points. If each point is identified with a treatment combination, then the design consisting of 8 : "

Transcription

1 Handout #14 - Regular fractional factorial designs An example of regular fractional factorial design was given in Section This handout presents a general theory of the construction of regular fractional factorial designs. Such designs are easy to construct, have nice structures and are relatively straightforward to analyze. When one only observes a fraction of the treatment combinations, not all the factorial effects can be estimated and they may be mixed up (aliased) in a complicated way. Under a regular design, the aliasing of the factorial effects can be determined in a straightforward manner. An algorithm for constructing a regular design under which certain required effects are estimable is presented. We also show that all regular designs are orthogonal arrays. The discussion in Section 13.4 explains how design (13.3.1) can be viewed as a threedimensional subspace of a six-dimensional finite Euclidean geometry. This is extended to the general construction of regular fractional factorial designs. Throughout this chapter, it is assumed that = is a prime number or power of a prime number, and only symmetric = 8 factorials are considered. Recall that in this case the = 8 treatment combinations can be identified with the points (or vectors) in an 8-dimensional Euclidean geometry EG Ð8ß=Ñ, and pencils of Ð8 "Ñ-flats can be used to define contrasts representing factorial effects Construction and defining relation For any positive integer : with : 8, an ÐÑ-flat in EG Ð8ß=Ñ contains = points. If each point is identified with a treatment combination, then the design consisting of " 8 all the = treatment combinations in an ÐÑ -flat is an = fraction of the complete = : factorial, and is called a regular = fractional factorial design. Recall that an Ð8 :Ñ-flat is the solution set of equations +Bœ-, 3 œ "ß âß :, Ð14Þ"Þ"Ñ 3 3 where + 1, âß + : are : linearly independent vectors in EG Ð8ß =Ñ, and -" ß âß -: GF Ð=Ñ. When -" œ â œ -: œ!, it is an ÐÑ-dimensional subspace. For the other -3's, (14.1.1) defines : parallel Ð8 :Ñ-flats in the same pencil. Any of the = Ð8 :Ñ-flats can be used as an = fractional factorial design. Without loss of generality and for notational convenience, unless otherwise stated, we shall assume that - œ â œ - œ! and write Ð14 Þ"."Ñ as where E œò+ â+ Ó. " : " : EB œ 0ß Ð14Þ"Þ#Ñ Let eðe Ñ be the space generated by + " ß âß+ :. Then the regular fraction defined by Ð14 Þ"Þ#Ñ consists of all the = treatment combinations B that satisfy + B œ! for all + eðeñ. The :-dimensional space eðeñ is called the defining contrast subgroup of the design

2 6 $ It follows from (13.4.4) that design Ð13Þ3 Þ"Ñ in Handout #13 is a regular 2 design defined by + 1 œ Ð"ß "ß!ß "ß!ß!Ñ, + # œ Ð!ß "ß "ß!ß "ß!Ñ and + 3 œ Ð1ß "ß "ß!ß 0ß 1 Ñ. The defining contrast subgroup consists of + 1, + #, + 3 and their four linear combinations # œð"ß!ß"ß"ß"ß!ñ, œð0ß!ß"ß"ß0ß1 Ñ, œð"ß!ß0ß0ß "ß1 Ñ, and # + $ œ Ð!ß "ß!ß "ß "ß "Ñ. When 0 and " are replaced by " and ", respectively, the seven equations +B œ 0, where + e Ð E Ñ, are equivalent to Ð 13 Þ 4 Þ$Ñ. If the six factors are labeled by the letters E, F, G, H, I and J, then (13.4. $ ), called the defining relation of the design, is usually written as M œ EFH œ FGI œ EFGJ œ EGHI œ GHJ œ EIJ œ FHIJ Ð14Þ"Þ$Ñ In this example, there are seven nonzero vectors in the defining contrast subgroup, each representing a factorial effect. These factorial effects are called defining effects or defining contrasts. When expressed as in Ð14 Þ"Þ$Ñ, they are also called defining words. In general, : there are 2 " defining effects (or words) in the defining relation of a 2 regular fractional factorial design. In the above example, if we choose the 3-flat defined by EB œ Ð"ß!ß!Ñ instead of EB œð0 ß!ß!Ñ, then Ð13Þ4 Þ"Ñ becomes B % œ BBß " # B & œbb # $ and B 6 œbbb " # $. The corresponding defining relation is written as Mœ EFHœFGIœEFGJ œ EGHIœ GHJ œeij œ FHIJ ) ( : In general, for regular 2 fractional factorial designs, the # combinations of signs for the : independent defining words correspond to the # : flats in the same pencil, or : equivalently, the # choices of the -3's in Ð14 Þ"Þ"Ñ. One can first write down the : independent defining words with appropriate signs, and then form all possible products to complete the # defining relation. When multiplying the words, again we follow the rule that \ œ " for any letter \. It can be seen that design (13.3.2) is a regular 2 design defined by B% œbbb " # $, B' œbbb " # &, B( œbbb " $ &, B) œbbb # $ &. If the eight factors are denoted by E, F, G, H, I, J, K and L, then EFGH, EFIJ, EGIK and FGIL are four independent defining words, generating a total of fifteen defining words. As an example of a three-level design, consider the 2-flat in EG Ð%ß $Ñ defined by B" B# #B$ œ! and B" #B# #B% œ!, i.e., B$ œ B" B# and B% œ B" #B#. Writing % # each treatment combination as a row, we have the following regular 3 fractional factorial design:

