Optimal blocking of two-level fractional factorial designs

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1 Journal of Statistical Planning and Inference 91 (2000) Optimal blocking of two-level fractional factorial designs Runchu Zhang a;, DongKwon Park b a Department of Statistics, Nankai University, Tianjin , People s Republic of China b Department of Statistics, Yonsei University, Wonju, , South Korea Received 26 January 1999; received in revised form 21 March 2000; accepted 30 March 2000 Abstract In this paper, the minimum aberration criterion is extended for choosing blocked fractional factorial designs. Ideally, one should seek a design that has minimum aberration with respect to both treatments and blocks. We prove the nonexistence of such a design. For this reason, it is needed to compromise between the wordlength pattern of blocks and that of treatments. By exploring the wordlength patterns of a two-level fractional factorial design, we introduce a concept of alias pattern and give accurate formulas for calculating the number of alias relations for any pair of orders of treatment eects as well as of treatment and block eects. According to the structure of alias pattern and the hierarchical principles on treatment and block eects, a minimum aberration criterion for selecting blocked fractional factorial designs is studied. Some optimal blocked fractional factorial designs are given and comparisons with other approaches are made. c 2000 Elsevier Science B.V. All rights reserved. MSC: 62K15; 62K10; 62K05 Keywords: Alias; Blocking; Fractional factorial design; Minimum aberration 1. Introduction Consider a regular 2 n p fractional factorial (FF) design, which has n factors, denoted by 1; 2;:::;n, and 2 n p runs and is uniquely determined by p independent dening words. The group formed by the p dening words is represented by G t = {I; w 1 ;:::;w 2 p 1} (1) and called the treatment dening contrast subgroup. Every element in G t not I is called a word, the number of letters in a word is called a wordlength and the vector W t =(A 1;0 ;:::;A n;0 ) (2) Corresponding author. address: zhrch@nankai.edu.cn (R. Zhang) /00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII: S (00) 转载

2 108 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) is called the wordlength pattern, where A i;0 denotes the number of words with length i in G t. As usual, we take the following two assumptions: (i) Lower-order interactions are more likely to be important than higher-order interactions. (ii) Interactions of the same order are equally likely to be important. According to this hierarchical principle, the minimum aberration (MA) (Fries and Hunter, 1980) has been used as an important criterion for selecting FF designs. Systematic sources of variations in experiments can be eectively eliminated by properly grouping the runs into blocks. This motivates people to study blocked FF designs. Usually, the block eects including their interactions exist signicantly, but the experimenter is not interested in estimating them. Let D(2 n p :2 k ) denote a 2 n p design in 2 k blocks of size 2 n p k (k n p). It can be viewed as a 2 (n+k) (p+k) FF design, where the factors are divided into dierent types: n treatment factors 1;:::;n and k block factors b 1 ;:::;b k. The 2 k combinations of the block factors are used to divide the 2 n p treatment combinations into 2 k blocks. In such a design, there are two types of words, which are called, respectively, treatment dening words and block dening words. To study the construction of optimal blocked designs, Bisgaard (1994) rst argued that it is no longer appropriate to dene the length of a block dening word as the number of letters it contains. He also treated the interaction of two or more block factors as a block eect. Sun et al. (1997) (henceforth abbreviated as SWC) made two additional assumptions which distinguish blocked FF designs from unblocked FF designs. The assumptions are: (iii) The interaction between two or more block factors has the same importance as a block main eect. (iv) Block factors have no interaction with treatment factors. Arranging a 2 n p FF design into 2 k blocks is equivalent to selecting k independent dening words for the k block factors b 1 ;:::;b k. Therefore, we can formally choose b 1 = v 1 ;:::;b k = v k (or write them as I = b 1 v 1 = :::= b k v k ) and call it a blocking scheme. The v i s and all their interactions v 1 v 2 ;:::;v 1 v k are respectively, confounded with the 2 k 1 block eects represented in notation by the b i s and their products b 1 b 2 ;b 1 b 3 ;:::;b 1 b k, and by assumption (iii) any one of these block eects is treated as a block main eect, denoted by b. By dropping all the b s from b i v i, i =1;:::;k, all the words v i ;v i v j ;:::;v 1 v k and the identity element form a group of size 2 k. We call the group {I; v 1 ;v 2 ;v 1 v 2 ;:::;v 1 v k } (or write it as {I; v 1 ;v 2 ;:::;v 2 k 1}); the block dening contrast subgroup, denoted by G b, and v 1 ;:::;v k are still called block dening words. By multiplying v i to the treatment dening contrast subgroup {I; w 1 ;:::;w 2 p 1}, we see that a block eect confounds the following set of eects: {v i ;v i w 1 ;v i w 2 ;:::;v i w 2p 1}:

