Indicator function and complex coding for mixed fractional factorial designs 1

Transcription

1 Indicator function and complex coding for mixed fractional factorial designs 1 Giovanni Pistone Department of Mathematics - Politecnico di Torino Maria-Piera Rogantin Department of Mathematics - Università di Genova Abstract In a general fractional factorial design the n levels of each factor are coded by the n-th roots of the unity. This device allows a full generalization to mixed designs of the theory of polynomial indicator function already introduced for binary designs by Fontana and the Authors (2000). Properties of orthogonal arrays and regular fractions are discussed. Key words: Algebraic statistics, Complex coding, Mixed designs, Regular fraction, Orthogonal arrays 1 Introduction Algebraic and geometric methods are widely used in theory of design of experiments. There is a variety of these methods: real linear algebra, Z p arithmetic, Galois Fields GF(p s ) arithmetic, where p is a prime number as introduced in Bose (1947). See, e.g., Raktoe et al. (1981) and the more recent books Dey and Mukerjee (1999) and Wu and Hamada (2000). Complex coding of levels is used by different authors in various contexts, see e.g. Bailey (1982), Kobilinsky and Monod (1991), Edmondson (1994), Kobilinsky and Monod (1995), Collombier (1996). Corresponding author. addresses: giovanni.pistone@polito.it (Giovanni Pistone), rogantin@dima.unige.it (Maria-Piera Rogantin ). 1 Partially supported by Italian grant PRIN03 coordinated by G. Consonni 1

2 The use of a different background, called Commutative Algebra or Polynomial Ring Algebra, has been first advocated by Pistone and Wynn (1996) and discussed in detail in Pistone et al. (2001). Other relevant basic references are Robbiano (1998), Robbiano and Rogantin (1998) and Galetto et al. (2003). In the present paper, we consider mixed designs with replicates. We code the n levels of a factor by the n-th roots of the unity, while the fraction itself is encoded in its indicator function. This is a generalization of the approach to the binary case discussed in Fontana et al. (1997) and Fontana et al. (2000). In Tang and Deng (1999) the coefficients of the polynomial indicator function were introduced independently with the name of generalized word length patterns. The representation of a fraction by its indicator polynomial function was generalized to designs with replicates in Ye (2003) and extended to not binary factors using orthogonal polynomials with an integer coding of levels in Cheng and Ye (2004). The coefficients of the indicator function encode in a simple way many interesting properties of the fraction: orthogonality between and among factors and interactions, projectivity (Sections 4.3 and 5), aberration, regularity (Section 6). In Section 2 we mention a further interpretation of our setting in the framework of Fourier Analysis on a finite group. Sections 1 to 3 are a self-contained introduction. The main results are in Sections 4 to 6. In Section 5 we discuss combinatorial orthogonality vs. geometrical orthogonality, and in Section 6 we discuss the equivalence between different definitions of a regular fraction. Some examples are shown in Section 7. The use of indicator function has important computational implications. It allows one to use new powerful algorithms available in symbolic computation systems to study, analyze and classify designs. In our examples we use the symbolic software CoCoA, CoCoATeam, while SAS ( c SAS Institute Inc.) is used for numeric computations in the modulus n arithmetics. An example is the generation of all fractions with a given orthogonality structure, as done in the binary case in Fontana et al. (2000) with 5 factors. A caveat is due: the high computational complexity of the general purpose algorithms used in Computer algebra currently limits the approach to small-to-medium size problems. 2 Coding of factor levels Let m be the number of factors of a design. We denote the factors by A j, j = 1,...,m, and by n j the number of levels of the factor A j. We consider only qualitative factors. 2

3 For each factor A j let u 0j,u 1j,...,u (nj 1)j with u 0j = 1 be an orthogonal basis of the real functions defined on A j, f : A j R. We denote by U j the matrix [u hj (a)] a Aj,h=0,...,n j 1. Here orthogonality means a A j u hj (a)u kj (a) = 0 with h k i.e. the columns of the matrix U j are orthogonal. We denote by D the full factorial design, D = A 1 A m, and by R(D) the space of all real responses (treatment effects) defined on D. R(D) has an orthogonal split into interaction spaces m R(D) = H 0 H i H ij i=1 i<j i<j<k H 12 m. For each subset I {1,...,m} of factor indices, an orthogonal basis of the interaction subspace H I is: u ij j : i j = 1,...,(n j 1). j I In the equivalent language of matrices the products are the elements of the Kronecker products of the matrices U j with j I. In the qualitative case, the relevant properties of a fraction F D are mainly combinatorial, e.g. orthogonality of factors means that all level combinations appear equally often. We shall show in Section 5 that combinatorial and geometric properties are actually connected in most cases. Moreover, as usually there is no notion of increasing complexity associated to the sequence u ij for increasing i, models are best specified by a list of interaction spaces, i.e. by including in the linear model all elements of the basis of the given interaction spaces. In others words, there is no meaningful notion of linear, quadratic etc. terms. For a discussion about confounding in this framework see Galetto et al. (2003). In some cases, it is of interest to code qualitative factors with numbers, especially when the levels are ordered. Classical examples of numerical coding with rational numbers a ij Q are: (1) a ij = i, or (2) a ij = i 1, or (3) a ij = (2i n j 1)/2 for n j odd and a ij = 2i n j 1 for n j even, see (Raktoe et al., 1981, Tab. 4.1). Of special importance is the second case, where the coding takes value in the additive group Z nj, i.e. integers modulus n j. In 3

4 fact, in such a coding we can define the important notion of regular fraction. The third coding is the result of the orthogonalization of the linear term in the second coding with respect to the constant term. The coding 1, +1 for binary factors has a further property in that it takes value in a multiplicative group. This fact was widely used in Fontana et al. (2000), Ye (2003) and Tang and Deng (1999). In the present paper, we take an approach aimed to parallel our theory for binary factors with coding 1, +1. We code the n levels of a factor by the complex solutions of the equation ζ n = 1: ω k = exp (i 2π ) n k, k = 0,...,n 1. (1) We denote by Ω n such a factor with n levels, Ω n = {ω 0,...,ω n 1 }. With such a coding, a complex orthonormal basis of the responses on the full factorial design is formed by all the monomials and a real basis for real responses can be derived from it, see Section 3.1. As a basic reference to the algebra of the complex field C and of the n-th complex roots of the unity we refer to Lang (1965); some useful facts are collected in Section 8 below. As α = β mod n implies ωk α = ω β k, it is useful to introduce the residue class ring Z n and the notation [k] n for the residue of k mod n. For integer α, we have (ω k ) α = ω [αk]n. The mapping Z n Ω n C k ω k (2) is a group isomorphism of the additive group of Z n on the multiplicative group Ω n C. In other words, ω h ω k = ω [h+k]n. We drop the sub-n notation when there is no ambiguity. We denote by: #D: the number of points of the full factorial design, #D = m j=1 n j. L: the full factorial design with integer coding {0,...,n j 1}, j = 1,...,m, and D the full factorial design with complex coding: L = Z n1 Z nm and D = D 1 D j D m with D j = Ω nj According to the map (2), L is both the integer coded design and the exponent set of the complex coded design; α, β,... : the elements of L: L = {α = (α 1,...,α m ) : α j = 0,...,n j 1,j = 1,...,m} 4

