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1 Represetatio of Fractioa Factoria Desig i Terms of (,)-~fatrices By D. A. ANDERSON* AND W. T. FEDERER Uiversity of Wyomig, Core Uiversity SUMMARY Let T deote a mai effect pa for the s factoria with N assembies, that is Tis an x matrix with eemets from the {,,...,s-. Deote by T, T,., T s- then x icidece matrices of,,...,s- respectivey, so that T it. ad T. = JN. Usig the x Hemert poyomias to defie sige degree of freedom mai effect cotrasts we write E{y} = X~, where X is the desig matrix correspodig to T.. XG = X* = [ ; T... : T A trasformatio G is obtaied for which. s- J thus givig a represetatio for the desig matrix directy i terms of the (,)-icidece matrices. t is show that ei = (s!)-(-)(s-) ad jx'x = (s!) 2 x*'x*. f Tis a saturated mai effect pa, the x = (s!)x*. Thus the determiat of the iformatio matrix is directy expressibe i terms of the determiat of a (,)-matrix. These resuts are exteded to icude the geera asymmetrica factoria s.. Upper bouds are obtaied for the determiat vaues of X* whe X* is square ad i geera for X*'X*. Oe importat aspect of this represetatio is that the costructio of mai effect pas ad a assessmet of their goodess via the determiat criteria ca be studied directy i terms of (,) matrices. A extesio to icude iteractio terms for the s factoria where s is a prime or prime power is give. Keywords: D OPTMAL DESGN: MAN EFFECT DESGN: HADAMARD BOUND. *First draft writte whie o eave from the Uiversity of Wyomig ad at Core Uiversity. ~J ~~~ ~f/

2 -2-. NTRODUCTON a factoria experimet ivovig factors with the ith factor at s. eves i =,2,...,, the tota umber of treatmet combiatios is is.. i= A desig is usuay represeted as a N x matrix T whose N rows deote the particuar treatmet combiatios ad whose coums correspod to the eves of the factors. The eemets of a coum of T correspodig to a factor at s. eves are itegers from {O,,...,s.-}, i deote the s. eves. the aaysis, the matrix T is amost uiversay repaced by,2,...,, to a matrix ~- -~XV which refects the v sige degree of freedom parametric cotrasts i the parametric vector ~ from the usua regressio equatio The orma equatios are X~X~ = X~y, ad soutios to the orma equatios provide best iear ubiased estimates of estimabe fuctios of the parameters i~ The matrix Xc is caed the iformatio matrix of the desig T, ad if X~X is osiguar the variacecovariace matrix of Sis proportioa to (X~X)- Most criteria of goodess of a desig deped upo some fuctio of (X~X)-, as for exampe, the determiat, trace, ad maximum root criteria. f X~X is siguar we cosider a coditioa iverse (X~X) ad restrict to estimabe fuctios of the S's. Sice the matrix X is obtaied directy from the matrix T, a of the iformatio cocerig the goodess of the desig (i terms of some fuctio of (X~X)-) is cotaied withi T. Thus for the purpose of costructig good desigs ad the compariso of desigs, the simpest ad most direct represetatio of this property i terms of T itsef woud be usefu. Raktoe ad Federer [97] obtaied such a represetatio directy i terms of the (,) matrix (:T) for mai effect pas for the 2 factoria,

3 -3- where deotes a vector with every eemet uity. Sectio 2 of this paper we preset a simiar represetatio for the. s. factoria, ad we p represet cx~x)- directy i terms of this represetatio. the third sectio a upper boud o the x~x is obtaied for both the symmetric ad asymmetric factorias, ad the miimum ozero vaue of this determiat is idicated. The importace of the represetatio preseted ies i the isight that may be gaied toward the costructio of fractioa factoria pas ad the assessmet of their goodess via the deterimat criteria. 2. REPRESENTATON OF MAN EFFECT PLANS N TERMS OF (,)-MATRCES Cosider first the s symmetrica factoria ad the correspodig mai effect pas for estimatig the v = + (s-) mea ad mai effects uder the assumptio that a two-factor ad higher-factor iteractios are zero. Let TN x be an x matrix, N ~ v, with eemets from the set {O,,...,s-} deotig such a mai effect pa. matrix of eemet i i T. correspodig eemet of T is i or ot. LetT., i = O,,,s-, be then x icidece That is, a eemet oft. is oe or zero as the The, s- L T = J ad T. i N x ' = s- LiT.. i=o Typicay mai effects are defied i terms of a set of orthogoa (2.) poyomias. For coveiece, we sha use the Hemert poyomias, - -2 (2.2).. -(s-) eve though ay set of orthoga poyomias may be used. The, if y deotes a N x observatio vector correspodig to T, et 2 E[y] = X~ ad Cov(y) = cr N,