3 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) Doing this for every elements in G b except for I, we get a set of eects, which are confounded with block eects. Its size is 2 p (2 k 1) and we denote the set by G b t. The wordlength pattern of G b t is dened by W b t =(A 1;1 ;:::;A n;1 ); (3) where A i;1 is the number of words with length i. The W b t indicates the inuence of block factors on the treatment eects. Up to now, we have not discussed an MA denition for a blocking scheme. Since judging the eectiveness of a blocking scheme depends on an FF design accompanied, we cannot directly give the denition of a minimum aberration on a blocking scheme, but instead we can give the denition of a minimum aberration on its wordlength pattern W b t. Let {w 1 ;:::;w p } denote p independent dening words of an FF design 2 n p. Take a blocking scheme b 1 = v 1 ;:::;b k = v k, {v 1 ;:::;v k } being independent and independent of {w 1 ;:::;w p }. Then we call {w 1 ;:::;w p ;v 1 ;:::;v k } a set of dening words of the blocked FF design. Actually, we can consider arbitrary p + k independent dening words in which n letters appear as a blocked FF design, as long as p words among them are specied to be treatment dening words and the rest to dene a blocking scheme. For simplicity, we still use {w 1 ;:::;w p } and {v 1 ;:::;v k } to denote them, respectively. For given n, p and k, let D denote an arbitrary blocked FF design D(2 n p :2 k ) with a choice of such {w 1 ;:::;w p } and {v 1 ;:::;v k }. Also, let W b t (D) denote its corresponding blocking scheme wordlength pattern. Generally, a set of dierent {w 1 ;:::;w p } and {v 1 ;:::;v k } corresponds to a dierent design D and a dierent W b t (D). According to the order of aberration of W b t (D) as usual, there is a minimum aberration design D in all such possible designs {D : {w 1 ;:::;w p }; {v 1 ;:::;v k }}. We call the wordlength pattern W b t (D ) an MA blocking scheme wordlength pattern for D(2 n p :2 k ) blocked FF designs, denoted by Wb t. Moreover, a blocking scheme with W b t is called an MA blocking scheme. Ideally, one should search for a design that has MA with respect to both treatments and blocks. In Section 2, however, based on a further analysis of the relationship between the two wordlength patterns W t and W b t we prove that there is no such design. So, we need to seek a criterion that compromises between W t and W b t.in this paper, we try to develop a new combining approach for choosing optimal blocked FF designs. In Section 3, we establish a criterion for choosing blocked FF designs. First, in Section 3.1, from the point of view of estimability we further explore the wordlength patterns of an FF design to introduce a concept of alias pattern for treatment eects and give an accurate formula for calculating the number of alias relations for any pair of orders of eects. And then, we give a formula for calculating the number of alias relations of any order treatment eects with block eects and deduce a minimum aberration criterion for blocked FF designs in Section 3.2. In Section 4, based on the criterion we give optimal D(2 n p :2 k ) blocked FF designs for some parameters p and k with run sizes 8, 16, 32, 64, and 128 and make some comparisons.

4 110 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) A relationship between wordlength patterns W t and W b t Let {w 1 ;:::;w p ;v 1 ;:::;v k } denote a blocked FF design with p treatment dening words and k block dening words. Similar to a regular FF design, the p+k independent words and I generate a group, denoted by G t+b, and we can dene its wordlength pattern and denote it by W t+b. Obviously, we have G t+b = G t + G b t and W t+b = W t + W b t : (4) On the other hand, for a given set of dening words {w 1 ;:::;w p ;v 1 ;:::;v k }, from the generated subgroup G t+b, we can take any set of p independent words for treatment factors and a set of k independent words independent of the selected p words for block factors. Clearly, a dierent selection gives a dierent blocked FF design. Example 1. Consider a design with 2 2 blocks. Take four dening words, presented by I = 1235 = 1246 = 134 = 234: Then the corresponding dening contrast subgroup G t+b is G t+b ={I; 12; 35; 46; 134; 136; 145; 156; 234; 236; 245; 256; 1235; 1246; 3456; } and its wordlength pattern is W t+b =(0; 3; 8; 3; 0; 1): We take two independent words 12; 35 as dening relations of two treatment factors, and two independent words 46; 136 for two block factors. Clearly, G t = {I; 12; 35; 1235} and G b t = G t+b G t = G t+b {I; 12; 35; 1235} = {46; 134; 136; 145; 156; 234; 236; 245; 256; 1246; 3456; }: Denote this design by D 1. We have W t =(0; 2; 0; 1; 0; 0); W b t =(0; 1; 8; 2; 0; 1): Now from G b+t take two more designs with dierent treatment dening relations and blocking schemes: D 2 : I = 234 = 156 and b 1 = 134, b 2 = 145, which has W t =(0; 0; 2; 0; 0; 1); W b t =(0; 3; 6; 3; 0; 0); and D 3 : I = 1235 = 1246 and b 1 = 134, b 2 = 234 (SWC 6-2.1/B2.1), which has W t =(0; 0; 0; 3; 0; 0); W b t =(0; 3; 8; 0; 0; 1): According to the aberration criterion in W b t, obviously, D 1 D 2 D 3, whereas D 3 D 2 D 1 in W t. Here D i D j means that D i is better than D j according to the aberration criterion.

