STATISTICS RESEARCH ARTICLE. s 4

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1 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: STATISTICS RESEARCH ARTICLE D-optmal saturated factoral desgns St. A. Chatzopoulos 1 * and F. Kolyva-Machrea 1 Receved: 0 October 017 Accepted: March 018 Frst Publshed: 9 March 018 *Correspondng author: St. A. Chatzopoulos, Department of Mathematcs, Arstotle Unversty, Thessalonk 51, Greece E-mal: cstavros@math.auth.gr Revewng edtor: Guohua Zou, Chnese Academy of Scences, Chna Addtonal nformaton s avalable at the end of the artcle Abstract: In ths paper, Resoluton III saturated s 1 s, s s 3 s factoral desgns and specally the cases (s k) s, s k, k = 0, 1 are studed, n order to obtan D-optmal plans. Subjects: Scence; Mathematcs & Statstcs; Statstcs & Probablty; Statstcs; Mathematcal Statstcs; Statstcal Computng; Statstcal Theory & Methods Keywords: D-optmal desgns; saturated desgns; desgn matrx AMS subject classfcatons: Prmary 6K05; 6K15 ; Secondary 11C0; 6J05 1. Introducton Saturated factoral plans s a very nterestng ssue n theory of exeprmental desgns, snce the reduced number of observatons s very usefull n practse especally n screenng experments, where are used to determne whch of many factors affects the measure of pertnent qualty characterstcs. In saturated desgns the number of observaton s equal to the number of parameters, so all degrees of freedom are consumed by the estmaton of parameters, leavng no degrees of freedom for error varance estmaton. The purpose of ths paper s to gve saturated resoluton III desgns, mnmzng the generalzed varance of the man effects and the general mean, that s, D-optmal desgns. In recent years, there has been a consderable nterest n optmal saturated man effect desgns wth two or three factors. Mukerjee et al. (1986) and Kraft (1990) showed all two-factor desgns are equvalent wth respect to D-optmalty crteron. Later Mukerjee and Snha (1990) ABOUT THE AUTHORS The problem of fndng optmal desgns under dfferent types of crtera preoccupes many researchers the last decades. Most of the work on constructng optmal desgns for the estmaton of parameters n fractonal factorals s concentrated on factors at two levels. Chatzopoulos, Kolyva-Machera, and Chatterjee (009), studed the optmalty of desgns whch are obtaned by addng p runs to an orthogonal array for experments nvolvng m factors each at s levels. Chatterjee, Kolyva-Machera, and Chatzopoulos (011), consdered the ssue of optmalty of fractonal factoral experments nvolvng m factors each at two levels. Percleous, Chatzopoulos, Kolyva-Machera and Kounas, study the problem of estmatng the standardzed lnear and quadratc contrasts n fractonal factorals wth k factors, each at 3 levels, when the number of runs or assembles s N = 3 and ntroduced a dfferent noton of Balanced Arrays. Chatzopoulos and Kolyva-Machera (005), studed the saturated m 1 m m 3 desgns and Chatzopoulos & Kolyva- Machera (008), consdered the problem of fndng D-optmal saturated m m 3 desgns. PUBLIC INTEREST STATEMENT An ssue of nterestng n expermental desgns s the saturated desgns. An expermental desgn s called saturated f all the degrees of freedom are consumed by the estmaton of the parameters wthout leavng degrees of freedom for error varance estmaton. The saturated factoral desgns, where the nterest s to estmate the general mean and the man effects whle all hgher order nteractons are neglgble (resoluton III plans), are commonly used n screenng experments. In recent years, there has been a consderable nterest n optmal saturated man effect desgns. Most researchers have dealt wth the case where two or three factors are nvolved n the experment on two levels. The problem s dfferent and becomes more dffcult when three or four factors are nvolved n the experment on three or more levels. 018 The Author(s). Ths open access artcle s dstrbuted under a Creatve Commons Attrbuton (CC-BY).0 lcense. Page 1 of 13