4 s- where_ _= (!J,S,...,s s,...,s.s,... f\ ) deotes the v X parameter vector of sige degree of freedom cotrasts as derived from Hemert poyomias ad where X is the desig matrix. The desig matrix may be writte i terms of the T. as X = [ s-2 L T. - (s-)t ] i= s- (2. 3) theorems 2. ad 2.2 we trasform the desig matrix X for a mai effect pa from the s factoria ito a (,)-matrix. The resuts are exteded i theorems 2.3 ad 2.4 for the geera ~ si asymmetrica factoria. The = importace of these resuts ceters aroud the facts that (i) cosiderabe theory is avaiabe o the costructio of mai effect pas for the 2 factoria ad o the vaues of the determiats of (,)-matrices, (ii) this theory ca ow be appied to the costructio ad to the cosideratio of optimaity of mai effect pas from the geera factoria, ad (iii) these resuts exted the resuts of Raktoe ad Federer [97] for the 2 factoria to the geera symmetrica ad asymmetrica factorias. Cosider the coum operatios o X resutig from postmutipyig by a matrix G as foows: XG =X*, (2.4) where G is the foowig V X v matrix: ' S- ' s.- ' S- ' s- - 2 (2. 5) G= - 2[3) 3 3 (4) 3 (4) - 4 s(s-) s(s-) s(s-)

5 -5- The foowig resuts ca be verified by direct cacuatio. Theorem 2.. Uder the trasformatio XG = X*, we have! Ts_J, that is, the trasformed desig (b) (c) matrix is a (,) -matrix composed of icidece matrices T.,. i =, 2,..., s-; ei= (s!)-(-)(s-); - G = (s-) si (d) X = X*G-; ad (e) x.. x = (s!)2ix*.. X*. Theorem 2.2 f N = v ad the desig is a saturated mai effect pa, the determiat of the resutig square matrix may be expressed as: The proof foows directy from parts (a), (b), ad (d) of theorem 2.. Exampe 2. Cosider the saturated mai effect pa for the 3 4 factoria derived from the foowig pair of orthogoa ati squares: ~j ~ 2 ad _2 2 2! ~J

6 -6- The, a o ~ o o i T = 2 o, T =, ad T 2 o. 2 a o!! ' 2 2 Q i! 2 2!!o!! 2 2 oi :o - ' :... _, _! o Sice T is a orthogoa array, 27 is the maximum vaue possibe for x* of a 3 4 saturated mai effect pa. The effect of the structure costraits (2.) o T, T, ad T 2 is apparet from exampe 2.. The vaue 27 is far beow the maximum vaue of the determiat of a (,)-matrix of size 9 with a eadig coum of oes. fact, for such matrices, Aderso ad Federer [974] obtaied the foowig vaues: a itegers ~ 33, 36, 4, 44, 48, ad 56, with o assurace that a vaues have bee obtaied or that 56 is the maximum vaue. Thus the argest possibe vaue for the determiat of form ~ T ~ T 2 j where T ad T 2 satisfy (2.) is a itermediate vaue amog a possibe vaues. The resuts for the s factoria are ow exteded to the geera asymmetrica factoria s x s 2 x. x s = ij s. Sice factors have.=. differet umbers of eves, it wi be coveiet to cosider the represetatio idividuay for each factor. Thus et the ith coum of T correspodig to the ith factor be deoted by ~i so that T = [~ ~ 2... ~]. Sice F. has s... eves here deoted by O,,...,s.-, the coum d.. -]_ cotais these symbos. Let the icidece matrix of the eve j i the ith coum d. be deoted by d.(j). The equatios (2.) are thus expressed -]_ -]_