3 !!!!! " " #! # # " "! " " " " #! " #! # #! # # # "! " # # "! Ð14Þ"Þ5Ñ The defining contrast subgroup consists of Ð"ß "ß #ß!Ñ, Ð"ß #ß!ß #Ñ, and their two linear combinations Ð"ß"ß#ß!Ñ Ð"ß#ß!ß#Ñ œð#ß!ß#ß#ñ and Ð"ß"ß#ß!Ñ 2Ð"ß#ß!ß#Ñ œ Ð0ß2ß#ß1 Ñ, where the multiplication and addition are reduced mod 3. The nine treatment % # combinations in this 3 fraction are solutions of the equations B" B# #B$ œ B" #B# #B% œ #B" #B$ #B% œ #B# #B$ B% œ!. To make the leading nonzero coefficients equal to 1, as discussed in Example 9.9.1, the last two equations are replaced by 2Ð#B" #B$ #B% Ñ œ #Ð#B# #B$ B% Ñ œ 0, i.e., B" B$ B% œ B# B$ 2B% œ 0. Then the four defining equations of the design are B" B# #B$ œ B" #B# #B% œb" B$ B% œb# B$ #B% œ!, which are usually written as # # # M œ EFG œ EF H œ EGH œ FGH #. Again, this is called the defining relation of the design. Each defining word appearing in the defining relation represents two degrees of freedom of factorial effects. In general, each nonzero vector + in the defining contrast subgroup eðeñ is called a defining word of the regular = fractional factorial design. The factorial effects represented by the defining words are called defining effects or defining contrasts. There are a total of : Ð= "ÑÎÐ= "Ñ defining words when vectors that are nonzero multiples of each other are considered as equivalent Aliasing and estimability Recall that with the two levels represented by 1 and, the 2 fractional factorial design Ð13ÞÞ"Ñ 3 was defined by B% œbbßb " # & œbb # $, B6 œbbb " # $. Interaction contrasts of the basic factors were used to define three additional factors. As a result, in the model matrix, assuming that all the factorial effects are present, the column corresponding to the main effect " ' $