2 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: consdered, for the two-factor case, the optmalty results on almost saturated man effect desgns. Pesotan and Raktoe (1988) worked also n the specal case for s factorals and a subclass of s 3 factorals. Chatterjee and Mukerjee (1993) were the frst who attempted to extend the two factor results to three factors. They consder s, s, s 3 s factoral to derve D-optmal saturated man effect desgns. Later Chatterjee and Narasmhan (00), usng technques from Graph Theory and Combnatorcs, clamed about the upper bound of the determnant of the saturated 3 s, s 3, s 3 s factorals when s s odd. Chatzopoulos and Kolyva-Machera (006) extend the results concernng D-optmal saturated man effect desgns for s to 3 s factorals, when 3 s 6 and s 3 s. Karaganns and Moyssads (005) and Karaganns and Moyssads (008) extend the Graph theoretc approach of Chatterjee and Narasmhan (00) and the results of Chatzopoulos and Kolyva-Machera (006), and gve the D-optmal saturated 3 s, s 3, s 3 s desgns. In ths paper, we study the D-optmalty for saturated s 1 s factorals. Moreover, we gve the upper bound of the determnant for the (s k) s, s k, k = 0, 1, saturated desgns and the correspondng desgn, whch attans ths bound. The paper s organzed as follows. Some notatons and prelmnares are frst presented n Secton. Secton 3 deals wth the man results of ths paper.. Notatons and prelmnares In ths paper, we follow the same notatons as n Chatzopoulos and Kolyva-Machera (006) adapted for four factors. Let us consder the setup of an s 1 s,s s 3 s saturated factoral experment, nvolvng four factors F 1, F, F 3 and F appearng at s 1, s, s 3 and s levels, respectvely, wth N = s 1 + s + s 3 runs. For 1 let the levels of F be denoted by τ and coded as 0, 1, s 1. Our nterest s to fnd D-optmal resoluton III desgns. There are altogether s 1 s s 3 s treatment combnatons denoted by τ 1 τ τ 3 τ, that wll hereafter be assumed to be lexcographcally ordered. Let, for 1, 1 be the s 1 vector wth each element unty, I the dentty matrx of order s, denotes the Kronecker product of matrces and P be an (s 1) s matrx such that (s ( 1 ) 1, P ) s orthogonal (A denotes the transpose of matrx A). The usual fxed effect model under the absence of nteractons s Y = Wβ + ε, where Y s the response vector of the experment, ε s the vector of uncorrelated random errors wth zero mean and the same varance σ and β s the vector of unknown parameters, s consder. In our case β =(μ, β, 1 β, β, 3 β ), where μ s the unknown general mean and the elements of the (s 1) 1 vectors β are unknown parameters representng a full set of mutually orthogonal contrasts belongng to the man effects F and W =[ , W 1, W, W 3, W ], where W 1 = P , W = 1 1 P 1 1 3, W = P 1 3 and W = P. It s easy to see that the D-optmal desgn does not depend on the choce of P, 1. Followng Mukerjee and Snha (1990) let X 0 =[ , X 1, X, X 3, X ], where X 1 = I , X = 1 1 I 1 3 1, X 3 = I 3 1 and X = I. We denote X (1), = 1,, 3 the matrces obtaned by deletng the frst column of X, = 1,, 3. Consder the u (s 1 + s + s 3) matrx U, whch s a submatrx of X 0 gven by U =[X (1), 1 X(1), X(1), X 3 ], whch has full column rank. The u rows of matrx U lke those of W, correspond to the lexcographcally ordered treatment combnatons. Moreover the columns of U span those of X 0 and hence those of W, whch also has full column rank. Hence, one may obtan W = UH, where matrx H s a nonsngular matrx of order s 1 + s + s 3. For any desgn d n the class of the saturated resoluton III desgns wth N = s 1 + s + s 3 runs, the desgn matrx s W d = H, where s a square matrx of order s 1 + s + s 3 such that for 1 j s 1 + s + s 3 f the -th run n d s gven by the treatment combnaton τ 1 τ τ 3 τ then the j-th row of s the row of U correspondng to the treatment combnaton τ 1 τ τ 3 τ. A desgn d s sad to be D-optmal n the class, f t maxmzes the Page of 13