7 -7- for each i as si- j=o The s.- ~.(j) ad d = -i Si- L; ( j=o J ~ i j). (2. 6) coums i the X matrix correspodig to the factor F. are give by si-2 [d.(o)-d.(): d.(o)+d.()-2d.(2)! ; L d. (j) j=o - (s.-)d (s -) -i -. (2. 7) ad we et Fiay, et * z. [d. () d. (2) d. (s.-)] (2.8) s} - G. H. j= /s. /s. -/2 /2 (3) -/3 /s.(s.~) /s. (s.-) /s. /s. /s.(s.-)-/s. deote the matrix G of (2. 5) writte oy for the ith factor at s. eves, ad for a factors et j; j s - s - s -, 2 : H G H2 (2. 9),- i : :. io H - - t ca be ascertaied that if X is the desig matrix for a mai effect pa from the asymmetrica factoria that the foowig two theorems hod: Theorem 2.3(a) XG=X* = []:_ : Z~! Z~! (,)-matrix with eadig coum ; (b) (c) ij = - G = (s.!) ad i= x--x = (s.!5x*"x* i= Theorem 2.4. f T is a saturated mai effect pa for the asymmetrica s. factoria, the x = _ ( s.! ) X*. =, j : Z*], where X* is a

8 -H- Note that the coums of X* are ordered so that a (s.-) coums correspodig to the ith factor appear together. This orderig is possibe ad sometimes preferred for the symmetric case aso. The coditios (2.6) are actuay structure costraits o X*. For exampe, each row of Z~, i =,2,...,, ca have at most oe vaue of oe, hece the ier product of ay two coums of Z~ is zero. 2 Exampe 2.2 A desig for a 2 x3x4 factoria i eight rus ad its correspodig (,) represetatio are give as ~ ~ ~! : 2 2i, i T =o 3: X* = : ' o; i i 2 f!! 2:! 3 ' ~ ~ ~ L t is easiy show that x* = 6 so from Theorem 2.4 x 2!2!3!4! BOUNDS ON THE DETERMNANTS OF NONSNGULAR DESGN MATRCES The trasformatio from X to X* provides a simpe proof that the determiat of x~x is ivariat to ay chage of eve desigatio for ay factor. Ay permutatio of the o zero eves resuts oy i a correspodig permutatio of coums i X* which of course does ot chage the vaue of the determiat. Likewise, ay o zero eve may be iterchaged with the zero eve for ay specified factor. The correspodig chage i X* is a iear combiatio of the first coum of a oes ad the coums of T, T 2,..,Ts- correspodig to that factor. Agai this does ot chage the determiat. This ivariace property is a we kow resut, see for exampe Paik ad Federer [97] ad Srivastava, Raktoe, ad Pesota [97], but the represetatio i terms of (,) matrices makes it more apparet.

9 -9- k Let o., k =,,2,..., s- deote the umber of treatmet combiatios s- f f. T h. h. h. th f '.. ' k - f o a racto w c cota t e actor at eve k. ue, k~o oi - N or each i. ay discussio ivovig til' determiat uf X, or of X*, -.c without oss of geeraity, assume that o. because of the ivariace property. : o. :2 may, s- :2 o. for each i Raktoe ad Federer [97] obtaied the foowig boud o X* usig Hadamard's theorem: (3.) Sice x*must be a iteger, we take the iteger part of the right had side of (3.) as the upper boud. We ow obtai a geeraizatio of their resut for X* matrices, ad cosequety X matrices, for saturated mai effect pas from the symmetrica s factoria. Theorem 3.. Let T be a saturated mai effect pa for the s factoria with N = (s-) +. f X* [:_ : T : T2 : : Ts- ], the x* ~iteger part of NN/ 2 s-s/ 2. (3.2) Whe s = 2, this reduces to equatio (3.). Proof: From theorem 2.2, (3.3) From Hadamard's determiat theorem, we kow that x x is ess tha or equa to the product of its diagoa eemets with equaity oy if X'X is a diagoa matrix. where we take. 2 o. 2 Usig equatios (2.5) ad (3.3), we obtai: 2-2 s- 2 k x* ~ (s!) N (o~ + o~ k o.), (3.4) s- 2 o. for each i. i= k= Expressio (3.4) wi be maximized wheever each of the iterior products is maximized; thus, we eed oy cosider (3. 5)