5 [α]: the m-tuple ( [α 1 ] n1,...,[α m ] nm ). The interest of the complex coding basically depends on the properties of the harmonic analysis of the finite group L as summarized below. For each ω D the mapping γ : L C α ω α = m j=1 ω α j j is called a character of the Abelian group L, that is: γ(α + β) = γ(α) γ(β) and γ(α) = 1. The basic facts are the discrete Fourier transform formulæ, see Rudin (1962): ˆf(ω) = f(α) ω α and f(α) = ω D ˆf(ω) ω α (3) where f is a generic complex valued response on the design L and ˆf is its Fourier transform. 3 Responses on the design In this section we discuss the responses on the design and the linear models. According to the generalization of the algebraic approach of Fontana et al. (2000), the design D is identified as the zero-set of the system of polynomial equations ζ n j j 1 = 0, j = 1,...,m. A complex response f on the design D is a C-valued function defined on D. It can be considered as the restriction to D of a complex polynomial. We denote by: X i ; the i-th component function, mapping a design point into its i-th component: X i : D (ζ 1,...,ζ m ) ζ i. The function X i is called simple term or, by abuse of terminology, factor. X α, with α = (α 1,...,α m ) L: the interaction term X α 1 1 Xm αm, i.e. the function X α : D (ζ 1,...,ζ m ) ζ α 1 1 ζ αm m, α L. 5

6 The function X α is a special response that we call monomial response or interaction term, in analogy with current terminology. In the following, we shall use the word term to indicate either a simple term or an interaction term. We say term X α has order (or order of interaction) k if k factors are involved, i.e. if the m-tuple α has k non-null entries. If f is a response on D then its mean value on D, denoted by E D (f), is: E D (f) = 1 #D f(ζ). ζ D We say that a response f is centered if E D (f) = 0. Two responses f and g are orthogonal on D if E D (f g) = 0. Note that the set of all responses is a complex Hilbert space with Hermitian product f g = E D (f g). Two basic properties connect the algebra with the Hilbert structure, namely (1) X α X β = X [α β] ; (2) E D (X 0 ) = 1, and E D (X α ) = 0 for α 0, see Section 8 Item (3). 3.1 Bases for the space of responses on the design The set of functions {X α, α L} is an orthonormal basis of the complex responses on the design D. In fact, from properties (1) and (2) above it follows: Moreover, #L = #D. E D (X α X β ) = E D (X [α β] 1 if α = β ) = 0 if α β Then, each response f can be represented as an unique C-linear combination of constant, simple and interaction terms: f = θ α X α, θ α C (4) where the coefficients are uniquely defined by: θ α = E D ( fx α). (5) 6

7 In fact f(ζ)x α (ζ) = β X ζ D ζ D β Lθ β X α (ζ) = θ β X β (ζ)x α (ζ) = #D θ α. β L ζ D We observe that a function is centered on D if and only if θ 0 = 0. Notice that the discrete Fourier transform formulæ (3) supply an equivalent interpretation of the Equations (4) and (5). As θ α = E D ( fx α ), the conjugate of the response f has representation: f(ζ) = θ α X α (ζ) = θ [ α] X α (ζ). A response f is real valued if and only if θ α = θ [ α], α L. In usual applications, the roots of the unity Ω n will be taken as a coding because of the mathematical convenience we are going to show, but we will be interested in the real valued responses, e.g. measurements, on the design points. Note that both the real vector space R(D) and the complex vector space C(D) of the responses on the design D have a real basis, see (Kobilinsky, 1990, Prop. 3.1). In particular, a special real basis common to both spaces is described in the following Proposition. Proposition 1 For each factor X j with n j levels, an orthogonal basis of the space of real functions consists of the constant 1 together with the n j 1 vectors defined by: 1 2 ( X k j + X k j ) = R(X k j ), 1 k n j /2 1 2 i ( X k j X k j ) = I(X k j ), 1 k < n j /2 (6) The full basis of R(D) is obtained via Kronecker product of each factor bases. Proof. Note first the range of the k s in the equation above. If n j = 2a, then n j /2 = a and there are a + (a 1) = n j 1 vectors. If n j = 2a + 1, then n j /2 = a+1/2, and there are a+a = n j 1 vectors. The limits on the ranges are due to the symmetric relations R ( X k) = R ( X n j k ) and I ( X k) = I ( X n j k ). 7

8 For each real function defined on the factor X j, we have f = f, then: n j 1 k=0 θ k X k j = n j 1 k=0 θ k X k j. If we split the real and imaginary part of each coefficient as θ k = a k + i b k, then θ k = a k i b k and each real function f can be written as f = a 0 + n j 1 a k k=1 = a k n j /2 ( X k j + X k j ) 2 α k R(X k j ) + + i n j 1 b k k=1 1 k<n j /2 ( X k j X k j ) 2 β k I(X k j ) with α k = a n a n k, k < n j /2, α k = a k for k = n j /2 (even case), β k = b k + b n k, k < n j /2. As the number of vectors in the generic linear combination of Equation(7), equals the dimension of the space, then they form a basis. Now we check the orthogonality of the elements of the basis. First, all the vectors are orthogonal to the constant. Then, let h and k be distinct integer numbers such that 1 k < h < n j /2. I(X h ) I(X k ) = = 1 2 = 1 2 ( X h j Xj n h 2 ( X h+k j ) ( X k j Xj n k ) 2 + X n h k j 2 Xh k j ( R(X h+k j ) R(Xj h k ) ) Its mean value is zero because neither h + k nor h k are 0. + Xj n h+k ) 2 (7) The cases I(X h ) vs R(X k ) and R(X h ) vs R(X k ) are similar. The elements of the basis of R(D) in Proposition 6, as functions of the integer lattice coding s = 0, 1, 2,...,n j 1, consist of the usual trigonometric functions, cfr. e.g. Riccomagno et al. (1997) and Bates et al. (1998). What we are discussing in this paper are in a sense the algebraic properties of Fourier regression. 4 Fractions A fraction F is a subset of the design, F D. We have two ways of describing in an algebraic way a fraction, namely the generating equations and the indicator polynomial functions. 8

9 4.1 Generating equations All fractions can be obtained by adding to the design equations X n j j 1 = 0, for j = 1,...,m, further polynomial equations, called generating equations, to restrict the number of solutions. For example, we consider a classical III regular fraction, e.g. (Wu and Hamada, 2000, Table 5A.1), coded with complex number according to the map (2): X 1 X 2 X 3 X ω 1 ω 1 ω 1 1 ω 2 ω 2 ω 2 ω 1 1 ω 1 ω 2 ω 1 ω 1 ω 2 1 ω 1 ω 2 1 ω 1 ω 2 1 ω 2 ω 1 ω 2 ω 1 1 ω 2 ω 2 ω 2 ω 1 1 It is defined by X 3 j 1 = 0, j = 1,...,4 together with the generating equations X 1 X 2 X 2 3 = 1 and X 1 X 2 2X 4 = 1. Such a representation of the fraction is classically termed multiplicative notation. In our approach, it is not question of notation or formalism, but the equations are actually defined on the complex field C. As the recoding mapping (2) from the field Z 3 into the field C is an homomorphism of the additive group on the multiplicative group, then it maps generating equations which are additive in Z 3 (of the form A + B + 2C = 0 mod 3 and A + 2B + D = 0 mod 3) into binomial equations in C. In this case, and in all the regular cases, the generating equations are binomial (polynomial with two terms). In the following, we consider general subsets of the full factorial design and, as a consequence, no special form for the generating equations is assumed. 9

