Handout #13: Fractional factorial designs and orthogonal arrays
When the number of factors is large, it may be feasible to observe only a fraction of all the treatment combinations. Under such a fractional factorial design, not all factorial effects can be estimated. In this handout, we introduce an important combinatorial structure called orthogonal arrays, and describe how they can be used to run factorial experiments. We discuss the estimability of factorial effects under an orthogonal array, and show that when some effects are assumed to be negligible, other effects become estimable and their estimates are uncorrelated. Several examples are given to illustrate different types of orthogonal arrays, in particular, the distinction between regular and nonregular designs. We briefly discuss Hadamard matrices, an important class of nonregular designs, and also present a useful design construction technique called foldover. In this and the next few handouts, we assume that the experimental units are unstructured and the experiment is to be conduced with complete randomization. Multi-stratum fractional factorial designs will be discussed in a later handout.
13.1. Model for completely randomized factorial experiments
When the experimental units are unstructured, a factorial design can be specified by the number of observations to be taken on each treatment combination. For a design ., let 8." ÐBßâ ßB 8 Ñ be the number of observations on treatment combination ÐBßâ" X ßB88Ñ and let 8." be the Ð= â= Ñ‚" vector whose entries are the 8 ."8 ÐBßâ ßB Ñ 's X arranged in lexicographic order. Then . is determined by 88".. We have . œR , where R is the run size. When 8." ÐBßâßB 8 Ñœ! or 1, 8. can also be viewed as the indicator function of the selected treatment combinations.
Let CßâßC"R be the R observed responses. Without loss of generality, we may assume that CßâßC"R are uncorrelated with constant variance, and if C 3 is an observation X on the treatment combination ÐBßâßB"8 Ñ , then
EÐC3"8 Ñ œ.α ÐB ß â ß B Ñ,
where αÐBßâ"8 ßB Ñ is the effect of treatment combination ÐBßâ "8 ßB Ñ . We have seen in Handout #9 how to parametrize αÐBßâ"8 ßB Ñ in terms of various factorial effects. In general, let and let XX , , be mutually orthogonal :"! œâ==""â=8 :"=αα :"â=8 " â= 8 " X treatment contrasts representing the factorial effects, with :3 α being a 5 -factor interaction contrast if the entries of :3 only depend on the levels of 5 factors. Then α can be expressed as , where and , , , X , TT"::""œÒ ã"= ãâã"â=8 " Ó œÐ""!" â"="" â=8 Ñ with ".".""œ" :X α. We shall absorb into and still denote as . Then a full 3!!!: # 3 ll3 model with all factorial effects present can be expressed as
-13.1- EÐC"\\TÑœ. XXα" œ , (13.1.1)
where C is the R‚" vector of observed responses. Often some of the factorial effects are assumed to be negligible; then the associated terms are dropped from ÐÑ 13.1.1 . This leads to a linear model
~ EÐC\UÑœ X " , (13.1.2)
where UT is obtained from by deleting the columns of T that correspond to negligible ~ factorial effects, and "" is the subvector of consisting of the nonnegligible effectsÞ
The contrasts ""â= , â , "=""8 can be constructed by using finite geometry or by the method as described in Theorem 9.6.3. In this Handout, we will use the latter method. D Then T is as in (9.6.18). We follow the notation in (9.6.17) to write each "3 as " , where D D D;−W"8 ‚â‚W . Let be the column of \UX corresponding to a nonnegligible " . D If the nonzero entries of D are D3" , âD , 35 , then the entry of ; corresponding to an observation on the treatment combination ÐBßâßB Ñ:X is equal to 5 34 as defined "8 #4œ" BßD3344 in ÐÑ9.6.12 .