5 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) Theorem 1. There is no minimum aberration design with respect to both treatments and blocks. Before giving the proof of the theorem, we prove two lemmas. Lemma 1. Let {w 1 ;:::;w p ;v 1 ;:::;v k } with p 1 and k 1 be a set of independent words for dening a 2 n p FF design with 2 k blocks and G t+b denote the generated subgroup. Suppose that in G t+b there is at least one subset of p independent words which includes all the n letters. Then in G t+b there are at least two subsets of p independent words with dierent wordlength patterns. Proof. First, we note the following fact. For any set of independent dening words (u 1 ;:::;u s ), let (A 1 ;:::;A n ) be the wordlength pattern of the group generated by these words and n be the number of letters appearing in the group. Then the following equations hold: Ai =2 s 1; iai = n2 s 1 (5) (Brownlee et al., 1948). Because of p 1, k 1 and the independence of the p + k words, there are at least two sets of p independent words from G t+b which generate two dierent subgroups. Let N denote the number of such dierent subgroups. According to the assumption, all the n letters appear in at least one subset of p independent words. Suppose that all the possible subgroups generated by dierent sets of p independent words chosen from G t+b have the same wordlength pattern. Then, every such subgroup satises (5) with s = p and hence the total number of appearance of words in the N subgroups is N(2 p 1), and the total sum of length of words appearing in the N subgroups is Nn2 p 1. So, the average length of the words appearing in all the subgroups is (Nn2 p 1 )=(N (2 p 1)). On the other hand, according to (5), the average length of words in G t+b is (n2 p+k 1 ))=(2 p+k 1)). Since by group properties each word in G t+b appears in the N subgroups with the same number of times, we have Nn2 p 1 N (2 p 1) = n2p+k 1 (2 p+k 1) ; which implies 2 p+k 1=2 k (2 p 1): (6) However, the right-hand side of (6) is odd and the left-hand side of (6) is even. This contradiction proves the lemma. Lemma 2. Let {w 1 ;:::;w p } be the dening words for a 2 n p FF design. If the design is an MA design; then all the n letters must appear in the dening words. Proof. If some letter, say i, does not appear in the dening words, we put i into any word in {w 1 ;:::;w p }, the wordlength pattern generated by the new dening words is

6 112 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) better than the original one. This contradicts the assumption of MA and proves the lemma. Proof of Theorem 1. Assume that, for given n; p and k, there exists a blocked design that has minimum aberration with respect to both treatments and blocks. Let {w 1 ;:::;w p ;v 1 ;:::;v k } denote the design, where {w 1 ;:::;w p } is the generators for treatment factors and {v 1 ;:::;v k } is the scheme for k block factors, and G t+b denote the generated group. Also, use G t to denote the subgroup generated by {w 1 ;:::;w p }.By Lemma 2, {w 1 ;:::;w p } includes all the n letters. By Lemma 1, we can take another p treatment generators from the group G t+b with the wordlength pattern dierent from G t. That is, the aberration of the new wordlength pattern will become more or less compared with G t. This means that the aberration in G b t becomes less or more from (4). In other words, as the aberration in W b t becomes less, aberration in W t becomes more and vice versa. The proof follows by contradiction. 3. Criteria for MA blocked FF designs There are some recent works on nding optimal blocked FF designs (Bisgaard, 1994; SWC, 1997; Sitter, Chen and Feder (henceforth abbreviated as SCF), 1997). Chen and Cheng (1997) has given a convincing review. From their analysis, we can be aware that to nd a good criterion for choosing optimal blocked FF design, the key point is to properly treat the two types of words. Two methods can be considered. One is to treat separately the two wordlength pattern vectors W t and W b t. However, by Theorem 1, it is impossible to nd an optimal design for both W t and W b t. The other is to combine the components of the two wordlength pattern vectors into one mixed wordlength pattern in an appropriate ordering according to the hierarchical assumptions. SCF (1997) used the second method. We also focus on this method to improve their ordering A further exploration to MA criterion of FF designs The MA criterion has been extensively studied to construct optimal FF designs for both symmetrical and asymmetrical cases (Franklin, 1984; Chen and Wu, 1991; Wu and Zhang, 1993). To extend the MA criterion to blocked FF designs, it is necessary to reveal further more essential natures of the criterion of MA designs. First, starting from the wordlength pattern we give a formula for calculating the number of alias relations for every pair of orders of eects in an FF design and introduce a concept of alias pattern. Second, in accordance with the hierarchical principle we deduce an appropriate ordering of the components in the alias pattern. Let us start with a simple example. Consider a design with the dening relations I = 124 = = 345:

7 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) Its wordlength pattern is W t =(2; 0; 1; 0). Calculate the number of alias relations for 2-factor interaction eects aliased with 3-factor interaction eects. From I = 124, we use 13; 15; 16; 23; 25; 26; 34; 45; and 46 to multiply, respectively, its both sides, getting 13 = 234, 15 = 245, 16 = 246, 23 = 134, 25 = 145, 26 = 146, 34 = 123, 45 = 125, and 46 = 126. Similarly, from I = 345, we get 13 = 145, 15 = 134, 14 = 135, 23 = 245, 24 = 235, 25 = 234, 36 = 456, 46 = 356, and 56 = 346. From I = 12356, we can get alias relations 12 = 356, 13 = 256, 15 = 236, 16 = 235, 23 = 156, 25 = 126, 26 = 135, 35 = 126, 36 = 125, 56 = 123: So, in total, the number of alias relations is equal to 28. Similarly, by symmetry, we can obtain the number of alias relations for 3-factor interaction eects aliased with 2-factor interaction eects, which is also equal to 28. Clearly, the number of alias relations indicates the degree of aliasing one type of eects by another type of eects. Now, let l C m denote the number of alias relations of l-factor interactions (l- s) which are aliased by m- s and let us deduce a general formula for calculating l C m. By symmetry of l C m, we only consider m l. Let 1 l and 1 m denote an l- and an m-, respectively. If the two eects are aliased each other in the design, then we have 1 l = 1 m or I = 1 l 1 m, where probably some i s are equal to some j s. It implies that by deleting the same i s and j s, the product 1 l 1 m must be a word in the dening contrast subgroup G t and its length is m + l 2u for some u, or written as m l +2k for some k. On the other hand, if 1 m l+2k is a word in G t, we can choose any k letters from the m l +2k letters of { 1 ;:::; m l+2k } (k6l), say 1 ;:::; k, and choose any l k letters from the rest n (m l +2k) letters of {1; 2;:::;n}\{ 1 ;:::; m l+2k }, say k+1 ;:::; l,we get 1 k k+1 l = 1 k k+1 l 1 m l+2k = k+1 l i1 im l+k ; where { i1 ;:::; im l+k } = { 1 ;:::; m l+2k }\{ 1 ;:::; k }. It gives an alias relation of the l- 1 k k+1 l and ( the m- k+1 )( l i1 ) im l+k. In total, from the word 1 m l+2k we can obtain n (m l+2k) m l+2k l k k dierent alias relations of the pair of l- and m-. Obviously, from dierent words we obtain dierent alias relations. Also, the number k can be and can only be 0; 1;:::;l. For any other words with length m l +2k 1 with any k, it is impossible to get an alias relation of the pair of l- and m-. Thus, we obtain the general formula for calculating lc m : ( )( ) lc m = l n (m l +2k) m l +2k A (m l+2k);0 ; l;m=1; 2;:::;n; k=0 l k k (7) where ( ( ) x 0) =1; x y = 0 for x y or x 0, and A i;0 = 0 for i62 ori n. We write the numbers of alias relations { l C m } of a design as a matrix C =( l C m ), which is symmetric, and call the matrix an alias pattern matrix (or alias pattern, for short).

