EXTENDED CONCEPT OF RESOLUTION AND THE DESIGNS DERIVED FROM BALANCED ARRAYS

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TRU Mathenlatics 20−2 〔1984）

EXTENDED CONCEPT OF RESOLiUTION AND THE DESIGNS

DERIVED FROM BAI、ANCED ARRAYS

Sumiyasu YAMAMOTO and Yoshifumi HYODO 〔Received November 5，1984〕 1． _{IntrOdnCtiOn and SUmmary．} Fractional factorial designs introduced first by Yates ［14］ and developed by Fimey［5］，Box and Hunter［1，2］and others have been finding increasi㎎ use in many fields of research and application． In the begiming， the theory was developed along the orthogonal fraction of symmetrical factorial designs in which the estimates of all factorial effects of interest are uncorrelated． The use of orthogonal fraction， however， is unecon㎝ic in that they involves more than the desirable number of ass㎝blies． Non−orthogonal or irregular fractions especially balanced fractions， have been investigated by Bose and Srivastava ［3，4］， Srivastava 【9］， Srivastava and Chopra 【10］，Yamamoto， ShirakUra and Kuwada ［12，13▲ and others． In their paper， Yamamoto， Shirakura and KUwada have introduced the con− cept of triangular t）rpe multid迦ensional association scheme among the facto− rial effects up to ∫し一factor interactions of 2m factorial design and investi− gated its algebraic structure． They have succeeded in obtaining an explicit expression of the infoτmation matrix of a fractiona1 2m factorial design de− rived fr㎝abalanced array〔B−array）of strength 22． They have sho、m thqt abalanced 2m fractional factoria1 〔 2m−BFF 〕 design of resolution 2A」＋1 is e（luivalent to the design derived fr㎝aB−array of strength 22 and m con− straints provided the infomation matrix is nonsingular． Subsequently， Shirakura［6，7］has shown that s㎝e B−arrays of strength 2￡yield 2m−BFF designs of resolution 22 l He has also given a necessary and sufficient condition for a B−array T to、 be a 2m−BFF design of resolution 2￡［8］． In this paper， the concept of resolution will be generalized． An attempt will be made to characterize the design derived from a two−symbol B−array of 341

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S．YAMAMOTO AND Y． HYODO ・trength 2￡・m・・n・trai・t・and i・d・x set｛pO・Vl・・°・・U22｝・・i・9・th・・XPli・it fbrmula of irreducible representation of the information matrix given in m Yamamoto and Hyodb ［11］．．Several examples of 2 −BFF designs are given in which all main effects and the general mean are estimable under the assump− tion that 4−factor or higher order interactions are negligible． Such designs may be superior to the resolution IV design in that confounding of the three factor interactions to the main effects can be avoided even though the former might exist．、 2． Preliminaries． C・n・id…f・a・ti・na・Pm f・・t・ri・・（2m−FF）d・・ign・i・h・fact・r・F、・F2・ …・Fm・ea・h・t．tw・1・v・1・・At・eatm・nt・。mbinati・n・r・n ・・sembly i・rePre− sent・d by・vect・r〔j 1・j2・…・jm）・・h・・e jk・th・1・v・1・f Fk・・qu・1・0・・1・ Consider the situation where a11 〔￡＋1〕−factor or higher order interactions can be assumed negligible for a fixed integer 2 〔1 ≦ 2 ≦m／2 〕， i．e．，the set of un㎞own factoriql effects is ass㎜ed to be 〔1）・、・・1。。θk 品e「e°・＝｛θ・、・、…・、1｛t・・t・・”◆・tk｝⊂｛1・2・’°◆・m｝｝andθφ・θ・、 and・’n gene「− a1・θt、t2…tk den°te the gene「al mean・the effect°f the fact°「F・、 and the k−fact°「’nte「act’°n°f the fact°「s Ft、・Ft、・’”・Ft、・「espect’vely・ Let T be a 2m−FF design with N asse油1ies， then T can be expressed as a （0，1）−array of size N x m whose rows denote asse凪）lies． Cbnsider an N x l °bse「vat’°n vect°「YT°f T曲゜Ge expectati°頑XT）and c°va「’ance mat「ix Var〔XT〕 are given by ET旦 andσ IN respectively・ Le・・〔2）ヱT ＝ ET旦＋皇 where皇denotes the error vector，旦denotes the v￡xlvector of unknown param− eters up to 27factor interactions arranged in an order of factorial effects， i・e・，（3）旦＝（θφ，旦i，旦2．，・．・，Q』L〕’， v、一・｛．。Φ，旦kd・n・t・・avect。・・f k−fact・r i…racti・n・respec・i…y fbr k＝ @1・2・°”・￡・and ET denotes the N x v2 design matrix of T・