X We define the Hadamard product of two vectors BCœÐBßâßB"8 Ñ and œ XX?@ ÐC"8 ßâßC Ñ to be B C;; œÐBC ""88 ßâßB C Ñ . Let and be two different columns of
\UX3, and ?ß""â? , 34<= and @ 4 ß â@ , be the nonzero entries of ?@ and , respectively. Then ;;?@ and can be expressed as
?/? ? / ;;ω3" 333"< ;< ,
@/@@ / ;;œâ4" 444"= ;= , where the 3 th entry of /3 is equal to 1, and all its other entries are zero; i.e., each column corresponding to an interaction contrast is the Hadamard product of the relevant main- effect columns. If each treatment combination is observed exactly once (that is, we have one replicate of the complete factorial), none of the factorial effects is negligible, and the entries of C\ and " are arranged in lexicographic order, then X is the identity matrix and X Uœœ TTT=="#ŒŒâŒ T = 8. In this case, the information matrix Ð \UXX Ñ \U XX X X œU\\UX X œœ UU TT is a diagonal matrix. As a consequence, the least squares estimates of the factorial effects are uncorrelated and can be calculated easily: "sDDœÐ" : ÑX C. :D # ll
A complete factorial experiment requires =â="8 runs. Cost and other practical considerations often call for observing only a fraction of the treatment combinations.
-13.2- Indeed if only a subset of the factorial effects are expected to be important, then observing a fraction of the treatment combinations would be sufficient Under such a fractional factorial design, not all the factorial effects are estimable. There are two interesting issues. Given a model where some of the factorial effects are assumed negligible, under what designs can the nonnegligible effects be estimated? Given a fractional factorial design, what models can it entertain? In the former question, one may also ask, under what designs are the estimators of the nonnegligible effects uncorrelated? In the next section, we introduce orthogonal arrays and provide some answers to this question.
13.2 Orthogonal arrays
Orthogonal arrays were first introduced by Rao (1946, 1947) for the case where all the factors have the same number of levels. Such arrays will be referred to as symmetric orthogonal arrays. Rao (1973) extended the definition to also cover asymmetrical (mixed-level ) orthogonal arrays . A useful reference is the book "Orthogonal Arrays: Theory and Applications" by Hedayat, Sloane and Stufken, Springer, 1999.
Definition 13.2.1. An orthogonal array OAÐRß="8 ‚â‚= ß>Ñ is an R‚8 matrix with =3 distinct symbols in the 3 th column, 1 Ÿ3Ÿ8ß such that in each R‚> submatrix, all combinations of the symbols appear equally often as row vectors. The positive integer > is called the strength of the orthogonal array. If =œâœ=œ="8 , then it is called a symmetric orthogonal array and is denoted as OAÐRß=8 ß>Ñ .
From the definition it follows immediately that if an OAÐRß=8 ß>Ñ exists, then R must be a multiple of => .
An OAÐRß="8 ‚â‚= ß>Ñ can be used to define a factorial design of size R for 8 factors with ="8 ßâß= levels: each column corresponds to one factor and each row represents a treatment combination. The following result explains the role of the strength and the utility of orthogonal arrays.
Theorem 13.2.2. An orthogonal array of strength > œ #5 ( 5 " ) can be used to estimate all the main-effect contrasts and all interaction contrasts involving up to 5 factors, assuming that all the interactions involving more than 5 factors are negligible. An orthogonal array of strength > œ #5 " ( 5 # ) can be used to estimate all the main- effect contrasts and all interaction contrasts involving up to 5" factors, assuming that all the interactions involving more than 5 factors are negligible. In both cases, all the estimators are uncorrelated.
Proof. Suppose all the interactions involving more than 5 factors are negligible. Let D \UX be the model matrix as in (13.1.2). Then each column of \UX is a ;D, where contains at most 5 nonzero entries.
-13.3- X Case 1. >œ#5: We claim that the information matrix ÐÑÐÑ\UXX \U is a diagonal X matrix; then ÐÑÐÑ\UXX \U is invertible and all the unknown parameters in the model are estimable with uncorrelated estimators.