10 4.2 Responses on the fraction, indicator and counting functions The indicator function F of a fraction F is a particular response defined on D such that 1 if ζ F F(ζ) = 0 if ζ D \ F. A polynomial function F is an indicator function of some fraction F if and only if F 2 F = 0 on D. Using the monomial basis, it is represented as F = b α X α and b α = b [ α] because F is real valued. If F is the indicator function of the fraction F, then F 1 = 0 is a generating equation of the same fraction. If f is a response on D then its mean value on F, denoted by E F (f), is defined as E F (f) = 1 f(ζ) = 1 F(ζ)f(ζ) = #D #F #F #F E D(Ff) ζ F ζ D where #F the number of runs (or points) of the fraction. Proposition 2 The coefficients b α of the indicator function of F satisfy the following properties: (1) b α = 1 #D ζ F (2) The coefficients b α are related according to: X α (ζ) and in particular b 0 = #F #D. (8) b α = β Lb β b [α β]. (9) (3) If F and F are complementary fractions and b α and b α are the coefficients of the respective indicator functions, then b 0 = 1 b 0 and b α = b α Proof. (1) We have: X α (ζ) = F X α (ζ) = b β X β (ζ)x α (ζ) ζ F ζ D ζ D β L = ζ D b α = #D b α. 10

11 (2) It follows from the relation F = F 2. In fact: b α X α = α β = α b β X β b γ X γ = b β b γ X [β+γ] = γ β,γ b β b γ X α = b β b [α β] X α. [β+γ]=α α β (3) It follows from F = 1 F. As suggested by Ye (2003) for binary case, the idea of indicator function can be generalized to fractions with replicates. We adopt the extension, but we prefer to maintain the original indicator function too, so we introduce the new name of counting function. A fraction with replicates, that we denote by F rep, can be considered a multi-subset of a design D, or an array with repeated rows. Definition 1 (Counting function) The counting function R of a fraction with replicates F rep is a response defined on the design such that for each ζ D, R(ζ) equals the number of appearances of ζ in the fraction. We denote by c α the coefficients of the representation of R as: R(ζ) = c α X α (ζ) ζ D. A fraction with replicates is fully described by its counting function. The mean value of a response f on F rep, denoted by E Frep, is: E Frep (f) = 1 f(ζ) = #D E D (R f) #F rep ζ F rep #F rep where #F rep is the total number of treatments of the fraction, #F rep = ζ D R(ζ). Proposition 3 The coefficients c α of the counting function of a fraction F rep are: c α = 1 X #D α (ζ) and in particular c = #F ζ F rep #D. (10) Proof. Same proof as in Proposition 2, Item (1). Note that Items (2) and (3) of Proposition 2 do not apply as such to counting functions. 11

12 4.3 Orthogonal responses on a fraction In this section, we discuss the general case with replicates. As in the full design case, we say that a response f is centered on a fraction F if E F (f) = 0 and we say that two responses f and g are orthogonal on F if E F (f g) = 0, i.e. the response f g is centered. Note that the term orthogonal refers to vector orthogonality with respect to an Hermitian product. Unfortunately, there is a clash with the standard use in orthogonal array literature where orthogonality of factors means that all level combinations appear equally often, e.g. Hedayat et al. (1999). In this case, we will talk of orthogonal factors, while in the vector case of orthogonal responses or terms. Vector orthogonality is affected by the coding of the levels, while factor orthogonality is not. This section and the next one are devoted to discuss how the choice of complex coding makes the two orthogonalities essentially equivalent. Proposition 4 Let R = c α X α be the counting function of a fraction F. (1) The term X α is centered on F if and only if c α = c [ α] = 0. (2) The terms X α and X β are orthogonal on F if and only if c [α β] = c [β α] = 0; (3) If X α is centered then, for each β and γ such that α = [β γ] or α = [γ β], then X β is orthogonal to X γ. (4) A fraction F is self-conjugate, that is R(ζ) = R(ζ) for any ζ D, if and only if the coefficients c α are real for all α L. Proof. 1-3 The first three Items follow easily from Proposition We have: R(ζ) = c α X α (ζ) = c [ α] X [ α] (ζ) = c α X [ α] (ζ) R(ζ) = c α X α (ζ) = c α X [ α] (ζ). Then R(ζ) = R(ζ) if and only if c α = c α. Note that the argument applies to all real valued responses. An important property of the centered responses follows from the structure of the roots of the unity as cyclical group. This will connect the combinatorial properties with the coefficients c α s through the following two basic properties which hold true on the full design D. 12

13 P-1 Let X i be a simple term with level set Ω n. Then the term Xi r takes all the values of Ω s, with s = n/gcd(r,n), equally often. P-2 Let X α = X α j 1 j 1 X α j k j k be an interaction term of order k and let X α j i j i take values in Ω sji. Then X α takes values in Ω s, with s = lcm{s j1,...,s jk }. Let X α be a term with level set Ω s on the design D. Let r k be the number of times X α takes the value ω k on F, k = 0,...,s 1. The generating polynomial of the sequence (r k ) k=0,...,s 1 is P(ζ): P(ζ) = s 1 k=0 r k ζ k. See Lang (1965), and in Appendix a review of properties of generating and cyclotomic polynomials. Proposition 5 Let X α be a term with level set Ω s on the full design D. (1) X α is centered on F, E F (X α ) = 1 #F s 1 k=0 r k ω k = 0 if and only if the generating polynomial has the form P(ζ) = Φ s (ζ)ψ(ζ) where Φ s is the cyclotomic polynomial of the s-roots of the unity and Ψ is a suitable polynomial with integer coefficients. (2) Let s be prime. Then the term X α is centered if and only if its s levels appear equally often: r 0 = = r s 1 = r (3) Let s = p h 1 1 p h d d, with p i prime, for i = 1,...,d. The term X α is centered on F if and only if the following equivalent conditions are satisfied. (a) The remainder H(ζ) = P(ζ) mod Φ s (ζ), whose coefficients are integer combination of r k, k = 0,...,s 1, is identically zero. (b) The polynomial of degree s P(ζ) = P(ζ) d s Φ d (ζ) mod (ζ s 1), whose coefficients are integer combination of the replicates r k, k = 0,...,s 1, is identically zero. The indices of the product are the d s that divide s. 13