10 -- Next itroduce the Lagrage mutipier correspodig to the costrait s- k s-2 s- a. - N =, ad take derivatives with respect too. ad a. k=o s-2 s- Equatig these two derivatives, we obtai a expressio i a. ad a. as foows: s-2. 2 (s-2) + (s-) (s-3) s-2 The equatios are satisfied whe a. s- a. s- a. 2N s(s-)(s-3) N/s. (3. 6) We may assume that s-2 s- s- a ~ a. ad that a. ~ N/s from the orderig previousy described. i s- s-2 s- Wheever a. < N/s, we have a. <a from (3.6). Hece, it foows that i s- s-2 o. = a. tota is N, we have Thus, N/s ad sice the smaest of the a~ equas N/s ad sice their s- a. N/s, i,2,...,. (3.7) s- s-n+}/2 ~ (s!)-{nn. k(k+) K= (s!)-nn/2[(s-)!s!] /2 NN/2 s -s/2. -(s-)/2 s Coroary 3. Let T be a mai effect pa for a s factoria experimet with N ~ (s-)+. The N(s-)+ -s X*~X* ~ iteger part of s. (3.8) Proof The proof of theorem 3. uses x'x ad the essetia steps do ot deped o N = (s-)+. Hece the proof is compete. Exampe 3. Cosider a set of t orthogoa ati squares of orders. This set may be regarded as a orthogoa mai effect pa for the st+ 2 factoria with N=s 2. f t = s-, which is possibe wheever sis a prime or prime power, the set forms a saturated mai effect pa. The s 2 x (+(t+2)(s-)) matrix X* is give by X*= [ T T 2... Ts_ J where

11 -- T. ' T. (s-) + J ad T. T. = J- i ;t j. Thus J X*'X* = 2 s s' s' s' s si+(j-) J- J- (3.9) s J- si+(j-). J- The determiat of X*'X* is x*'x* s J- J- si+(j-. s(t+2)(s-2)+2 which equas the (3 ) Whe t = s-, the desig is saturated. boud give cora ary.. The x*'x* = ss(s-) ad x* i theorem 3.. ss(s-)/ 2 which equas the boud give Exampe 3.2 Suppose T is a orthogoa array of size N, costraits, s eves, of stregth 2, ad idex A deoted by (N,,s,2). That is, Tis a fractio for a s factoria i N rus such that for ay pair of factors 2 each of the s possibe combiatios of eves occurs exacty A times. Ceary N = As 2 ad the matrix X*'X* for this fractio is exacty A times the matrix (3.9) i exampe 3.. The determiat of X*'X* aso attais the upper boud give i coroary 3.. The upper boud for the geera asymmetrica factoria may be proved i a simiar maer sice the maximizatio is essetiay for a sige factor at a time. The resuts are cotaied i the foowig theorem ad coroary. Theorem 3.2 Let T be a saturated mai effect pa for a geera asymmetrica J s factoria with N = + ~ (s.-) rus ad et X* be the = (,)-matrix of theorems 2.3(a) ad 2.4. The -s X* jj :> iteger part of NN 2 i~ si i/ 2 Coroary ~~ Let T be a mai effect pa for a geera asymmetrica ij = s. factoria with N ~ + ~ (s.-) rus ad et X* be the correspodig = (,)-matrix represetatio. The v iteger part of N i~ x*'x* :> -si s. ' v +:(s.-).

