Designing Two-level Fractional Factorial Experiments in Blocks of Size Two

Sankhyā : The Indian Journal of Statistics 2004, Volume 66, Part 2, pp 325-340 c 2004, Indian Statistical Institute Designing Two-level Fractional Factorial Experiments in Blocks of Size Two P.C. Wang National Central University, Chung Li, Taiwan Abstract We need extra runs to design two-level factorial experiments in blocks of size two to estimate all the available effects, as is possible in experiments without blocking. The number of runs suggested is (n p)2 n p for 2 n p fractional factorial experiments. In designing such an experiment, two issues need to be considered. First, the precision of estimates is usually different because different numbers of observations are used for estimation in the analysis of the resulting data. It is important to have more precise estimates of the effects with which we are most concerned. Second, the trade-off between runsize reduction and the possibly negligible effects is of significance, especially when the number of factors is large. To deal with these two issues, several assignment rules are suggested for designing good experiments. AMS (2000) subject classification. 62K05, 62K15. Keywords and phrases. Precision, blocking, factorial designs, orthogonal arrays. 1 Introduction Blocking is an important tool to increase precision of an experiment. With blocking, usually experimental runs are obtained based on the defining relations or the assignment of factors in an orthogonal array, and then partitioned equally into blocks of a given size. The block size can be varied depending on experimental situations. Sun, Wu and Chen (1997) discussed optimal blocking schemes for two-level factorial experiments in blocks of all possible sizes. In industrial experiments for improving quality of products, experimental processes in the laboratory might involve using a small oven in which only two experimental units can be put. Due to various temperatures at different times, we need to consider the oven as a block of size two. Also many industrial processes might yield different products at different times. If only two experimental runs are allowed to be executed at a time, considering time as a block of size two is a good choice. For more cases of blocks of

326 P.C. Wang size two, see Bisgaard(1994). In this paper, we consider fractional factorial experiments in blocks of size two. A major concern in designing a factorial experiment with blocks is the confounding of factorial effects with block effects. The confounding causes inaccurate estimates of the corresponding effects. If only main effects of factors are considered important, the mirror-image pair of runs given in Box, Hunter and Hunter (1978) can be used to estimate main effects free of block effects. If all the factorial effects are to be investigated, Draper and Guttman (1997) suggested using n2 n runs for n complete factorial experiments and (n p)2 n p runs for 2 n p fractional factorial experiments. Based on the analysis of previous experimental data, some high-order interactions might not be as important as main effect or low-order interactions. We might sacrifice some high-order interactions to reduce the run size; otherwise any experiment with a large number of factors becomes huge and costly when using Draper and Guttman s (1997) method. In addition, even if all the effects are estimable, the degrees of confounding in different effects may be different. That means some estimates of effects might be more precise than others. The precision of estimates depends on the blocking method used for experimental runs. In this article, we propose methods to deal with the precision problem and the size problem. To understand these problems, we use an example with detailed explanations in the next section. In section three, in order to propose procedures for obtaining more precise estimates for main effect and low-order interactions, two approaches to establish experimental runs in blocks are discussed and compared. The orthogonal array approach is shown to be better than the traditional approach of contrasts in dealing with the precision problem and the size problem. Wang and Jan (1995) also used the orthogonal array approach to set up rules for designing fractional factorial experiments when the run order is important. In section four, using the fixed blocking scheme, we suggest assignment rules to obtain experimental runs in blocks of size two for complete factorial experiments. Based on these suggestions, we establish several good designs for experiments in various replicates. In sections five and six, we suggest two appropriate assignment rules for 2 n 1 fractional factorial experiments and two for 2 n p fractional factorial experiments respectively. Finally conclusions and discussion follow in Section 7. 2 2 3 Designs in Blocks of Size Two Suppose we run a 2 3 factorial experiment in blocks of size two. Draper and Guttman (1997) suggested 24 runs for exploring three main effects, three

