AN EXTENSION OF THE CONCEPT OF S IMI工，ARITY PRESERVIED IN
THE COMPOSITION PROCESS OF EXPER］｝正INTN DESIGNS
Sumiyaslu YAMAMOTO and Yoshiftmi HYOCO 〔Received May 7， 1984） 1． Introduction ．， ’ In his theoretical work on experimental designs， Ya皿amoto ［4］，one of the present authors， has formUlated the concept of simi larity and partial simllarity between an、 algebraic structure represented by a relationship algebra over a vector space and its image algebra induced naturally by a linear mapPing from a vector space to another in the comPosition process of experimental designs。 He introduced further several concepts related to the composition of relationship algebras on the basis of the mapping． He and his colleagues ［5］ have constructed systemqtically several series of associ− atlon algebras， such as， trianguユar， nested or generalized group divisible， factoria1， hyper−cubic， and orthogonal Latin−square series of assQciation schemes． He also discussed in ［6］ the concept of we11−designed designs in a sense such that the structure of relevant parameters expressed by a relationship algebra is mapped similarly or partially similarly to the space of eXperimental mits upon e1㎞inating the effects of irrelevant parameters． Factorial designs， randomized block designs， Latin−square designs， balanced and partially balanced incolTIF）1ete block designs， and so on， can be shown to be in the class of well−designed designs of the above sense． In those cases， however， relationship algebras introduced among rele− vant and／or irrelevant parameters are co㎜utative and their ideals are a11 1inear・ The deconl［）osition of the sum of squares corresponding to the de− composition of algebra into the direct sum of irreducible two−sided ideals is preserved with similarity or partial similarity by the mapping． In a s）㎜・t・ical fracti…12m fact・ri・1 desigh・・w・1ユ・・a・）㎜・t・ical fracti・n・1・m fact・・i・l d・・ig・i， h・w・ver， the re1・ti・n・hip。19。bra， whi。h i、 introduced among the space of treatment effects up to， say， ￡−factor interac− gions due to the s）㎜etry of factors， is not co㎜utative． Some modification or extension of the concept of similarity or partial similarity of the mapping is， therefore， needed in order to characterize the wel1−designedness of the 163S． YAMAMOTO AND Y． ｝rYODO balanced fractiona1 2m as well as−sm factorial designs． In this paper， an atte㎎）t is， theref（）re， made to generalize the concept of partial si皿ilarity with respect to a linear mapping of a relationship algebra． In Theorem 