4 of factor H is identical to that corresponding to the two-factor interaction EF. It implies that these two effects are completely mixed up and cannot be estimated at the same time, and we say that they are aliases of each other. Likewise, the main effect of factor I is an alias of the two-factor interaction FG and the main effect of J is an alias of the three-factor interaction EFG. Complete information about the aliasing among factorial effects can be determined from the defining relation as follows. The defining relation Ð13Þ4 Þ$Ñ or Ð14 Þ"Þ$Ñ implies that the columns of the model matrix corresponding to the factorial effects EFH, FGI, EFGJ, EGHI, GHJ, EIJ and FHIJ are the same as the vector of 1's. Thus these effects are aliases of the grand mean and cannot be estimated. If each term in Ð13Þ4 Þ$Ñ is multiplied by B ", then we obtain B" œbb # % œbbbb " # $ & œbbb # $ ' œbbb $ % & œbbbb " $ % 6 œbb & ' œbbbbb " # % & '. This implies that the columns of the model matrix corresponding to the eight factorial effects E, FH, EFGI, FGJ, GHI, EGHJ, IJ and EFHIJ are identical, and so these eight factorial effects are completely mixed up. We say that they constitute an alias set, and write E œ FH œ EFGI œ FGJ œ GHI œ EGHJ œ IJ œ EFHIJ. These aliased effects can also be obtained by multiplying each term in Ð14 Þ"Þ$Ñ by E, again # using the rule that \ œ " for any letter \. Other alias sets can be determined similarly. ' Among the sixty-three factorial effects in a 2 factorial, seven appear in the defining relation, and the other effects are divided into seven alias sets each containing eight effects as follows: E œ FH œ EFGI œ FGJ œ GHI œ EGHJ œ IJ œ EFHIJ F œ EH œ GI œ EGJ œ EFGHI œ FGHJ œ EFIJ œ HIJ G œ EFGH œ FI œ EFJ œ EHI œ HJ œ EGIJ œ FGHIJ H œ EF œ FGHI œ EFGHJ œ EGI œ GJ œ EHIJ œ FIJ I œ EFHI œ FG œ EFGIJ œ EGH œ GHIJ œ EJ œ FHJ J œ EFHJ œ FGIJ œ EFG œ EGHIJ œ GH œ EI œ FHI EGœFGHœEFIœFJ œhiœehj œgij œefghij (14.2.1) : In general, in a regular # fractional factorial design, 2 " factorial effects appear in the defining relation, and the other effects are divided into # " alias sets, each : containing 2 effects. It will be shown later in this section that if all but one effect in the same alias set are negligible, then the nonnegligible effect is estimable

5 % # Under the $ design Ð14Þ"Þ5 Ñ, adding B" to each term in the four defining equations B" B# #B$ œ B" #B# #B% œb" B$ B% œb# B$ 2 B% œ!, we conclude that B" œ#b" B# #B$ œ#b" #B# #B% œ2 B" B$ B% œb" B# B$ #B%. Ð14Þ#Þ2Ñ Similarly, adding 2 B " to each term in the defining equation= yields #B" œb# #B$ œ#b# #B% œb$ B% œ#b" B# B$ #B%, i.e., B" œ #B# B$ œ B# B% œ 2B$ 2B% œ B" 2B# 2B$ B% Ð14Þ#Þ3Ñ It follows from Ð14 Þ#Þ#Ñ, Ð14 Þ#Þ$Ñ and the convention of making the leading nonzero # # # # coefficient equal to 1 that the contrasts represented by E, EF G, EFH, EG H, EFGH, # # # FG, FH, GH and EF G H coincide on all the treatment combinations in the chosen fraction. Therefore they are completely aliased and cannot be estimated at the same time. This determines the aliases of the main effect of factor A. The other alias sets can be determined similarly. There are a total of four alias sets in this example. In general, let. be the regular = fractional factorial design consisting of the treatment combinations satisfying Ð14Þ"Þ1 Ñ. If, is a nonzero vector in EG Ð8ß=Ñ that appears in the defining relation of., i.e.,, eðe Ñ, then since, is a linear combination of + 1, âß+ :, Ð14Þ"Þ1 Ñ implies that, B is a constant for all B.. Thus all the treatment combinations in. fall entirely in one of the Ð8 "Ñ-flats defined by,, and therefore have the same coefficient in any contrast defined by the pencil TÐ, Ñ. It follows that the contrasts defined by the pencil TÐ, Ñcannot be estimated. If, is a nonzero vector in EG Ð8ß =Ñ that does not belong to eðe Ñ. Then, and + 1, âß + : are linearly independent. Therefore, partitions., which is itself an Ð8 :Ñ-flat, into = disjoint Ð "Ñ-flats according to the values of, B, B.. If," and,# are two nonzero vectors in EG Ð8, =Ñ Ï eðeñ such that," -, # eðeñ for some nonzero - GF Ð=Ñ, then," B œ -,# B for all B.. This implies that," and,# give identical partition of. into = disjoint Ð "Ñ-flats. In this case, we cannot separate the treatment contrasts defined by TÐ, " Ñ from those defined by TÐ, # Ñ. These contrasts cannot be estimated at the same time and we say that the pencils TÐ, " Ñ and TÐ, # Ñ (or the treatment contrasts defined by them) are aliased with each other. The set of all pencils that are aliased with one another is called an alias set. If TÐ, " Ñ and TÐ, # Ñ are in different alias sets, i.e.,," -, #  eðe Ñ for all nonzero - GF Ð=Ñ, then,",,# and + 1, âß + : are linearly independent. Let 0, âß = " be the partition of. according to the values of, " B and Y!, âß Y= " be the partition of. according to the values of, 2 B. Then each 3 Y4 contains = # treatment combinations since it is an

