ACDEpv ACEF ABEG 7736 ACpq BCpr BDsv ADtv ABpu ABCD ABD ACE ABCF Apqr 6 8 3 6 (GMC) Cqs BCpt Bqu ABCpqv BCqw ABCD ACE ABF ABpqr 6 8 3 6 (MA) BCqs ACpqt BCpu Bpv Cqw ACDE ABDF CDpq ACpr 6827 BDps ABpt ABCDpu BCpv ADpw BCD ACE ABpr Aqs Cpqt 5 9 2 7 (GMC) Bqu ABCpqv ACqw BCqx BCD ACE Bpqr ABCqs 5 9 2 7 (MA) Apqt BCpqu ACpqv ABqw Cpx ABDE Apq ACDpr BCDps 5 9 1 8 (GMC) ABDpt ABCpu Bpv Cpw BDpx ABDE ACDpq Apr Bps 5 9 1 8 (MA) ABCpt ABDpu BCDpv Dpw BCpx ABCD Bpqr ACqs Apqt 4 10 1 8 ABCpqu Cpqv ACpw BCpx BCqy Bpq Apr BCDps ABDpt 4 10 0 9 (GMC) ACDpu Cpv ABCpw Dpx ABpy BCpq BDpr ACps ADpt 4 10 0 9 (MA) CDpu ABpv ABCDpw Dpx ABCpy ABC ABpqrs Bqrz qrt Aru 3 11 1 8 (GMC) Bprv Bqw Apqx ABry ABC Aprx Apqt Bpu Bqz 3 11 1 8 (MA) Aqrv ABprs ABqry ABpqw ACqr ACps BCqt Bpqu Apqv 3 11 0 9 BCpw Cpqx ABqz ABpy Apqs Aprt pqru Bpqv 2 12 0 9 Bqrw Aqrx Bpry ABqz ABrp1 Aqst Apru Apsv prsw Apqrsx 1 13 0 9 Aqry Arsz Apqp1 qrsq1

Sci China Math

April 2010

(0) # # # 1C2 ; 2C2 ; 2(s)C(w)

Vol. 53

WLP

No. 4

SWLP

Clear effects

A3 , . . . , A6 B1 , . . . , B4

ME 2fi(WP, WS, SP)

4,39,32,48

0,0,17,55

4

4(0,1,3)

4,8,0 ,1; 12,0 ,36,30; 78

4,38,32,52

0,0,17,55

4

4(0,1,3)

2,03,12; 0,24,0,48,0,12,7; 24

16,45,64,72 0,1,12,66

2

0(0,0,0)

3,02,8,3; 0,6,51,16,0,18; 33

12,41,64,72 0,3,33,165

3

0(0,0,0)

4,8,02,1; 12,02,48,0,18; 78 2

2

4,0,8,0,2; 0,24,0,48,0,12,7; 40 8,45,64,72

0,6,60,271

4

0(0,0,0)

2,4,6,0,2; 4,20,0,36,25,6; 45

8,43,64,80

0,10,90,365

2

0(0,0,0)

1,12,04,1; 13,03,60,18; 45

6,55,40,96

0,10,90,365

1

1(0,1,0)

14; 05,84,7; 48

0,77,0,168

0,15,120,435 14

2,03,9,3; 4,20,0,36,25,6; 60

17,43,61,80 0,10,75,250

2

4 ( 0,4,0 )

0,4,3,3,3,1; 4,6,54,12,15; 65

12,35,64,88 0,5,47,196

0

0(0,0,0)

14; 05,84,7; 49

0,77,0,168

0(0,0,0)

0,21,147,476 14

0(0,0,0)

4,0,8,0,2; 0,24,0,48,0,12,7; 68 8,45,64,72

0,8,64,230

4

0(0,0,0)

2,4,6,0,2; 4,20,0,36,25,6; 69

8,43,64,80

0,7,60,228

2

0(0,0,0)

14; 05,84,7; 48

0,77,0,168

0,28,168,490 14

0(0,0,0)

4,0,8,0,2; 0,24,0,48,0,12,7; 69 8,45,64,72

0,12,80,246

4

0(0,0,0)

1,12,04,1; 13,03,60,18; 71

6,55,40,96

0,10,74,248

1

1(0,1,0)

2,4,6,0,2; 4,20,0,36,25,6; 45

8,43,64,80

0,36,180,486 2

0(0,0,0)

0,8,4,0,2; 8,16,0,32,35; 45

8,42,65,84

0,36,180,486 0

0(0,0,0)

14; 05,84,7; 73

0,77,0,168

0,12,84,262

0(0,0,0)

2,4,6,0,2; 4,20,0,36,25,6; 40

8,43,64,80

0,45,180,480 2

0(0,0,0)

0,8,4,0,2; 8,16,0,32,35; 40

8,42,64,85

0,45,180,480 0

0(0,0,0)

0,5,41,133

4

0(0,0,0)

6,55,40,96

0,5,39,129

1

1 ( 0,1,0 )

0,77,0,168

0,15,93,263

14

0(0,0,0)

0,77,0,168

0,5,44,139

14

0 ( 0,0,0 )

0,77,0,168

0,0,22,77

14

0(0,0,0)

4,0,8,0,2; 0,24,0,48,0,12,7; 83 8,45,64,72 4

3

1,12,0 ,1; 13,0 ,60,18; 83 5

14; 0 ,84,7; 73 5

14; 0 ,84,7; 85 5

14; 0 ,84,7; 91

14

Note: In the tables, the marks (GMC) and (MA) are used to denote GMC-MSA-FFSP designs (or GMC-FFSP designs) and MA-MSA-FFSP designs respectively. No mark after the parameters of a design means that the design is both GMC-MSA design and MA-MSA-FFSP design. In the column of generators, capital letters A, B, . . . are used to denote WP factors and lowercase letters p, q, r, s, . . . , and p1 and q1 are used to denote SP factors. The column M E lists the number of clear ME’s, the column 2fi lists the number of clear 2fi’s and clear WP2fi’s, and clear WS2fi’s and clear SP2fi’s are listed in the bracket followed, respectively.