1， it is shown that a set of generalized conditions， similar to that of Theorem l in ［4］， is necessary and sufficient fOr the newly defined extended concept of partial si皿ilarity of the mapping． As an application， it is shown that a necessary and sufficient condition fO・th・t th・de・ign mat・i・・f・fracti・n・12m fact・ri・1 design indUces a par− tially si皿ilar mapping of a triangular multidi皿ens．ional association algebra， which is defined naturally among the factorial effects up to ￡−factor inter− actions， to the space of asseπi）1ies or experimental units， is that the design is obtained from a balanced array of strength 22， in so far as the resolution of the design is 22＋1 0r the corresponding information matrix is nonsingular． With respect to the fractiona13m factorial designs of resolution V， the results given by I（uwada ［1］ show that the use of three−symbol balanced arrays of strength 4 is necessary and sufficient for preserving the partial similarity of the mapping of parameter algebra． The results by KuLvada and Nishii ［2］ also show that， in the fractiollal sm factorial designs of resolution 22＋1， the use of s−symbol balanced arrays of strength 2￡ is also necessary and sufficient fbr preserving the partial similarity． 2． Similar and partially similar mappings Let R be a皿atrix algebra generated by a finite number of real sy㎜etric matrices of order m． It is㎞om that R is semi−simple， completely reducible and can be uniquely decomposed into the direct sしm of irreducible two−sided ideals， say，
（1） R＝R1θR2θ゜’°㊥Rk
apart froln the order．of the ideals． Each component algebra Ri is isomorphic：da鵬：器xR；g蕊、愁P芸。lrlζ1，Lllvml㌻㌔、lh諜「
・dea・b・・is sati・fy・…EC）・，Sl〕一・ij・mv・6）・Th…、一・：±、・Sa）飢d E・Σ至．、Ei are@the principal idempotents of Ri and R・respectively・ The ranks of tllose ・d・mp・t・n・・are・（・、）・βi・・、・、・…〔・）・Σ1．、βi・m，・especti・…．SIMILARITY PRESERVED IN EXPERIMENTAL DESIGNS Let F be an n x m real matrix which maps a column vector of m−dimensional Eucl idean space Vm t・that・f n−dim・n・i・na1 Vn・i・…
（2〕 F・Vm∋9−一旦★＝Fg・Vn・
The mapping F induces naturally a mappingσof R， i．e．， 〔3〕 σ ：R∋A−一一→A★＝FAF’∈R★＝｛A＊lA★＝FAF’， A（R｝ or σ：R−一一一→・R★＝FRF’＝σ〔R〕． DEFINITIoN：Alinear mappingσof R induced by F is said to be partially similar if it satisfies the following two conditions： 〔・〕 〔・）・〔・、）・〔・j〕・δij・：三、Σ三、・El〕・〔・EC）〕・・h，j・・，・，…，・， 品ere these c・eff・・・・・・・・…ce・C（’〕・・ll・島）ll are・・㎜・t… positive definite or zero and not all zeXo simultaneously． ．〔・） 〔ii）lf C〔’〕・・p・・i・i・・d・f・・…，・h・n・（・（・、〕）・r〔・、〕． Th・m・pP・・g is ca…d・・…ar・fC〔’）・c・。．