3 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: quantty det(w W d d ). Snce matrx H s nonsngular a desgn s D-optmal f t maxmzes the quantty det( ), where: =[Z (1) 1, Z(1), Z(1) 3, Z ]. (1) The matrces Z (1), 1 3 and Z are obtan.ed from the matrces X (1) and X n a smlar way, as s obtaned from U. Defnton.1 For 1 3, f the -th factor enters the experment at level 0 then the correspondng row of the matrx Z (1) s a row vector wth s 1 elements zero. On the other hand f the -th factor enters the experment at level p, 1 p (s 1), then the correspondng row of the matrx Z (1) equals the p-th row of the dentty matrx of order (s 1). Smlarly, f the fourth factor enters the experment at level p, 0 p s 1, then the correspondng row of the matrx Z equals to the (p + 1)-th row of the dentty matrx I s. Let n p, 0 p (s 1), denote the number of these rows. It holds that (s 1) N = n p, = 1,, 3,. Defnton. For 1 j and 0 p (s 1), 0 q (s j 1), let n pq j, denote the number of runs where the -th factor appears at level p and the j-th factor appears at level q. It holds that n p = p=0 (s j 1) q=0 n pq j, for j = 1,, 3,, j, (s 1) n q = n pq, for = 1,, 3,, j, j j N = p=0 (s 1) p=0 (s j 1) q=0 n pq j, for 1 j. Defnton.3 For 1 j k and 0 p (s 1), 0 q (s j 1), 0 r (s k 1) let n pqr, denote the number of runs where the -th factor appears at p level, the j-th factor appears at level q and jk the k-th factor appears at level r. It holds that (s k 1) n pq = n pqr, for k = 1,, 3,, k j, j jk n pr k n qr jk N = r=0 (sj 1) = q=0 (s 1) = p=0 (s 1) p=0 n pqr, for j = 1,, 3,, k j k, jk n pqr, for = 1,, 3,, j k j, jk (s j 1) q=0 (s k 1) r=0 n pqr, for 1 j k. jk Remark.1 It holds that n p desgn has full column rank. 1, 1, 0 p (s 1), snce the desgn matrx of a saturated Remark. By the choce of the labels for the levels one can always assume, wthout loss of generalty (w.l.g), that n 0 = max{n p,0 p s 1, 1 }. The followng lemmas are crucal for the man results of our paper and can be founded n Chatterjee and Mukerjee (1993) and Chatzopoulos and Kolyva-Machera (006). Page 3 of 13

4 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: Lemma.1 Consder the saturated s 1 s desgn e, s, wth N = s 1 + s 1 runs and correspondng matrx X e. It holds that det(x e ) = 1. () Proof See Chatterjee and Mukerjee (1993). Lemma. Consder the saturated s desgns d (), s 3 s, wth N = s runs and correspondng matrx U (). It holds that d det(u () d ) s. (3) Proof See Chatterjee and Mukerjee (1993). Lemma.3 Consder the saturated s 1 s s k desgn d. If s and n p = 1 for some 0 p s 1 then det( ) = det( ), where d s a saturated s 1 (s 1) s k desgn. Proof See Chatzopoulos and Kolyva-Machera (006), lemma.1. Corollary.1 Consder the saturated s 1 s, s s 3 s desgn d wth N = s 1 + s + s 3 runs. If w = s (s 1 + s 3) 0, then usng the pgeonhole prncple we can easly verfy that n p = 1, for some 0 p s 1 at least w tmes. Applyng, w tmes, lemma.3, we get det( ) = det( ), where d s a saturated s 1 s (s 1 + s 3) desgn. Remark.3 For s 1 = s = we have to study only the cases where 0 s s 3 1, that s the cases s, s and (s 1) s, s 3. Lemma. Let d be a saturated s 1 s, s 3 s desgn. If det( ) = s 1 p=1 np and n 0 n 1 n (s 1) n 0 1, = 1,, 3, then d s D-optmal. Proof See Chatzopoulos and Kolyva-Machera (006), theorem Man results Lemma 3.1 The determnant of the matrx =[Z (1), 1 Z(1), Z(1), Z 3 ], gven n (1), whch corresponds to a s 1 s, s s 3 s saturated factoral desgn d s left nvarant by nterchangng the levels of the factors. Proof For the fourth factor, we can nterchange the columns whch correspond to two levels and the proof s obvous. Smlarly, for the nonzero levels of the frst, second and the thrd factor, nterchangng the columns p and q, the levels (p 1) and (q 1) are nterchanged. Moreover, for the frst (or second or thrd) factor, addng all the columns of matrx Z 1 (or 1 Z1 or Z1 3 ) to the column whch corresponds to level p, subtractng the sum of all columns of matrx Z and multplyng the resultng column by ( 1) the levels 0 and p are nterchanged. Lemma 3. Let d be a s 1 s, s s 3 s saturated factoral desgn wth correspondng matrx =[Z (1), 1 Z(1), Z(1), Z 3 ] as gven n (1). Let npqr = w > 1 for some 1, j, k and some jk 0 p s 1, 0 q s j 1, 0 r s k 1. Then det(u d (13)) for(, j, k) =(1,, 3), det(u det( ) = d (1)) for(, j, k) =(1,, ), det( (13)) for(, j, k) =(1, 3, ), det( (3)) for(, j, k) =(, 3, ), Page of 13