6 Ð #Ñ-flat. Let 7 and 9 be two treatment contrasts defined by," and,#, respectively. Suppose 7 has coefficient - 3 for each treatment combination in 3 and 9 has coefficient. 4 for each treatment combination in Y4. Then the inner product of the coefficient vectors of 7 and # 9, restricted to., is equal to = = " = " 3œ! 4œ! , which is zero. Thus, if we assume that in each alias set of pencils of Ð8 "Ñ-flats, the treatment contrasts defined by all but one pencil are negligible, then the columns of the model matrix associated with. are mutually orthogonal. We summarize this in the following theorem. : Theorem The treatment contrasts defined by each of the = " nonzero vectors in the defining contrast subgroup of a regular = fractional factorial design cannot be estimated. Two pencils TÐ, " Ñ and TÐ, # Ñ are in the same alias set if and only if," -, # belongs to the defining contrast subgroup for some nonzero - GFÐ=Ñ. There are a total of : Ð= "ÑÎÐ= "Ñ alias sets, each containing = pencils of Ð8 "Ñ-flats. In each alias set, if the treatment contrasts defined by all but one pencil are negligible, then the nonnegligible treatment contrasts are estimable. Furthermore, the least squares estimators of orthogonal nonnegligible treatment contrasts are uncorrelated Analysis ' $ In Section 13.3, the 2 design Ð13Þ3 Þ"Ñ was constructed by first writing down the complete factorial of three basic factors. There is a complete factorial of three factors embedded in design Ð13Þ3 Þ"Ñ. We shall show in Section 14.5 that this is true in general: there is a complete factorial of factors embedded in each regular 2 fractional factorial design. Therefore the design can be constructed by using the interactions of the basic factors to define : additional factors. An algorithm for identifying basic factors from the defining relation is also presented in Section Once we have found a set of basic factors, we can proceed with the calculation of estimates of their factorial effects in the usual way, e.g., by employing the Yates algorithm, tentatively ignoring those of the added factors. However, due to aliasing, what we are getting are actually estimates of combinations of the factorial effects of basic factors and some other effects involving added factors. Suppose we have E ÐC Ñ œ \""" \ #"#, where "" consists of factorial effects of the basic factors, and " 2 consists of all other factorial effects. The least squares estimators of "" when " is ignored can be computed as " s # " œð " " Ñ " \ \ \" C. Under the full model, we have s" " " " " " EÐ Ñ œ EÒÐ\ \ Ñ \ CÓ œ Ð\ " "\" Ñ \" Ò\ """ \ #"# Ó " œ " Ð\ \ Ñ \ \ ". " " " " # # So what " s " actually estimates is " Ð\ \ Ñ \ \ ". " " " " " # #

7 Consider, for example, two-level designs. Define the factorial effects as in Then for a 2 design, \ \ œ# M. Therefore we have " " EÐ " s " Ñ œ " \ # " \ " " " # # Each entry of \" \# is the inner product of a column of \" and a column of \#. Since the main-effect column of each added factor is a Hadamard product of columns of \ ", possibly with a sign change, all the columns corresponding to the factorial effects of the added factors and the interactions of basic and added factors are Hadamard products of columns of \ ", possibly with sign changes. Furthermore, the Hadamard product of any two columns of \ " is also a column of \". Therefore for each column? of \#, either? or? is a column of \". It follows that for any of \ and any column? of \, " œ #, if œ œ, i.e., if the corresponding effects in "" and "# are in the same alias set; 0, otherwise. Therefore for each factorial effect in " ", say ", we have E Ðs" Ñ œ " " ~. ~ ~ " : " is in the same alias set as " 6 $ Example Under the 2 design defined by (14.1.4), it follows from the defining relation that an estimate of the main effect of factor E actually estimates E FH EFGI FGJ GHI EGHJ IJ EFHIJ. It is an estimate of E if FH, EFGI, FGJ, GHI, EGHJ, IJ and EFHIJ are negligible. Note that this design can be constructed from the complete factorial of E, F, G by making H œ EF, IœFG and JœEFG. To compute estimates of the factorial effects, we run the Yates algorithm with the data arranged in the Yates order of the treatment combinations of the basic factors E, F and G. Each estimate obtained from the Yates algorithm estimates the sum of a string of effects aliased with the corresponding effect of the basic factors. For example, the estimate of EF when the added factors are ignored is an estimate of EF H EGI GJ FGHI EFGHJ FIJ EHIJ. If EF, EGI, GJ, FGHI, EFGHJ, FIJ and EHIJ are negligible, then it is an estimate of H. For data collected from fractional factorial designs, half-normal probability plots are used for identifying significant effects. When some contrasts are found significant but cannot be attributed to specific factorial effects due to the presence of nonnegligible aliases, one has to perform follow-up experiments to resolve the ambiguity. This is called de-aliasing