〔…〕・・r…，・，…，・， 1 ・・here l。 d・n・t・・th・血it， m・t・i・．・f・rde・… Note that the definエtion of partial similarity first defined in ［4ユ is ・・r・re・t…ti・・i・・b・・C③・c、・。．，品ere c、，・are…・g・t・ve c・n・t・…and l ・ not all zero si皿ultaneously． ・ T田……OREM 1． AZineCt？maρPinσσof the aZ4ebrαR〈fefined by F isραrtiαz乙y simiZay if and onZy if F sαtisfies the oondition： ・ ・〔・） ・’F−・亭．、Σ1三、Σ：三、C岳〕・￡〕・Z，吻・・c（’〕・ll・岳〕1い・・，・，…，・，膨一劫・d・蹄・・・・…⊇…．・・t
αZZ aepo3 and Z isαMα亡rix satisfyinθEZE＝0． ’・uρpbse u鋤・，鋤Zン・a・a。・・and。‥。吻C〔i）・。α。。魎ti。。 d。fi。it。 fove i＝1，2，… ，hαnd geroプ’or i＝h＋1，… ，k． Then， the imαgeαzgehra R★＝σ〔R〕 is α乙SOεθ励一8⑭Zθαnd eanわθdeeOiuposed int・the di”ee t szen of i？redueihZe tz“o− sidθd idθαZs su已h α6： 〔7〕 R★＝σ（R〕rσ（R1〕㊥σ〔R2〕㊥… ㊥σ〔Rh〕・ llve？y oonzl）onentαzgθz）アασ（Ri） o王R★is isomorρhic カo Ri an〈I the mztztipZioity of the ”ep・・…t・力伽z・αz…i・Th・id・aZ b・・is・了・（Ri〕z・9加・・々S． YAMAM）TO AND Y． HYODO （・〕 ・畠〕・・1三、Σll、b品〕・le〕・（・EC）〕，’…，・，…，・，，，，・・，・，…，・、， 吻訓・￡〕II・Ai i／2Q、・nd・Q、・・α・・励・，・u・・’mα・・…s・t・・f、・・g ．Q、c（’）Q三・A、・d・・g〔・｛’），・S〕，…，・9））．”．． 1 th…r”e・P・nding d・・卿・・iti・n・f・prineipaZ翻卿・t・nt Eジ励励・αZZg
°「鞠・mZ pW吻・Z鋤ρ・t・nt・ 9‡・τ輪』Z・i・
（9） E＊＝El＋E》＋…・Eζ・ 』・E至・Σ1三、・畠〕・・1：、Σ1三、Σ：：、b品〕・￡〕・・岳〕F．’．田、隠∵1；置：、竃；㌶）㌘，ll 、，W二llli。ご鑑1∴〔。〔，、〕，
・・〔・E、Fつ一r〔・、F’FE、〕・r（・1三、Σ：1、・￡）・5e〕）−r〔・、）． Th・・i。。ar m。pP・。、。 defined by F is， therefore， partially similar． Conversely， ass㎜e thatσis partially s迦ilar， i．e．，〔4）and〔5）hold． W・…m assum・，・ith・ut 1・ss・f g・nerali・y， th・t C〔i〕・， are p。si・i。。 d。fi。i，。 for i＝1，2，… ，h 〔1≦h≦k） and zero otherwise， we have 〔・〕FE、F’FE、F’・Σ：1、Σel、・Sl）・・Se〕・・f・r…，・，…，・， （b〕FEiF’FEiF’ ＝ O f・r i−h・1，…，k， （10） ’ （c）FEiF’FEjF’＝°f・r i・j・i・j＝1・2・…・k・and ． ．〔d）・〔FEiFつ＝r（Ei）．f・r i・1，2，…，h・ Since Ei’・ar・㎜t・・11y・rth・g・n・1・卿・t・i・id・叩。t・nt・， there exi・ts．a・・ °「th°g°nal mat・i・Pwhi・h di・g・n・1i・e・Ei’・ ・im・lt・n・・u・1y，・u・th ・1・ 〔11） P’E．P＝ 1 0 °．｝β1＋…・βi．1 ● ．01． ｝β． ° 1 ’10 ・ ● ． 0 ＝ ei， or Ei＝PeiP’ for i＝1，2，・・●，k， whe「eβi＝r〔Ei）＝・mi・・i・SIMILARITY PRESERVED IN EXPERIMENTAL DESIGNS Put FP＝K， （a，） （b’） 〔10’） （ct） 〔d’〕 Put Y＝ （10） may be expressed as ・・iK’・・、K’・Σ：1、Σ：三、・艮〕KP’・￡）・K’・・r…，・，…，・， K・iK’K・iK’＝0’f・r i＝h＋1・…・k・ ・・iK’・・j・’“・f・r・・j…j…2・…・k； ・・d ・〔K・iKつ＝・（・i〕f・ri＝1・2・…・h・ Bl・B2・…・βk and 1・t K1・nd K2 be submat「ices°f K c°mp°sed of first Y and the remaining m−Ycolumns， respectively・i・e・・K＝［K1：K21・ ・・n・t・，fu・th・r，・h・f…tY・YP・…ip・・・・・・…f・i・P’・、p and P’・5e〕P ・・f、・…烏），respect・…y，・・e・， ＝ ・− e コ f．