12 -2- Examp~ } :]_. Proportioa frequecy desigs ad sets of orthogoa F squares as discussed by Hedayat ad Seide [97] provide exampes of orthogoa mai effect pas for the asymmetric factoria. t ca be show that the structure of X*'X* for these desigs is simiar to (3.9). Theorem 3.3. The cass of saturated mai effect desigs for the.t s..=. factoria cotais desigs for which jx*j =. ozero vaue is aways attaiabe. as That is the miimum possibe Proof. The famiiar "oe at a time desig" has a (,) represetatio X* v- = f= (si-), whose determiat is ceary oe. The proof is compete sice oe desig is exhibited for every case. Coroary 3.3. f T is a saturated mai effect pa for the.t s. factoria, 2.= the miimum possibe vaue of x'x is ig (si!) ad this vaue is aways attaiabe. Thus for ay saturated desig the jx'x is a mutipe of this miimum vaue. Proof. This foows directy from theorems 2.3 ad 3.3. Thus for saturated mai effect pas the smaest vaue of the determiat of X, or X*, ca aways be attaied. The upper boud o the determiat of X, or X*, wi be attaied wheever a orthogoa saturated mai effect desig with equa umbers of repetitios o the eves of each factor is obtaied. the 3 series, for exampe, this wi occur with = 4 ad N = 9 yiedig x* = 3 3 ; the ext orthogoa saturated mai effect pa occurs for = 3 ad N = 27 yiedig x* = 3 2 cases where a orthogoa desig does ot exist the upper boud wi ot be attaied.

13 -3-4. ON THE CONSTRUCTON OF MAN EFFECT PLANS The costructio of mai effect pas for the symmetric ad asymmetric factoria is ow directy reated to costructios of (,) matrices with certai costraits o the coums. Thus the body of kowedge ad deveoped theory of (,) matrices ca be directy brought to the costructio of mai effect pas. costructios. this sectio we iustrate this with a few types of Reca from (2.) that for a factor at s eves there must be a correspodig set of (s-) coums i X* with a pairwise ier products zero ad amog these coums at east oe row must be a zero. Exampe~ Circuat Matrix Costruct. Let c 2 be a 2 x 2 circuat matrix whose first row cotais oes ad zeros such that the ith ad (+i)th coordiates are ot both oe. The remaiig rows of C are of course just cycic permutatios of the first row. X* J -< ~ x2 2x2 Let X* be This X* matrix is appropriate for a 3 saturated mai effect pa, ad sice the theory of circuats is we kow the determiat is easy to evauate. To iustrate we ist the first row of a suitabe C matrix for the 3 factoria with = 3,4,5,6 ad 7 ad give the correspodig determiat of X*. First row of C Det. of X* 3 ( ) 4 4 ( ) 27 5 ( ) 88 6 (. ) 28 7 ( ) 42 A simiar costructio for the s factoria woud require a (s-) x (s-) circuat matrix with at most oe oe i the i, +i, 2+i,...,(s-)+i coums i =, 2,...,.

14 -4- Exampe~ Sum Compositio- Let T,T 2,...,Ts- be x matrices of oes ad zeros, ad et (s-)ts- Aderso ad Federer [974] cosidered possibe vaues for the determiat of (,)-matrices ad used te methods of costructio to obtai may of the possibe vaues. Here we preset a possibe determiat vaues attaiabe by the above method of costructio for saturated mai effect pas from the 3 series for = 3,4,5,6, ad 7. 3: 4: = 5: 6: = 7: 3 6 [,,2,4] 4 6 [,,2,3,6,9] 5 = [,,2,3,4,5,6,8,9,,2,5,2,25] [,,2,3,4,5] 6 6 [,,2,...,9] 2 = 6 6 [a possibe products of itegers,,...,9] 6 7 [a itegers ~ 8,2,24,32] 2 where the itegers withi a square bracket represet possibe vaues for the determiat of X*. t shoud be oted that this costructio is restrictive ad does ot provide a possibe vaues of jxj. For exampe, for = 3, ad for aother costructio, it is possibe to obtai a desig for which X = 6 3 (3) ad which is ot obtaied via the above costructio. Eve though this method