8 14.4 Resolution Box and Hunter (1961) defined the resolution of a regular fractional factorial design as the length of the shortest defining word, where the length of a defining word is the number of letters it contains, or equivalently, the number of nonzero entries in the corresponding vector in eðeñ. According to this definition, design is of resolution three and design is of resolution four. If the resolution is V, then there is at least one defining word of length V. It implies that for each + such that 1 Ÿ + V, at least one factorial effect involving + factors is aliased with an effect involving V + factors. However, since there are no defining words of lengths shorter than V, none of the effects involving + factors are aliased with effects involving fewer than V + factors. Thus, for example, in a design of resolution three, no main effect is aliased with another main effect, but some main effects are aliased with two-factor interactions. Under such a design, all the main effects are estimable if the interactions are assumed negligible. In a design of resolution four, not only the main effects are not aliased with one another, they are also not aliased with two-factor interactions; therefore all the main effects are estimable if the three-factor and higher-order interactions are negligible. In a design of resolution five, none of the main effects and two-factor interactions are aliased with one another. Therefore all the main effects and two-factor interactions are estimable if the three-factor and higher-order interactions are negligible. Let EÐ.Ñ 3 be the number of 3-factor interaction pencils appearing in the defining relation of a regular = fractional factorial design.. Then t he resolution of. is the smallest integer 3 such that E3 Ð.Ñ!. The sequence ÐE" Ð.Ñß âß E8Ð.ÑÑ is called the word-length pattern of.. The word length pattern plays an important role in design selection which will be discussed later Regular fractional factorial designs are orthogonal arrays We show that there is complete factorial of factors embedded in each regular = fractional factorial design. As an important corollary, we show that all regular fractional factorial designs are orthogonal arrays. Theorem In a regular = fractional factorial design defined by Ð14 Þ"Þ"Ñ, there exist factors such that the design contains each combination of these factors exactly once. Without loss of generality, suppose they are the first factors, then there are : vectors," ßâß, : EG Ðß=Ñ such that all the = treatment combinations B œðb" ßâßB8Ñ in the design satisfy B œ,?, where? œ ÐB âßb Ñ, 1 Ÿ 4 Ÿ :. 4 4 "

9 T<990. We shall give a proof due to Franklin and Bailey (1977). The proof is essentially an algorithm that can be used to identify a set of basic factors. Without loss of generality, we assume that - œ â œ - œ!. " : First of all, we note that if there are : factors, say factors 3" ßâß3:, and a basis Ö-1, âß -: of e ÐEÑsuch that for each 1 Ÿ4Ÿ:, among the 3" th, â, and the 3: th entries of - 4, only the 3 4 th entry is nonzero, then the conclusion of the theorem holds. This is because if - 4 œð- "4 ßâß- 84 Ñ and is the only one among ßâß- 3 4 that is nonzero, then since 4 " : " -4 B œ! for all B in the design, we have B3 œ Ð-3 4 Ñ?ÂÖ3 ßâß3 -?4 B? ÞIt follows that 4 4 " : B3 ß âß B3 are linear functions of ÖB3 À 3 Á 3" ß âß 3 : for all B in the design. In other words, 1 : factors 3ßâß3 " : can be defined by the other factors; so they are added factors and the other factors form a set of basic factors. The design must contain each combination of the basic factors exactly once. We also note that if Ö-1, âß -: is a basis of eðeñ, then after replacing a -3 with -3, -4, where 4Á3 and, is a nonzero element of GF Ð=Ñ, it is still a basis. We now describe an algorithm to construct a basis Ö-1, âß -: satisfying the condition described in the previous paragraph. We first set 4œ" and - 1 œ+ 1 ßâß- : œ+ :. At each step 4œ"ßâß:, we select a positive integer 34 such that the 34th entry of -4 is nonzero. For each 6 œ 4 ", âß:, if the 34th entry of -6 is nonzero, then we re-set -6 to be the sum of the original -6 and,-4, where, is an element of GF Ð=Ñ chosen so that the 3 4 th entry of the new - 6 is zero. This is possible since the 34 th entry of -4 is nonzero. When this is done, we re-set 4 to be 4 ", and continue until 4œ: ". At the end of the above procedure, we will have chosen : factors 3" ßâß3: and a basis Ö-1, âß -: of e ÐEÑsuch that for all 1 Ÿ4Ÿ:, the 34th entry of -4 is nonzero, and all the 34th entries of - 6 with 6 4are zero. Next we start with 4œ: and go backwards. At each step 4œ:ß: "ßâß#, for each 6such that 4 " 6 ", if the 34th entry of -6 is nonzero, then we re-set -6 to be the sum of the original -6 and,-4, where, is an element of GF Ð=Ñ chosen so that the 34th entry of the new -6 is zero. Again this is possible since the 34 th entry of -4 is nonzero. After this is done, we re-set 4to be 4 " and continue until 4 œ ". When the algorithm ends, we will have successfully chosen a basis Ö-1, âß -: of eðe Ñ with the desired property. An important consequence of Theorem is that every regular fractional factorial design can be constructed by first writing down all the level-combinations of : basic factors, and then use different interaction contrasts of the basic factors to define : additional factors. The algorithm described in the proof of this theorem can be used to identify =

