Chapter 2 OCDs in Completely Randomized Design Set-Up

Transcription

1 Chapter OCDs n Completely Randomzed Desgn Set-Up.1 Introducton We consder n ths chapter the one-way lnear model wth v treatments, c covarates and a total of n expermental unts. We work under the lnear model y = τ + c t=1 γ t z (t) + e, 1 j n, 1 v. (.1.1) where n (> 1) s the number of tmes the th treatment s replcated; clearly v n = n. (.1.) =1 For 1 j n,1 v, here y s the observaton arsng from the jth replcaton of the th treatment, τ effect due to the th treatment. In matrx notaton the above model can be represented as ) (Y, Xτ + Zγ, σ I n, (.1.3) where, Y s an observaton vector and X s the desgn matrx correspondng to vector of treatment effects τ v 1 and Z = ((z (t) )) s the desgn matrx correspondng to vector of covarate effects γ c 1 = (γ 1, γ,...,γ c ). Ths s referred to as one-way model wth covarates (wthout the general mean). Troya Lopes (198a, b) studed the nature of optmal allocaton of treatments and covarates n the above set-up for smultaneous estmaton of the (fxed) treatment effects (n the absence of the general effect) and the covarate effects wth maxmum effcency n the sense of mnmum generalzed varance. Ths s to note that the nformaton matrx wth respect to model (.1.3) s gven by σ I(η), where Sprnger Inda 015 P. Das et al., Optmal Covarate Desgns, DOI / _ 13

2 14 OCDs n Completely Randomzed Desgn Set-Up I(η) = ( X XX ) Z Z XZ Z (.1.4) and η = ( τ, γ ). The problem s to suggest an optmal allocaton scheme (for gven desgn parameters n, v,c) for effcent estmaton of the treatment effects as well as the covarate effects by ascertanng the values of the covarates for each one of them, assumng that each one s controllable and quanttatve wthn a stpulated fnte closed nterval. The nformaton matrx of γ s gven by σ I (γ) = Z Z Z X(X X) X Z (.1.5) where (X X) s a generalsed nverse of X X satsfyng X X ( X X ) X X = X X (cf. Rao 1973, p. 4). It s evdent that Z X(X X) X Z s a postve sem-defnte matrx. So from (.1.5), t follows that σ I (γ) Z Z (.1.6) n the Loewner order sense (vde Pukelshem 1993) where for two non-negatve defnte matrces A and B, A s sad to domnate B n the Lowener order sense f A B s a non-negatve defnte matrx. Equalty n (.1.6) s attaned whenever X Z = 0. (.1.7) If Z satsfes (.1.7), then treatment effects and covarate effects are orthogonally estmated. Agan under condton (.1.7), the nformaton matrx I(γ) reduces to I(γ) = Z Z.Thez-values are so chosen that Z Z s postve defnte so that from (.1.6) Var( γ t ) σ n v =1 j=1 z (t) σ n as z (t) [ 1, 1];, j, t. Now equalty n (.1.8) holds for all f and only f the Z-matrx s such that and (.1.8) z (s) z (t) = 0 s = t. (.1.9) z (t) =±1 (.1.10)

3 .1 Introducton 15 Condton (.1.7) mples that the estmators of ANOVA effects parameters or parametrc contrasts do not nterfere wth those of the covarate effects and condtons (.1.9) and (.1.10) mply that the estmators of each of the covarate effects are such that these are parwse uncorrelated, attanng the mnmum possble varance. Thus the covarate effects are estmated wth the maxmum effcency f and only f Z Z = ni c (.1.11) along wth (.1.7). The desgns allowng the estmators wth the mnmum varance are called globally optmal desgns (cf. Shah and Snha 1989, p. 143). Henceforth, we shall only be concerned wth such optmal estmaton of regresson parameters and by optmal covarate desgn, to be abbrevated as OCD hereafter, we shall only mean globally optmal desgn, unless otherwse mentoned. It s clear that condtons (.1.7) and (.1.11) hold smultaneously f and only f z s are necessarly +1or 1 and that condton (.1.7) s satsfed. It s dffcult to vsualze the Z-matrx satsfyng condtons (.1.7) and (.1.11).In the set-up of the model (.1.3), t transpres from Troya Lopes (198a) that optmal estmaton of the treatment effects and the covarates effects s possble when the treatment replcatons are all necessarly equal, assumng that n s a multple of v, the number of treatments. We set n = bv, where b s the common replcaton of treatments, henceforth. Das et al. (003) had represented each column of the Z- matrx by a v b matrx W wth elements of ±1, where the rows of W correspond to the v treatments and the columns of W correspond to dfferent replcaton numbers. Condton (.1.7) mples that the sum of each row of W should vansh. Agan, condton (.1.11) mples that the sum of products of the correspondng elements,.e. the Hadamard product of W (s) and W (t), defned n (.1.13) should also vansh, 1 s < t c. The above two facts can be represented n the followng schematc forms through the row totals and Hadamard product. Row Totals: Tr. Repl. no. Row 1... b Totals 1 0 W (s) = 0 (.1.1). (±1). v 0 Hadamard product of W (s) and W (t) (cf. Rao 1973, p. 30): W (s) W (t) = Tr. Repl. no b 1. v (w (s) w (t) ) (.1.13)