5 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: where d (13), d (1), d (13) and d (3) s s 1 s (s w + 1), s 1 s (s 3 w + 1) s, s 1 (s w + 1) s 3, (s 1 w + 1) s, saturated factoral desgn, respectvely. Proof Let as assume that n pqr = w > 1, 0 p s 1, 0 q s 1, 0 r s , whch means that the saturated desgn s 1 s contans the runs pqrx 1, pqrx,, pqrx w. By subtractng the row correspondng to run pqrx 1 from the other rows whch correspond to runs pqrx,, pqrx w, addng the columns correspondng to levels x,, x w to the column correspondng to level x 1 and expandng det( ) along the (w 1) rows whch contan levels x,, x w, we get det( ) = det( (13)), where d (13) s s 1 s (s w + 1) saturated desgn. The proof s smlar for n pqr = w, npqr = w 1 13 and n pqr = w D-optmalty of (s 1) s saturated desgns Lemma 3.3 Consder the saturated (s 1) s desgn d wth N = s runs and correspondng matrx as gven n (1). For the D-optmal desgn t holds that: n p = s, = 1,, p = 0, 1. n 0 = 3 n1 = 3, 3 np =, p (s ). 3 =, 0 p (s 1). n p Proof Expandng det( ) along ts frst column, we have that det( ) = n1 1 det(u =1 d ), where d (), () = 1,,, n 1 are (s 1) s saturated desgns wth correspondng matrces U 1 d. Let det(u () d ) = max { det( ()), = 1,,, n 1 }. Then det( ) n 1 det( ). () From lemma 3.1, by nterchangng the levels 0 and 1 of the frst factor, t also holds det( ) n 0 det( ). (5) From ()-(5), we get det( ) n0 1 +n1 1 det(u d ) = s det( ). Accordng to lemma., n order to fnd the D-optmal desgn d, t must hold that n 0 n 1 n s 1 n 0 1, =, 3,, whch mples n 0 = n1 = s, n0 = 3 n1 = 3, 3 np =, p (s ), 3 np =, 0 p (s 1). Expandng det(u d ) along ts second column, and followng the same procedure we get n 0 = 1 n1 = s. 1 Lemma 3. Consder the saturated (s 1) s desgn d wth N = s runs and correspondng matrx as gven n (1). For the D-optmal desgn t holds that: n 0q = n1q = 1, = 1,, 0 q (s 1). Proof get n 0q 1 Suppose, w.l.g, that n 0p = for some 0 p (s 1). Then from the pgeonhole prncple, we 1 = for some q, 0 p q (s 1). From lemmas 3.1 and 3., we may assume that n a desgn d the followng treatment combnatons exst: 000τ p, 01τj 3 τp, 11τk 3 τq, 10τ 3 τq. Consder now matrx U, d whch corresponds to the desgn d, as gven n (1). Subtract the row correspondng to the treatment combnato0τ p from the row correspondng to the treatment combnaton 01τj 3 τp. The row correspondng to the treatment combnaton 01τ j 3 τp s now r =(0, 1, 0,, 0, 1, 0,, 0, 0,,0), where the second ace s at the (j + )-th column of r, that s at the j-th column of Z (1). Then, subtract the row r 3 from the row correspondng to the treatment combnaton 11τ k 3 τq. Contnue by addng the column of Z (1), whch corresponds to 3 τj to the column of Z(1) whch corresponds to 3 3 τk 3. Consequently, treatment combnaton 10τ k 3 τp s now n the poston of treatment combnaton 11τk 3 τp. Add the row correspondng to the treatment combnato0τ p to row r. The resultng row corresponds to the treatment combnaton 01τ j 3 τp. Hence, the desgn d contans the treatment combnatons 000τp, 01τj 3 τp, 10τk 3 τq, 10τ 3 τq, whch mples n10q =. Then, proceedng as n lemma 3., we have that det(u ) = det(u 1 d d ), Page 5 of 13

