2-level fractional factorial designs which are the union of non trivial regular designs

2-LEVEL FRA CTIONAL F A CTORIAL DESIGNS WHICH ARE THE UNION OF NON TRIVIAL REGULAR DESIGNS ROBER TO FONT ANA AND GIOV A NNI PISTONE Abstra ct. Every fraction is a un ion of p oints, which are trivial reg- ular fractions. T o characteri ze non trivial decomp osition, w e derive a condition for the inclusion of a regular fraction as follo ws. Let F = P α b α X α b e the indicator p olynomial of a generic fraction, see F on tana et al, JSPI 2000, 149-172. R egular fractions are characterized b y R = 1 l P α ∈L e α X α , where α 7→ e α is an group homeomorphism from L ⊂ Z d 2 into {− 1 , +1 } . The regular R is a subset of th e fraction F if F R = R , whic h in turn is eq u iv alen t to P t F ( t ) R ( t ) = P t R ( t ). If H = { α 1 . . . α k } is a generating set of L , and R = 1 2 k (1 + e 1 X α 1 ) · · · (1 + e k X α k ), e j = ± 1, j = 1 . . . k , the inclusion condition in term of the b α ’s is (*) b 0 + e 1 b α 1 + · · · + e 1 · · · e k b α 1 + ··· + α k = 1 . The last part of the p aper will discuss some examples to inv estigate the practical applicability of th e previous condition (*). This p ap er i s an offspring of the Alc otr a 158 EU r e se ar c h c ontr act on the pl anning of se quent ial designs for sample surveys in tourism statis- tics. 1. Int roduction W e consider 2-lev el fractional designs with m factors, wh ere the lev els of eac h factor are co ded − 1 , +1. The full factorial d esign is D = {− 1 , +1 } m and a fraction of the fu ll d esign is a sub set F ⊂ D . According to the algebraic description of designs, as it is discussed in [7], [6], the fr actio n ide a l Ideal ( F ), also ca lled d esign ideal, is th e set of all p olynomials with real co efficien ts that are zero on all p oints of the fraction. Tw o p olynomials f and g are aliased by F if and only if f − g ∈ Id eal ( F ) and the quotien t space defined in such a wa y is the vect or space of real resp onses on F . The fraction ideal is generated b y a finite n um b er o f its elemen ts. This finite se t of p olynomials is called a b asis of the id eal. b ases are not un iquely determined , unless ve ry sp ecial conditions are met. A Gr¨ obner b a sis of th e fraction ideal can b e d efi ned after the assignmen t of a total order on monomials called monomial or der . If a monomial order is giv en, it is p ossible to iden tify the le a ding monomial of eac h p olynomial. As far as applications to statistics are concerned, a Gr¨ obner basis is c haracterized b y the follo wing prop erty: the set of all monomials that are are not divided b y an y of the leading term of the Date : Presented by R . F ontana at t he DAE 2007 Conference, The Un ive rsity of Mem- phis, Nov em b er 2, 2007. 1 2 R. FONT ANA AND G. PISTONE p olynomials in the basis form a linear b asis of the quotien t v ector space. A general reference to the r elev ant computational comm utativ e algebra topics is [2]. The ring of p olynomials in m indeterminates x 1 . . . x m and r ational co- efficien t is d en oted b y R = Q [ x 1 . . . x m ]. The design id eal Ideal ( D ) has a unique ‘minimal’ b asis x 2 1 − 1 , . . . , x 2 m − 1, w hic h happ en s to b e a Gr¨ obner basis. T h e p olynomials that are add ed to this basis to generate the ideal of a fraction are called gener ating e quations . An ideal with a basis of bi- nomials with co efficien ts ± 1 is called binomial ide al . Ind icator p olynomials p olynomials of a fraction w ere in tro duced in [3], see also [9]. An ind icator p olynomial has the form (1) F = X α b α x α , α = ( α 1 , . . . , α m ) ∈ { 0 , 1 } , x α = x α 1 1 · · · x α d d and it satisfies the conditions F ( a ) = 1 if a ∈ F , F ( a ) = 0 otherwise. If necessary , w e distinguish b et w een the ind eterminate x j , the v alue a j and the mapping X j ( a ) = a j . How to mo v e b et w