・0 1◆ ● ■ ● ● ・ ● ■ 0 ・0 ● ， ・’・EC〕・・ コ f（i〕：O uv ． ■ ● ● ◆ ● ・ ■ ・ ロ 0 ・ 0 ： for i＝1，2，… ，k． Then， fr㎝（1門we have 〔a・r）・、f、Ki・、f、Ki・Σ：±、Σ1：、・SC〕・、fGl）・i・・r…，・，…，・， 〔b「T〕KlfiKiKlfiKi ＝ O f・r i＝凪…・k・ （’°”〕 @〔・tT）・、f、Ki・、fj・i・・f・r・・j…j・・・…◆・…and 〔d”〕・（KlfiKi〕＝r（fi）f・r i＝1・2・…・h・ ・…（・，’〕…・ii…av・K、f、〔・、f、〕一＝…hu・w・h・v・・、・、・・ for i＝h＋1，… ，k． The latter shows that all the elements of the last Bh．1・…＋βk・・1・㎜・・f Kl are・er・・D・n・t・th・・ubm・t・ix c°mp°sed°f fi・・tρ一β1＋B2＋…＋βh・・1㎜・・f Kl by L・th・nw・have 〔12〕 ．Kl＝［L I O］・KlKi ‘ LL’・KiK1 ＝ の ピL・ 0 ● り り ’ ■ ： 0 ・ 0 ● From （bt’〕， （ピ’〕，and 〔d’T〕，we have （・3〕 ・〔・’L〕−r（LLつ・r（・、Ki〕・r（・、（・至．、f、〕・i〕・ΣL、・〔・、f、Ki〕 一Σ㌧、・〔fi〕・Σ』βゴρ・
168 S． YAMAUvK）TO AND Y． HYODO The matrix L’L is， therefore， nons ingular． Using 〔10’っ and （12），we have （・4） LL’LL’・・．・K、Ki・、Ki・K、〔・至．、f、）・i・、〔・至。、f、〕・i ・Σ1．、Σ：三、Σ：三、・C）・、fC〕・i−L（・：．、Σ1三、Σ：三、・Sl）富：〕〕・’， 曲ereる：〕・・th・f…t・・ρP・・…p・・・…r・f fGC〕・・fU・t・p・y・・g th・f…t −1 −1 L’ from the．left and by L〔L’L） frOlnand the last members of 〔14） by 〔L’L） the right， we have
（・5）・’…：．、Σ1三、Σ：1、・餌：〕． ．
Thus we have （・6）・i・、・Σ：．、Σ：三、Σ鵬、・SC）・C〕・Σ1．、Σ：：、Σll、・S3〕・’・EC〕・， and consequently， （17〕 ・’・一・〈。、Σ：1、Σll、・￡）・’・GC）・・ 0：KfK2 ■ ■ ロ ロ ロ エ ■ ロ ： K三K1：KEK2 ・ Hence， if we put the last member P’ZP， then｝ザe have ． （・8〕・’・・Σr．、Σ：三、Σ1三、・￡）・Ee）・Z and Z satisfies EZE＝0． This cornpletes the proof of the if al〕d only if part of the Theorem． The r㎝aining Part of the Theorem may be proved as follows・ sinceσ is a linear mapPing of R andσ（R）σ〔R〕Eσ〔R） holds under （6〕， σ〔R） is an algebra． Mbreover，σ（R） is semi−simple since it can be generated by symmetric matrices of order n・ As Ri＝EiR Ei・RRi E Ri and RiR⊆Ri hold・ it is easy to show thatσ〔R）σ（Ri〕Eσ（Ri）andσ（Ri〕σ〔R）⊆σ〔Ri） for i；1，2，… ，h・ The imageσ〔Ri）＝FRiF’ of two−sided ideal Ri is also a two−sided ideal or nu・・acc。rdi。g as C〔i〕i， p。，i・i。。 d。fi。i・。・r・e・・． ・・i…：三、Σ：三i・品〕・IC）・GC）一・pq， we can・h…h・tSIMILARITY PRESERVED IN EXPERIMENTAL DESIGNS 〔・9〕 ・畠）・9〕・δq。・ll）・・・…，・，…，・・ Thi・m・an・th・t（8〕i・th・ideal b・・i・・f・（Ri）・Th・p・i・・ip・1 id㎝P・tent・ E窒 of the ideals σ（Ri） and E★ ofσ〔R〕 are given by 〔9〕． 3． S㎝e apPIications C・n・id・・fi・・t 2m fact・ri・1 d・・ign・with re・pect t・mfact。r， ea、h at two levels and assume・that丸＋l or more facto士 interactions are negligible． ASsume further ￡≦m／2 for the simplicity of description． Amo㎎those factorial effects up to見一factor interactions， i．e．，general m ・θﾓ・Φ・・’・・ffect・・θt、・〔；〕加・−fact・r’・…ac・i・n…t、t2・…・・nd Φ見一fact・r’・teract’・n…t、t、…・、・・・…ngU・ar皿…dpt・n・i・na・par・i…y balanced （丁卜㊥PB） association algebra is introduced as a relationship algebra expresslng a natural relationship among parameters （see［7］〕， i．