6 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: where d s (s ) (s 1) saturated desgn. Corollary 3.1 For the D-optmal saturated (s 1) s desgn d wth N = s runs, f there exsts the treatment combnatoτ 3 τ (01τ 3 τ ) then there exsts the treatment combnaton 11τ 3 τ (10τ 3 τ ). Lemma 3.5 Consder the saturated (s 1) s desgn d wth N = s runs and correspondng matrx as gven n (1). For the D-optmal desgn t holds that: n pq = 1, = 1,, 0 p 1, q (s ). 3 Proof The proof s smlar as n lemma 3.. Corollary 3. Let us now consder the saturated (s 1) s desgn d wth N = s runs. Then, from lemma 3. and usng the pgeonhole prncple, we get: n pq 1, 0 p, q (s 1), 3 n pq =, 1, 0 p 1, 0 q 1. 3 Corollary 3.3 For the D-optmal saturated (s 1) s desgn d wth N = s runs t holds that: n pq 1 = s f s 0 mod, 0 p, q 1, = 1 n11 = s+1 1 = 1 n11 = s 1 1 and n 01 = 1 n10 = s 1 1 and n 01 = 1 n10 = s+1 1 or } f s 1 mod. Lemma 3.6 Consder the saturated (s 1) s desgn d wth N = s runs and correspondng matrx as gven n (1). For the D-optmal desgn t holds that: 0 13 = n = n = n = n = n = 1. (6) Proof From lemma 3.3, we have n p = 3, p = 0, 1 and from lemma 3. and corollary 3. the D-optmal 3 desgn ncludes one of the followng sets of treatment combnatons: (00pτ, 01pτj, 11pτk) or (00pτ, 01pτ j, 10pτk) or (00pτ, 10pτj, 11pτk) or (11pτ, 01pτj, 10pτk ). Applyng lemma 3.1, we can always choose, w.l.g. the treatment combnato0τ, 010τj, 110τk. Ths choce, usng pgeonhole prncple, mples the exstence of the treatment combnatons 001t q, 111tr, 101tt and the proof of (6) s obvous. Theorem 3.1 Consder the saturated (s 1) s desgn d wth N = s runs and correspondng matrx as gven n (1). It holds that: det( ) { s s 1 fs 0 mod, fs 1 mod. (7) Proof Let u = = 1 n11 1. So, from corollary 3.3, we have u = s f s 0 mod or u =(s + 1) f s 1 mod. Moreover, from lemmas and corollares , w.l.g., the D-optmal saturated desgn (s 1) s can be wrtten as: Page 6 of 13

7 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: Let 0 1 k be a 1 k vector wth all elements equal to zero, I k be the dentfy matrx of order k. For m < k, t can be easly seen that: I k = e 1k e k e kk = e 1m e mm 0 1 m 0 1 m 0 1 (k m) 0 1 (k m) e 1(k m) e (k m)(k m), (8) Matrx as gven n (1), can be wrtten as: Page 7 of 13