10 a set of basic factors from the defining relation. The factors that are not chosen by the algorithm form a set of basic factors, and those that are chosen are added factors. From the proof, we see that at each step since - 4 may have more than one nonzero entry, the choices of 3 4 may not be unique. This may lead to different choices of basic factors. Example Consider design (14.1.3) defined by M œ EFH œ FGI œ EFGJ œ EGHIœGHJ œeij œfhij. Among the three independent defining words EFH, FGI and EFGJ, we see that EFH is the only word that contains H, FGI is the only word that contains I, and EFGJ is the only word that contains J, so we can write H œ EF, IœFGand JœEFG. Thus the design can be constructed by using E, Fand Gas basic factors. In this example, referring to the proof of Theorem , we have :œ$, H, I, and J correspond to factors 3" ß 3# and 3$, and the defining words EFH, FGI and EFGJ correspond to -" ß -# and -$. In general, it may not be easy to spot a set of defining words satisfying the condition specified in the beginning of the proof. However, the algorithm described in the proof can be used to find a set of such independent defining words from an arbitrary set of three independent defining words. We now give a step-by-step illustration. Starting from the three independent defining words EFH, FGI and EFGJ, first we choose an arbitrary letter from the first word EFH, say E. The second word FGI does not contain E, so nothing is done; but since the third word EFGJ contains E, we replace it with EFHÐEFGJ Ñ œ GHJ to eliminate E. Now we end up with a new set of independent defining words EFH, FGI and GHJ in which only the first word contains E. Next we choose a letter from the second defining word FGI, say F. Then since the third defining word GHJ does not contain F, again nothing is done. Then we choose a letter from GHJ, say G. At this point we go backwards to eliminate G from the second defining word FGI by replacing it with ÐFGIÑÐGHJ Ñ œ FHIJ. Nothing needs to be done to the first word EFH since it does not contain G. Now we have independent defining words EFH, FHIJ and GHJ in which only the first word contains E and only the third word contains G, but F appears in both the first and second words. The last step is to eliminate F from the first word EFH by replacing it with ÐEFHÑÐFHIJ Ñ œ EIJ. The algorithm ends up with the three independent defining words EIJ, FHIJ and GHJ in which only the first word contains E, only the second word contains F and only the third word contains G. Therefore we have EœIJ, FœHIJ and GœHJ. This shows that H, I and J also form a set of basic factors, and the added factors E, F, G can be defined by interaction contrasts of H, I and J. Remark By Theorem , a saturated OA ÐRß# ß#Ñ can be obtained by deleting one column of 1's from a Hadamard matrix of order R. In view of Remark and " " Theorem , when this is applied to the 5-fold Kronecker product Œ Œ " " â " " " ", the resulting array is a saturated regular 5 fractional factorial design of size 2 for 2 5 " factors. In the proof of Theorem , at each step we can choose any factor that is involved in the interaction defined by. Suppose ÖE ßâßE is a set of factors such that none of - 4 " ; R "