15 -5- of costructio gave the argest vaue obtaied for = 3, it is expected that this wi hod for arger. Whe :: 4, the orthogoa saturated desig i exampe 2. yieds a desig for which jxj = 6 4 (27), which is three times arger tha the argest vaue obtaied from this sum compositio. The spectrum of possibe vaues or eve the argest possibe vaue of jxj is ukow at preset. The trasformatio of X to X*, i.e., a (,)-matrix, is cosidered to be oe step toward the resoutio of these probems. Exampe 4.3. The costructio of exampe 4.2 ca be exteded to the geera mai effect pas. Let T, T 2,.,Ts- be (,)-matrices of order N x, N 2 x,..,n x, respectivey, for N. ~, such that [ T~] coud be regarded as a s- - mai effect pa foowig desig The, for the 2 factoria with N. + rus. s- for the s factoria with N = + i= T = [ T - 2T 2... (s-)t ] s-. O' Now, cosider the N. rus: X* T s- ad N T - T - 2 T s- X* X*= T 2- T _ o-- T T s- s- Give the T., i =, 2,.., s-, it is a reativey simpe matter to compute X* X*. To cocude, we suggest oe additioa costructio for mai effect pas from the s factoria. This method makes use of a (,)-matrix T ad its compemet (J-T) ad by arragig these matrices to satisfy costraits (2.) ad (2.2). We

16 -6- the 4 iustrate the procedure for the 3, series, ad the for the s series. Exampe 4.4. Let T be a N x (,)-matrix of fu rak with N?. T t >ad T2 For the 3 series, cosider the pa defied by with 3N rus. Each of the three eves of each factor occurs N times. For this desig, 3N N' N' X*'X* N T 'T+(J-T) '(J-T) (J-T) 'T N T' (J-T) T 'T+(J-T) '(J-T)_ f T itsef is a structured matrix, the X*'X* has a simpe structure. For exampe, if T =, the 3 ' ' X*'X* i+(-2)(j-) (J-) (J-) i+(-s) (J-) ad = 3 ( -3+3). For the 4 series, the costructio is give by T J - T T J ~ J- T -- - geera, for the s factoria we ett., i =,Z,...,s-, be sn x matrices whose ith ad (i+)st bocks are T ad J- T, respectivey, with the remaiig bocks composed of zero matrices. For this costructio, we have: sn N' N' N' N' N A B N B A B X*'X* N B A N where A= T'T + (J-T)'(J-T) ad B = (J-T)'T.

17 -7-5. EXTENSON TO NCLUDE NTERACTON TERMS f s is a prime or prime power, it is possibe to icude iteractios i the (,) represetatio of the s factoria. This represetatio is i terms of the geometric defiitio of the factoria effects. this defiitio of the factoria effects the symbo F. F.a a~ EGF(s) is used J to deote (s-) degrees of freedom beogig to the iteractio of the. th d. th f -- a J-- actors. As a rages over the s- ozero vaues of the Gaois fied of order s,gf(s), a (s-)(s-) degrees of freedom for the iteractio betwee F. ad F. are idetified. J with (s-) degrees of freedom is deoted by a 2, a 3,..., ak are ozero eemets of GF(s). k- over a possibe ozero vaues, a (s-) th A geera ~ order compoet (s-) degrees of freedom associated with this kth order iteractio are idetified. f T deotes a desig for the s factoria, et the coums of T be deoted as T = [~ i2... d ]. To icude the iteractio betwee factors F. ad F. i the mode, adjoi tot the (s-) coums J d. +a d. - -J a ;t E GF(s), (5.) where a cacuatios are i the fied GF (s). Each of these coums ceary cotais oy the eemets of GF(s) ad hece have the same form as the coums of T. For higher order iteractios, say F. Fa2 i2 we adjoi to T coums of the form. ~i + a i d i a. d. 2-2 k -k ai,a.,... ai ;to EGF(s) 2 k (5.2) Let D deote the N x(+m) matrix with m coums adjoied to T for a desired iteractios. The matrix D has eemets from GF(s) ad as i (2.), we et D. deote the Nx(+m) icidece matrix of i i D, iegf(s). The L: D iegf(s) i J Nx(+m) ad D i D. (5.3)