8 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: Usng relaton (8), and after permutaton of columns matrx can be wrtten as: Now subtract the (u)-th column from the (u + 1)-th column and add the last s u columns to the (u + 1)-th column. Matrx, as gven n (9), can be wrtten as: = ( Ud1 0 u (s u) A U ). Matrx 1 s the desgn matrx of the saturated u u desgn d 1 wth N 1 = u runs. Matrx U s not desgn matrx as ts frst column s (,,, 0,,0), but det(u ) = det( ), where matrx s the desgn matrx of the saturated (s u) (s u) desgn d wth N = (s u) runs. Hence, det( ) = det(1 ) det( ). From (3), we get det( ) u (s u). Recallng that u = s f s 0 mod, or u =(s + 1) f s 1 mod, we get that (7) holds. Theorem 3. Let u =(s 3 + 1) f s 1 mod, or u = s f s 0 mod. The saturated, s > s 3 desgn d wth the followng N = s 3 + s + 1 treatment combnatons 00 (0 u 1), 110(u 1), 11( + 1) (0 u ), 101u, 10( + 1) (u s 3 1), 010s 3, 01 (u s 3 1), 01(s 3 1) (s s 1) s a D-optmal desgn n the class of all, s > s 3 saturated desgns. Proof The proof s obvous from lemma.3, lemmas and theorem D-optmalty of s saturated desgns Lemma 3.7 Consder the saturated s desgn d wth N = s + 1 runs and correspondng matrx as gven n (1). For the D-optmal desgn, t holds that: n 0 = s + 1 and n 1 = s, or n 0 = s and n 1 = s + 1, = 1,. n 0 = 3, n p =, = 3,, 1 p s 1. Proof The proof s smlar as lemma 3.3. Page 8 of 13

9 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: Corollary 3. For the saturated s desgn d wth N = s + 1 runs, from lemma 3. and usng the pgeonhole prncple we get: n pq s + 1, 0 p, q 1, 1 n pq, 0 p, q s 1, 3 n pq, 1, 0 p 1, 3 j, 0 q s 1. j Lemma 3.8 Consder the saturated s desgn d wth N = s + 1 runs and correspondng matrx as gven n (1). For the D-optmal desgn, t holds that: n 0q = n1q = 1, = 1,, 1 q (s 1). j =, or n 10 j =, = 1,, 3 j. (10) (11) Proof For the proof of relaton (10) see lemma 3.. The proof of relaton (11) s obvous, snce n 1 = s + 1 or n 0 = s + 1. Corollary 3.5 For the D-optmal saturated s desgn d wth N = s + 1 runs t holds that: 1 = n01 1 = n10 1 = s 1 = n01 1 = n11 1 = s 1 = n10 1 = n11 1 = s n 01 1 = n10 1 = n11 1 = s and n 11 = s + 1 or 1 and n 10 = s + 1 or 1 and n 01 = s + 1 or f s 0 mod. 1 and = s = n01 1 = n10 1 = s+1 1 = n01 1 = n11 1 = s+1 1 = n10 1 = n11 1 = s+1 n 01 1 = n10 1 = n11 1 = s+1 and n 11 1 = s 1 and n 10 1 = s 1 and n 01 1 = s 1 and 1 = s 1 or or f s 1 mod. or Theorem 3.3 Consder the saturated s desgn d wth N = s + 1 runs and correspondng matrx as gven n (1). It holds that: det( ) s(s + 1). Proof If s 0 mod, then u = s = = 1 n01 = 1 n11 1 and n10 1 = u + 1, whle f s 1 mod, then u =(s 1) = n 01 1 and n00 = 1 n10 = 1 n11 1 = u + 1. From lemmas and corollares , the D- optmal saturated s desgn, w.l.g., can be wrtten as: (1) Page 9 of 13

10 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: By nterchangng the levels 0 and (s 1) of the -th factor, accordng to lemma 3.1, the determnant of the matrx s left nvarant. Hence the D-optmal saturated s desgn, w.l.g., can be wrtten as: Matrx as gven n (1), can be wrtten as: Page 10 of 13