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﹁ i t δ 惰り〒 r EXT日NDED CONCEPT OF RESOa TION OF A D旺ESIGN （4） The nomal equation fbr estilnating旦is given by A Mia ＝ Eicar・

曲・・eMr＝唱舗it i・ca11・d th・桓fbm・ti。・m・t・血・f th・design T・

Antt）ng th°se sets°f fact°「ia’ effects｛θφ｝・｛θt、｝・｛θt、・、｝・°”・画｝ atriangular type multidimensional partially balanced （TMDPB ）｛θ

tlt2…t兄

ass㏄’ati°n s』is def血ed・in a way such that et、…・。鋤θti・…↓a「e the α一th associates if and only if （5〕 1｛t1・∵’・t。｝・｛ti・…・t↓｝1＝ min〔…）’・・ The so called TMDPB association algebra A generated by those （兄◆1〕（兄＋2）・（…3）／・ass。・・・…n腿・・ice・D5u・v〕（…，・，…，…（・，・）・・，…，・，…，・〕 representing the above sch㎝e is s㎝i−simple and coinpletely reducible． It is dec・・r…ab・・・・・・…加・一・id・d・dea・・AB・［・lu・v〕”・u…β・・……・・］・ …，・，…，2・・here ・lu・v〕”’・・a…fy・5u・s〕㌔lw・v）＃・・，。・。，・lu・v）”・〔・lu・v）tt）1・Dlv・u）書飢・・a・・c〔・lu・v）〃）・（器）一〔，T、）・ぴ・・d・a・Aβi・i・・− morphic to the complete 〔兄一β＋1） x 〔￡一β＋1） matrix algebra and its multiplicity

：φs＝s望ll。〔當’芸1°iih：h：。麗1、：慧：1：nD㌻㌶si蕊．

・i・DEu・v）”㎝，。f・・ur・e， b・・een i・［・3］・ 1・is s㎞川・・，・2，・3】・』t th・i・f・・ma・i・n胆・riX Mr・f・Pm−FF・de・ign T derived frorn a balanced array（B−array）of strength 2いconstraints and i・d・xset｛・0・μ1・…・μ2￡｝b・1・ng・t。 th・㎜PB ass・ci・ti・n a19・bra ・nd hence it can be expressed as （・）叫・Σ：．。Σt。Σ：三3（u・v〕・、。一。1．、。Dlu・v〕， or equivalently，（7）

where

（8）叫・Σき．。Σ｛二1考二1・；・j・1β◆’・β＋j）＃・・、・・；：。Σ；．。〔一・）P（1）（j鶴〕Uj

_for

i＝0，1，… ，2￡， 343

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翅

S．YAM㎜AND Y． HYODO

and，：9）・；・j・・1・’・1・；．。Σ：b6j−’（一・）「（；〕〔m−2t’＋b）｛（m〕雪’）（｛）｝1／2（2bζ曽’）・（2￡一（2β＋i＋j）2＋r−（β＋j ＋b）〕22（β＋’−b）！（j−9’b）】・、・・1．、［・；．。ΣibL6j−’（一・）「（；）（m’2t’＋b）｛樗’）〔｛〕｝112 ・〔2b？一’）〔㌶；1鷲∼））22〔β＋”b）／（j−；＋b）］ ■ ・｛・兄．s・（一・）〕−1・兄．s｝・ fbr O≦i≦j≦兄一β；β＝0，1，…，兄． The irreducible representation of Mr with re・Pe・t t・eaCh・ide・1 AB i・give・by飢（兄一β＋1）・（兄一β◆1）mt虫KB such ’hat （10〕

_KB＝

1 ，

11

β ，，・●●一

〇β−β 兄RF

KK K

O

，

00

Rピ

，，⋮一

〇β−凸β 兄β

KK

K ● ● ● ● ● ● ● ■ ● 0，鳥一β Kβ 1，兄一β Kβ … 兄一β，兄一β Kβ and the皿1tiplicity of the representation is φβ・ 3・Pm−FF d・・ign・・f re・。1・ti・n R（・｜Ω兄〕・ C。。，ider a “−FF d・・ign T孤d ass鵬t｝・・t th・b・・i・1㎞ear㎜1・1（2〕de− Pends・n・he・et。f f㏄・・ri…ffec・・Ω、・・1．。・k， i…，…㈹一輪・・。・ higher order interactions are assumed to be negligible・ D・finiti・・ 1． A・2M−FF d・・ig・is said t・b・・f・e・・1・ti・・R〔kXSeqlΩ2）if （・・）〔i）・lk・k）旦・・ave・t・r・f k−fact・・in…act・。・・is est・励・・ fbr every kεS⊂｛0，1，・●．，貝」｝，〔・・）〔ii）・8h・h）旦・r・a・ve・t・r・f h−fa・t・r・interacti・ns・…t・・t・励1・ for e鴨ry h∈｛0，1，●・・，￡｝−S．