11 their interactions appears in the defining relation. Then when we apply the algorithm, at each step, each of the linearly independent vectors -1, âß -: defines an interaction that involves at least one factor other than EßâßE " ;. Therefore at each step we can choose a factor other than E" ßâßE;. Since the chosen factors are added factors, this shows that E" ßâßE; can be part of a set of basic factors. Therefore we have the following result. Theorem Suppose EßâßE " ; is a set of factors such that none of their interactions appears in the defining relation of a regular = fractional factorial design. Then these factors can be part of a set of basic factors. As a consequence, the design contains each combination of these ; factors = ; times. The following is an important corollary of Theorem Theorem A regular = fractional factorial design of resolution V is an orthogonal array of strength V ". Proof. Since the design has resolution V, there is no defining word of length shorter than V. Thus for any set of V " factors, none of their interactions can appear in the defining relation. By Theorem , any set of V " factors can be part of a set of basic factors. Therefore the design contains each of their combinations the same number of times. In other words, the design is an orthogonal array of strength V ". We note that for a regular = fractional factorial design of resolution V, V " is also its maximum strength. It is clear that the design does not contain all the combinations of the V factors that appear in a defining word of length V. Note that Theorem shows that for each regular fractional factorial design, there is at least one set of factors such that the design contains each combination of these factors once. But not every set of factors can be basic factors. For example, even though the design defined by (14.1.3) contains all the eight combinations of factors E, F and G, since EFH is a defining word, it contains only four combinations of E, F and H, each with two replications. By Corollary and Theorem , if there is a regular design of resolution three or higher, then 8 Ÿ ÐR "ÑÎÐ= "Ñ. The construction of a two-level saturated regular resolution III design is described at the end of Remark For convenience, we say a design is of resolution III+ if it has resolution III or higher = 14.6 Foldovers of regular fractional factorial designs of odd resolution Theorem shows that the foldover of a two-level orthogonal array of even strength > is an orthogonal array of strength > ". The following is its counterpart for regular designs

12 Theorem The foldover of a regular 2 fractional factorial design of odd resolution Ð8 "Ñ : V is a regular 2 design of resolution V " Proof. Represent the two levels by 0 and 1 and consider the regular 2 design consisting of all the solutions B œðbßâßb " 8Ñ of (14.1.2), where + 3 EG Ð8ß#Ñ. For each + œ 8 Ð+ " ßâß+ 8Ñ eðeñ, we have 3œ" +B 3 3 œ! for all B in _.. Note that each treatment combination in the foldover of. is of the form B œðb" ßâßB8ßB8 " Ñ, where ÐB" ßâßB8Ñ., and B8 " œ! or Ð1 B" ßâß1 B8Ñ. and B8 " œ 1. For each +œð+ " ßâß+ 8Ñ eðeñß let _ Ð+ ßâß+ ß!Ñ, if + has an even number of nonzero entries, + œ œ " 8 Ð14Þ6Þ"Ñ Ð+ " ßâß+ 8ß1 Ñ, if + has an odd number of nonzero entries. Then it is clear that _ +Bœ! for all Bin the _ foldover of.. Thus the foldover of. is a regular design whose defining words are those in Ö+ À + eðeñ. Since V is odd, by Ð14Þ6 Þ"Ñ, the resolution of the foldover of. is V ". The following is obvious from Ð14 Þ&Þ"Ñ. Corollary All the defining words of the foldover of a regular 2 design have even lengths. Draper and Mitchell (1967) called regular designs in which all defining words have even lengths even designs. Thus the foldovers of regular 2 designs are even designs. The foldover construction (13.6.1) includes an additional factor that is at a constant level before being folded over. Suppose such a factor is not included; that is, let \ \ œ \, (14.6.2) where is a 2 design. Then the same argument as in the proof of Theorem shows 8 Ð: "Ñ that the defining words of the 2 design \ are precisely those defining words of \ that have even lengths. So the conclusions of Theorem and Corollary still hold for \. Notice that if \ is a resolution III design that has no defining word of length four, then ~ the design \ in (14.6.2) has resolution VI or higher, but the foldover \ of \ as in (13.6.1) has resolution IV. \ It can be shown that the converse of Corollary is also true. That is, we have the following result: Theorem If the two levels are denoted by 1 and ", then a regular two-level even \ design must be of the form, where \ is either a regular design of resolution III or is \ such a design supplemented by a column of 1's.