e．，θ tlt2°・◆tu andθti・》…・↓a「e a，th ass°dates’f and°n’y’f （2°） 1｛t・・t・・’°’・t。｝・｛ti・t2・◆°’・・↓｝1＝mi・（…）一・： Note that this relation df association is equivalent to the introduction of the maxi皿al invariant function with respe（三t to the symmetric group of the permutation of m factors． The’algeb士a， denoted by A， is semi−simple， completely reducible， and is decompQsed into di・rept suih of 2＋1 two−sided idea1・Ao・A1・…・・nd A兄・曲ere Aβi・i・・m・rphi6・・th・e・mpl・t・m・吐 algebra of order ∫し一β＋1． …v、…Φ・〔》〕・…・！；）・nd・・n・ider・、・・、・…ices wh・se r・w・ ．apd・・1㎜・a・・n・mbered・by th・ ・ub・et・｛t1・t2・…・t。｝・haracteri・i・9 the fact・r・…ffect・θt、・、…・u arran・・d・n a・・pP・・P・・・…rd・r…t ・！u’v〕 be su・h・m・t・iX ・yh・・e｛t1・t2・…・t。｝th・・w・・d｛ti・t2・…・t↓｝th・・1㎜・nt・y ’s皿’ ey’f and°nly’fθt、t、…・。 and eti・2…・↓a「eα一th ass°c’ates f°「a fixed ordered pair of u and v， and zero otherwise． Then， it is shown in ［7］
S． YAMAMOTO AND Y． HYODO that TMDPB association algebra A is given by the linear closure of those （…）（…）（…3）／…t・・ces ・t！”・v〕・・一・，・，…，…（・，・〕，・，…，・，…，・． 1…a・…h・wn・・［・］・h・t th・・d・a・b・・i・Dlu・v〕＃・・，…，…，…，・，
・fABi・gi・・nby
〔・・） ・lu・v〕＃・（・lv・u〕＃）’・・9。。・巴，。）・！’・v〕，・…u・v，where
・巴，。〕一・19・v）｛〔；）一〔，！］、）｝／・ΦΦ（晶）｝，・nd ・i9・v）一・9．。〔一・）α一b〔u；β〕〔：：：〕（m’uBβ÷b〕｛〔m｛｝lir｛1β）〔y：：）｝1／2／〔v’9＋b）』． In・fracti・n・12m fact・ri・1 desig・T・・mp・・ed・f・assembli・・with 「espect t°mfa・t・r・・the e・rPect・ti・n lT・f th・・bserv・ti・n vect・r野i・ given by ． 〔22〕 nT＝E（ZT）＝ETθ 曲e「eθ＝（θφ・｛θt、｝・｛θ・、t、｝・’°’・｛θt、t、…・、｝〕’and ET’s an n×v・des’gn matrix determining a mapping ofθ to the space of experimental units， or pre− cisely speaking・t・the e・rPect・ti・・nT・f・th・・b・erv・ti・n vect・r XT・Th・ normal equation for estimating θ is given by〔23〕 MrO＝EセT’
・he「e叫＝EflET・A1ユf・rt・ri・1・ffect・up t・￡−fact・r i・teracti・n・are慧盤潔。1211、ごB㌦’1，三nlln麗byT罵gl。’蕊。1：．be°f
Af・acti・n・12m fact・ri・1 d・・ig・with・assembli・・can be regard・d・・a tw°←s）mb°1 n×marray T・In an n×marray T・if in every・ubarray TO・・㎎・・ed °ftc°1㎜・・f T・・v・ry・vect・r・f w・ight i・ccu・・exactly Vi t血es a・r・W・・f TO・ then T is called a balanced array of strength t， size n， m constraints， tw°『s帥・l and i・d・x・et｛μ0・V1・…・μt｝・ It has been shown in Theorem 3．4 0f ［7］ that a necessary and sufficient c°nditi・・fb・th・t th・i・f・m・ti・n m・t・i・Mr・EfET can be e・rPressed・・SIMI LARITY PRESI…RVED・IN BXPERIMENTAI． DESIGNS 〔24） ・評，・Σ＆．。Σ（ll．。Σ：三言（u・v〕・1。−vl．2。・8u・v〕 ・r Mr・E｛］ET・Ai・th・t T i・ab・1・n・ed array・f・trength 22・1・thiS ca・e・ the infomation matrix can also be e）甲ressed as〔see［3］，［8］）’ （25〕 and it ・宇，・Σ1。。