Designing experiments in blocks 327 two-factor interactions and one three-factor interaction. Let A, B and C be three two-level factors. Using Draper and Guttman s (1997) method, Table 1 shows twelve blocks, each with two runs, for investigating seven effects. It is clear that all the main effects and interactions can be estimated using some comparisons from 12 blocks. To estimate the main effect A free of block effects, only eight runs in blocks 1, 2, 3 and 4 can be used. However, to estimate the interaction of A and B, we can use sixteen runs in blocks 1, 2, 3, 4, 5, 6, 7 and 8. These two estimates differ in their precision due to using different numbers of observations. Table 1. A 2 3 factorial design in three groups of twelve blocks of size two Group 1 Group 2 Group 3 Block Runs Block Runs Block Runs 1 (1), a 5 (1), b 9 (1), c 2 b, ab 6 a, ab 10 a, ac 3 c, ac 7 c, bc 11 b, bc 4 bc, abc 8 ac, abc 12 ab, abc In fact, when partitioning these twelve blocks into three groups as given in Table 1, we see clearly which effects can be estimated by observations in each group. Four effects that can be estimated free of block effects using eight runs in the first group are main effect A, and the interactions AB, AC and ABC. Runs in the second and third groups are used to estimate B, AB, BC, ABC, and C, BC, AC, ABC, respectively. Notice that we can estimate the threefactor interaction with 24 observations, but every main effect with only eight observations in the above design. It is not a good design and we need to look for designs with more observations for estimating main effects. Table 2. A good design for 2 3 factorial experiment in blocks of size two Group 1 Group 2 Group 3 Block Runs Block Runs Block Runs 1 a,bc 5 ac,bc 9 ab,bc 2 c, ab 6 (1), ab 10 (1), ac 3 b, ac 7 a, b 11 a,, c 4 (1), abc 8 c, abc 12 b, abc A better design arranges the 24 runs in three groups of four blocks given in Table 2. The design in the first group gives estimates of main effects A, B, C and interaction ABC free of block effects. This eight-run design contains mirror-image pairs in blocks. Clearly, the design in the third group offers estimates of main effects A and C and interactions AB and BC, while

328 P.C. Wang that in the second group gives estimates of main effects A and B and interactions AC and BC. Now, if the proposed 24 runs are designed in the same way as given in Table 2, we would estimate A with 24 observations, B,C and BC with 16 observations and the rest with eight observations. This design is considerably better than the one given in Table 1. This fact is obvious by observing the cube-plots in Figures 1(a)&1(b) given by two designs. Corners of the cube stand for eight experimental runs and solid lines in the cubes are blocks. It is obvious that more main effects are confounded with block effects in Figure 1(a) than in Figure 1(b). Cube-plots in Figure 1(b) contain the top/bottom pattern and the front/back pattern, but not the left/right X pattern. They have a 3D star pattern instead for the estimation of interaction ABC. With the left/right X pattern, we can only estimate main effects B and C and interactions AC and AB. Further, we consider the size problem with the sacrifice of information on interactions. If interaction ABC is not considered, only 16 runs in the second and third groups are required for the investigation of the remaining effects. This 16-run design is better than that given by statistical software MINITAB and does not appear in Bisgaard (1994). If interactions are not active, eight runs in the first group are enough to estimate main effects. This eight-run design is the same as the mirror-image design given in Box, Hunter and Hunter (1978). Figure 1. Cube plots of two three-replicate 2 3 designs in blocks indicated by solid lines

Designing experiments in blocks 329 3 Orthogonal Arrays The traditional method to establish a good design in Table 2 is to find three sets of contrasts for three replicates and figure out which runs are in the same block. These sets are {AB, AC}, {AB, ABC} and {AC, ABC} respectively. Surely these contrasts would give us a good design by listing the contrast columns or using defining contrasts. The confounding effects with block effects are obvious. However, when the number of factors increases, the number of contrasts increases and contrasts become more complicated. These cause difficulty to find experimental runs in the same block. The difficulty is even more in fractional factorial experiments. Another method is to use an orthogonal array in Table 3. Assigning factors A, B, andc to columns seven, six and five respectively, we obtain experimental runs bc, a, ac, b, ab, c, (1) and abc, one from each row. Using the first column in the table for blocking, we get two blocks of four runs each. When the second column is added for blocking, we have eight runs in four blocks: each block contains two adjacent runs. In fact the resulting design is the design given in group one of Table 2. Different blocking columns give different designs. The design in group two of Table 2 can be obtained using columns one and four for blocking, while that in group three is obtained using columns two and four. These three blocking schemes are called G 1,G 2 and G 3 respectively. Assume the arrays like Table 3 are available and the blocking scheme is fixed. When the number of factors increases, we assign factors and obtain runs in blocks as easily as obtaining those in Table 2 after the assignment is completed. Table 3. Orthogonal array OA 8 (2 7 ) Column 1 2 3 4 5 6 7 1-1 -1 1-1 1 1-1 2-1 -1 1 1-1 -1 1 3-1 1-1 -1 1-1 1 4-1 1-1 1-1 1-1 5 1-1 -1-1 -1 1 1 6 1-1 -1 1 1-1 -1 7 1 1 1-1 -1-1 -1 8 1 1 1 1 1 1 1 Constructors X 1 X 2 X 1X 2 X 3 X 1X 3 X 2X 3 X 1X 2X 3 Blocking G 1 Blocking G 2 Blocking G 3