14 (4) If the vector of replicates is a combination with positive weights of indicators of subgroups or laterals of subgroups of Ω s, then X α is centered. Proof. (1) As ω k = ω1, k the assumption k r k ω k = 0 is equivalent to P(ω 1 ) = 0. From Section 8, items 4 and 5, we know that this implies P(ω) = 0 for all primitive s-roots of the unity, that is P(ζ) is divisible by the cyclotomic polynomial Φ s. (2) If s is a prime number, then the cyclotomic polynomial is Φ s (ζ) = s 1 k=0 ζk. The generating polynomial P(ζ) is divided by the cyclotomic polynomial, P(ζ) and Φ s (ζ) have the same degree, then r s 1 > 0 and P(ζ) = r s 1 Φ(ζ), so that r 0 = = r s 1. (3) The divisibility shown in Item 1 above is equivalent to the condition of null remainder. Such remainder is easily computed as the reduction of the the generating polynomial P(ζ) mod Φ s (ζ). By the same condition and the Equation (15), we obtain that P(ζ) is divisible by ζ s 1, then also it equals 0 mod ζ s 1. (4) If Ω p is a prime subgroup of Ω s, then ω Ω p ω = 0. Now assume the replicates be 1 on a primitive subgroup Ω pi. Then ω Ω pi ω = 0 by Equation in Item (3). The same in case of the laterals and sum of such cases. See also the remark below. Remark The set of non-negative integer vectors (r 0,...,r s 1 ) such that 1 s 1 #F k=0 r kω k = 0 as in Item (1) of Proposition 5 is actually a semi-group. In general, such a set has an unique minimal set of generators. This will be illustrated in an example below. The general discussion of semi-group bases is outside the scope of the present paper. The existence and algorithmic construction of the unique finite set of generators is discussed e.g. in Sturmfels (1996). Example We consider the case s = 6. This situation occurs in case of mixed factorial designs with both three level factors and binary factors. In this case, the cyclotomic polynomial is Φ 6 (ζ) = ζ 2 ζ + 1 whose roots are ω 1 and ω 5. The 14

15 remainder is 5 H(ζ) = r k ζ k mod Φ 6 (ζ) k=0 = r 0 + r 1 ζ + r 2 ζ 2 + r 3 ζ 3 + r 4 ζ 4 + r 5 ζ 5 mod ( ζ 2 ζ + 1 ) = (r 1 + r 2 r 4 r 5 )ζ + (r 0 r 2 r 3 + r 5 ) The condition H(ζ) = 0 implies the following relations about the numbers of replicates: r 0 + r 1 = r 3 + r 4, r 1 + r 2 = r 4 + r 5, r 2 + r 3 = r 0 + r 5, where the first one follows summing the second with the third one. Equivalently: r 0 r 3 = r 4 r 1 = r 2 r 5. (11) We consider the replicates corresponding to the sub-group {ω 0,ω 2,ω 4 } and we denote by m 1 the min{r 0,r 2,r 4 }. Then we consider the replicates corresponding to the lateral of the previous sub-group {ω 1,ω 3,ω 5 } and we denote by m 2 the min{r 1,r 3,r 5 }. We consider the new vector of the replicates: r = (r 0,r 1,r 2,r 3,r 4,r 5) = (r 0 m 1,r 1 m 2,r 2 m 1,r 3 m 2,r 4 m 1,r 5 m 2 ) = r m 1 (1, 0, 1, 0, 1, 0) m 2 (0, 1, 0, 1, 0, 1) The vector r satisfies Equation (11). As at least one among r 0, r 2 and r 4 is zero, then the common value in Equations (11) is zero or negative. Moreover, as at least one among r 1, r 3 and r 5 is zero, then the common value in Equations (11) is zero or positive. Then the common value is zero and r 0 = r 3, r 1 = r 4, r 2 = r 5 and r = r 0(1, 0, 0, 1, 0, 0) + r 1(0, 1, 0, 0, 1, 0) + r 2(0, 0, 1, 0, 0, 1) Then a term is centered if the vector of the replicates of ω 0,...,ω 5 is of the form: (r 0,...,r 5 ) = a(1, 0, 0, 1, 0, 0) + b(0, 1, 0, 0, 1, 0) + c(0, 0, 1, 0, 0, 1) + d(1, 0, 1, 0, 1, 0) + e(0, 1, 0, 1, 0, 1) with a,b,c,d,e non negative integers. There are 5 generating integer vectors of the replicate vector. Notice that if the number of levels of X α is not prime, then E F (X α ) = 0 does not imply E F (X rα ) = 0. In the previous six-level example, if X α is centered, then the vector of replicates of X 2α is of the form (2a + d + e, 0, 2b + d + e, 0, 2c + d + e, 0) and X 2α is centered only if a = b = c. 15

16 5 Orthogonal arrays In this section we discuss the relations between the coefficients c α, α L, of the counting function and the property of being an orthogonal array. Let OA(n,s p 1 1,...,s pm m,t) be a mixed level orthogonal array with n rows and m columns, m = p p m, in which p 1 columns have s 1 symbols,..., p k columns have s m symbols, and with strength t, as defined e.g. in (Wu and Hamada, 2000, p. 260). Strength t means that, for any t columns of the matrix design, all possible combinations of symbols appear equally often in the matrix. Definition 2 Let I be a non-empty subset of {1,...,m}, and let J be its complement set, J = I c. Let D I and D J be the corresponding full factorial designs over the I-factors and the J-factors, so that D = D I D J. Let F be a fraction of D and let F I and F J be its projections. (1) A fraction F fully projects on the I-factors if F I = s D I, that is the projection is a full factorial design where each point is replicate s times. (2) A complex coded fraction F is a mixed orthogonal array if it fully projects on all the I-factors with #I = t. Using the notations of Definition 2, for each point ζ of F we consider the decomposition ζ = (ζ I,ζ J ) and we denote by R I the counting function restricted to the I-factors of a fraction, i.e. R I (ζ I ) is the number of points in F whose projection on the I-factors is ζ I. We denote by L I the sub-set of the exponents restricted to the I-factors and by α I an element of L I : L I = {α I = (α 1,...,α j,...,α m ), α j = 0 if j J}. Then, for each α L and ζ D: α = α I + α J and X α (ζ) = X α I (ζ I )X α J (ζ J ). We denote by #D I and #D J the cardinalities of the projected designs. Proposition 6 (1) The number of replicates of the points of a fraction projected on the I- factors is: R I (ζ I ) = #D J α I c αi X α I (ζ I ). (2) A fraction fully projects on the I-factors if and only if R I (ζ I ) = #D J c 0 = #F #D I for all ζ I 16