?@ Let ;; and be two different columns of \UX3, and ?ß""â? , 34<= and @ 4 ß â@ , be the nonzero entries of ?@ and , respectively. Then ÐÑ ;;?@X is equal to the sum of the ?@? //?@ /@ / entries of ;; œÐ ;34""34" â ;33<=< ÑÐ ;" â ; 44= Ñ . Since <=Ÿ5, , <= is no more than the strength of the orthogonal array. Therefore each row of the ?34//34?@33//@ 44 R‚Ð<=Ñ matrix Ò; """ ãâã;;<=< ã" ãâã ;= Ó is replicated ==â=Î"# 8 R times ?34//34?@33//@ 44 in the Ð==â=Ñ‚Ð<=Ñ"# 8 matrix Ò: """ ãâã::<=< ã" ãâã := ] Þ It follows that ÐÑ;;?@XX œR ÐÑ :: ?@ œ!. ==â="# 8
D Case 2. > œ #5 " : Partition \UX" as Ò YãÓYY#", where consists of all the ; 's where ? DY;? contains at most 5" nonzero entries, and 2 consists of all the 's where contains 5 nonzero entries. Then the information matrix for the effects involving up to 5" XXXX X factors is equal to YY""##"##"ÐÑ YY YY YY . Similar to the proof of case 1, YY " " XXXXXX is a diagonal matrix, and YY"""##"#"##""œÐÑœ ! . Thus YY YY YY YY YY is a diagonal matrix.
In either case, the least squares estimators of the factorial effects of interest are uncorrelated and can be calculated easily: "sDDœÐ1 ; ÑX C . ;D # ll For orthogonal arrays of odd strengths, one can prove a result somewhat stronger than that in Theorem 13.2.2. Under an OA of strength #5 " , we may not be able to estimate all interaction contrasts involving 5 factors. However, consider the model which contains all the main-effect contrasts, all interaction contrasts involving up to 5" factors, and all the 5 -factor interaction contrasts involving a given factor . In this X case, the information matrix ÐÑÐÑ\UXX \U for all the nonnegligible effects is also diagonal. This is because, as in the proof of Theorem 13.2.2, each off-diagonal entry of X ÐÑÐÑ\UXX \U is the sum of all the entries of the Hadamard product of relevant main- effect columns of \UX . The total number of factors involved in each of these Hadamard products is no more than #5 " , since a common factor is involved in all the nonnegligible 5 -factor interactions. Then the same argument as in the proof of Theorem X 13.2.2 shows that all the off-diagonal entries of ÐÑÐÑ\UXX \U are zero since the strength of the array is #5 " . We state this result in the following theorem.
Theorem 13.2.3. An orthogonal array of strength > œ #5 " ( 5 " ) can be used to estimate all the main-effect contrasts, all interaction contrasts involving up to 5" factors, and all the 5 -factor interaction contrasts involving a given factor, assuming that all the other interaction contrasts are negligible.
-13.4- By counting the numbers of degrees of freedom for the estimable orthogonal treatment contrasts, from Theorem 13.2.2 and Theorem 13.2.3, we obtain the following general lower bounds on the run sizes of orthogonal arrays due to Rao (1947).
Theorem 13.2.4. If there exists an OAÐRß=8 ß>Ñ , then 5 8 3 (i) for > œ #5ß R 3 Ð= "Ñ ; 3œ! ˆ‰ 5" 88"35 (ii) for >œ#5" , R 35" Ð="Ñ Ð="Ñ . 3œ! ˆ‰ ˆ ‰
The following result follows easily from Theorem 13.2.4.
Corollary 13.2.5.
(i) If there exists an OAÐRß =8 ß #Ñ , then 8 Ÿ ÐR "ÑÎÐ= "Ñ . In particular, if there exists an OAÐRß 28 ß #Ñ , then 8 Ÿ R " .
(ii) If there exists an OAÐRß 28 ß 3 Ñ , then 8 Ÿ RÎ# .
An orthogonal array achieving the bound in Corollary 13.2.5(i) can accommodate the maximum possible number of factors for a given run size, and is called a saturated orthogonal array of strength two.