330 P.C. Wang Define PQ =(p j g j ) to be the direct product of vectors P =(p j )and Q = (g j ). When C 1 and C 2 are blocking columns, so is column C 1 C 2. This means G 1,G 2 and G 3 all have three columns for blocking. When the column for a contrast of interest is one of the blocking columns, the effect represented by the contrast is confounded with block effects. With a good blocking scheme as G i above, we can tell confounding effects and word length pattern for blocking (its definition will be reviewed later) easily. The array in Table 3 can be constructed by direct products of columns one, two and four. When factors A, B and C are assigned to columns seven, six and five and G 1 is used, the columns for contrasts AB, AC and BC are blocking columns and so two-factor interactions are all confounded. All these ideas and results can be extended to the case with more factors. To obtain a general orthogonal array OA N (2 N 1 ) with N rows and N 1 columns for an integer N =2 k, we multiply different numbers of N 1 vectors X 1 =( 1, 1,..., 1, 1, 1,...,1) T, X 2 =( 1,..., 1, 1,...,1, 1,..., 1, 1,...,1) T,..., and X k =( 1, 1, 1, 1, 1,..., 1, 1) T. For simplicity, the direct product of any number of these columns is represented by their low indices such as 235 for X 2 X 3 X 5. Array [1, 2, 12, 3, 13, 23, 123, 4,..., 1234... k] is an OA N (2 N 1 ) orthogonal array. A set of k columns that construct an orthogonal array OA N (2 N 1 ) by multiplications of its columns is called a set of constructors. To obtain a 2 n p fractional factorial design with different runs, we should assign each of k = n p factors to a distinct member in a set of constructors and the rest to other columns in the array. When p =0, we obtain a complete factorial design. When factor A i is assigned to column C(A i ) in the orthogonal array for i =1, 2,...,m, column m i=1 C(A i) is the column for contrast A 1 A 2...A m. The estimate of the effect of a factor or interaction assigned to the column can be obtained by dividing the inner product of the corresponding column and response vector by a constant N/2. We say that the column represents main effect or interaction. In a complete factorial design, each column in the array represents one and only one effect, either main effect or interaction, no confounding appears. In the fractional factorial design, one column represents several effects, i.e., these effects are confounded with each other. Of course, in both designs, the representation of a column depends on the assignment of factors in the array. Use the convention that column k + 1 is a vector of one s. The blocking scheme used to obtain good designs in Table 2 is generalized to G i which is a set of k 1 blocking columns {1, 2,...,k i, k i+2,..., k} for an integer i in orthogonal array OA N (2 N 1 ). G 1 puts two adjacent runs in the same block starting from the first run, G 2 does every other runs, G 3 does every four other runs,... and so on. When