17 and all the coefficients of the counting function involving only the I- factors are 0, that is c αi = 0 with α I L I, α I (0, 0,...,0) and especially the levels of a factor X i appear equally often on F if and only if all the power X r i are centered, for r = 1,...,n j 1. (3) If there exists a subset J of {1,...,m} such that the J-factors appear in all the non null elements of the counting function, then the fraction fully projects on the I-factors, with I = J c. (4) A fraction is an orthogonal array of strength t, if and only if all the coefficients of the counting function up to the order t are zero: Proof. (1) We have: c α = 0 α of order up to t, α (0, 0,...,0). R I (ζ I ) = R(ζ I,ζ J ) = c α X α (ζ I,ζ J ) ζ J D J ζ J D J = c α X α I (ζ I )X α J (ζ J ) ζ J D J = α I L I c αi X α I (ζ I ) The thesis follows from: ζ J D J X α J (ζ) + α L I c α X α I (ζ) X α J 0 if α J (0, 0,...,0) (ζ J ) = ζ J D J #D J if α J = (0, 0,...,0) ζ J D J X α J (ζ). (2) The number of replicates of the points of the fraction projected on the I-factors, R I (ζ I ) = #D J α I c αi X α I (ζ I ), is a polynomial and it is a constant if all the coefficients c αi, with α I (0, 0,...,0), are zero. (3) The condition imply that the c αi s are zero, if α I (0, 0,...,0), and the thesis follows from the previous item. (4) It follows from the previous items and the definition. Proposition 7 Let I be a subset of indices of cardinality t, I = {i 1,...,i t }. The following statements are equivalent: (1) A fraction fully project on the I-factors. (2) c αi = 0 for each i I and 0 < α i n i 1 and for f 1,...,f t functions: E F (f 1 (X i1 ) f t (X it )) = E F (f 1 (X i1 )) E F (f t (X it )). 17

18 (3) c αi = 0 for each i I and 0 < α i n i 1 and R I (ζ I ) #F = R i (ζ i ) i I #F. Proof. Each function of the simple terms can be uniquely written as: f k (X i ) = a ik + n i 1 α i =1 a αi X α i i. Then f 1 (X i1 ) f t (X it ) = a i1 a i2 a it + c α1 α t X α 1 1 X αt α 1 α t t. If c αi = 0, then E F (X α I ) = 0 and E F (f 1 (X i1 ) f t (X it )) = a i1 a i2 a it = E F (f 1 (X i1 )) E F (f t (X it )). Vice-versa, if f k (X ik ) = X α i i k, then item (2) implies E F (X α I ) = E F ( X α 1 i 1 X αt i t ) = EF ( X α 1 i 1 ) EF ( X α t i t ) = 0 because α i = 0 for at least one i. Remarks (1) If a fraction fully projects on the I-factors, then its cardinality must be equal or a multiple of the cardinality of D I. (2) If the numbers of levels are all prime, then the condition c αi = 0 for each i I and 0 < α i n i 1 in the Items (2) and (3) of previous Proposition, simplify to E F (X i ) = 0, by Proposition (5.2). 6 Regular fractions We first review the theory of regular fractions from the view point of the present paper. Various definitions of regular fraction appear in the literature, e.g. in the books (Raktoe et al., 1981, p. 123), (Collombier, 1996, p. 125), (Kobilinsky, 1997, p. 70), (Dey and Mukerjee, 1999, p. 164), (Wu and Hamada, 2000, p. 305). To our knowledge, all definitions are known to be equivalent if all factors have the same prime number of levels, n = p. The definition based on Galois Field computations is given for n = p s power of a prime number. Most definitions assume symmetric factorial designs, i.e. all factors have the same number of levels. 18

19 Regular fraction designs are usually considered for qualitative factors, then the level coding is arbitrary. The integer coding, the GF(p s ) coding, the roots of the unity coding, as introduced by Bailey (1982) and used extensively in this paper, can all be used. Each of the coding is associated with specific ways to characterize a fraction, and more important for us, a specific basis for the contrasts. One of the possible definitions of regular fraction refers to the property of non-existence of partial confounding, and this property has to be associated to a specific basis of contrasts, as clearly pointed out in Wu and Hamada (2000). Before proceeding with our approach we review the relevant definitions following (Dey and Mukerjee, 1999, p. 3). For each treatment effect τ : F R (F with integer coding) the value of the linear parametric function l is l(τ) = l j τ j j F Without restrictions l is uniquely extendable by 0 outside F to a vector ˆl whose elements ˆl j are defined for each j = (j 1,...,j m ) L. The parametric function ˆl has an unique polynomial representation ˆlj = θ α X α (ω j1,...,ω jn ) Notice that, given a polynomial representation P of l on F, then ˆl has polynomial representation PF, where F is the indicator function of the fraction. We shall use ˆl to denote both the original parametric function and its polynomial representation, as each of them is obtained from the other by recoding. The following statements are easily proved: (1) l is a treatment contrast if and only if θ 0 = 0 (2) l belongs to the factorial effects J, J = {j 1,...,j g } {1,...,m} if θ α = 0 if the support of α is not J. In our approach we use polynomial algebra with complex coefficients, the roots of the unity coding, the idea of indicator polynomial function, and we make no assumption about the number of levels. In the specific coding we use the indicator polynomial is actually a discrete Fourier transform and the monomial effects are contrasts and orthogonal in the full factorial design. We refer to such a basis to state the no-partial confounding property. In Proposition 8 below we are able to show the equivalence of various definitions, so we could have taken any of them as a basic definition. We have chosen to start with the one we think more traditional, i.e. defining equations in multiplicative form. The results presented below are a generalization of our previous study of the binary case in Fontana et al. (2000). A first draft of the ideas was presented in the GROSTAT V 2003 Workshop and in the technical report 19

20 Pistone and Rogantin (2003). Some of the results of the Proposition 8 were obtained independently by Ye (2004). Careful examination of the proof shows that the symmetry assumption is actually not used. However, we have chosen not to depart from such an usual assumption. We consider a fraction without replicates where all factors have n levels. We recall that Ω n is the set of the n-th roots of the unity, Ω n = {ω 0,...,ω n 1 }. Let L be a subset of exponents, L L = (Z n ) m, containing (0,...,0) and let l be its cardinality (l > 0). Let e be a map from L to Ω n, e : L Ω n. Definition 3 A fraction F is regular if (1) L is a sub-group of L, (2) e is an homomorphism, e([α + β]) = e(α) e(β) for each α, β L, (3) the equations X α = e(α), α L (12) define the fraction F, i.e. they are a set of generating equations, according to Section 4.1. The Equations (12) are called the defining equations of F. If H is a minimal generator of the group L, then the Equations X α = e(α) with α H L are called minimal generating equations. Notice that we consider the general case where e(α) can be different from 1. Notice also that form items (1) and (2) it follows that a necessary condition is the e(α) s must belong to the subgroup spanned by the values X α. For example for n = 6 an equation like X 3 1X 3 2 = ω 2 can not be a defining equation. In the fraction of Section 4.1, we have: H = {(1, 1, 2, 0), (1, 2, 0, 1)} and e(1, 1, 2, 0) = e(1, 2, 0, 1) = ω 0 = 1. The set L is: {(0, 0, 0, 0), (0, 1, 1, 2), (0, 2, 2, 1), (1, 1, 2, 0), (2, 2, 1, 0), (1, 2, 0, 1), (2, 1, 0, 2), (1, 0, 1, 1), (2, 0, 2, 2)}. Examples of regular fractions will be discussed in some detail in Section 7. Proposition 8 Let F be a fraction. The following statements are equivalent: (1) The fraction F is regular according to definition 3. (2) The indicator function of the fraction has the form F(ζ) = 1 l e(α) X α (ζ) ζ D. where L is a given subset of L and e : L Ω n is a given mapping. (3) For each α,β L the parametric functions represented on F by the terms X α and X β are either orthogonal or totally confounded. 20