een the ideal r ep resen tation and the ind icator function repr esen tation, is d iscussed in [5]. The defi nition and c haracterization, from the algebraic p oin t of view, of regular fractional factorial designs (briefly regular d esigns) is discussed in [3], see also [9]. In p articular, the last p ap er r eferred to considers mixed factorial design, but this case is outside the scop e of the presen t pap er. Orthogonal arra ys as are d efined in [4] can b e c haracterized in the previous algebraic framew ork, see [9] and [1], as follo ws. A fraction F with indicator p olynomial F is orthogonal with strength s if b α = 0 if 1 ≤ | α | ≤ s , | α | = P j α j . The notion of indicator p olynomial can b e accommo dated to cases with replicated design p oin ts by allo wing inte ger v alues other than 0 and 1 to F , see [11]. In such a case, w e prefer to call F a c ounting p o lynomial of the fraction. A systematic algebraic searc h of orthogonal arra ys with r eplications is discussed in [1 ]. F o r sak e of easy reference in S ection 5 b elow, w e quote a couple of sp ecific result ab out orth ogonal arra ys. In fact, considerin g m = 5 factors and strength s = 2, it is shown in [1, T able 5.2] that th ere are 192 O A’s with 12 p oin ts and n o r eplicatio ns, and there are 32 OA’s w ith 12 p oint s, one of them r eplicated. This p ap er is organized as follo ws. In Section 2 the algebraic theory is review ed and in Section 3 it is applied to the problem of findin g fr actions that are union of regular fr actions. In Section 5 the imp ortan t case of Plac k ett-B urm an designs is considered. 2. Re gular fr actions According to the definitions in [3] and [8] a regular fraction is defin ed as follo ws. Let L b e a subset of L = Z m 2 , w h ic h is an additive group. Let Ω 2 b e the multiplicat iv e group {− 1 , +1 } Definition 2.1. L e t e b e a map fr om L to Ω 2 . A non-empty f r action F is regular if UNION OF REGULAR FRACTIONS 3 (1) L ⊂ L s a su b-gr oup ; (2) the e quation s X α = e ( α ) , α ∈ L define the fr action F , i.e. ar e a set of generating equations . In such a c ase, e is a gr oup home omorphism. Other known d efinitions are shown to b e equiv ale nt to this one by the follo wing prop osition. Theorem 2.1. L et F b e a fr action. The fol lowing statements ar e e quivalent: (1) The fr a ction F is r e gular ac c or ding to definition 2.1 . (2) The indic at or function of the fr action has the form F ( ζ ) = 1 l X α ∈L e ( α ) X α ( ζ ) , ζ ∈ D . wher e L is a given subset of L and e : L → Ω 2 is a give n mapping. (3) F or e ach α, β ∈ L the inter actions r epr esente d on F by the terms X α and X β ar e either ortho gonal or total ly aliase d. (4) The Ideal ( F ) is binomial. (5) F is either a sub gr oup or a later al of a sub gr oup of the multiplic a tive gr o up D Pr o of. Most of the equiv alences are either wel l kn o wn or pro v ed in the cited literature. W e prov e the equiv alence of (4) The ideal of a regular design is generated by the basis of the full d esign and b y generating p olynomials of the form X α − e α , where e α = ± 1; all these p olynomials are binomials. Vicev ersa, if th e v ariet y of a b inomial ideal is a fraction of D , then all the p olynomials x 2 i − 1 are con tained in its id eal, and eve ry other binomial in the basis, sa y x α − ex β , e = ± 1, is equiv alen t to the generating p olynomial x α + β + e .  W e will sho w some examples of app licatio n of suc h theorem b elo w. W e first will p ro v e t w o p rop ositions that c haracterize the simple cases of 1-p oint and 2-p oints regular fractions. Prop osition 2.1. Every 1-p oint fr a ction is r e gular Pr o of. W e can pro v e the statemen t using design ideals. A single generic p oint is a = ( a 1 , . . . , a m ) ∈ D . A binomial basis is { x i − a i , i = 1 , . . . , m } and, therefore, F ≡ { a } is regular. Equiv ale ntly w e can us e indicator functions. Indeed the ind icator function of a single p oin t a is F a = 1 2 m (1 + a 1 x 1 ) · · · · · (1 + a m x m ) and F a meets the requirement s for b eing an ind icator function of a regular design.  