8 114 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) Example 2. Continue the design above. By calculating the numbers of alias relations of all pairs, we obtain its alias pattern: C = : From formula (7), we see that l C m is a function of A i;0 s but not a one-to-one function of A i;0 s. From the denition we can see that, the larger the number of alias relations between l- s and m- s, the more serious the aliasness between l- s and m- s. Especially, for the lower order l- s and m- s, the larger the numbers, the more serious the aliasness of eects for a design. Also, we need to note that an l- probably aliases with more than one m-. For example, for the design above, the main eect 4 is aliased by two relations with 2-: 4 = 12 and 4 = 35. It implies that the design with the larger number of alias relations of l- s and m- s may not have the larger number of dierent l- s which are aliased with at least one m-. But, we can consider this in another way: if one l- is aliased with more m- s, then it is more serious for the l- to be aliased with m- s. From the point of view of estimability, because of this, the l- has fewer possibilities to be estimated by using this design. Therefore, the alias pattern can be used as a measurement to evaluate the aliasness of a design. Furthermore, how do we use the alias pattern to judge an FF design to be good or not? According to the hierarchial principle, we can rank the components of the alias pattern into an appropriate ordering of importance as follows: ( 1 C 2 ; 2 C 2 ); ( 1 C 3 ; 2 C 3 ; 3 C 3 ); ( 1 C 4 ; 2 C 4 ; 3 C 4 ; 4 C 4 ); ( 1 C 5 ; 2 C 5 ; 3 C 5 ; 4 C 5 ; 5 C 5 );:::;( 1 C j ; 2 C j ;:::; j C j );:::; (8) In this ordering, the numbers in every parenthesis form a subset, all the subsets are ranked, and the numbers in every subset are also ranked. According to the ordering, for a concrete design and experiment we only need to consider a part of the numbers. For example, when three-factor and higher-order interactions are negligible, the only numbers we need to consider are ( 1 C 2, 2 C 2 ). Obviously, among ( 1 C 2, 2 C 2 ), 1 C 2 is more important than 2 C 2, because the estimability of main eects is to be considered rst and then consider the estimability of two-factor interaction eects. Also, the smaller the number 1 C 2, the better. If the numbers 1 C 2 for two designs are the same, then the design with smaller number 2 C 2 is better, because they have the same number of estimable main eects but the one with smaller number 2 C 2 has more estimable two-factor interactions. When four-factor and higher-order interactions are negligible, we do not need to consider the numbers ( 1 C 4, 2 C 4, 3 C 4, 4 C 4 ) and the others after them. By the same reasoning, to choose optimal FF designs we should sequentially minimize

9 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) the numbers 1 C 2, 2 C 2, 1 C 3, 2 C 3, 3 C 3. Generally, we obtain that, to choose optimal FF designs we should sequentially minimize the numbers 1C 2 ; 2 C 2 ; 1 C 3 ; 2 C 3 ; 3 C 3 ;:::; 1 C j ; 2 C j ;:::; j C j ;:::: (9) From formula (7), we know that there is a functional relationship between i C j s and A i;0 s as well as between the orderings of i C j s and A i;0 s. It is easy to nd out that the ordering of i C j s is equivalent to the ordering of A i;0 s: A 3;0 ;A 4;0 ;A 5;0 ;:::;A n;0 ; (10) which is just the rule of the minimum aberration criterion for FF designs. That is to say, to choose optimal FF designs we just sequentially minimize A 3;0 ;:::;A n;0. From the above analysis we can see that the alias pattern gives more details about the MA criterion, so that the ordering of components of wordlength pattern vector in the criterion has a more natural explanation Minimum aberration criteria for blocked FF designs Now, we rst apply the concept of alias pattern to W b t =(A 2;1 ;:::;A n;1 ). We notice that, in G b t, all the identity relations before dropping b i s are of the form I = t tb, where the number of t is at least two because of A 1;1 = 0. For any relation I = t tb, multiplying both sides by t s or b cannot get any alias relation between treatment interaction eects. Also, under assumptions (iii) and (iv), all the interactions of the form t tb with at least one t are negligible. Therefore, the only alias relation we need to consider from W b t or G b t is the one between the treatment interaction t t and the block eect b. So we only need to calculate the number of alias relations of treatment l- s and the block eect and denote the number by l C b (or b C l ). Then we simply have lc b = A l;1 ; for l =2; 3;:::;n: (11) Next, let us consider how to arrange l C b to an appropriate place in the ordering (8). First, we consider 2 C b. According to the hierarchial principles, since 2 C b does not involve the main eects of the aliasness of treatment but 1 C 2 does, 1 C 2 is more important than 2 C b. Comparing 2 C b with 2 C 2, they both involve the estimabality of treatment 2- s, but the block eects more likely exist than a treatment 2- does, and hence the number 2 C b is more serious than 2 C 2. Therefore, 2 C b should be ranked after 1C 2 and before 2 C 2. In accordance with the same reasoning, for any l l C b should be located after (l 1) C l and before l C l. As a result, we obtain the ordering of the mixed components of treatment and block alias patterns: ( 1 C 2 ; 2 C b ; 2 C 2 ); ( 1 C 3 ; 2 C 3 ; 3 C b ; 3 C 3 ); ( 1 C 4 ; 2 C 4 ; 3 C 4 ; 4 C b ; 4 C 4 ); ( 1 C 5 ; 2 C 5 ; 3 C 5 ; 4 C 5 ; 5 C b ; 5 C 5 );:::;( 1 C j ; 2 C j ;:::; (j 1) C j ; j C b ; j C j );:::: (12)