4 16 OCDs n Completely Randomzed Desgn Set-Up where * denotes Hadamard product. For orthogonalty of sth and tth columns of v b Z, t s requred that w (s) w (t) = 0. =1 j=1 The schematc representaton (.1.1), (.1.13) of Das et al. (003) s a breakthrough n the sense that handlng of Z-matrx has been made much easer and t has been followed throughout the monograph. Troya Lopes (198a) frst studed the nature of optmal allocaton of treatments and covarates n the above set-up when n v s an nteger. It may be noted that whenever condton (.1.7) s ensured, presence of the covarates n model (.1.3) does not pose any threat to the usual optmal treatment allocaton problem. In Sect.., followng Troya Lopes, we ntend to dscuss about the avalablty of Z-matrces satsfyng (.1.7) and (.1.11) when the treatment allocaton matrx X corresponds to equal allocaton number,.e. n stuatons where n s a multple of v. We wll wrte n = vb so that b s the common allocaton number of the v treatments under nvestgaton. The stuatons where (.1.7), (.1.11) and b = n v = nteger are satsfed, are dentfed as regular cases. Otherwse t s called a non-regular case. If the stuaton s nonregular, then t s not possble to allocate smultaneously the treatments and covarates optmally. For non-regular stuaton, effcent allocaton of treatments and covarates smultaneously can be done by usng other specfc optmalty crtera. Dey and Mukerjee (006) and Dutta et al. (014) consdered ths problem n non-regular stuatons and found D-optmal desgns n ths context. Detals are presented n Sect..3. It has been seen that Hadamard matrx plays a key role for constructng OCDs. Defnton of Hadamard matrx (cf. Hedayat et al. 1999, p. 145) s gven below: Defnton.1.1 A Hadamard matrx H t of order t s a t t matrx wth elements ±1 satsfyng H t H t = ti t.. Covarate Desgns Under Regular Cases Consder the case when n s a multple of v, that s n = vb where b s such that H b, Hadamard matrx of order b, exsts. We shall also consder some cases where b s even. Then ANOVA parameters as well as the covarate effect-parameters can be estmated orthgonally and/or most effcently. Ths holds smultaneously for c covarates and one can deduce maxmum possble value of c for ths to happen. As already mentoned, the most effcent estmaton of γ-components s possble when (.1.7) and (.1.11) are smultaneously satsfed and these condtons reduce, n terms of W-matrces defned n above, to C 1, C where