21 Proof. First we prove the implication (1) (2). Let F be a regular fraction and let X α = e(α) be its defining equations with α L, L a sub-group of L and e a homomorphism. If and only if ζ F: 0 = X α (ζ) e(α) 2 = (X α (ζ) e(α)) (X α (ζ) e(α)) = ( X α (ζ)x α (ζ) + e(α)e(α) e(α)x α (ζ) e(α)x α (ζ) ) ( = 2 l e(α)x α (ζ) e(α) X α (ζ) = 2 l e(α) X α (ζ) ) then 1 l e(α) X α (ζ) 1 = 0 if and only if ζ F. The function F = 1 l e(α) X α is an indicator function, because we can show that F = F 2 on D. In fact, L is a sub-group of L and e is an homomorphism; then: F 2 = 1 e(α) e(β) X [α+β] = 1 e([α + β]) X [α+β] = l 2 l 2 β L = 1 l e(γ) X γ = F l 2 γ L β L Then F is the indicator function of F, and b α = e(α) l, for all α L. We now prove (2) (1). Note that an indicator function is real valued, then F = F. 1 X α (ζ) e(α) 2 0 on F = 2 F(ζ) F(ζ) = 2 2F(ζ) = l 2 on D \ F. Then the equations X α = e(α) with α L define the fraction F as the generating equations of a regular fraction. L is easy to seen to be a group. In fact if γ = [α + β] / L then there exists one ζ such that X γ (ζ) = X α (ζ)x β (ζ) = e(α)e(β) Ω n and the value e(α)e(β) depends only on γ. Applying the previous argument and the uniqueness of the polynomial representation of the indicator function we get a contradiction. We prove (1) and (2) (3). The non-zero coefficients of the indicator function of F are of the form b α = e(α)/l.. 21

22 We consider two terms X α and X β with α,β L. If [α β] / L then X α and X β are orthogonal on F because the coefficient b [α β] of the indicator function equals 0. If [α β] L then X α and X β are confounded because X [α β] = e([α β]); then X α = e([α β]) X β. We prove (3) (2). Let L the set of exponents of the terms confounded with a constant: L = {α L : X α = constant = e(α), e(α) Ω n }. For each α L, b α = e(α) b 0. For each α / L, because of the assumption, X α is orthogonal to X 0, then b α = 0. Corollary 1 Let F be a regular fraction with X α = 1 for all defining equations. Then F is self-conjugate and a multiplicative subgroup of D. Proof. It follows from Prop. 4 Item 4. The following proposition extends a result presented in Fontana et al. (2000) for the binary case. Proposition 9 Let F be a fraction with indicator function F. Let us collect all the exponents α such that bα b 0 = e(α) Ω n : F(ζ) = b 0 e(α) X α (ζ) + b β X β (ζ) ζ F, L K =. β K Then L is a subgroup and the equations X α = e(α), with α L, are the defining equations of the smallest regular fraction F r containing F restricted to the factors involved in the L-exponents. Proof. The coefficients b α, α L, of the indicator function F are of the form b 0 e(α). Then, from the extremality of n-th roots of the unity, X α (ζ) = e(α) if ζ F and X α (ζ)f(ζ) = e(α)f(ζ) for each ζ D and L is a group. We denote by F r the indicator function of F r. For each ζ D we have: F(ζ)F r (ζ) = 1 l F(ζ) e(α) X α (ζ) = 1 e(α) X α (ζ) F(ζ) = l = 1 e(α) e(α) F(ζ) = 1 l l l F(ζ). The relation F(ζ)F r (ζ) = F(ζ) implies F F r. The fraction F r is minimal because we have collected all the terms confounded with a constant. 22

23 We conclude this paragraph on regular fraction by discussing how to reduce the Galois Field construction to our definition via the introduction of pseudofactors or dummy-factors. Proposition 10 Let F be generated by the following equations in [GF (p s )] m : m a ij z j = b i i = 1,...,r a ij,z j,b i GF (p s ) j=1 Let g be an irreducible polynomial such that GF (p s ) is isomorph to the field Z p [x]/ < g(x) >, g(x) = x s + s 1 i=1 g i x i. Let ω be a primitive p-th root of the unity. Under the recoding: GF (p s ) z z 0 + z 1 x + + z s 1 x s 1 (ω z 0,ω z 1,...,ω z s 1 ) (Ω p ) s F is a regular fraction with m s factors, where each of the original factors splits into s pseudo-factors coded with the p-th roots of the unity. Proof. Without loss of generality, we assume r = 1 and a single generating equation: m a j z j = b mod g (13) We have: j=1 a j z j = ( a j0 + a j1 x + + a j(s 1) x s 1) ( z j0 + z j1 x + + z j(s 1) x s 1) = = = s 1 u,v=0 s 1 a ju z jv x u x v s 1 a ju z jv u,v=0 k=0 s 1 s 1 x k s 1 z jv k=0 v=0 u=0 f uvk x k a ju f uvk. Then, denoting s 1 u=0 a ju f uvl by h jvk, the Equation (13) is equivalent to the system in Z p : m s 1 h jvk z jv = b k k = 1,...,s 1 j=1 v=0 and, under the complex recoding, m s 1 j=1 v=0 (ω z jv ) h jvl = ω b k k = 1,...,s 1. The defining equations of the complex coding fraction in the pseudo-factors are: m s 1 X α jvl jv = ω b k k = 1,...,s 1 j=1 v=0 23

24 Then F is regular. 7 Examples 7.1 A regular fraction with n = 3 We consider the classical fraction of Section 4.1. Its indicator function is: F = 1 ( 1 + X2 X 3 X 4 + X 2 9 2X3X X 1 X 2 X3 2 + X1X 2 2X 2 3 ) +X 1 X2X X1X 2 2 X4 2 + X 1 X 3 X4 2 + X1X 2 3X 2 4. We observe that the coefficients are all equal to 1. The minimum order of 9 interactions appearing in the indicator function is 3, then the fraction is an orthogonal array of strength 2. All the defining equations are of the form X α = 1, then the fraction is self conjugate. We analyze on this fraction the effects of level transformations. For instance, we permute the levels ω 0 and ω 1 of the last factor X 4 using the transformation: (ζ 1,ζ 2,ζ 3,ζ 4 ) (ζ 1,ζ 2,ζ 3,ω 1 ζ 2 4). The indicator function of the fraction image is: F = 1 ( 1 + ω1 X 2 X 3 X4 2 + ω 2 X 2 9 2X3X X 1 X 2 X3 2 + X1X 2 2X 2 3 +ω 1 X 1 X2X ω 2 X1X 2 2 X 4 + ω 2 X 1 X 3 X 4 + ω 1 X1X 2 3X4) 2 2. Then it is a regular fraction and a set of generating equations is: X 2 X 3 X4 2 = ω 2 and X 1 X 2 X 3 = Regular fractions with n = 4 We consider two factors with 4 levels and we compare the regular fractions as in Definition 3 and Proposition 8 and fractions constructed via Galois Field and pseudo-factors. (1) Fractions as in Definition 3 and Proposition 8. All the regular inequivalent fractions with generating equations of the 24