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臥㎜田㎝㏄口OF㎜L田1㎝OFAD硲1㎝

民fi・i・i。・2・A2m−FF desi四・f・e・・…i・・R（k2SθklΩ兄〕is said・・b・ b・1旬・記即dd・n・・ed by “−BFF・design・f re・・…i・・R（k2Sθk1Ω￡〕if・h・ c。vari孤ce m・t・ix・f th・B皿・f kUSek i・i悩ari孤t・nder・ny p・㎜t・ti・n・f m factors．

。d、二ee蕊。：蒜1：’：1。1｛：is隠ぼ：鷺蕊1裟llli三篭，

correspond respectively to the resolution 2￡＋1 and the resolution 2兄． In this section， we shall give a necessary and sufficient condition for a 2m−FF design to be of resolution R〔k2SOklΩ兄）・ Le皿胆1． A8θカo了kparurretr・iσゴ『鋤σ庇oπ6 C旦i8 estimabte if and onZy if

吻re・痂力・・k・・兄随』X…み微力）CM， −C・M・・・・・…if b・th Mr and C

bθZong カo the TMDPBα880（元αがoηαz≦7eb㌘αA then X amz be σ九〇sθπ80 カ』カXEA・ Proof． If C皇 is estimable， then there exists a matrix A such that E（叫）＝叫旦＝C旦h・1d・id・ntically in旦・i・…A・ati・fi・・AEr ＝C・ Hence r辿E｝＝rank［叫・C’】・［［hi・is equiva1・nt t°「ank ir＝「剛Mr：C’］・ Thus there exist5 a k x v∫L matrix X sudh that XM士＝ C．

C。nve・sely・f㎜（2）we hav・EO㌣r〕＝X牌＝）叫＝〔Q・

1⇒孤dcb・1。・g t。 A・th・n there exi・ts an・rth・g…1皿・t・i・P・・ch that P’xPKロムholds・曲ere K＝ P’MTp＝diag（Ko：K1’°°K1：°°°°’°：Kピ゜°K2） and △’P’CP 8 diag〔△0・△1…△1：……・△兄…△兄）・ The matrix x can， therefore， be chosen in A so that P，XP＝diag（Xδ：Xi．・・Xi：°・°°・°：X蔓．°°X蔓） holds． This completes the proof． The follorwing Theorem is an inmediate consequence of the Lem囮a 1．血…㎝2・Ti・・♂−FF 4・⑳・f・…Z・伽R（kどSθklΩ、）

・2m一即吻伽T・・尻・鐸・8訪・回Zα吻9・α肋鋤・

（13）（14） ifαncl onZy if

（i）励・…』・・、・・、ma・… Xk・…伽猶・D8k・k）f・・

θりθヱΨ kεS，

〔ii）』・⊇・…、・・、 m…in Xh・a…f・…Xh叫・Dlh・h）fo・

θ∂e121 hξ｛0，1，●・・，兄｝−S． 345

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w

S．YA㎜AND Y． HYOOO

4． 21m−BFF designs deτived from balanced arrays． In this section， we shall restrict ourself to those 2m−FF designs derived fr㎝two−sy戚）01 balanced arrays of strength 2兄， m constraints釦d index set ｛・。，・、，…，・、、｝・…h・・ca・叫・A and ・lk・k〕・Σ5．。DSk・k）＃・A…ev・ry k＝ 0，1・… ，￡，there exists an orthogonal matrix P of older v兄 sudh that （15） P’ｩP＝K＝diag〔Ko：K1°°°K1：°°”°°：Kピ・°K兄〕・

and

（・6） Pl・8k・k）・・r、・d・ag（・lk）・r｛k）…・｛k〕・…・…rlk）…・lk）〕・

吻・・eth・皿・tip・i…i…fb・thい・・lk）areφβ・（・ee・・・…mP・・nd返・f

Shirakura ［6］）． Here ・lk〕・d・・g（・（、．β）．（、．β）・…〔、．、）．（、一、））・・r・・・……・・一・・ k＝1，2，…，鳥一1，・lk）・…g〔…（、．β）．（、．β）〕f・…k・k・・……・・一・・・Ek）・・（、一，．、〕．（、．β．、）飴…k………k・・・・・・・…一・・・1兄）・d・・g〔・（、．β）．〔、．β）・・）飴・β一・……・・一・・飢・・1￡）… グ砲ere％・q d・n・tes a nul1 mat「iX°f°「de「pxq・