5 . Covarate Desgns Under Regular Cases 17 C 1. Each of the c W-matrces has all row-sums equal to zero; C. The grand total of all the entres n the Hadamard product of any two dstnct W-matrces reduces to zero. Now we defne optmum W-matrces for covarate desgns n CRD set-up. (..1) Defnton..1 Wth respect to model (.1.3), the c W-matrces correspondng to the c covarates are sad to be optmum f they satsfy condtons C 1 and C of (..1). In ths context, the followng results were deduced n Troya Lopes (198a). Theorem..1 Let c be the maxmum number of covarates that can be optmally accommodated. Then a lower bound to c s gven by (a) b 1 when v = odd, H b exsts; (b) (b 1) when v (mod4),h b exsts; (c) 4(b 1) when v 0(mod4),H b exsts; (d) 3v when b 0(mod4),H v exsts; (e) v when b (mod4),h v exsts. Proof Hadamard matrx H b s gven to exst and we wrte t as H b = (h 1, h,...,h b 1, 1). (..) The choce of optmum W-matrces s ndcated below one by one. The verfcaton of (..1) s mmedate and we leave t to the reader. The Kronecker product of two matrces s formally defned n Chap. 5 (Defnton 5.1.1) and t s used n the constructons below. (a) W ( j) v b = 1 v h j, 1 j b 1; (..3) (b) W ( j) v b = (1, 1) 1 v h j, 1 j b 1; } W (b 1+ j) v b = (1, 1) 1 v h j, 1 j b 1. (..4) (c) W ( j) v b = (1, 1, 1, 1) 1 v 4 h j, 1 j b 1; W (b 1+ j) v b = (1, 1, 1, 1) 1 v 4 h j, 1 j b 1; W ((b 1)+ j) v b = (1, 1, 1, 1) 1 v 4 h j, 1 j b 1; W (3(b 1)+ j) v b = (1, 1, 1, 1) 1 v 4 h j, 1 j b 1. (..5) (d) Let us represent a Hadamard matrx H v of order v as H v = ( h 1, h,...,h v). (..6)

6 18 OCDs n Completely Randomzed Desgn Set-Up W ( j) v b = (1, 1, 1, 1) 1 b h j, 1 j v; 4 W (v+ j) v b = (1, 1, 1, 1) 1 b h j, 1 j v; (..7) 4 W (v+ j) v b = (1, 1, 1, 1) 1 b h j, 1 j v. 4 (e) W ( j) v b = (1, 1) 1 b h j, 1 j v. (..8) Remark..1 In case (c), we can assume exstence of H v for all practcal purposes as v 0 (mod 4). So n ths case, an optmal desgn for maxmum possble v(b 1) optmum W-matrces can easly be constructed as W ((b 1)( 1)+ j) = h h j, = 1,,...,v, j = 1,,...,b 1. (..9) Ths was obtaned n (Rao et al. 003) where t was observed that OCDs n CRD and RBD have one to one correspondences wth mxed orthogonal array (MOA) (defnton gven n Chap. 3). Ths fact wll be dscussed n Sect. 3.3 of Chap. 3 n some further detals..3 Covarate Desgns Under Non-regular Cases Now we examne the stuatons where at least any one of the condtons (.1.7), (.1.11) and b = n v = nteger s volated. In that case, t s not possble to estmate smultaneously ANOVA parameters and γ-parameters orthogonally and/or most effcently. Thus we consder D-optmalty crteron to gve an effcent allocaton of treatments and covarates n Set-up (.1.1). Dey and Mukerjee (006) and Dutta et al. (014) have consdered ths stuaton and found D-optmal desgn. Here we dscuss ther contrbutons n ths drecton n detals. The vector of parameters θ, where θ = (μ 1, μ,...,μ v, γ 1,...,γ c ) (.3.1) s assumed to be estmable. The nformaton matrx for θ s gven by σ I(θ), where I(θ) = ( ) N T T Z, (.3.) Z N = Dag(n 1, n,...,n v ), (.3.3) T = (T 1, T,...,T v ), T = 1 n Z, (.3.4)

7 .3 Covarate Desgns Under Non-regular Cases 19 Z n c = (Z 1, Z,...,Z v ) (.3.5) and Z n c = z (1) 1 z (1) z() 1... z(c) 1 z()... z(c).. z (1) n z () n z (c) n. (.3.6) For D-optmalty, we have to maxmze the determnant of I(θ), denoted as det(i(θ)), wth respect to the desgn varables {z (t) } satsfyng z(t) [ 1, 1], 1 j n, 1 v and n s satsfyng (.1.). From (.3.) t s easy to see that ( v ) det(i(θ)) = n det(z Z T N 1 T) =1 ( v ) = n det(z Z =1 = det(n)det(c), n 1 T T ) (.3.7) where C = Z Z n 1 T T. (.3.8) Note that C s the nformaton matrx for the regresson coeffcents γ 1, γ,...,γ c. The maxmzaton of det(i(θ)) s done n two stages. In the frst stage, the maxmzaton s done for varyng z-values for fxed n s. Ths leads to an upper bound for det(i(θ)) obtaned through completely symmetrc C-matrces. At the second stage, maxmzaton s done for varyng n s subject to n = n, and ths leads to a suffcently small class N of contendng n = (n 1, n,...,n v ) s wheren the overall upper bound to det(i(θ)) belongs..3.1 Frst Stage of Maxmzaton Maxmsaton of I(θ) wth respect to z (t) [ 1, 1] s based on the followng lemma. Lemma.3.1 A necessary condton for maxmzaton of det(c) of(.3.8) wth respect to z (t) [ 1, 1], for fxed n s s that z (t) = ±1, j and t.