13.3 Examples of orthogonal arrays
We list a few examples of orthogonal arrays. The first three are symmetric two- level arrays in which the two levels are denoted by 1 and " . The fourth is asymmetrical with seven three-level factors and one two-level factor. The first two designs are examples of the so called regular fractional factorial designs, while the last two designs are nonregular. Regular designs are discussed in the next section and Handout #14.
Example 13.3.1. an OAÐ)ß#' ß#Ñ :
" " " "1 " 1" " 111 "111 " " 11"""" ÐÞÞ"Ñ13 3 " "11 " " 111" " " "111 " " 111"" 1
-13.5- Example 13.3.2 an OAÐ"'ß #) ß $Ñ :
"""""""1 " " " " " " " " " " " " " " " " """""""" """" """" " " " " " " " " """""""" """""""" ÐÞÞ#Ñ13 3 """""""" """""""" " " " " " " " " """""""" """""""" " " " " " " " " " " " " " " " " " " " " " " " "
Example 13.3.3. an OAÐ"#ß #"" ß #Ñ :
""""""""""" """"""""""" """"""""""" """""""""" " " " " " " " " " " " " " " " " " " " " " " " ÐÞÞ$Ñ13 3 """"""""""" """"""""""" """"""""""" """"""""""" """"""""""" """""""""""
-13.6- Example 3.3.4. an OAÐ")ß 2 ‚ 3( ß #Ñ :
!!!!!!!! !!"""""" !!###### !"!!""## !"""##!! !"##!!"" !#!"!#"# !#"#"!#! !##!#"!" ÐÞÞ%Ñ13 3 "!!##""! "!"!!##" "!#""!!# ""!"#!#" """#!"!# ""#!"#"! "#!#"#!" "#"!#!"# "##"!"#!
13.4 Regular fractional factorial designs
The difference between regular and nonregular designs lies in their construction. Design 13.3.1 is constructed in the following simple manner. The first three columns in this design consist of all the eight treatment combinations of the first three factors. Column 4 is the Hadamard product of the first two columns, column 5 is the Hadamard product of the second and third columns, and column 6 is the Hadamard product of the first three columns. If each row (treatment combination) is denoted by ÐB" ßâßB6 Ñ , where Bœ"3 or " , then the eight treatment combinations in the design satisfy
B% œBBßB "# & œBB #$, B6 œBBB "#$ . Ð 13 ÞÞ"Ñ 4
In other words, since the design has 8œ#$ runs, we first write down all the combinations of three factors (in this case, factors 1, 2, and 3); these factors are called basic factors . Then we use ÐÞÞ"Ñ 13 4 to define three additional factors, called added factors .
Note that ÐÞÞ"Ñ 13 4 is equivalent to
BBB"#% œ"BBB, #$& œ" and BBBB "#$' œ". Ð 13 ÞÞ#Ñ 4
-13.7- If BBB"#% , BBB #$& and BBBB "#$' are all equal to 1, then their products ÐBBBÑÐBBBÑ "#% #$& œBBBB"$%&, ÐBBBÑÐBBBBÑœBBB "#% "#$66 $%, ÐBBBÑÐBBBBÑœBBB #$& "#$' "&' and ÐBBBÑÐBBBÑÐBBBBÑœBBBB"#% #$& "#$' #%&' are also equal to 1. Thus the eight treatment combinations in this fraction are solutions of the following system of equations
"œBBB"#% œBBB #$& œBBBB "#$' œBBBB "$%& œBBB $%6 œBBB "&' œBBBB #%&'. ÐÞÞ$Ñ13 4
Suppose the two levels are represented by the two elements 0 and 1 of ™# . Specifically we replace 1 and " with 0 and 1, respectively. Then Ð 13 Þ 4 Þ#Ñ is equivalent to
BBBœ!ßBBBœ!"#% #$& and BBBBœ! "#$' . Ð 13 ÞÞÑ 4 4 In other words, the eight treatment combinations in this fraction are the solutions of a system of three independent linear equations. They are those in the principal block when the 64 treatment combinations in a complete 26 factorial are divided into eight blocks of size eight by confounding the interactions of factors 1, 2, 4, factors 2, 3, 5 and factors 1, 2, 3, 6. These eight treatment combinations also constitute a three-dimensional subspace of EGÐ'ß #Ñ when each treatment combination is considered as a point in EG Ð'ß #Ñ . In general, when =Ð8ß=Ñ is a prime or prime power, subspaces (or flats) of EG are called regular fractional factorial designs. This implies that the run size of a regular design must be a power of = . Construction and properties of regular designs will be discussed in more details in Handout #14.