The follo wing result lo oks less trivial. Prop osition 2.2. Every 2-p oints fr a ction is r e gular. 4 R. FONT ANA AND G. PISTONE Pr o of. Let 1 = (1 , · · · , 1) b e the n ull elemen t of D . W e observe that ev ery subset F of D made up of t w o elemen ts, sa y a and b with a 6 = b is a s u bgroup or a coset of a subgroup. Ind eed if a = 1 or b = 1 then F is a subgroup. If a 6 = 1 and b 6 = 1 then F is the coset aH where H is the su bgroup  1 , a − 1 b  .  2.1. Remark. W e can also pro v e the result comparing the n umber of 2- p oint s s ubsets with the num b er of subgroups of order 2. The num b er of 2-p oin ts fr actions of D is  2 m 2  = 2 m · ( 2 m − 1) 2 = 2 m − 1 · ( 2 m − 1) On the other s id e, ev ery regular fraction is a sub group of D or a coset of a subgroup of D ([3]). I n particular the n umber of r egular fractions of size 2 is equiv alen t to th e num b er of subgroups of order 2 m ultiplied by the n umber of cosets of a s ubgroup, that is 2 m − 1 . The num b er of subgroup s of order equal to 2 is 2 m − 1. Ind eed every set { 1 , p } with 1 = (1 , · · · , 1) and p ∈ D , p 6 = 1 is a subgroup of order equal to 2. It follo ws that the n umber of r egular fractions of s ize 2 will b e equal to 2 m − 1 · ( 2 m − 1) that is the n umb er of 2-p oint s fraction. If we consider 2 k -p oin ts fractions ( k ≥ 2) a similar argumen t is not v a lid as will b e clear in th e next sections. It also follo ws that every 3-p oin ts fraction can b e considered as th e u n ion of a 1-p oint fraction and a 2-p oin ts fraction. 3. Union of re gular de signs In this section we consid er the union of regular designs. T o simplify form ulæ we w ill in tro du ce th e follo wing notation: X α ≡ X α 1 1 · · · · · X α m m = X ¯ α where ¯ α is the set for wh ic h α i 6 = 0, { i ∈ { 1 , . . . , m } : α i 6 = 0 } . W e will also write α in place of ¯ α with a small ab u se of notation. As an example let’s consider m = 4 and α = (0 , 1 , 1 , 0). It follo ws that X α = X 2 X 3 will b e written as X 23 . Let F 1 and F 2 t w o r egular fractions, b oth in clud ed in D . The ind icator functions of F 1 and F 2 , sa y F 1 and F 2 resp ectiv ely , allo w to easily determine the indicator fu nction of the u nion of F 1 and F 2 , F = F 1 ∪ F 2 as F = F 1 + F 2 − F 1 × F 2 In general, the union of tw o (disj oin t) regular fractions is not a r egular fraction. As an example let’s consid er m = 2 factors, D = {− 1 , +1 } × {− 1 , +1 } and F 1 = { ( − 1 , − 1) } an d F 2 = { ( − 1 , + 1) , (+1 , − 1)) } . Both F 1 UNION OF REGULAR FRACTIONS 5 and F 2 are regular fractions, according to the prop ositions of the previous sections. Indeed their indicator f unctions meet the requ ir emen ts for regular fractions: F 1 = 1 4 (1 − X 1 ) · (1 − X 2 ) and F 2 = 1 2 (1 − X 1 · X 2 ). Ho we v er, the union F = { ( − 1 , − 1) , ( − 1 , + 1) , (+1 , − 1)) } , is not a r egular fraction, b ecause its in dicator function is F = 3 4 − 1 4 X 1 − 1 4 X 2 − 1 4 X 1 · X 2 . The same conclusion can b e obtained considering design ide als related to fractional designs. Giv en F 1 ⊂ D , F 2 ⊂ D and F = F 1 ∪ F 2 the asso ci- ated ideals will b e Ideal ( F 1 ), Ideal ( F 2 ) and Ideal ( F ). In general, the fact that Ideal ( F 1 ) and Id eal ( F 2 ) are binomial ideals b y Theorem ?? do esn ’t imply that Ideal ( F )) is a b inomial ideal . In deed, for the previous exam- ple, the Gr¨ obner bases B 1 , B 2 and B of Id eal ( F 1 ), Ideal ( F 2 ) and Ideal ( F ) resp ectiv ely , are: B 1 = { X 1 + 1 , X 2 + 1 } B 2 =  X 2 2 − 1 , X 1 + X 2  B =  − 1 / 4 X 1 X 2 − 1 / 4 X 1 − 1 / 4 X 2 − 1 / 4 , X 2 2 − 1 , X 2 1 − 1  It results that Ideal ( F 1 ) and Ideal ( F 2 ) are binomial ideals wh ile Ideal ( F ) is not. 3.1. Remark. More generally , let’s consider tw o disjoin t regular fr actions, namely aG and bH , wh er e G and H are subgroups of D and a / ∈ G and b / ∈ H . Let’s tak e ag and bh . In order to hav e ( ag )( bh ) ∈ aG we should ha v e bg h ∈ G or, equiv ale ntl y , bh ∈ G . 