8 0 OCDs n Completely Randomzed Desgn Set-Up Proof From (.3.8), C can be expressed as where C = Z MZ = Z Z (.3.9) Z = MZ, M = dag(m 1, M,...,M v ), M = (I n n 1 1 n 1 n ). (.3.10) It s known that (cf. Gall and Kefer 1980; Wojtas 1964), det(z Z )smaxmum at the extreme entres of Z.Agan,asz (t) s are lnear n z (t) s, the determnant s maxmum at the extreme values of z (t) s for all, j and t. Hence the lemma follows. Theorem.3.1 For fxed {n } s satsfyng (.1.), ( v ) det(i(θ)) n {a + (c 1)b}(a b) c 1 (.3.11) =1 where a = n δ, b = ξ δ (.3.1) δ = v =1 n 1 δ, δ = 1(0) f n = odd(even) (.3.13) δ f both of n, δ are odd or even ξ = ξ(n, δ) = δ + 1 fn = odd, δ = even or n=even, δ = odd (.3.14) δ =greatest nteger less than equal to δ. Proof Because of Lemma.3.1, we restrct z (t) to the class χ ={z (t) : z (t) =±1}. From the Eq. (.3.8), we note that, c t,t,the(t, t ) th element of the C-matrx s gven by c t,t = { j z (t) ) z(t j z (t) j n z (t ) }, 1 t, t c. (.3.15) It follows from Wojtas (1964) that det(c) s maxmum when C s completely symmetrc wth all the dagonal elements equal to a and all off-dagonal elements equal to b where a and b are gven by max c tt and mn c 1 t c 1 t =t tt respectvely. Agan as c

9 .3 Covarate Desgns Under Non-regular Cases 1 z (t) =±1, j and t, for fxed n s, t can be deduced that max c tt = n δ = a, 1 t c mn c 1 t =t tt = ξ δ =b (.3.16) c where δ and ξ are gven n (.3.13) and (.3.14) respectvely. Therefore the theorem follows..3. Second Stage of Maxmzaton In vew of Theorem.3.1, we now consder the problem of maxmzng ( v ) g(n) = g(n 1, n,...,n v ) = n {a + (c 1)b}(a b) c 1 (.3.17) wth respect to n s subject to =1 v n = n, where a and b are gven by (.3.1) =1 (.3.14), so as to fnd the overall upper bound of det(i(θ)). The followng lemma helps to reduce the class N of n s where n = (n 1, n,...,n v ), satsfyng (.1.), to a subclass n whch maxmum of g(n) les. Lemma.3. Let n = ( n 1, n,...,n v) be a maxmzer of g(n) of (.3.17) subject to the condton (.1.). Then n cannot have () two unequal odd elements; () two even elements that dffer by more than ; () an even and an odd element that dffer by more than 1. Proof () Wthout loss of generalty t s assumed that n 1 and n be odd and n 1 n. Defne ñ = (ñ 1, ñ,...,ñ v ), where ñ 1 = n 1 + 1, ñ = n 1 and ñ = n = 1,. Note that ñ s satsfy condton (.1.). Then by Eq. (.3.17), g(ñ) g(n ) = ( v ) ( v ñ / ) ( ) {ã+(c 1) b}(ã b) c 1 {a +(c 1)b }(a b ) c 1 n =1 =1 = (n 1 +1)(n 1) {ã+(c 1) b}(ã b) c 1 n, 1 n {a +(c 1)b }(a b ) c 1 (.3.18) where, ã = n v =1 ñ 1 δ = n v =1 n 1 δ + 1 n n = a + 1 n n (.3.19)