13.5 Hadamard matrices
Design 13.3.3 can be constructed from a Hadamard matrix of order 12. A Hadamard matrix of order RR‚R is an matrix L of 1's and " 's such that
X LLœR MR ,
where MR is the identity matrix of order R . If we multiply all the entries in the same row or the same column of a Hadamard matrix by " , then the resulting matrix is still a Hadamard matrix. Therefore, without loss of generality, we may assume that all the entries in the first row and/or the first column of L are equal to 1.
We have the following equivalence between a Hadamard matrix of order R and an OAÐRß#R" ß#Ñ .
Theorem 13.5.1. Suppose L is a Hadamard matrix of order R# such that all the entries in the first column are equal to 1. Then the matrix obtained by deleting the first column of L is an OAÐRß#R" ß#Ñ . Conversely, adding a column of 1's to an OAÐRß#R" ß#Ñ results in a Hadamard matrix of order R .
-13.8- Proof. It is trivial that an OAÐRß#R" ß#Ñ supplemented by a column of 1's is a Hadamard matrix. Thus it is enough to prove the converse. Suppose L is a Hadamard X matrix such that all the entries in the first column are equal to 1. Let 23"3œÒ2 ßâß2R3 Ó X and 2L4"4œÒ2 ßâß 2R4 Ó be the 3 th and 4 th columns of , where 3Á"ß4Á" and 3Á4. We need to show that each of Ð"ß"ÑßÐ"ß"ÑßÐ"ß"Ñ and Ð"ß"Ñ appears RÎ% times among the R pairs Ð2"3 ß 2 "4 Ñßâß Ð2R3 ß 2 R4 Ñ. Suppose Ð"ß "Ñ appears + times, Ð"ß "Ñ appears , times, Ð"ß "Ñ appears - times, and Ð"ß "Ñ appears . times. Then since 2234 is orthogonal to , and they are both orthogonal to the column of 1's, we have
+,-.œRß
+,-.œ!ß
+,-.œ!ß and +,-.œ!.
Solving these equations, we have +œ,œ-œ. .
We shall call an OAÐRß#R" ß#Ñ constructed from a Hadamard matrix of order R as described in Theorem 13.5.1 a Hadamard design. Design 13.3.3 can be obtained by deleting a column of 1's from a Hadamard matrix of order 12. In view of Corollary 13.2.5, all Hadamard designs are saturated.
An immediate consequence of Theorem 13.5.1 is that if there exists a Hadamard matrix of order R# , then R must be a multiple of 4. Hadamard designs are not regular when RR is not a power of 2; when is a power of 2, they may or may not be regular.
The following are Hadamard matrices of orders 1 and 2:
T" œÒ"Ó,
"" T# œ . ”•"" It has been conjectured that a Hadamard matrix of order RR exists for every that is a multiple of 4Þ This is called the Hadamard conjecture .
It is easy to see that if LO and are Hadamard matrices of orders 78 and , respectively, then the Kronecker product LOŒ78 is a Hadamard matrix of order . Applying this result to T# repeatedly, we conclude that there exists a Hadamard matrix of order 28 for every positive integer 8 .