13 ' $ 7 $ Example The foldover of the regular 2 design defined by (14.1.3) is a regular 2 design of resolution IV. According to the proof of Theorem , all the defining words of length four remain defining words of the foldover design, and each of the defining words of length three is supplemented by the additional factor (G). This yields the defining relation M œ EFHK œ FGIK œ EFGJ œ EGHI œ GHJK œ EIJK œ FHIJ for the foldover design when the two levels are represented by 1 and ". That for this example the signs of all the defining words are positive can be determined from the fact that the foldover design contains the combination with all factors at level 1. By Corollary and Theorem , if there is a regular two-level design of resolution four or higher, then 8ŸRÎ#. This upper bound is attained by the foldover of a saturated regular design of resolution III. A design is said to be of resolution IV+ if it has resolution IV or higher. 14. ' Construction of regular fractional factorial designs to estimate certain required effects Franklin and Bailey (1977) and Franklin (1985) proposed an algorithm for selecting independent defining contrasts to construct regular designs under which certain required effects can be estimated. Suppose the factorial effects are divided into three classes: effects E3's whose estimates are required, effects F4's which are not negligible but whose estimates are not required, and effects G 5 's which are negligible. The objective is find the smallest regular design under which the required effects E 3 's can be estimated. Franklin and Bailey (1977) considered the two-level case and Franklin (1985) extended the algorithm to the case in which the number of levels is a prime power. First one needs to determine the effects that cannot appear in the defining relation. These are called ineligible effects. It is easy to see that they are all the required effects E 3 and all the pairwise products of the forms EE 3 4 and EF 3 4. Then the algorithm starts with a smallest possible run size as suggested by some theoretical lower bound derived in Franklin and Bailey (1977). But often a simple lower bound on the run size can be obtained by counting the required number of degrees of freedom. A set of 5 basic factors is chosen if the initial size is set to be = 5. Independent defining words are then selected, one for each of the remaining (added) factors, subject to the constraint that none of their generalized interactions (products) is ineligible. If the search fails to find a suitable set of defining words, then we change to another set of basic factors. When all choices of basic factors are exhausted without producing a suitable design, the run size is increased to the next power of the number of levels. This is basically an exhaustive search, but Franklin and Bailey (1977) also provided a theoretical result to cut unnecessary searches in certain cases. If we run the algorithm with a particular set T of basic factors such that none of their interactions are eligible, and it fails to produce a solution, then there is no need to try any other set of basic factors of the same size. Any solution of the same size can be generated by using the factors in T as basic factors. This

14 is a consequence of Theorem since all the interactions of the factors in T are ineligible and cannot appear in the defining relation of any solution. Example Suppose there are six 2-level factors E, F, G, H, I and J. We would like to construct the smallest regular fractional factorial design that allows the estimation of all the six main effects and nine specific two-level interactions EG, EH, EJ, FG, GH, GI, GJ, HI, IJ, assuming that all the other interactions are negligible. In this case, the set of ineligible effects consists of all the 15 required effects and their pairwise products. It can be verified that these are all the main effects, all the two-factor interactions, the three-factor interactions EFG, EGH, EGI, EGJ, EHI, EIJ, EFH, EFJ, FGH, FGI, FGJ, FHI, FIJ, GHI, GIJ, EHJ, GHJ, HIJ, and the four-factor interactions EGHI, EGIJ, EFGH, EGHJ, EHIJ, EFGJ, FGHI, FGIJ, GHIJ. Since there are 15 effects to be estimated, we need at least 16 runs. We start with E, F, G and H as the basic factors, and set up the following table of eligible effects, with the ineligible effects crossed out: I J M E F EF EFI G EG FG EFG EFGI H EH FH FHJ EFH EFHI EFHJ GH EGH FGH FGHJ EFGH EFGHI EFGHJ We need to choose one effect from each column to define the two added factors I and J. We start with EFI and FHJ, the effects on the top of the two columns. If both EFI and FHJ are defining effects, then their product EHIJ must also be a defining effect. Since EHIJ is ineligible, this choice is ruled out. Then we keep EFI and go down the second column to choose EFHJ, which still does not lead to a solution since ÐEFIÑÐEFHJ Ñ œ HIJ is also ineligible. The next choice, FGHJ, however, provides a solution because ÐEFIÑÐFGHJ Ñ œ EGHIJ is eligible. If we continue the search, we can see that this is the only solution with E, F, G, H as basic factors. Since all the interactions of these four factors are ineligible, we can conclude that no other 16-run solutions can be found with other choices of four basic factors. So the only 16-run solution is the design defined by MœEFIœ

15 FGHJ œ EGHIJ. On the other hand, if we were not able to find a solution using E, F, G, H as basic factors, then we could immediately jump to 32-run designs without searching through other choices of four basic factors