4. Decompos ing a fraction into regular f ractions In this part of the w ork w e would lik e to explore the in v erse path, i.e. to analyze the decomp osition of a giv en F ⊂ D int o the un ion of d isjoin t regular fractions. W e will in dicate with R the generic regular fraction. Let’s ind icate with F and R the indicator functions of F ⊂ D and R ⊂ D resp ectiv ely . Under which cond ition R will b e a sub set of F ? Theorem 4.1. L et F b e the i ndic ator fu nc tion of a generic fr actional design F ⊂ D , F = P α b α X α . L et R the indic ator function of a r e gu lar fr a ctional design R ⊂ D , R = 1 l P α ∈L e α X α = 1 2 k (1 + e 1 X α 1 ) · · · (1 + e k X α k ) . The fol lowing statement holds: R ⊆ F ⇔ b 0 + e 1 b α 1 + · · · + e 1 · · · e k b α 1 + ··· + α k = 1 Pr o of. F or R to b e a subset of F it m ust h app en that the n umber of p oint s of R m ust b e equal to the n umber of p oin ts of R ∩ F . In term s of indicator functions the equalit y R = R ∩ F b ecomes P t F ( t ) R ( t ) = P t R ( t ) b eing t ∈ D . W e ha v e 6 R. FONT ANA AND G. PISTONE F R = ( X α b α X α ) · 1 2 k (1 + e 1 X α 1 ) · · · (1 + e k X α k ) = 1 2 k X α b α X α + 1 2 k X α b α X α e 1 X α 1 · · · + 1 2 k X α b α X α e 1 · · · e k X α 1 + ··· + α k It follo ws that X t F ( t ) R ( t ) = 1 2 k 2 m b 0 + 1 2 k 2 m e 1 b α 1 + · · · 1 2 k 2 m e 1 · · · e k b α 1 + ··· + α k On the other hand X t R ( t ) = 1 2 k 2 m It follo ws b 0 + e 1 b α 1 + · · · + e 1 · · · e k b α 1 + ··· + α k = 1  Corollary 4.1.1. A ne c essary, b u t not sufficient, c ond ition for a r e gular fr a ction R to b e c o ntaine d in D is b 0 + | b α 1 | + · · · + | b α 1 + ··· + α k | ≥ 1 4.1. A small e xa mple. L et’s consider the 3-p oin ts fraction F ⊂ D = {− 1 , +1 } × {− 1 , +1 } that w e hav e int ro du ced in the pr evious section: F = { ( − 1 , − 1) , ( − 1 , +1) , (+1 , − 1)) } The indicator function F of F is F = 3 4 − 1 4 X 1 − 1 4 X 2 − 1 4 X 1 · X 2 , that is b 0 = 3 4 , b 1 = − 1 4 , b 2 = − 1 4 , b 12 = − 1 4 . It follo ws b 0 − b 1 = 1 b 0 − b 2 = 1 b 0 − b 12 = 1 b 0 − b 1 − b 2 + b 12 = 1 b 0 − b 1 + b 2 − b 12 = 1 b 0 + b 1 − b 2 − b 12 = 1 UNION OF REGULAR FRACTIONS 7 F rom eac h relation, us ing theorem 4.1, we can obtain the indicator fu n c- tions of the r egular fractions that are conta ined in to F . These are F 1 = 1 2 (1 − X 1 ) F 2 = 1 2 (1 − X 2 ) F 3 = 1 2 (1 − X 1 · X 2 ) F 4 = 1 4 (1 − X 1 ) · (1 − X 2 ) F 5 = 1 4 (1 − X 1 ) · (1 + X 2 ) F 6 = 1 4 (1 + X 1 ) · (1 − X 2 ) Therefore the corresp onding regular fractions are, r esp ectiv ely F 1 = { ( − 1 , − 1) , ( − 1 , +1)) } F 2 = { ( − 1 , − 1) , (+1 , − 1)) } F 3 = { ( − 1 , +1) , (+1 , − 1)) } F 4 = { ( − 1 , − 1) } F 5 = { ( − 1 , +1) } F 6 = { (+1 , − 1) } 5. Plackett-Burman designs Another example can b e obtained considering the w ell-kno wn “Plac k ett- Burman” designs [10]. In p articular the Plack e tt-Burman design for 11 v a riables and 12 ru ns is built according the follo wing pro cedur e: (1) the firs t ro w, namely th e ke y , is giv en: + + − + + + − − − + − (2) the second ro w up to the elev en th ro w are built s hifting th e key of one p osition eac h time (3) the 12th r o w is set equal to − − − − − − − − − − − The Plac k ett-Burman design f or elev en parameters b ecomes 8 R. FONT ANA AND G. PISTONE N A B C D E F G H I J K 1 + + − + + + − − − + − 2 − + + − + + + − − − + 3 + − + + − + + + − − − 4 − + − + + − + + + − − 5 − − + − + + − + + + − 6 − − − + − + + − + + + 7 + − − − + − + + − + + 8 + + − − − + − + + − + 9 + + + − − − + − + + − 10 − + + + − − − + − + + 11 + − + + + − − − + − + 