25 form X α = 1 are: X 1 X 2 ω 0 ω 0 ω 1 ω 1 ω 2 ω 2 ω 3 ω 3 X 1 X 2 ω 0 ω 0 ω 1 ω 3 ω 2 ω 2 ω 3 ω 1 X 1 X 2 ω 0 ω 0 ω 0 ω 2 ω 1 ω 1 ω 1 ω 3 ω 2 ω 0 ω 2 ω 2 ω 3 ω 1 ω 3 ω 3 X 1 X 2 ω 0 ω 0 ω 0 ω 2 ω 2 ω 1 ω 2 ω 3 X 1 X 2 ω 0 ω 0 ω 1 ω 2 ω 2 ω 0 ω 3 ω 2 Their indicator functions are respectively: 1 ( ) 1 + X1 X2 3 + X 3 1 ( 3 1X 2, 1 + X1 X 2 + X 3 3 1X2) 3 1 (, 1 + X X2) 2 1 ( 1 + X1 X2 2 + X 3 3 1X2) 2 1 ( ), 1 + X1 X2 3 + X 3 3 1X 2. We notice that the last two fractions do not fully project on both factors. (2) Using Galois Fields and pseudo-factors. The Galois Field GF(2 2 ) can be represented as the field of equivalence classes mod x 2 + x + 1 of polynomials of degree 1 whose coefficients belong to Z 2 : GF(2 2 ) z z 0 + z 1 x z 0,z 1 Z 2 We consider the regular fractions generated by a 1 z 1 + a 2 z 2 = 0, with a 1,a 2,z 1,z 2 GF(2 2 ). As in Proposition 10 this equation becomes: (a 10 + a 11 x)(z 10 + z 11 x) + (a 20 + a 21 x)(z 20 + z 21 x) = 0 mod (x 2 + x + 1) with a ij,z ij Z 2 or, equivalently: a 10 z 10 + a 11 z 11 + a 20 z 20 + a 21 z 21 = 0 mod 2 a 11 z 10 + (a 10 + a 11 )z 11 + a 21 z 20 + (a 20 + a 21 )z 21 = 0 mod 2 The generating equations of the complex coded fraction in the pseudofactors are: X a X a X a X a = 1 X a X a 10+a X a X a 20+a = 1 For non trivial choices of the a ij s coefficients, we have the following generating equations, written in polynomial-galois notation and in multiplicative notation respectively: 25

26 z 1 + z 2 = 0 and X 10 X 20 = 1, X 11 X 21 = 1 xz 1 + xz 2 = 0 and X 11 X 21 = 1, X 10 X 11 X 21 X 20 = 1 (x + 1)z 1 + (x + 1)z 2 = 0 and X 10 X 11 X 20 X 21 = 1, X 10 X 20 = 1 z 1 + xz 2 = 0 and X 10 X 21 = 1, X 11 X 20 X 21 = 1 xz 1 + (x + 1)z 2 = 0 and X 11 X 20 X 21 = 1, X 10 X 11 X 20 = 1 We observe that the first three fractions have the same defining equations in multiplicative notation and so the last two. Then there are two inequivalent fractions and their points in the two notations are: Z 1 Z x 1 + x 1 1 x x 0 0 X 10 X 11 X 20 X Z 1 Z x x x x X 10 X 11 X 20 X The first fraction corresponds to the first fraction in Item (1), but the latter is not equivalent to any fraction listed in Item (1). 7.3 A regular fraction with n = 6 We consider a 6 3 design. From property [P-2] of Section 4.3 the terms X α take values either in Ω 6 or in one of the two subgroups {1,ω 3 } and {1,ω 2,ω 4 }. Let F be a fraction whose generating equations are: X 3 1X 2 3 = ω 3 and X 4 2X 4 2X 2 3 = ω 2. In this case we have: H = {(3, 0, 3), (4, 4, 2)} and e(3, 0, 3) = ω 3, e(4, 2, 2) = ω 2. The set L is: {(0, 0, 0), (3, 0, 3), (4, 4, 2), (2, 4, 4), (1, 4, 5), (5, 2, 1)}. The full factorial design has 216 points and the fraction has 36 points, listed below in three blocks. 26

27 X 1 X 2 X 3 ω 0 ω 0 ω 1 ω 0 ω 1 ω 5 ω 0 ω 2 ω 3 ω 0 ω 3 ω 1 ω 0 ω 4 ω 5 ω 0 ω 5 ω 3 ω 1 ω 0 ω 2 ω 1 ω 1 ω 0 ω 1 ω 2 ω 4 ω 1 ω 3 ω 2 ω 1 ω 4 ω 0 ω 1 ω 5 ω 4 X 1 X 2 X 3 ω 2 ω 0 ω 3 ω 2 ω 1 ω 1 ω 2 ω 2 ω 5 ω 2 ω 3 ω 3 ω 2 ω 4 ω 1 ω 2 ω 5 ω 5 ω 3 ω 0 ω 4 ω 3 ω 1 ω 2 ω 3 ω 2 ω 0 ω 3 ω 3 ω 4 ω 3 ω 4 ω 2 ω 3 ω 5 ω 0 X 1 X 2 X 3 ω 4 ω 0 ω 5 ω 4 ω 1 ω 3 ω 4 ω 2 ω 1 ω 4 ω 3 ω 5 ω 4 ω 4 ω 3 ω 4 ω 5 ω 1 ω 5 ω 0 ω 0 ω 5 ω 1 ω 4 ω 5 ω 2 ω 2 ω 5 ω 3 ω 0 ω 5 ω 4 ω 4 ω 5 ω 5 ω 2 The indicator function is: F = 1 6 ( 1 + ω3 X 3 1X ω 4 X 4 1X 4 2X ω 2 X 2 1X 2 2X ω 1 X 1 X 4 2X ω 5 X 5 1X 2 2X 3 ) 7.4 An OA(2, 3 7, 2) We consider the following fraction of a design with 18 runs. It is taken from (Wu and Hamada, 2000, Table 7C.2) and recoded with complex levels. X 1 X 2 X 3 X 4 X 5 X 6 X 7 X ω 1 ω 1 ω 1 ω 1 ω 1 ω ω 2 ω 2 ω 2 ω 2 ω 2 ω 2 1 ω ω 1 ω 1 ω 2 ω 2 1 ω 1 ω 1 ω 1 ω 2 ω ω 1 ω 2 ω ω 1 ω 1 1 ω 2 1 ω 1 1 ω 2 ω 1 ω 2 1 ω 2 ω 1 ω 2 ω 1 1 ω ω 2 ω 2 1 ω 2 ω 1 1 ω ω 2 ω 2 ω 1 ω ω ω 2 ω 2 ω ω 2 ω 1 ω ω 2 1 ω 1 1 ω 1 ω 2 1 ω 2 ω 1 1 ω 1 ω 1 ω 2 1 ω 1 1 ω 2 1 ω 1 ω 2 1 ω 1 ω 2 ω ω 2 1 ω 2 ω 1 ω 2 1 ω 1 1 ω 2 ω 1 1 ω 2 1 ω 1 ω 2 1 ω 2 ω 2 ω 1 1 ω 1 ω