10 OCDs n Completely Randomzed Desgn Set-Up Agan, v b = ξ =1 ñ 1 v δ ξ =1 ( 1 n 1 δ + n + 1 ) ( 1 1 n b + n + 1 ) 1 n. (.3.0) We consder the two cases b b and b > b separately. (a) Let b b. Then, as by (.3.19), ã > a,tfollowsthatg(ñ) >g(n ), whch s mpossble. (b) Let b > b and let b assume the hghest possble value gven n (.3.0). Then from (.3.18) (.3.0), t s seen that g(ñ) g(n ) > (n 1 + 1)(n 1) n > 1 (.3.1) 1 n whch s agan a contradcton. As the nequalty (.3.1) s true for the hghest value of b, t wll be true for all values of b n [b, b + 1 n n ] as ã > a. () If possble, let n have two even elements, say n 1 < n whch dffer by more than. Then as n () above, we reach at a contradcton by ncreasng n 1 by two and decreasng n by two. () If possble, let n have an even element n 1 and an odd element n whch dffer by more than 1. Case A: Let n 1 > n. Satsfyng (.1.), defne ñ = (ñ 1, ñ,...,ñ v ), where ñ 1 = n 1, ñ = n + and ñ = n = 1,. Then by Eq. (.3.17), we have g(ñ) g(n ) = (n 1 )(n + ){ã + (c 1) b}(ã b) c 1 (n 1 n ){a + (c 1)b }(a b ) c 1 (.3.) where, ã = n ( ñ 1 δ = n ) ( 1 n 1 δ + n 1 ) ( 1 n + = a + n 1 ) n + (.3.3) ( 1 b = ξ δ (ξ δ ) + n 1 ) ( 1 n b + n 1 ) n. (.3.4) We consder two cases when b b and b > b.for b b,tfollows,from(.3.) that g(ñ) >g(n ) as ã > a.agan,for b > b, we assume ts hghest value vz. b + ( 1 n 1 n ) from (.3.4) and use t n (.3.). It s seen that g(ñ) >g(n ), whch obvously holds for all other values of b > b as ã > a. So we reach at a contradcton that n s a maxmzer of g(n).

11 .3 Covarate Desgns Under Non-regular Cases 3 Case B: Let n 1 < n (.e. n 1 n 3), then we have the followng two cases: (a) n s not the only odd element of n. (b) n s the only odd element of n. For (a), let n have another odd element n 3. Then by part () of ths lemma, n =n 3. Defne ñ = (ñ 1, ñ,...,ñ v ), where ñ 1 = n 1 +, ñ = ñ 3 = n 1 and ñ = n = 1,, 3. Then by (.3.17) g(ñ) g(n ) = (n 1 + )(n 1) {ã + (c 1) b}(ã b) c 1 (n 1 n n 3 ) {a + (c 1)b }(a b. (.3.5) ) c 1 where, ã = n ( ñ 1 δ = n ) n 1 δ + n = a + n (.3.6) b = ξ δ (ξ δ ) + n b + n. (.3.7) If b b, then from (.3.5) and (.3.6) g(ñ) >g(n ) whch s a contradcton. If b > b, the above contradcton also holds by the same reasons as gven n Case A. For (b), let us defne ñ = (ñ 1, ñ,...,ñ v ) satsfyng (1.) where ñ 1 = n 1 +, ñ = n and ñ = n = 1,. Proceedng as before, t can be proved that ( 1 ã = a + n 1 ) ( n, b = 1 1 ) n. (.3.8) Usng (.3.8) n(.3.17), t s seen that g(ñ) g(n ) = (n 1 + )(n ) n 1 n ( n + c 1 ( n + c 1 c n ) c n Ths s agan a contradcton. Therefore the lemma follows. ) > 1 as(n n 1 ) 3. From Lemma.3., we get the followng theorem whose proof s mmedate. Theorem.3. Let o be an odd nteger, where o = n v or n v +1 accordng as n v s odd or even and n = ( n 1, n v),...,n be a maxmzer of g(n) of (.3.17) subject to n = n. Then n {o 1, o, o + 1}.