Designing experiments in blocks 331 the columns representing factors or interactions can be constructed by the columns for blocking, these factors or interactions are confounded with block effects. This allows us to find which factors or interactions are confounded with block effects after the assignment is completed. In the next three sections, we use orthogonal arrays to propose several assignment rules with G i for obtaining good designs. 4 Complete Factorial Designs in Blocks of Size Two The array used in this section is OA N (2 N 1 )withk = n. After n factors are assigned to a set of constructors in the array, we obtain N treatment combinations in an order, the row order. When G 1 is used for blocking these combinations into N/2 blocks, all the columns containing column X n are orthogonal to the blocking columns. This means that all the effects represented by those columns be free of block effects. From now on, a main effect is called one-factor interaction for convenience. Let X = X 1 X 2...X n and j denote XX j for j =1, 2,..., n. Assign n factors A 1,...,A n to columns, X, 1, 2,..., (n 1) (1) When G 1 is used, all the odd-factor (1-factor, 3-factor, 5-factor... ) interactions can be estimated free of block effects and the even-factor (2-factor, 4-factor,.....) interactions are confounded with block effects in the resulting design. In fact, this assignment generalizes the assignment used to obtain the design in group 1 of Table 2. The resulting design is equivalent to the mirror-image pairs in blocks. When G i is utilized, all the effects given by those columns containing column X n i+1 are free of block effects. The assignment (1) with G i results in a design in which the effects from oddfactor interactions involving A n i+2 and even-factor interactions not involving A n i+2 are confounded with block effects. Based on this result, we can find the number of runs needed for estimating all the effects in a complete factorial experiment when the experiment is conducted in blocks of size two. Using G 1 for the first N runs, one can estimate all the odd-factor interactions. When another N runs are considered and blocked using G 2, there are 2 n 2 more distinct effects that can be estimated. They are even-factor interactions involving A n because the columns representing these interactions contain column X n 1. Continue this process to runs with G i,only2 n i more effects which are even-factor interactions involving A n i+2, but not involving A n i+3,a n i+4,...,ora n can be estimated. This means that up to in runs, only 2 n 1 +2 n 2 + +2 n i effects can be estimated. If all the effects from complete factorial experiment are required to be estimated, we need n2 n

332 P.C. Wang runs using the above blocking scheme as N 1=2 n 1 +2 n 2 + +2 1 +2 0. This gives the same results as given in Draper and Guttman (1997). To see the advantage of our assignment with G i, the word-length pattern of a blocking scheme defined by Sun, Wu and Chen (1997) is utilized. For a blocking scheme B in a n-factor design, W b = W b (B) =(g 1 (B),g 2 (B),..., g n (B)) is the word-length pattern of B, where g j (B) is the number of j-factor interactions confounded with block effects. Let c n m denote the combination. Assignment (1) with G 1 has word-length pattern while those with G i for i>1allhave W b (G 1 )=(0,c n 2, 0,c n 4, 0,c n 6,...), W b (G i )=(1,c n 1 2,c n 1 2,c n 1 4,...). Notice that the resulting designs from assignment (1) with G 1 are the same as those given in Table 2 of Sun, Wu and Chen (1997). Designs from assignment (1) with G i for all i>1 are used to increase the number of estimable effects. In fact, each of these designs sacrifices the estimate of main effect A n i+2 for estimating all the two-factor interactions with A n i+2. When we carry out n replicates of 2 n runs with each replicate blocked by one G i,onemain effect, i.e. A 1, can be estimated with n2 n runs and all the other main effects with (n 1)2 n runs. In addition, when some effects are negligible in a complete factorial experiment, the number of runs can be reduced and the reduction depends on the number of negligible effects. This assertion is true for any assignment of factors to a set of constructors. The practitioners must choose the right blocking procedures for reduction. Our suggestion is to use assignment (1) with different G i for different replicates of runs. For convenience, an n-tuple of columns stands for the assignment of the j th factor to the column in the j th component for j =1, 2,...,n. For example, (X, 1, 2,..., (n 1)) stands for assignment (1). Also, (C 1,C 2,...,C n )ing means a design with the assignment of factor A j to column C j for all j and its resulting 2 n runs blocked by G. Now factors A 1,A 3,A 5,... are called odd factors and A 2,A 4,A 6,... are even factors. The assignment (n, 1, (2)n, (3)n 2,...,(n 1)n n 2 ) (2) will be useful for estimating two-factor interactions. We list several useful designs and their estimable effects in Table 4 based on assignments (1) and (2).