12 − − − − − − − − − − − W e co nsider the case with m = 5 fact ors and, f r om t he “Plac k ett-B urm an ” for 11 factors we randomly s elect the follo wing F , corresp ond ing to columns A,B,F,H and I of the original design. The p lus sign ’+’ h as b een cod ed with ’1’ and the min us sign ’ − ’ with ’ − 1’. F = N X 1 X 2 X 3 X 4 X 5 1 1 1 1 1 1 2 1 1 − 1 − 1 1 3 1 − 1 − 1 − 1 1 4 − 1 1 − 1 1 1 5 − 1 − 1 1 1 1 6 − 1 − 1 1 − 1 1 7 1 1 1 − 1 − 1 8 1 − 1 1 1 − 1 9 1 − 1 − 1 1 − 1 10 − 1 1 1 − 1 − 1 11 − 1 1 − 1 1 − 1 12 − 1 − 1 − 1 − 1 − 1 The indicator function F of F is 3 8 + 1 8 X 345 + 1 8 X 245 − 1 8 X 235 − 1 8 X 234 + 1 8 X 2345 − 1 8 X 145 − 1 8 X 135 + 1 8 X 134 + 1 8 X 1345 + 1 8 X 125 + − 1 8 X 124 + 1 8 X 1245 + 1 8 X 123 + 1 8 X 1235 + 1 8 X 1234 It follo ws that F is not regular. No w we start to search for regular fractions that are con tained in F . UNION OF REGULAR FRACTIONS 9 Of course the fir st constrain t concerns the size of the regular f raction. It m ust b e less or equal to 12, the num b er of p oint s of F . Being R a regular fraction, it follo ws that the size of R could b e 2 0 = 1 or 2 1 = 2 or 2 2 = 4 or 2 3 = 8. W e already k n o w, from the prop ositions of section 2 that • all th e 12 p oint s that constitute R are 1-p oint regular fraction; • all th e  12 2  = 66 are 2-p oin ts regular fraction. Let’s study 4-p oin ts and 8-p oin ts su bsets of F . The corolla ry of theorem Th. 4.1 allo w s u s to immediately exclude 8- p oint s r egular fractions. I ndeed the follo wing condition should b e true for a prop er c hoice of e 1 , e 2 and α 1 , α 2 b 0 + e 1 b α 1 + e 2 b α 2 + e 1 e 2 b α 1 + α 2 = 1 But b 0 = 3 8 and the absolute v alue of b i is 1 8 , ∀ i and so it is n ot p ossib le that the left side of the previous equation sums up to 1. No 8-p oints regular fraction is conta ined into F . Finally we inv estiga te 4-p oints regular fr actions. F or a p rop er c hoice of e 1 , e 2 , e 3 and α 1 , α 2 , α 3 the follo wing relation should hold b 0 + e 1 b α 1 + e 2 b α 2 + e 3 b α 3 + e 1 e 2 b α 1 + α 2 + e 1 e 3 b α 1 + α 3 + e 2 e 3 b α 2 + α 3 + e 1 e 2 e 3 b α 1 + α 2 + α 3 = 1 A subgroup of order eigh t will b e { 1 , a, b, ab, c, ac, bc, abc } with a 6 = 1, b 6 = 1 , c 6 = 1 and a 6 = b , a 6 = c and b 6 = c . W e can c ho ose a , b and c in  31 2  · (31 − 3) different w a ys. T he num b er of different s ubgroups is obtained dividing this num b er b y  7 2  · 4. W e get 155 different subgroups. Ev ery su bgroup of order 8, S (8) i = < α 1 i , α 2 i , α 3 i >, i = 1 , . . . , 155 defines 8 regular f r actions of size 4 (the su bgroup orthogonal to S (8) i and its cosets). T o find the regular fr actions emb edded into F we must solve the follo wing systems of equ ations ( i = 1 , . . . , 155) 8 > > < > > : e 2 1 − 1 = 0 e 2 2 − 1 = 0 e 2 3 − 1 = 0 b 0 + e 1 b α 1 i + e 2 b α 2 i + e 3 b α 3 i + e 1 e 2 b α 1 i + α 2 i + e 1 e 3 b α 1 i + α 3 i + e 2 e 3 b α 2 i + α 3 i + e 1 e 2 e 3 b α 1 i + α 2 i + α 3 i − 1 = 0 T o do it w e generate the 155 subgroup s of D of order eigh t (for exa mple usin g the p ac k age GAP [ ? ]). As an example let’s consider S 1 = < { 1 } , { 2 } , { 3 } > . Being b 0 = 3 8 , b 1 = b 2 = b 3 = b 12 = b 13 = b 23 = 0 and b 123 = 1 8 the 10 R. FONT ANA AND G. PISTONE corresp onding system of equation is        e 2 1 − 1 = 0 e 2 2 − 1 = 0 e 2 3 − 1 = 0 3 8 + 1 8 e 1 e 2 e 3 − 1 = 0 The system do esn’t hav e any solution. Let’s no w consider S 2 = < { 4 } , { 12 } , { 135 } > . Being b 0 = 3 8 , b 4 = b 1 2 = 0, b 135 = b 124 = b 235 = − 1 8 and b 1345 = b 2345 = 1 8 the corresp onding system of equation is        e 2 1 − 1 = 0 e 2 2 − 1 = 0 e 2 3 − 1 = 0 3 8 − 1 8 e 3 − 1 8 e 1 e 2 + 1 8 e 1 e 3 − 1 8 e 2 e 3 + 1 8 e 1 e 2 e 3 − 1 = 0 The system has the follo wing solution e 1 = − 1 , e 2 = 1 , e 3 = − 1 that defines the f ollo wing indicator function F (1) 1 8 (1 − X 4 )(1 + X 12 )(1 − X 135 ) The corresp on d ing s et of p oin ts F (1) is N X 1 X 2 X 3 X 4 X 5 2 1 1 − 1 − 1 1 6 − 1 − 1 1 − 1 1 7 1 1 1 − 1 − 1 12 − 1 − 1 − 1 − 1 − 1 T o pro ceed into the decomp osition of F we r emov e the p oin ts of F (1) . The indicator function of the new set w ill b e F − F (1) : 1 4 + 1 8 X 4 − 1 8 X 12 + 1 8 X 345 + 1 8 X 245 − 1 8 X 234 + − 1 8 X 145 + 1 8 X 134 + 1 8 X 125 + 1 8 X 1245 + 1 8 X 123 + + 1 8 X 1235 + 1 8 X 1234 W e n o w searc h for the regular f r actions cont ained in to F − F (1) . A r egular fraction R to b e cont ained in to F − F (1) m ust b e cont ained in to F . W e can therefore limit our search to the solution that w e h a v e identified in the first part. Let’s no w consider S 3 = < { 12 } , { 35 } , { 245 } > . UNION OF REGULAR FRACTIONS 11 Being b (1) 0 = 1 4 , b (1) 35 = 0 b (1) 245 = b (1) 134 = b (1) 1235 = 1 8 and b (1) 234 = b (1) 145 = b (1) 12 = − 1 8 the corresp ondin g system of equation is        e 2 1 − 1 = 0 e 2 2 − 1 = 0 e 2 3 − 1 = 0 1 4 − 1 8 e 1 + 1 8 e 3 + 1 8 e 1 e 2 − 1 8 e 1 e 3 − 1 8 e 2 e 3 + 1 8 e 1 e 2 e 3 − 1 = 0 The system has the follo wing solution e 1 = − 1 , e 2 = − 1 , e 3 = 1 that defines the f ollo wing indicator function F (2) 1 8 (1 − X 12 )(1 − X 35 )(1 + X 245 ) The corresp on d ing s et of p oin ts F (2) is N X 1 X 2 X 3 X 4 X 5 3 1 − 1 − 1 − 1 1 4 − 1 1 − 1 1 1 8 1 − 1 1 1 − 1 10 − 1 1 1 − 1 − 1 If w e remo v e this set of p oin ts from F − F 1 w e get the follo wing in dicator function F (3) = F − F (1) − F (2) : 1 8 + 1 8 X 4 + 1 8 X 35 + 1 8 X 345 + 1 8 X 125 + 1 8 X 1245 + 1 8 X 123 + 1 8 X 1234 or, equiv alen tly , 1 8 (1 + X 4 )(1 + X 35 )(1 + X 125 ) and the corresp onding set of p oin ts F (3) N X 1 X 2 X 3 X 4 X 5 1 1 1 1 1 1 5 − 1 − 1 1 1 1 9 1 − 1 − 1 1 − 1 11 − 1 1 − 1 1 − 1 F (3) meets the requir emen ts to b e an indicator fun ction of a regular de- sign. W e ha v e therefore d ecomp osed F in to three regular designs, F = F 1 ∪ F 2 ∪ F 3 . 5.1. Decomp osition of the given ‘Plac k ett-Burman’ design in to al l the unions of 4-p oin ts regular designs. In this p art w e fi nd all the p ossible decomp ositions of the giv en “Plac k ett-Burman” design. As de- scrib ed in the previous section, w e consider all the 155 s ubgroups of order 8, S (8) i = < α 1 i , α 2 i , α 3 i >, i = 1 , . . . , 155 and we s earch for the solution of the follo wing systems of equations 12 R. FONT ANA AND G. PISTONE 8 > > < > > : e 2 1 − 1 = 0 e 2 2 − 1 = 0 e 2 3 − 1 = 0 b 0 + e 1 b α 1 i + e 2 b α 2 i + e 3 b α 3 i + e 1 e 2 b α 1 i + α 2 i + e 1 e 3 b α 1 i + α 3 i + e 2 e 3 b α 2 i + α 3 i + e 1 e 2 e 3 b α 1 i + α 2 i + α 3 i − 1 = 0 15 of these 155 systems of equ ations ha v e a non-empty set of solutions. Eac h of these non-empt y sets define an ind icator f unction R j , j = 1 , · · · , 15: R 1 = 1 8 (1 − X 4 )(1 + X 12 )(1 − X 235 ) R 2 = 1 8 (1 + X 1 )(1 + X 23 )(1 + X 245 ) R 3 = 1 8 (1 + X 1 )(1 − X 45 )(1 − X 235 ) R 4 = 1 8 (1 − X 2 )(1 + X 34 )(1 − X 145 ) R 5 = 1 8 (1 + X 2 )(1 + X 15 )(1 − X 345 ) R 6 = 1 8 (1 − X 23 )(1 − X 45 )(1 − X 135 ) R 7 = 1 8 (1 − X 3 )(1 + X 25 )(1 − X 145 ) R 8 = 1 8 (1 + X 3 )(1 + X 14 )(1 − X 245 ) R 9 = 1 8 (1 − X 14 )(1 − X 25 )(1 + X 345 ) R 10 = 1 8 (1 − X 15 )(1 − X 34 )(1 + X 245 ) R 11 = 1 8 (1 − X 5 )(1 + X 13 )(1 − X 234 ) R 12 = 1 8 (1 + X 4 )(1 + X 35 )(1 − X 125 ) R 13 = 1 8 (1 + X 5 )(1 + X 24 )(1 − X 134 ) R 14 = 1 8 (1 − X 12 )(1 − X 35 )(1 + X 245 ) R 15 = 1 8 (1 − X 13 )(1 − X 24 )(1 + X 345 ) T o build a generic decomp osition of F w e s tart from one of these indicator function, let’s sa y R 1 that identify the regular fraction R 1 . W e hav e no w to c ho ose another indicator functions in the set made u p b y R 2 , . . . , R 15 , let’s say R k , with the cond ition that the corresp ondin g r egular fraction R k do esn’t intersect R 1 : R 1 ∩ R k = ∅ . W e ha v e tw o p ossible choic es, R 12 and R 14 . If we c hoose R 12 the only p ossible r emaining is R 14 and, vicev ersa, if w e c ho ose R 14 the only p ossible remaining is R 12 . Repeating the same pro cedure for all the R i and considering only the differen t decomp ositions, w e get that F can b e considered as the union of three regular 4-p oin ts designs F = R 1 ∪ R 12 ∪ R 14 F = R 2 ∪ R 3 ∪ R 6 F = R 4 ∪ R 5 ∪ R 10 F = R 7 ∪ R 8 ∪ R 9 F = R 11 ∪ R 13 ∪ R 15 The decomp osition that has b een found in the previous section is F = R 1 ∪ R 12 ∪ R 14 . 