-13.9- Remark 13.5.2. As shown in Section 9.6 in Handout #9, the 28 " columns of the 8 - fold Kronecker product "" "" ŒŒâ , ”•"" ”• "" #8 except the first one, constitute an orthogonal basis of ‘ ‹K0, representing the various factorial effects in a 28 complete factorial. Notice that the Hadamard design obtained by "" "" "" deleting the first column of ŒŒ is the same as the array ”•”•”•"" "" "" in Ð 9 Þ'Þ 3 Ñ after the seven columns of the array are rearranged in the lexicographic order (1, 2, 12, 3, 13, 23, 123). In fact, the regular fractional factorial design in Example 13.3.1 consists of six of these seven columns. It can be seen from ÐÞÞ"Ñ 13 4 that the construction of this design amounts to using interaction contrasts of the basic factors to define the added factors. This construction will be treated in more details in Handout #14.
Although the Hadamard conjecture has not been proved, so far no counterexamples have been found, and Hadamard matrices have been constructed for many orders that are multiples of four. This provides more flexibility than regular designs in terms of the run sizes. Plackett and Burman (1946) were the first to propose the use of Hadamard designs in factorial experiments. The Hadamard designs constructed in their paper are referred to as Plackett-Burman designs .
A method of constructing Hadamard matrices used by Plackett and Burman was due to Paley (1933). Suppose RR" is a multiple of 4 such that is an odd prime power. Let ;œR" and let ααα"#; œ!ß ßâß denote the elements of KJÐ;Ñ . Define a function ; À KJ Ð;Ñ Ä Ö!ß "ß "× by
# Ú "œCC−KJÐ;Ñß, if " for some ;"ÐÑœ !œ, if " 0, Û ", otherwise. Ü
Let E be the ;‚; matrix Ò+Ó34 , where + 34 œ;αÐ34 α Ñ for 3, 4 œ "ß #ß âß; , and
X ""; TR œß ÐÞ"Ñ13 5. ”•"EM;;
where "M;; is the ;‚ 1 vector of 1's and is the identity matrix of order ; . Then T R is a Hadamard matrix. For a proof see Hedayat, Sloane and Stufken (1999). We shall call the OAÐRß#R" ß#Ñ obtained by deleting the first column of Ð 13 Þ 5 Þ"Ñ the Paley design of order R.
There is also a connection between Hadamard matrices and balanced incomplete block designs (BIBD). Suppose L is a Hadamard matrix of order R . Without loss of generality, assume that all the entries in the first row and first column of L are equal to 1. Delete the first row and first column from L ; then we have an ÐR "Ñ ‚ ÐR "Ñ matrix L ‡ of 1's and " 's. Define a block design . with R " treatments and R "
-13.10- blocks such that the 34Ð3ß4Ñ th treatment appears in the th block once if the th entry of L ‡ is equal to 1; otherwise, it does not appear. Then . is a balanced incomplete block design. The block size is RÎ#" since there are RÎ#" 1's in each column of L ‡ . Conversely, given a balanced incomplete block design .R" with treatments and R" blocks of size RÎ#" , write down an ÐR"Ñ‚ÐR"Ñ incidence matrix L ‡ of 1's and " 's such that the Ð3ß 4Ñ th entry of L ‡ is equal to 1 if and only if the 3 th treatment appears in the 4. th block of . Supplement L ‡ by a row and column of 1's; then we obtain an R‚R matrix which can be shown to be a Hadamard matrix.
For example, the OAÐ"#ß 2"" ß #Ñ displayed in Ð 13 Þ 3 Þ$Ñ can be constructed by applying the method described in the previous paragraph to a balanced incomplete block design with 11 treatments and 11 blocks of size 5. The first 11 rows of the array come from the incidence matrix of the BIBD which can be developed from the initial block Ö"ß $ß %ß &ß *× in a cyclic manner. Thus the associated incidence matrix is a circulant matrix. One row of 1's is then added at the bottom to produce an orthogonal array. If we also add a column of 1's, then a Hadamard matrix of order 12 is obtained.