・一・．The・…アk働・r i・』・吻…Dlk・k）旦i…timb…f・and

⑭ザ鰍陥・rank［KB・rEk）］h・・⑭・⑳・・y・・・………

…。・．・・nce Dlk・k）皇・s e・t皿b…f釦d・n・y・f・here ex・・t・x、・A・u・h ・h・・X凸・Dlk・k）・rrhi・i・明・i・・…tt・・h・t th・re e…t・XilB・u・h・h・t ・it，K，・rlk〕・r r孤・K，・rank【K，・rlk）］・・r・v・ry・・・……・・・…sc㎝一 Pletes the proof．

・一・・Dlh・h）皇・・n・…t血・…f鋤・…y・f趣陥・rank［KB・rlh〕］

for sαne Bε｛0，1，… ，h｝． This means that not all h−factor interactions are esti皿able．． Thus we have the fo11㎝ing Theor㎝． Theorem 4． Considerα 2m−FF design T derivedゴ『1りmα 加o−symboZ B−ar？ray ・ア・t・・ng・h・・．・i…m−BFF・輌・f・・…』R（・1；91・k i ・・、〕・ヂ・』・y tf T satisfies the ずb Z Zoω老ηg oenditions：

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EXTENDED CCNCEPT OF RESOIUrlON OF A DBSIGN Th● covariance matrix of co叫めsed of v ら v8r（馬 the proof． 5・E・i・・㎝・e。f評一BFF d・・ign・・f re・・…i・n R（・。・θ、1Ω3）・ With respect to the existence of new resolution designs， we have the following Theorem： Theorem 5． The foZZoロing series o了d2signs d6rfVedデ㌔oη7 B−arrays of

轡。6。l l4＝．32㌔：竺lll＝：：8。；°．1。1；麟。；。ll，Ω，）、

provided y， z・Oand x＋u・0・ ”

Proof． Fr㎝〔9〕，we have ・⑪・°・x・・2y・・5・・u，・⑪・1− 5z−u〕，（・7）（・）r・nk k・r・n・［K，3・rEk〕］f・・⑳・均…，・，…，・・ k＝0，1，…，兄一2，（18）（ii）・ank KB t rank［KB・rl￡’1）］f・…・β・｛・，・，…，・一・｝，碗

（・9）（iii〕・蝕陥・剛陥・rl兄）］f・……｛・，・，…，・｝・

Proof． Since T is derived fr㎝atwo−s）mbol B−array of strength 2兄， m Le㎜a 1， Theor㎝2and L㎝旧3sh㎝that T is a 2 −FF design of resolution ・（・lt；1・kl・、）if孤・・n・y・f・he c・・d・・・…〔・〕，〔ii）肌d〔iii）…d． The remaining Part of the proof is to show that such 2m−FF design T is へ

蒜：：’．，㌻；・：。曇；・。還lili；：二；蕊竃：：、：㌻：：Ch

・he・）・，・Cr・Σ：二；・lu・u〕，・h・・c・var・ance・ma・・iX・f・：二；・lu・u〕G・・g・・㎝by Var（・：二；・5u・u）亘）・Var（頑）・Var（…i“er）・岬，・2・A．へ

、一、。。、一、誌ご。lhi、：’；1二1：芸蕊二1：1：r

）is invariant under any permtation of m factors． This completes 昆・2 ・8・2・・1・1・・1・3・・1・3・．61／2（。一・51／2（・・4y−z・u〕，・1・3・一・・51／2〔・・3・ −u），・（・…6y・…3・〕，・；・2 ・一・・1！2〔・・… 一・u〕，