10 116 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) For the rst several components we list them as follows: 1C 2 =3A 3;0 ; 2C b = A 2;1 ; 2C 2 =6A 4;0 ; 1C 3 =4A 4;0 ; 2C 3 =3(n 3)A 3;0 +10A 5;0 ; 3C b = A 3;1 ; 3C 3 =6(n 4)A 4;0 +20A 6;0 ; 1C 4 =(n 3)A 3;0 +5A 5;0 ; 2C 4 =4(n 4)A 4;0 +15A 6;0 ; 3C 4 = 3 2 (n 3)(n 4)A 3;0 + 10(n 5)A 5;0 +35A 7;0 ; 4C b = A 4;1 ; 4C 4 =3(n 4)(n 5)A 4;0 + 20(n 6)A 6;0 +70A 8;0 : Also, we can indicate the alias pattern as a matrix C =( i C j ), where i C j = i C j if i j and i C i =( i C b ; i C i )ifi = j. From relationships (7) and (11), it is easy to show that sequentially minimizing the numbers in (12) is equivalent to sequentially minimizing the numbers A 3;0 ;A 2;1 ;A 4;0 ;A 5;0 ;A 3;1 ;A 6;0 ;A 7;0 ; A 4;1 ;A 8;0 ;:::;A 2j 1;0 ;A j;1 ;A 2j;0 ;:::: (13) For convenience, we denote the equivalent ordering vector (13) by W bt =(u 1 ;u 2 ;:::; u 2n 3 ), where u 1 = A 3;0 ;u 2 = A 2;1 ; and so on, and call W bt a combined wordlength pattern of a blocked FF design. The ordering vector (13) appeared rst in Chen and Cheng (1997) in the form A 3;0 ;A 2;1 }{{} ;A 4;0;A 5;0 ;A 3;1 }{{} ;A 6;0;A 7;0 ;A 4;1 }{{} ;A 8;0;::: and they considered A 3;0 and A 2;1, A 5;0 and A 3;1, and other pairs with under-brace to be at the same level of importance, and proposed a combined blocking wordlength pattern W b =(A b 3 ;Ab 4 ;Ab 5 ;Ab 6 ;:::) (in their notation), i.e. their criterion of minimum aberration for blocked factorials sequentially minimizes 3A 3;0 + A 2;1 ;A 4;0 ; 10A 5;0 + A 3;1 ;A 6;0 ;:::; etc. Our wordlength pattern only is the ordering vector (13). We treat separately components A 3;0, A 2;1, A 5;0, A 3;1, and so on, so our criterion has a dierence to theirs. Also, because of this, our discussion on resolution at the end of this section has a dierence from theirs. Using a simple symbol, we can express ordering (13) as follows: ttt ttb tttt ttttt tttb tttttt ttttttt ttttb ; (14) where the notation means more important than, more serious than or more undesirable than.

11 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) Comparing with SCF s ordering: ttt ttb tttt tttb ttttt ttttb tttttt tttttb ; (15) the new ordering (14) has some changes. From a comparison in Section 4 we nd that the new criterion is more reasonable than SCF s. In addition, based on the wordlengths of the dening contrast subgroups and a consideration on sequential experiments, Cheng and Wu (1997) introduced two dierent criteria with two dierent orderings. Their rst ordering is the same as (14) and the second one is the following: ttt tttt ttb ttttt tttttt tttb ttttttt tttttttt : (16) They proposed that, under the assumption that follow-up experiments will be conducted to de-alias some eects, the rst ordering is a more appropriate criterion; if there are no follow-up experiments, the second one is considered more appropriate. Although ordering (14) is the same as the rst one of Cheng and Wu (1997), the assumption and reasoning method are dierent so that the implication of optimality is dierent. In our approach, the assumption is very general without relating follow-up experiments. Also, our approach can quantitatively compare any two designs using the alias pattern to nd out how one of them is better than the other. We call this criterion Criterion 1. Thus, the denition of minimum aberration for blocked FF designs in accordance with Criterion 1 can be addressed as follows: Denition 1. Suppose D 1 and D 2 are two blocked FF designs with combined wordlength pattern W bt (D 1 ) and W bt (D 1 ), respectively. Let r be the smallest value such that u r (D 1 ) u r (D 2 ). Design D 1 is said to have less aberration than D 2 if u r (D 1 ) u r (D 2 ). If there is no design with less aberration than D 1, then D 1 is said to have the minimum aberration. Let us dene the resolution of a blocked FF design according to Criterion 1. Applying the concept of the resolution for the usual FF designs to the current situation, we can determinate the resolution of a blocked FF design as follows. Let r denote the smallest j such that u j 0inW bt =(u 1 ;u 2 ;:::;u 2n 3 ) of a design. If u r refers to A k;0 for some k, then the design is said to have resolution k. Ifu r refers to A j;1 for some j, then the design is said to have resolution (2j). Obviously, for the former case, it can be explained as the resolution of usual FF designs. Now, let us explain the latter case. Since we have assumptions (iii) and (iv), in this case all j 1 or lower-order s are not aliased with any lower than (j + 1)-order s and are estimable without any more assumptions. It basically meets the denition of resolution at least 2j. By a more meticulous consideration, for a usual resolution 2j design there are some j- s which are aliased with some other j- s, while for the current case, there are some j- s that are confounded with some block factors. So it seems that the resolution of the current case is a little weaker than the resolution 2j of the usual FF