Designing experiments in blocks 333 Table 4. Useful designs and estimable effects from them Design Estimable Effects 1 Assignment (1) in G 1. All the main effects 2 Assignment (1) in G i. 2-factor interactions with A n i+2 and all the main effects except A n i+2 3 Combination of designs 1 & 2. All main effects and the two-factor interactions with A n i+2 4 Assignment (2) in G 1. Main effects of odd factors and the two-factor interactions of an odd factor and an even factor. 5 (n 1) replicates: assignment (1) All the two-factor interactions in G i for each i>1. and main-effects. Design (1) which is a mirror-image design, is the best design as suggested in Sun, Wu and Chen (1997). All the main effects are estimable in this design. Design (2) emphasizes the estimation of two-factor interactions involving a specified factor. When three-factor and higher interactions are assumed not active, this design seems better because it estimates n 1 main effects and n two-factor interactions, more than just n main effects in design (1). Design (3) with 2N runs combines designs (1) and (2) in order to estimate all main effects and the two-factor interactions involving A n i+2. Also, in this design, one can estimate odd-factor interactions not involving A n i+2 using 2N runs and the remaining odd-factor interactions using N runs. Two-factor interactions with A n i+2 and main effect of A n i+2 can be estimated with just N runs. Using design (4), we can estimate more interactions and fewer main effects than using design (2). If we combine this design with design (1), we obtain a design with more estimable two-factor interactions than design (3). We sacrifice the precision of some main effects for the advantage. Design (5) is a combination of n 1 replicates of design (2). This design allows us to estimate all main effects and two-factor interactions. It uses (n 1)N runs to estimate main effect A 1 and two-factor interactions with A 1,(n 2)N runs for the remaining main effects and2n runs for the remaining two-factor interactions. When n = 3, we obtain 16 runs in groups two and three in Table 2. However, design (5) requires many runs and we might find a better one with fewer runs based on design (4) when n is greater than or equal to 4. In fact we need only Int((n+1)/2) replicates for estimating all two-factor interactions, where Int(y) is the largest integer less than or equal to y. The first replicate uses assignment (2), the second replicate uses the assignment obtained from switching the third and fourth components in assignment (2), the third replicate uses the assignment ob-

334 P.C. Wang tained from switching the fifth and sixth components in assignment (2), and the last replicate uses the assignment obtained from switching the last component and the one next to it if n is odd and all the assignments are in G 1. Adding one more replicate with assignment (1) in G 1,wehaveadesign that can estimate all main effects and two-factor interactions. Clearly this design is better than design (5) in terms of size. We list several specified cases blocked by G 1 in Table 5 for reference. Table 5. Complete two-level designs for estimating all the two-factor interactions The number of factors Design 4 (4, 1, 24, 3 ), (4, 1, 3, 24) 5 (5, 1, 25, 3, 45), (5, 1, 3, 25, 45), (5, 1, 25, 45, 3) 6 (6, 1, 26, 3, 46, 5), (6, 1, 3, 26, 46, 5), (6, 1, 26, 3, 5, 46) 7 (7, 1, 27, 3, 47, 5, 67), (7, 1, 3, 27, 47, 5, 67), (7, 1, 27, 3, 5, 47, 67), (7, 1, 27, 3, 47, 67, 5) 8 (8, 1, 28, 3, 48, 5, 68, 7), (8, 1, 3, 28, 48, 5, 68, 7), (8, 1, 28, 3, 5, 48, 68, 7), (8, 1, 28, 3, 48, 5, 7, 68) Sacrificing a little precision of some estimators to reduce the size is worthwhile when the number of factors is large. The size of an experiment in blocks of size two depends on experimental requirements and its setup. Based on the above discussions, we conclude (1) estimates of different effects in a design might have different precision and (2) the precision of estimates of the same effect under different assignments and/or different blocking can be different. The precision depends on assignment of factors and blocking procedures. The appropriate assignment and blocking would result in more precision for more effects of interest. 5 2 n 1 Fractional Factorial Designs in Blocks of Size Two For fractional factorial designs, the assignment of factors determines defining relations by observing which columns for factors are multiplied to a vector of ones. The defining relations then give the resolution and degree of aberration for the resulting design. In this section, we consider 2 n 1 fractional factorial designs and so k = n 1. To achieve the same goal as in a complete factorial design, we need (n 1)2 n 1 runs with our blocking scheme for 2 n 1 fractional factorial experiments. This can be justified by the same