13.6 Foldover designs
The orthogonal array displayed in ÐÞÞ#Ñ 13 3 has strength three. One can see that the first eight rows of this design constitute the Hadamard matrix "" "" "" Œ Œ , which is also the OA Ð)ß 2( ß #Ñ in Ð 9 Þ'Þ 3 Ñ ”•”•”•"" "" "" supplemented by a column of 1's after the other columns are rearranged in lexicographic order. The last eight rows of array ÐÞÞ#Ñ 13 3 are obtained from the first eight rows by interchanging the two levels. We say that array ÐÞÞ#Ñ 13 3 is the foldover of (the rearranged) array ÐÞ'ÞÑ 9 3 . In general, given an OA ÐRß#ß>Ñ8 \ where the two levels are represented by 1 and " , the following array is called the foldover of \ : ~ 1 \ \ œÐÞÞ"Ñ,136 ”•"\ where 1 is the R‚" vector of 1's. The foldover design is of size 2 R and has 8" two- level factors Compared with the original design, one factor is added and the run size is doubled.
The method of foldover was first proposed by Box and Wilson (1951) for regular fractional factorial designs. The following result for general two-level orthogonal arrays is due to Seiden and Zemach (1966).
Theorem 13.6.1. The foldover of a two-level orthogonal array of even strength > has strength >".
Theorem 13.6.1 follows from the following lemma.
-13.11- Lemma 13.6.2. Suppose \ is an OAÐRß 28 ß >Ñ with 8 > 1 in which the two levels are denoted by 1 and " . Let ]\ be an R ‚ Ð> "Ñ submatrix of . Then there exist > two nonnegative integers α" and with α"œRÎ# such that all the vectors B œÐB" , BâB#>""#>"3, , Ñ with BBâBœ 1, where Bœ 1 or 1, appear α times as row vectors of ] , and each of those with BB"# âB >" œ 1 appears " times.
Proof. For each BBBœÐBß"3âß B>" Ñ with B œ" or " , let 0Ð Ñ be the number of times appears as row vectors of ]\ . Then since has strength > , we have
0ÐBß""âßB> ßB >" Ñ0Ð Bß âßBßB > >" Ñœ-. Also,
0ÐBß""âßB>>" ßB Ñ0Ð Bß âßBßB >>" Ñœ-.
It follows from these two equations that 0ÐBß"" âßBßB> >" Ñœ0Ð Bß âßBßB > >" Ñ. Repeating the same argument, we see that any two rows BC and differing in an even number of components appear the same number of times as row vectors of ] . Thus all the vectors B œÐBBâB"# , , , >" Ñ with BBâBœ "# >" 1 appear the same number of times, say α times, as row vectors of ] , and each of those with BB"# âB >" œ 1 also appears the same number of times, say " times. Since \ has strength > , we have α"œRÎ#>.
Now we prove Theorem 13.6.1. Let \ be an OAÐRß 28 ß >Ñ , where > is even, and ~~~~ let ]\]\ be a 2R‚Ð>"Ñ submatrix of . If contains the first column of as displayed in ÐÑ 13.6.1 , then since \ has strength > , it is clear that all the Ð>"Ñ -tuples of ~ 1's and " 's appear the same number of times as row vectors of ] . Suppose
~ ] ] œ , ”•] where ]\ is an R‚Ð>"Ñ submatrix of . Then by Lemma 13.6.2, there exists a nonnegative integer αα such that each B œÐBBâBÑ"# , , , >" appears either or RÎ#> αα times as a row vector of ]B . Since >" is odd, if appears times, then B must appear RÎ#> αα times. It follows that if B] appears times in , then it appears ~ RÎ#>>> ααα times in ]], and thus appears ÐRÎ# Ñ œ RÎ# times in .
It is also clear from the above proof why the foldover method does not increase the strength when it is applied to an orthogonal array of odd strength.
Corollary 13.2.5 shows that an OAÐRß#8 ß#Ñ must have R 8" and an OAÐRß #8 ß 3 Ñ must have R 2 8 . It follows from Theorem 13.6.1 that the foldover of a saturated two-level orthogonal array of strength two achieves the lower bound on the run size of an orthogonal array of strength three.
-13.12-