18 -8- The X matrix for the mode cotaiig iteractio terms has the same form as (2. 3) ' that is = : X [._ Do D Do +~ 2 D2. s-2 r Di (s-) Ds-J (5.4) i=o t is ow apparet that with v = + (+m)(s-), the v x v matrix G of equatio (2.5) may be mutipied by XNx(+m) exacty as i (2.4) to produce a (,) represetatio of X. This observatio is expicity stated i Theorem 5.. Theorem 5.. With X as i (5.4) ad the trasformatios XG = X*, we have. s- (a) X* [ D ; D2.... ; D ], (b) ei (c) tx = X* (d) x'x = ( ') -(+m\ ~+m)(s-) s. - ' - G ' ad (s!) 2 (+m) x*'x*. Proof. The theorem foows directy from theorem 2. t shoud be oted that i the asymmetric factoria Tis. that iteractios. betwee factors with the same (prime power) umber of eves may be icuded i the mode exacty as i the discussio above. For factors with differig umbers of eves or with o prime power umber of eves the coveiet fied of order s does ot exist. There may be a correspodig (,) represetatio which icudes iteractio terms for the geera asymmetric factoria reative to some other formuatio of the iteractio cotrasts. Pesota ad Raktoe [975] show that the (,) represetatio does ot exted i a atura way if the product defiitio of the effects is used. They do show that such a represetatio does exist i terms of (-,,) matrices, ad exhibit suitabe casses of desig matrices T ad sets of factoria effects such that a atura (,) represetatio does exist.

19 -9- Exampe 5.. Cosider a case where there are three factors each at 3 eves ad it is desired to icude i the mode the iteractio of F with each of F 2 ad F 3, but the F 2 by F 3 iteractio ad the three factor iteractio are to be excuded. For ay desig T, we woud thus adjoi the four coums. (mod 3). The matrix D thus has seve coums ad the correspodig X* matrix of theorem 2. has + 2(7) = 5 coums. ACKNOWLEDGEMENT This research was partiay supported by NSERC grats No. A8776 ad No. AO 724.

20 -2- ANDERSON, D.A. ad FEDERER, W.T.(973). Represetatio ad Costructio of mai effect pas i terms of (,) matrices. Paper umber Bu-499-M i the Biometrics Uit Mimeo Series, Core Uiversity. ANDERSON, D.A. ad FEDERER, W.T.(975). Possibe absoute determiat vaues for square (,)-matrices usefu i fractioa repicatio. Utiitas Mathematica, 7, HEDAYAT, A. ad SEDEN, E.(97). F-square ad orthogoa F-squares desig: A geeraizatio of ati square ad orthogoa ati squares desig. A. Math. Statist JONER, J.R.(973). Simiarity of desigs i fractioa factoria experimets. Ph.D. Thesis, Core Uiversity, August. PAK, U.B. ad FEDERER, W.T.(97). A radomized procedure of saturated mai effect fractioa repicates. A. Math. Statist PESOTAN, H. ad RAKTOE, B.L.(975). Remarks o extesios of the Aderso Federer (,)-Matrix procedure for mai effects of a higher degree. Comm. i Stat. ±(9), RAKTOE, B.L. ad FEDERER, W.T.(97). A characterizatio of optima saturated mai effect pas of the 2 factoria. A. Math. Statist SRVASTAVA, J.N., RAKTOE, B.L., ad PESOTAN, H.(976). O ivariace ad radomizatio i fractioa repicatio. A. Statist. ±

Discrete Fourier Transform

Discrete Fourier Transform Discrete Fourier Trasform ) Purpose The purpose is to represet a determiistic or stochastic siga u( t ) as a fiite Fourier sum, whe observatios of u() t ( ) are give o a reguar grid, each affected by a