Designing experiments in blocks 335 argument as given in the last section of Wang and Liu (2000). The important consideration in the 2 n 1 fractional factorial design is the assignment of factors for maximum resolution, estimable main effects and precision of effects of interest. To obtain a 2 n 1 fractional factorial design, we need to assign k factors to a set of k constructors, and then the last factor to one of the remaining columns. The resolution of the design depends on the assignment of the last factor. Assume that factors A 1,A 2,..., and A n 1 are assigned to a set of k constructors {C(A 1 ),C(A 2 ),...,C(A n 1 )}. When factor A n is assigned to column m j=r C(A j), the resolution of the resulting design is (m r +2). To achieve maximum resolution, we assign factor A n to column n 1 j=1 C(A j). However, when blocking is required, we might assign it to a different column to estimate all main effects. Taking n = 4 as an example, we assign factors A 1,A 2,A 3 and A 4 to columns 123, 23, 13 and 3 (columns seven, six, five and four in Table 3) respectively for a resolution IV design. All the main effects can be estimated if three-factor interactions are negligible. Further when G 1 is used, main effects can still be estimated without any confounding because columns 123, 23, 13 and 3 are orthogonal to the blocking columns {1, 2, 12}. This means that using the above assignment in G 1 for the case of four factors in eightrun experiments, we obtain a design with estimable main effects. Extend this idea to the general n. Suppose G 1 is used. When n is odd, we assign the first n 1factors A 1,A 2,..., and A n 1 to columns X = 12...k, 1, 2,..., and (k 1). Because column k j=1 C(A j)= k can be constructed by columns in G 1, assigning factor A n to this column for maximum resolution would result in the confounding of factor A n with block effects. We might assign it to column k = k j=2 C(A j) instead. The resulting design does not have maximum resolution, but estimates of main effects under the design are free of block effects. When n is even, assign A 1,A 2,...,A n 1 as before and A n to k = k j=1 C(A j) for maximum resolution. This leads to the following assignment. The assignment (X, 1, 2,..., (k 1), k) (3) is suggested for establishing 2 n 1 fractional factorial experiments in blocks of size two. Its defining relation is either A 1 A 2 A 3...A k = I or A 2 A 3...A k = I depending on whether n is even or odd. Let W t =(t 1,t 2,t 3,t 4,...) be the word-length pattern for a fractional factorial design, where t j is the number of j factor letters in the defining relations. Note that t 1 and t 2 are always

336 P.C. Wang equal to zero. The word-length pattern W t of assignment (3) is (0, 0,...,0, 1) when n is even or (0, 0,...,0, 1, 0) when n is odd. Now when G 1 is utilized, the word length pattern for blocks is W b =(0,c n 2, 0,c n 4, 0,c n 6,...) W t The resulting designs from assignment (3) in G 1 are the minimum aberration designs as given in Sun, Wu and Chen(1997). When G i is utilized, the word length pattern for blocks, W b, becomes (2c n 2 0,c n 2 2 + c n 2 0, 2c n 2 2,c n 2 4 + c n 2 2, 2c n 2 4,...) W t In fact, 2 n 1 designs with assignment (3) in G 1 can estimate all main effects, while those in G i only estimate (n 2) main effects. The main effects confounded with block effects are A n i+2 and A n. Considering n 1 replicates of N runs as suggested by Draper and Guttman (1997), we use assignment (3) for all the replicates and block each replicate by G i for i =1, 2, 3,...,k respectively. Different effects are estimable in different replicates. As in the complete factorial design, some effects can be estimated with more replicates and are thus more precise. When three and higher-order interactions are negligible, main effect A 1 is estimated most precisely since it uses (n 1)2 n 1 observations. The estimate of main effect A n is least precise, using 2 n 1 observations, while the remaining main effects are estimated with (n 2)2 n 1 observations. Except A 1 A n, there are at most N observations available to estimate any other two-factor interactions involving A 1, for example, A 1 A n 1 in the second replicate and A 1 A n 2 in the third replicate. In fact, A 1 A n can be estimated using (n 2)N observations. It is clear that the precision of these estimates is distinct. Different defining relations would result in different alias structures of effects. This suggests using appropriate defining relations (that is, appropriate assignment of factors to the columns of the orthogonal array) on the establishment of N experimental runs to estimate more effects and/or execute fewer runs. If only one replicate of N runs is considered, assignment (X, 1, 2,..., (k 1),k)inG 1 is the best as suggested in Sun, Wu and Chen(1997). To estimate more two-factor interactions, we consider a modified version of assignment (2), that is, (k, 1, (2)k, (3)k 2,...,Xk k ), (4) Using this sequence for assignment, we need only Int((n+1)/2) replicates to estimate all the two way interactions. The first replicate uses assignment (4), the second replicate uses the assignment obtained from switching