12 118 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) designs, since a block factor eect exists more frequently (or likely) than a j- does. As a result, we say that the design in the latter case has resolution (2j) and call it weak resolution 2j. 4. Optimal blocked FF designs and comparisons To nd out optimal designs for Criterion 1, comparing with SCF s criterion, we note that the rst three components of both wordlength patterns are the same. So, all the optimal blocked designs with resolution not greater than IV for both the criteria are the same, but the wordlength patterns are dierent and the resolution III.5 in SCF s criterion corresponds to the resolution IV in Criterion 1. For example, for design D(2 5 1 :2 2 ), for both the criteria there is the same optimal design with dening relations I = 1235=134b 1 =234b 2, but for the wordlength pattern, W SCF =(0; 2; 1; 4; 0) and W bt =(0; 2; 1; 0; 4); and the resolution for SCF s criterion is III.5 and the resolution for Criterion 1 is IV. Since the components after the third one of the two wordlength patterns are dierent, the optimal designs with resolution greater than IV for the two criteria are probably dierent and hence need to search. By using an exhaustive method, for Criterion 1 we found out all the optimal blocked FF designs (up to an isomorphic) with the same parameters as SCF s. There are ve optimal designs which are dierent from SCF s with run size 32, 64, and 128. Since most of the optimal designs of Criterion 1 are the same as those of SCF s criterion, which has been listed in Tables 2 4 of SFC (1997) and a correction (to appear in Technometrics), so we only need to list the dierent ones. The ve dierent designs are D(2 6 1 : 2 1 ), D(2 7 1 : 2 2 ), D(2 8 1 : 2 1 ), D(2 8 1 : 2 2 ), and D(2 9 2 : 2 3 ), which are listed in Table 1 with a simple comparison with SCF s (for the usage of Tables 1 and 2 in Section 5 see SCF, 1997). These designs were also obtained by S.W. Cheng in his thesis proposal. We make a detailed comparison for the ve dierent designs listed in Table 1. For explaining the table, let us consider the following example. Example 3. Consider design D(2 7 1 : 2 2 ). According to SCF s criterion, they give the optimal design, denoted by D 1, with dening relations: I = = 125b 1 = 2356b 2 : According to Criterion 1, one of the optimal design, denoted by D 2 Table 1, has the dening relation: and shown in I = = 123b 1 = 145b 2 : Their wordlength patterns are listed in Table 1 (for simplicity, denote Criterion 1 by the short notation Cr.1).

13 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) Table 1 A comparison of some dierent optimal designs according to SCF s and Criterion 1 Design & Column Column W SCF =(A 3;0 ;A 2;1 ; W bt =(A 3;0 ;A 2;1 ; Resolution criterion (Trt) (Bl) A 4;0 ;A 3;1 ;A 5;0 ;A 4;1 :::) A 4;0 ;A 5;0 ;A 3;1 ;A 6;0 :::) :2 1 SCF s IV.5 Cr.1 s VI :2 2 SCF s IV.5 Cr.1 s VI :2 1 SCF s VI Cr.1 s VIII :2 2 SCF s V.5 Cr.1 s VIII :2 3 SCF s ::: ::: IV.5 Cr.1 s ::: ::: VI Consider their alias pattern matrices (part): (0; 0) (0; 0) C1 = 0 10 (0; 0) 20 0 ; C2 = 0 0 (0; 0) 35 0 : (2; 0) (3; 0) (3; 0) (3; 0) From the alias pattern matrices we can see that, for i63 and j63, all i C j = 0 for D 2, but not for D 1. For D 1, 2 C 3 = 10. This means that there are 10 two-factor interaction eects which are aliased with the three-factor interaction eects. But for D 2 there are no two-factor interaction eects which are aliased with any three-factor interaction eects. In addition to this, for D 1, there are ve main eects which are aliased with the four-factor interaction eects, but for D 2 there are no such aliases. So under the assumption that four-factor or higher-order interaction eects are negligible, all the main, two-, and three-factor interaction eects can be estimated for design D 2. But for design D 1 only the main eects can be estimated. If we only assume that ve-factor or higher-order interaction eects are negligible, then, for design D 2 all the main and two-factor interaction eects can be estimated, but for design D 1 neither the main nor the two-factor interaction eects can be estimated. In this case, obviously design D 2 is better than design D 1. A similar discussion applies to designs D(2 6 1 : 2 1 ), D(2 8 1 :2 1 ), D(2 8 1 :2 2 ), and D(2 9 2 :2 3 ).