・…1／2〔x−z・u），・；・2・・5…by・z・・5・，

−2・31！2（・x−z−5u），・1・3・4〔…3・…）， 347

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S．YAM㎜）AND Y． HYODO

0 1 0

ーー

馬Lと2陥塩

K K K

・8〔，…），・1・1・・6・，・1・2・8・61／2y・・・6（・y・・〕，・｝・2・・，・｝・2・48y・・・6・，・；・1・・6・・1！2・，・；・1・32…1・°… ．49、52晦2、・2・48・晦2。・8・92・⑳・u・・22SS・y2・u・・，＝ 24576y2z ＞ o， lK21＝ o， rank K2 ＝ 1 and lK31＝ o． Therefore， we have 〔i）（ii〕〔iii〕 This co叫）1etes the pro◎f．・ank k・r・nk［Ko・r5°）］，・ank k・rank［K，，・r5’）］・・ank K、・剛K、・rl1）］，・ank K、・剛K、・rl2）］，

・ank K、・剛K、・rl3〕］…rank K，・剛陀・rl3）］・

The folloWing numerical examples for m＝ res・1・ti・n R〔00・011Ω3〕・

N

66777888993333333333

句1212312323 性1111111111

7匂0000000000

＝

m力0000000000

_{㌦1111111111 馬2222222222 ％2132143243}

7 and 8 can be shown to be of

N

55666777884444444444

句2323423434 性1111111111

8％0000000000

＝

m当0000000000

_{㌦1111111111 鴨3333333333 ％4354365465}

ACKNCiWLE…DGMENrs ’n，e 、uth。rs express their thank・t。世． Masahid・K㎜・la f・r hi・valuable CO㎜nents．

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349 ㎜口）00N（EPT OF RESO［UrlON OF A m鵬IGN 11】［2］［3］［4］ 15］［6］［7］｛8】［91 ［10】［11］【12］【13］［14］ Box， G．1…．P． signs， Part Box， G．E．P． signs， Part Bose， R．C． and Srivastava， J．N．（1964〕． fractiolls． 5㎞妨〃α （A） 26， 117−144． Bose， R．C． and Srivasta照， J．N．（1964）． anced designs and their analysis，＿ anced factorial fractions． 5白η肋〃α 〔A） Fimey， D．J．（1945）． Aηη．吻【カカ． 5勧力i8t． 28， 993−1002． Shirakura， T．（1976）． REFEREN（路鋤｝㎞ter， J．S．（1961）．∩。2k’p fracti。皿1飽。t。ri。1由一二溜叉θ驚8蘭1’蓋：’、k−P・fra。，i。皿、 fa。，。r、。、 d。． II． Tec』励8 3， 449−458． Analysis of irregular factoria1 MUltidimensional partially ba1一 with apPlications to partially bal一 26， 145−168． The relationship algebra of an experimental design． Balanced fractiona1 2m factorial designs of even resolution obtained flrom balanced arrays of strength 2兄with index ｜」夏」＝0・ A万n・ Stczti8t・ 4， 723−735・ ShirakUra， T．（1977）．（bntributions to balanced fractiona1評factorial designs derived fr㎝balanced amys of strength 2見． Hirosh鋤ぬヵλ．♂． 217−285．’

S㌫鵠・、㍊C°L擁s：工瓢。器f鑑鵠n夢t認壽1。提霊，

、：…濃㌶31‘当1㌶品，鑑篇㌶」1〕P61fa’li、。皿、血。t。。、。、、esign，．

ε．ハ7． Roy Me，nor，iaZ 肋Z砺， 689−706．こ仇￠り． o了North CalvZinaαπ411砿鋤 Stati8ti■α2 1n8titutθ．

S：蕊蹴：三蒜鰹：la蓼Dgi堪1㌶盟隠蒜e註s；…：。蕊。°｛，

with applications．4朋． dOth． Stctti8t． 42，722−734．

Y灘｝°痘芸㌫霊，1ふ11器ll：。オ霊蒜＝蒜：ラ％“篇；．

to appear・

Y霊霊：孟；元・よ6’蒜：桓T濃、盟血臨蕊1’、㌫：竺㌃a鷲

Statiet．ハ12th． 279 143−157．

Y蒜篇。i；・。i罐↓よ，霊麟≧，M；，（盟躍：：綴’函

factorial designs of higher 〔2兄＋1） resolution． ”Essαys in Pro励iZity and Stati8tie8” （Bds． Ikeda， S． et a1．〕 Birthday volume in honor of Professor J． Oga鳩，73−94・ Yates， F．〔1937）．The design and analysis of factorial experilnents． lirper・必Z Bu？eau o了Soiz SOI enee， TechnieaZ Convnunteaカions 35．

庇P駅㎜OF肌IED順酬岱

SCIENCE UNIVERSITY OF TOKYO and DEPARTMENT OFハ帆THEMATICS SCIENCE UNIVERSITY OF TOKYO

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直 ← 、会、（ ■ ・ ’