Designing experiments in blocks 337 the third and fourth components in assignment (4), the third replicate uses the assignment obtained from switching the fifth and sixth components in assignment (4), and the last replicate uses the assignment obtained from switching the last component and the one next to it if n is odd, and all the assignments are in G 1. Table 6 lists several cases. Adding one replicate with assignment (1) in G 1 allows us to estimate all main effects and two-factor interactions when higher-order interactions are negligible. Notice that the precision of different estimates is different. Consider the design with n = 5 factors (A, B, C, D, E) in Table 6 as an example. We can estimate interaction AB using 48 runs, BD using 32 runs and AE using 16 runs. With 2N runs, we might use assignment (3) in G 1 for the first replicate and assignment (4) in G 1 for the second replicate, permitting the estimation of more two-factor interactions than using assignment (3) in G 1 and G 2. Table 6. 2 n 1 designs for estimating all the two-factor interactions n Design 5 (4, 1, 24, 3, 1234 ), (4, 1, 3, 24, 1234), (4, 1, 24, 1234, 3) 6 (5, 1, 25, 3, 45,1234), (5, 1, 3, 25, 45, 1234), (5, 1, 25, 3, 1234, 45) 7 (6, 1, 26, 3, 46, 5, 123456), (6, 1, 3, 26, 46, 5, 123456), (6, 1, 26, 3, 5, 46, 123456), (6, 1, 26, 3, 46, 123456, 5) 8 (7, 1, 27, 3, 47, 5, 67, 123456), (7, 1, 3, 27, 47, 5, 67, 123456), (7, 1, 27, 3, 5, 47, 67, 123456), (7, 1, 27, 3, 47, 5, 123456, 67) 6 2 n p Fractional Factorial Designs in Blocks of Size Two The number of runs needed in the 2 n p fractional factorial experiment in blocks of size two for estimating the same number of effects in the 2 n p fractional factorial experiment without blocking is k2 k with k = n p. This claim can also be justified with the same arguments as in Section 4. The assignment of factors to columns in an orthogonal array is more difficult for 2 n p fractional factorial experiments in blocks of size two. After the assignment of factors, the same blocking procedure is used here with k = n p. To estimate all main effects, an appropriate assignment is needed. Without loss of generality, N runs with G 1 is considered. Assign all the factors to those columns containing X k when the number of factors is less than 2 k 1. This guarantees that estimates of main effects are free of block effects, since all these columns are orthogonal to the blocking columns. Letting

338 P.C. Wang X = (123...k), we propose the following sequence for assignments of factors: X, 1, 2,... (k 1), 12,... 1(k 1), 23,..., (k 2)(k 1),..., k (5) We assign n factors to the first n columns in the sequence and block the resulting runs by G 1. The resulting designs are minimum aberration designs in blocks of size two as listed in Sun, Wu and Chen(1997). Its word-length pattern W t depends on the number of factors and can be computed when defining relations are obtained from the assignment. The word-length pattern for blocks W b is equal to (0,c n 2, 0,c n 4, 0,c n 6,...) W t Notice that when p = 1, assignment (3) and assignment (5) give two different designs: one with maximum or next to maximum resolution, and the other with resolution IV. Consider a 2 8 3 fractional factorial experiment in blocks of size two. Use sequence (5) to assign eight factors (A 1,A 2,...,A 8 ) to columns X = 12345, 1, 2, 3, 4, 12, 13 and 14. We can not find any three of these columns whose direct product is equal to a vector of one s. Thus when three-factor interactions are negligible, main effects are all estimable. In fact, the defining relations in this assignment are I = A 1 A 2 A 3 A 6 = A 1 A 2 A 4 A 7 = A 1 A 2 A 5 A 8. When the design is blocked by G 1, its two word-length patterns are W t =(0, 0, 0, 6, 0, 0, 0, 1) and W b =(0, 28, 0, 64, 0, 28, 0, 0). In fact this is exactly the 47 th design in Table 5 of Sun, Wu and Chen(1997). In addition, A 1 A 3 A 4, A 1 A 3 A 5, A 1 A 4 A 5, A 2 A 3 A 4, A 3 A 4 A 5, A 2 A 3 A 5, A 2 A 4 A 5, and A 4 A 5 A 6 can be estimated when the corresponding aliases are negligible. However all two-factor interactions are confounded with block effects. When G i instead of G 1 is considered, the word-length pattern of the resulting design might be difficult to find. It depends on n and p. Here we just explore the cases with 1 <i<kand 2k i 1 <n. Two main effects, A k i+2 and A 2k i and interaction A k i+2 A 2k i are confounded. But the two-factor interactions involving exact one of A k i+2 and A 2k i, are not confounded with block effects. Note that these interactions might be confounded with each other. The word length pattern for blocks in this design is (2,c n 2 2 + c n 2 0, 2c n 2 2,c n 2 4 + c n 2 2, 2c n 2 4,...) W t Take a 2 8 3 fractional factorial experiment in blocks of size two as our example again. Use assignment (12345, 2345, 1345, 1245, 1235, 345, 245,