5.2. Decomp osition of al l the “Plac k ett-Burman” designs with m=5 and 12 differen t runs in to al l the unions of 4-p oin ts regular de - signs. Using an ad-ho c soft w are routine written in SAS IML we consid er all UNION OF REGULAR FRACTIONS 13 the  11 5  = 462 differen t d esigns that can b e obtained c ho osing 5 columns out of the 11 of the original designs. W e get the follo wing table where the first column con tains an id en tification of the design, the second column the n umber of designs that are equ al to th e d esign and the third column the n umber of differen t runs cont ained in the design. F or example, the design F that we hav e considered in the previous sections, b elongs to the class “69”. There are 11 d esigns that are equal to F and eac h h as 12 p oin ts. I D N S I Z E 1 8 12 2 7 12 3 6 12 4 8 12 5 5 12 6 7 11 7 2 12 8 13 12 9 6 12 10 11 11 11 7 12 12 7 12 13 5 12 14 7 11 15 10 12 16 6 12 17 7 12 18 3 12 19 7 12 20 11 12 21 5 12 22 8 12 23 4 12 24 7 12 25 2 12 26 5 12 27 6 11 I D N S I Z E 28 6 12 30 10 12 32 6 11 35 6 12 37 3 12 39 4 12 44 11 12 45 7 12 46 6 12 49 2 12 51 7 12 52 9 12 53 5 12 54 4 11 55 4 12 57 3 11 58 6 12 61 6 12 63 4 12 64 3 12 65 8 12 66 5 12 67 2 12 68 7 12 69 11 12 70 13 12 71 6 12 I D N S I Z E 72 5 11 73 6 12 74 5 12 82 6 12 84 2 12 85 9 11 87 7 12 89 4 12 94 6 12 98 7 12 100 3 12 101 8 12 102 3 11 103 7 12 110 2 12 116 5 12 117 1 12 128 2 12 134 5 12 140 3 12 146 5 11 147 3 12 149 4 12 154 6 12 159 1 12 167 2 12 184 1 12 It follo ws that the 462 designs can b e partitioned in to 81 classes: • there are 70 classes wh ere eac h d esign con tains 12 runs • there are 11 classes wh ere eac h d esign con tains 11 runs W e limit to designs with 12 differen t r uns. W e rep eat the pro cedu r e describ ed in the p revious s ection for all the 70 different designs. First of all w e determine the indicator fu nctions of all the 70 d esigns. Every indicator function has th e follo wing form: 3 8 + a 345 X 345 + a 245 X 245 + a 235 X 235 + a 234 X 234 + a 2345 X 2345 + a 145 X 145 + a 135 X 135 + a 134 X 134 + a 1345 X 1345 + a 125 X 125 + a 124 X 124 + a 1245 X 1245 + a 123 X 123 + a 1235 X 1235 + a 1234 X 1234 where the co efficien ts a 345 , . . . , a 1234 are equal to ± 1 8 . W e decomp ose ev ery fraction in to three 4-p oin ts regular d esign. As for the design considered in the p revious example w e h a v e th at eve ry design • con tains 15 “4-p oin ts regular design” 14 R. FONT ANA AND G. PISTONE • can b e considered as the un ion of three regular d esigns in 5 d ifferen t w a ys W e ha ve examined the decomp osition structure of all the 70 designs. If w e indicate with R 1 , R 2 and R 3 the in dicator fun ctions of the regular designs con tained into one of the d esign, we get R 1 = 1 8 (1 + e 1 X α 1 + e 2 X α 2 + e 1 e 2 X α 1 + α 2 + e 4 X α 4 + e 1 e 4 X α 1 + α 4 + e 2 e 4 X α 2 + α 4 + e 1 e 2 e 4 X α 1 + α 2 + α 4 ) R 2 = 1 8 (1 − e 1 X α 1 + e 3 X α 3 − e 1 e 3 X α 1 + α 3 + e 5 X α 5 − e 1 e 5 X α 1 + α 5 + e 2 e 5 X α 2 + α 5 + e 1 e 3 e 5 X α 1 + α 3 + α 5 ) R 3 = 1 8 (1 − e 2 X α 2 − e 3 X α 3 + e 2 e 3 X α 2 + α 3 + e 6 X α 6 − e 2 e 6 X α 2 + α 6 − e 3 e 6 X α 3 + α 6 + e 2 e 3 e 6 X α 2 + α 3 + α 6 ) where, b eing | α | = P j α j , • | α 1 | , | α 2 | and | α 3 | are all less than thr ee • all the others, i.e. | α 1 + α 2 | , . . . , | α 2 + α 3 + α 6 | are all greater