Σ｛二1掛；・」・IB＋’・B＋」〕＃・ is nonsingu〔Lar if and only if 1Kβi 0，0 cβ 1，0 Kβ ⋮ Jl，一・β，O Kβ 0，1 Kβ 1，1 Kβ ⋮ ◆ ● ● ・ ． ● ● ● ● 1 ， β ’ 兄DP K 0，Sl，一β Kβ 1，2一β Kβ ●●● ￡一β，R・一β Kβ ＞0 for a11 β＝0，1，… ，￡． Combining those results with Theorem 1， we have the followirlg： T肥0肥M2． ln・鋤・力鋤・Z 2m伽力・・’i・Z de吻・・τ・…励・・22・1，・ z拠・・卿㌦9・fA d・fi・・d by ET i・P・・ti・ZZy・i・iZ・・ if・nd・晦if・T i・α bαZaneed avrαy of strength 22． rn paアtZouZαア． the maρping is similα？ ifαnd oアzZy if T isαn o？thOgonaZαrrαy o了strength 22． For fractiona1 3m factorial designs of resolution V， those results given in Kuwada ［1］ show that the mapping of parameter algebra to the experimental space is partially si皿ilar if and only if the design is a three−symbol balanced …ay・f・trength f・u・． Similar argum・nt・a1・・h・1d f・r th・fractiの・1・m fac− torial designs of resolution 22＋1 by using those results given in Kuwada and Nishii ［2］． Balanced design of resolution 2￡＋1 0btained by a balanced array of strength 22， therefore， belongs to the class of we11−designed designs in such a sense that it preserves partially simi！arly the algebraic structure of parameters in the space of e）q）erimental units．
』S．YAMAW〔）TO AND Y． HYOI〕0 R正iFERENCES ［1］ 12］ ［3］ ［4］ ［5］ ［6］ ［7］ ［8］ Kuwada， M． 〔1979〕．Balanced arrays of strength 4 and balanced fractiona1 3m factorial designs．」． Stαtist． PZaηning Inア． 3，347−360． K㌫；麟1、器謡’そ。Se，｛1：：1〕9・㍗。：，：霊cll：呈、：：lw鷲昆駕甑． 5ヒ）o． 9， 93−101． Shirakura， T．（1977〕．Contributions to‘balanced fractidna12m factorial designs derived from balanced arrays of strength 2見． Hiroshima hath．」． 7， 217−285． Yamamoto， S．（1964〕．S㎝e aspects for the composition of relationship algebras of experimental designs．」． Soi． M？oshimα Univ． ・4−1． 28， 167− 197． YalRarnoto， S．，FUj i i， Y． and Hamada， N． 〔1965）．Colnposition of some series of association algebras．」． Soi． Hiroshima 乙lniv． ・4−Z． 29， 181−215． YalTlamoto， S． and Fuj ii， Y．（1969）．Relationship algebra and the analys is of experi皿ental designs （in Japanes’e）．Sugαku 21， 264−274． Yamamoto， S．，Shira㎞ra， T． and Kuwada， M．〔1975）．Balanced arrays of strength 2k and balanced fractiona12m factorial designs． Ann． lnst． Stαti＄t． hath． 27， 143−157． S．，Shirakura， T． and I（uwada， M． 〔1976）．CharacteristicYa皿amoto， polynomials of the info㎝atioh matrices of balanced fractiona1 2m factorial designs of higher （22＋1） resolution． ’｛Essays in l2robaZ）iZity αη48カα協髭cs”（Eds． Ikeda， S． et a1．〕Birthday vo1㎜e in honor of Professor J． Ogawa， 73−94． DEPARTMENT OF APPLIED MATHEMATICS SCIENCE UNIVERSITY OF TOKYO and DEPARTMENT OF MAT田三MATICS SCIENCE UNIVERSITY OF TOKYO