Designing experiments in blocks 339 235) in G 3. This time main effects A 4,A 7 and interaction A 4 A 7 are confounded with block effects, and so are interactions not involving them since G 3 ={1, 2, 4, 5}. There are 16 two-factor interaction effects confounded with block effects. This design is not better than the previous one in G 1 in terms of aberration, but we can estimate some interactions when others are negligible. When we have 2N runs for the experiment, using this design in the second replicate seems to be an alternate choice. To estimate some two-factor interactions when three and higher order interactions are not active, we propose another sequence for assignment: X, 1, 2,...k 1, 12,...,1(k 1),...,(k 2)(k 1),...,12...(k 1). (6) This sequence is not useful for designs without replicates. When n factors are assigned to the first n columns in (6) and block the resulting runs by G 1, we can estimate first main effect and all two-factor interactions involving the first factor. In the case of a 2 8 3 fractional factorial experiment, assign (A 1,A 2,...,A 8 ) to columns 12345, 1, 2, 3, 4, 12, 13 and 14 in sequence (6). The defining relations become I = A 2 A 3 A 6 = A 2 A 4 A 7 = A 2 A 5 A 8.All the estimates of two-factor interactions involving A 1 are computed based on those columns containing column X k = X 5 and thus are free of block effects. When 2N runs are available, we can use assignment (5) in G 1 for the first replicate and use assignment (6) in G 1 for the second. The design is better than the one given in the last paragraph because more two-factor interactions can be estimated. In addition, it is also better than any design with 2N =2 n p+1 runs from a 2 n p+1 experiment. 7 Conclusions and Discussion Using orthogonal arrays to set up factorial experiments, either complete or fractional, in blocks of size two, one can easily determine which effects are free of block effects. Careful choice of interactions to be estimated in the experiment and appropriate assignment would reduce the size of the experiment and increase the precision of estimators of main effects and some low-order interactions. We have presented blocking procedures and several rules for assignment on the orthogonal array to design experiments in blocks of size two. When there are 64 runs available for designing a five-factor experiment in blocks of size two, we have a choice between two replicates of complete 32-run experiments and four replicates of 2 5 1 experiments. The latter one would be better because, with an appropriate design, it can estimate all

340 P.C. Wang main effects and two-factor interactions when higher-order interactions are negligible. The design combines assignment (3) in G 1 with three designs in G 2 for n = 5 in Table 6. There is no way to estimate all main effects and two-factor interactions with two replicates of complete 32-run experiments. Our conjecture is that 2 m replicates of 2 n p fractional factorial experiments with appropriate designs is the optimal when 2 m+n p runs are available for designing n-factor experiments in blocks of size two. Acknowledgements. I thank the referees and the editor for their helpful comments which greatly improved the presentation of the article. The support of this research under grant nsc90-2118-m008-015 by the National Science Council in Taiwan is acknowledged. References Bisgaard, S.(1994). Blocking generators for small 2 k p design. J. Quality Tech., 26, 288-296. Box, G.E.P., Hunter, W.G. and Hunter, J.S. (1978). Statistics for Experimenters. Wiley, New York. Sun, D.X., Wu, C.F.J. and Chen, Y. (1997). Optimal blocking schemes for 2 n and 2 n p designs. Technometrics, 39, 298-307. Draper, N.R. and Guttman, I. (1997). Two-level factorial and fractional factorial designs in blocks of size two. J. Quality Tech., 29, 71-75. Wang, P.C. and Jan, H.W. (1995). Designing two-level factorial experiments using orthogonal arrays when the run order is important. Statistician, 44, 379-388. Wang, P.C. and Liu, H.B. (2000). To design S-level factorial experiments in blocks of size S. Unpublished manuscript. P.C. Wang Institute of Industrial Management National Central University Chung Li, Taiwan 32054 E-mail address: pcwang@cc.ncu.edu.tw Paper received: January 2003; revised December 2003.