28 Here X 1 takes value in Ω 2, X i, with i = 2,...,8, and their interactions take value in Ω 3, and the interactions involving X 1 take values in Ω 6. All the 4374 terms X α of the fraction have been computed in SAS using Z 2, Z 3 and Z 6 arithmetic. Then, for each terms, the replicates of the values in the relevant Z k have been computed. We found: (1) 3303 centered responses. They are characterized by Proposition 5. Actually the replicates are of the type: (9, 9), (6, 6, 6), (3, 3, 3, 3, 3, 3) and (9, 0, 0, 9, 0, 0). We have: (a) the two-level simple term; (b) 1728 terms involving only the three level factors (14 of order 1, 84 of order 2, 198 of order 3, 422 of order 4, 564 of order 5, 342 of order 6 and 104 of order 7); (c) 1574 terms involving both the binary factor and the three level factors (14 of order 2, 66 of order 3, 188 of order 4, 398 of order 5, 492 of order 6, 324 of order 7 and 92 of order 8). (2) 9 terms with corresponding coefficients b α equal to b 0 = 18 = 3 5 ; (3) 1062 terms with corresponding coefficients different from zero and from b 0 : 450 terms involving only the three level factors (80 of order 3, 138 of order 4, 108 of order 5, 100 of order 6 and 24 of order 7) and 612 terms involving both the binary factor and the three level factors (18 of order 3, 92 of order 4, 162 of order 5, 180 of order 6, 124 of order 7 and 36 of order 8). From the previous results we can find some statistical properties of the fraction. (1) Analyzing the centered responses we observe that: (a) All the 15 simple terms are centered. (b) All the 98 interactions of order 2 (84 involving only the three level factors and 14 involving also the binary factor) are centered. This implies that both the linear terms and the quadratic terms of the three level factors are mutually orthogonal and they are orthogonal to the binary factor. The fraction is a mixed orthogonal array of strength 2. (c) The fraction fully project on the following subsets of factors: {X 1,X 2,X 3 }, {X 1,X 2,X 4 }, {X 1,X 2,X 5 }, {X 1,X 2,X 6 }, {X 1,X 3,X 6 }, {X 1,X 3,X 7 }, {X 1,X 4,X 5 }, {X 1,X 4,X 8 }, {X 1,X 5,X 8 }, {X 1,X 6,X 7 }, {X 1,X 6,X 8 }. In fact all the terms of order 1, 2 and 3 involving the same set of factors are centered. 28

29 (d) A basis of the responses on the fraction is: 1, X 1, X 2,X 2 2, X 3,X 2 3, X 4,X 2 4, X 5,X 2 5, X 6,X 2 6, X 7,X 2 7, X 8,X 2 8, X 1 X 2,X 1 X 2 2. (14) In fact this is a hierarchical orthogonal system of 18 terms because the terms involving all the couple of elements of the basis are centered. Other bases, for instance, are those formed by the constant, by the 15 terms of order 1 and by the two interactions involving the binary factor and one of the three level factors: X 1 X j,x 1 X 2 j, j = 3,...,8. The previous basis (14) has a remarkable special property, see Galetto et al. (2003) in a different coding: all the spaces of the simple terms H 1,H 2,...,H 8 and the interaction space H 12 are controlled, that is the basis of the fraction contains a basis of each subspaces. The recodings Ω 2 Z 2 and Ω 3 Z 3 are respectively, see Equation (16): k = 1 2 ζ k = 1 3 ζ2 (ω 2 1) ζ(ω 1 1) + 1. All subspaces H 1,H 2,...,H 8 and H 12 are mapped in the corresponding spaces in the new coding. In particular, this implies that (14) is a basis in both codings. (2) The minimal regular fraction containing our fraction restricted to the three-level factors has the following defining relations: X 2 2X 2 4X 5 = 1, X 2 X 4 X 2 5 = 1, X 2 X 3 X 2 4X 6 X 7 X 8 = 1, X 2 2X 2 3X 4 X 2 6X 2 7X 2 8 = 1, X 2 2X 3 X 2 5X 6 X 7 X 8 = 1, X 2 X 2 3X 5 X 2 6X 2 7X 2 8 = 1, X 3 X 4 X 5 X 6 X 7 X 8 = 1, X 2 3X 2 4X 2 5X 2 6X 2 7X 2 8 = 1. (3) The not centered terms have the following different configuration of level replicates and coefficients of the indicator function (N is the number of the points of the full factorial design, N = 4374). (a) With levels in Ω 6 : (i) (6,6,0,3,3,0): 54 terms with b α coefficients 3/N (ω 0 + ω 1 ) (6,0,3,3,0,6): 54 terms with b α coefficients 3/N (ω 0 + ω 5 ) (3,0,6,6,0,3): 27 terms with b α coefficients 3/N (ω 2 + ω 3 ) (3,3,0,6,6,0): 27 terms with b α coefficients 3/N (ω 4 + ω 3 ) 29

30 Fig. 1. OA(2, 3 7, 2): replicates for non centered terms with levels in Ω 6 (ii) (3,3,6,0,0,6): 27 terms with b α coefficients 3/N (ω 0 + ω 1 ) (3,6,0,0,6,3): 27 terms with b α coefficients 3/N (ω 0 + ω 5 ) (6,0,0,6,3,3): 54 terms with b α coefficients 3/N (ω 4 + ω 5 ) (6,3,3,6,0,0): 54 terms with b α coefficients 3/N (ω 2 + ω 1 ) (iii) (6,6,3,0,0,3): 54 terms with b α coefficients 6/N (ω 0 + ω 1 ) (6,3,0,0,3,6): 54 terms with b α coefficients 6/N (ω 0 + ω 5 ) (3,0,0,3,6,6): 27 terms with b α coefficients 6/N (ω 4 + ω 5 ) (3,6,6,3,0,0): 27 terms with b α coefficients 6/N (ω 2 + ω 1 ) (iv) (9,3,0,3,0,3): 54 terms with b α coefficients 9/N ω 0 (3,0,3,9,3,0): 18 terms with b α coefficients 9/N ω 3 (3,0,3,0,3,9): 18 terms with b α coefficients 9/N ω 5 (3,9,3,0,3,0): 18 terms with b α coefficients 9/N ω 1 (v) (9,9,0,0,0,0): 9 terms with b α coefficients 9/N (ω 0 + ω 1 ) (9,0,0,0,0,9): 9 terms with b α coefficients 9/N (ω 0 + ω 5 ) The configurations of the previous five kinds of replicates are in Figure 1. (b) With levels in Ω 3 : (i) (12,3,3): 180 terms with b α coefficients 9/N ω 0 (3,12,3): 45 terms with b α coefficients 9/N ω 1 (3,3,12): 45 terms with b α coefficients 9/N ω 2 (ii) (9,0,9): 90 terms with b α coefficients 9/N (ω 0 + ω 2 ) (9,9,0): 90 terms with b α coefficients 9/N (ω 0 + ω 1 ) The configurations of the previous two kinds of replicates are in Figure 2. 30