14 120 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) Table 2 A comparison of some dierent optimal designs according to SCF s and Criterion 2 Design & Column Column W SCF =(A 3;0 ;A 2;1 ; W bt =(A 3;0;A 2;1 ; Resolution criterion (Trt) (Bl) A 4;0 ;A 3;1 ;A 5;0 ;A 4;1 :::) A 4;0 ;A 3;1 ;A 5;0 ;A 6;0 :::) :2 2 SCF s V.5 Cr.2 s VII 5. Some discussions 1. The establishment of a criterion closely depends on the assumptions given. Criterion 1 is introduced under assumptions (i) (iv) and the assumption that although the block eects are signicant, the experimenter is not interested in estimating them. If the experimenter is interested in estimating them, Criterion 1 will not be suitable. It is suggested that any block eect is considered to be less important than the treatment main eects but more important than 2- s. In this case, the ordering of the components for the alias pattern will be changed as follows: ( 1 C 2 ; b C 2 ; 2 C 2 ); ( 1 C 3 ; b C 3 ; 2 C 3 ; 3 C 3 ); ( 1 C 4 ; b C 4 ; 2 C 4 ; 3 C 4 ; 4 C 4 ); ( 1 C 5 ; b C 5 ; 2 C 5 ; 3 C 5 ; 4 C 5 ; 5 C 5 );:::;( 1 C j ; b C j ; 2 C j ;:::; (j 1) C j ; j C j );:::: (17) By this ordering, from (7) we obtain the following combined wordlength pattern: A 3;0 ;A 2;1 ;A 4;0 ;A 3;1 ;A 5;0 ;A 6;0 ;A 4;1 ;A 7;0 ; A 8;0 ;A 5;1 ;A 9;0 ;:::;A 2j;0 ;A j+1;1 ;A 2j+1;0 ::: (18) twhich is a criterion for the blocked FF designs under the new assumptions, denoted by W bt and called it Criterion 2. We can similarly dene the resolution of a design under this criterion. For this criterion, since the rst ve components in (18) are the same as the rst ve components in SCF s criterion, all the optimal designs with resolution not greater than V listed in SCF (1997) are the same optimal designs for Criterion 2. Among the ve designs with resolution V.5 in SCF, only one design D(2 8 1 :2 2 ) is dierent, which is listed in Table 2. Under the assumption that four-factor or higher-order interaction eects are negligible, all up to three-factor interaction eects are estimable for the design, but not for the optimal design of SCF s criterion because 3 C 3 =20 0. So, the optimal design of Criterion 2 is better than the one of SCF s criterion even if block eects need to be estimated. 2. In principle, it is not hard to extend this approach to the case of S-level, where S is a prime or prime power.

15 R. Zhang, D. Park / Journal of Statistical Planning and Inference 91 (2000) Acknowledgements This work was completed while the authors were visiting the Department of Statistics, University of Michigan, in The authors would like to thank Professor C.F.J. Wu for discussions and help and to acknowledge discussions with S.W. Cheng. Also, thanks go to Dr. D.X. Sun for his complete list of blocked fractional factorial designs sent by . Zhang s research was supported by National Education Committee Foundation for Research Abroad 1996 and NNSF project of China. Park s research was supported by Korean Research Foundation supporting for Faculty Research Abroad. References Bisgaard, S., A note on the denition of resolution for blocked 2 n p designs. Technometrics 36, Brownlee, K.A., Kelly, B.K., Loraine, P.K., Fractional replication arrangements for factorial experiments with factors at two levels. Biometrika 35, Chen, H., Cheng, C.S., Theory of optimal blocking of 2 n m designs. Preprint. Chen, J., Wu, C.F.J., Some results on S n k fractional factorial designs with minimum aberration or optimal moments. Ann. Statist. 19, Cheng, S.W., Wu, C.F.J., Optimal blocking schemes for 3 n and 3 n k designs. Joint Research Conference on Statistics in Industry and Technology, June 2 4, 1997, New Brunswick, NJ. Franklin, M.F., Constructing tables of minimum aberration p n m designs. Technometrics 26, Fries, A., Hunter, W.G., Minimum aberration 2 n p designs. Technometrics 22, Sitter, R.R., Chen, J., Feder, M., Fractional resolution and minimum aberration in blocked 2 n p Designs. Technometrics 39, Sun, D.X., Wu, C.F.J., Chen, Y.Y., Optimal blocking schemes for 2 n and 2 n p designs. Technometrics 39, Wu, C.F.J., Zhang, R.C., Minimum aberration designs with two-level and four-level factors. Biometrika 80,