or equal to 3 This evidence has suggested the follo wing pro cedure. (1) W e ha v e b uilt all the α 1 , . . . , α 6 that satisfy the previous r equire- men t, N α 1 α 2 α 3 α 4 α 5 α 6 1 1 23 45 245 234 12 4 2 1 24 35 235 234 12 3 3 1 25 34 234 235 12 3 4 2 13 45 145 134 12 4 5 2 14 35 135 134 12 3 6 2 15 34 134 135 12 3 7 3 12 45 145 124 13 4 8 3 14 25 125 124 12 3 9 3 15 24 124 125 12 3 10 4 12 35 135 123 13 4 11 4 13 25 125 123 12 4 12 4 15 23 123 125 12 4 13 5 12 34 134 123 13 5 14 5 13 24 124 123 12 5 15 5 14 23 123 124 12 5 (2) F or ev ery c hoice of α 1 , . . . , α 6 w e ha v e built the 64 indicato r fu nctions that corresp ond to all the v alues of e 1 , . . . , e 6 , b eing e i = ± 1 , i = 1 , . . . , 6. According to this pro cedure w e ha v e generated 15 × 64 = 960 ind icator functions. If w e limit to the different ones we get 192 indic a tor functions . This num b er is the same that has b een f ound in [1], as the total num b er of orthogonal arra ys of strength 2. 5.3. Remark. It is interesting to p oint out that the “understand ing” of the mec hanism und erlying th e Plac k ett- Burman d esigns (m=5, 12-run s) has allo w ed to build al l the orthogonal arra ys of strength 2. UNION OF REGULAR FRACTIONS 15 6. Concl usions • The problem to determine regular designs that are cont ained in a giv en fraction has b een faced. • A condition in terms of th e co efficien ts of the p olynomial ind icator function has b een found . • The decomp osition of a give n fr action into regular designs seems useful for fractional factorial generation. Referen ces [1] Enrico Carlini and Giov anni Pistone. H ib ert bases for orthogonal arrays. Journal of Statistic al The ory and pr act ic e . Accepted 29-05-2007. Preprint arX iv :math/06112 76 . [2] David Cox, John Little, and Donal O’Shea. I de als, varieties, and algorithms . Under- graduate T exts in Mathematics. Springer-V erlag, New Y ork, second edition, 1997. An introduction to computational algebraic geometry and commutativ e algebra. [3] Rob erto F ontana, Gio v anni Pistone, and Maria Piera Rogantin. Classification of tw o- leve l factorial fractions. Journal of Statistic al Planning and Inf er enc e , 87(1):149–172, Ma y 2000. [4] A. S. H eda yat, N. J. A. Sloane, and John Stufken. Ortho go nal arr ays. The ory and applic ations . Springer S eries in St atistics. Springer-V erlag , N ew Y ork , 1999. [5] Rob erto Notari, Ev a Riccomagno, and Maria-Piera Rogantin. Two p olynomial rep- resen tations of exp erimental design. Journal of Statistic al The ory and Pr actic e , 2007. arXiv:0709.29 97 v 1 (in press). [6] Gio v anni Pistone, Eva R iccomagno, and Maria Piera Rogantin. Algebraic statistics for the design of exp eriments. In Luc Pronzato and Anton y A. Ziglja vsky , editors, Se ar ch for Optimali ty in Design and Statistics: Algebr a ic and Dinami c al System Metho ds , pages 95–129. Springer-V erla g, 2007. [7] Gio v anni Pistone, Ev a Riccomagno, and Henry P . Wynn. Algeb r aic Statistics: Com - putational Commutative Algeb r a in Statistics . Chap man & Hall, 2001. [8] Gio v anni Pistone and Maria-Piera Rogantin. Comparison of different definitions of regular fraction. Rapp orti Interni 2007/2, Politecnico di T orino DIMA T, 2007. [9] Gio v anni Pistone and Maria Piera R ogan tin. In dicator function an d complex cod ing for mixed fractional factorial designs. Journal of Statistic al Planning and I nfer enc e , 2007. Receiv ed 5 May 2005; revised 4 December 2006; accepted 8 F ebruary 2007. Av ailable online 12 Marc h 2007. [10] R.L. Plack ett and J.P . Burman. The design of opt imum multifactori al exp eriments. Biometrika , 33:305–325, 1946. [11] Kenny Q. Y e. Indicator function and its applicatio n in tw o-lev el fa ctorial designs. The Ann als of Statistics , 31(3):984–994 , 2003. DIMA T Politecnico di Torino E-mail addr ess : { giovanni.p istone|roberto .fontana } @polito.it