11 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: Usng relaton (8), and after a sutable permutaton of columns of the matrx, n order to make the left bottom block of the matrx a zero matrx, matrx can be wrtten as: Analytcally, the permutaton s: the frst u columns of matrx are the frst u columns of matrx Z (1) 3, columns (u + 1) u of matrx are the frst u columns of matrx Z, (u + 1)-th column of matrx s matrx Z (1) 1, (u + )-th column of matrx columns of matrx are the rest columns of matrces Z (1) 3 and Z. s matrx Z(1), whle the remanng (s + 1) (u + ) Page 11 of 13

12 Chatzopoulos & Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: Now subtract the (u + )-th column from the (u + 1)-th column. Then, matrx s a block trangular matrx. It holds that det( ) = det(u 1 ) det( ), where matrx U 1 s a (u + 1) (u + 1) matrx, whch does not correspond to any desgn and matrx s the desgn matrx of the saturated (s u) (s u) desgn d wth N = (s u) runs. From (3), we get det (s u). Moreover, expandng det(u 1 ) along ts last column we get that det(u 1 ) u+1 j=1 det(x ej ), where, after some manpulatons, X ej correspond to two factor saturated desgns e j, wth, accordng to lemma.1, det(x ej ) = 1. Hence, det( ) (u + 1)(s u). Recallng that u = s f s 0 mod, or u =(s 1) f s 1 mod, we get that (11) holds. Theorem 3. Let u = s f s 0 mod, or u =(s 1) f s 1 mod. The saturated s, s desgn d wth the followng N = s + 1 runs 100(u 1), 101(s 1), 10( + 1) (1 u ), 10u0, 01 (1 u 1), 01u(s 1), 110u, 11 (u + 1 s ), 11(s 1)0, 0000, 00( + 1) (u s ), s a D-optmal desgn n the class of all s, s saturated desgns. Proof From lemma.3, lemmas 3.1, 3., 3.7, 3.8 and theorem 3.3, the proof s obvous. Acknowledgements We would lke to thank the referees and the journal edtoral team for provdng valuable advce that mproved the qualty of the orgnal manuscrpt. Fundng The authors receved no drect fundng for ths research. Author detals St. A. Chatzopoulos 1 E-mal: cstavros@math.auth.gr F. Kolyva-Machrea 1 E-mal: fkolyva@math.auth/gr 1 Department of Mathematcs, Arstotle Unversty, Thessalonk 51, Greece. Ctaton nformaton Cte ths artcle as: D-optmal s3 s saturated factoral desgns, St. A. Chatzopoulos & F. Kolyva-Machrea, Cogent Mathematcs & Statstcs (018), 5: References Chatterjee, K., & Mukerjee, R. (1993). D-optmal saturated man effect plans for s factorals. Journal of Combnatorcs, Informaton & System Scences, 18, Chatterjee, K., & Narasmhan, G. (00). Graph theoretc technques n D-optmal desgn problems. Journal of Statstcal Plannng and Inference, 10, Chatzopoulos, S. A., & Kolyva-Machera, F. (006). Some D-optmal saturated desgns for 3 m m 3 factorals. Journal of Statstcal Plannng and Inference, 136, Karaganns, V., & Moyssads, C. (005). Constructon of D-optmal s 1 s factoral desgns usng graph theory. Metrka, 6, Karaganns, V., & Moyssads, C. (008). A graphcal constructon of the D-optmal saturated, 3 s man effect, factoral desgn. Journal of Statstcal Plannng and Inference, 138, Kraft, O. (1990). Some matrx representatons occurrng n two factor models. In R. R. Bahadur (Ed.), Probablty, statstcs and desgn of experments (pp ). New Delh: Wley Eastern. Mukerjee, R., Chatterjee, K., & Sen, M. (1986). D-optmalty of a class of saturated man effect plans and alled results. Statstcs, 17(3), Mukerjee, R., & Snha, B. K. (1990). Almost saturated D-optmal man effect plans and alled results. Metrka, 37, Pesotan, H., & Raktoe, B. L. (1988). On nvarance and randomzaton n factoral desgns wth applcatons to D-optmal man effect desgns of the symmetrcal factoral. Journal of Statstcal Plannng and Inference, 19, Page 1 of 13

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