12 4 OCDs n Completely Randomzed Desgn Set-Up Lemma.3.3 If f, f and f + be the frequences of o, o 1 and o+1 respectvely, then the followng relatons f + f + f + = v; of + (o 1) f + (o + 1) f + = n, (.3.9) mnmze consderably the search for optmum n, forwhchg(n) s a maxmum. Let N ( N ) denote the class of n s satsfyng Theorem.3. and Lemma.3.. Remark.3.1 For gven n, vand c, let g(n ) be the maxmum of g(n) of (.3.17) over n = (n 1, n,...,n v ) subject to n = n. Then by Theorem.3.1 det(i(θ) g(n ). (.3.30) If a choce of {z (t) } exsts correspondng to n, such that equalty n (.3.30) holds, then n together wth {z (t) } gves a D-optmal desgn. Remark.3. If all n s are even, so that all the T s of (.3.4) may be made equal to zero, then t s possble to estmate the regresson parameters γ s orthogonally to the μ s. In that case, γ s are estmated most effcently wth the mnmum possble varance when Z Z = ni c. Remark.3.3 If n = n v = an even nteger for all, the stuaton reduces to regular case and then Remark.3.1 s n full agreement wth Troya Lopes (198a) and n that case γ s can be estmated most effcently so that each estmator has mnmum possble varance when Z Z = ni c. Remark.3.4 If the v levels of the sngle factor set-up are assumed to be the v level combnatons of m factors F 1,...,F m havng s 1,...,s m levels, respectvely (v = s ), then the optmum desgn for the sngle factor set-up s also optmum for the estmaton of γ and all effects up to m-factor nteractons whch can be obtaned through an orthogonal transformaton of γ and the mean vector μ correspondng to the v level combnatons..3.3 Examples Now we consder followng examples to llustrate the above method. Example.3.1 Let us consder the one-way set-up wth n = 1, v = 4. It follows that N ={(3, 3, 3, 3), (, 3, 3, 4), (,, 4, 4)} {(3 4 ), (, 3, 4), (, 4 )}.

13 .3 Covarate Desgns Under Non-regular Cases 5 (a) For c = 1, n = (3 4 ) s the unque maxmzer of g(n) and ths n together wth Z 1 = (1, 1, 1), Z = (1, 1, 1), Z 3 = (1, 1, 1), Z 4 = (1, 1, 1) gves a D-optmal desgn. (b) For c = both n = (, 3, 4) and (, 4 ) are maxmzers of g(n). () n = (, 4 ) and ( ) ( ) Z 1 =, Z 1 1 =, Z = , Z 4 = , gve a D-optmal desgn. () n = (, 3, 4) and ( ) Z 1 =, Z 1 1 = 1 1, Z 3 = 1 1, Z 4 = , also gve a D-optmal desgn. (c) For c = 3, 4, n = (, 4 ) s the unque maxmzer of g(n). Example.3. In one-way set-up wth n = 9, v= 3, c = 3, D-optmal desgn should be searched wthn the set {(, 3, 4), (3 3 )} of n. It s seen that for n = (3 3 ) and D 1 : Z (1) = 1 1 1, Z () = and Z (3) = 1 1 1, C = dag(8, 8, 8) and g(n) = Butforn = (, 3, 4), and ( ) D : Z (1) =, Z () = and Z (3) = It can be seen that C = 8I J 3, where J 3 s a 3 3 matrx contanng elements one only. Also g(, 3, 4) whch s equal to 15360, attans the upper bound n (.3.14) and g(, 3, 4) >g(3, 3, 3) mplyng that D s D-optmal. Agan for n = 9,v = 3, c = 4, t s noted that n = (, 3, 4) together wth ( ) D 3 : Z (1) =, Z () = and Z (3) = maxmzes g(n) of (.3.17) and hence gves a D-optmal desgn.

14 6 OCDs n Completely Randomzed Desgn Set-Up Remark.3.5 It s seen from the examples that the choce of optmum n depends on the number of the covarates used apart from the number of cells v n the set-up. Agan t s noted from (.3.7) that det(i(θ)) depends on two factors vz. det(n)(= n ) and det(c). Determnant of N ncreases as the homogenety between the n s ncreases subject to n = n. On the other hand det(c) ncreases, apart from c, wth the largeness of a and the smallness of b, whch agan are acheved by ncluson of maxmum number of even n s closed to n v. The number of odd n s subject to n = n, n between the even ones wth proper homogenety, actually strkes a balance between det(n) and det(c). It s also seen that, when c s small, det(n) s the domnant factor, whle, f c s large det(c) becomes the domnant factor. Incdentally, the above analyss s based on the work n Dutta et al. (014) and t mproves over what was acheved n Dey and Mukerjee (006). References Das K, Mandal NK, Snha BK (003) Optmal expermental desgns wth covarates. J Stat Plan Inference 115:73 85 Dey A, Mukerjee R (006) D-optmal desgns for covarate models. Statstcs 40: Dutta G, Das P, Mandal NK (014) D-Optmal desgns for covarate models. Commun Stat Theory Methods 43: Gall Z, Kefer J (1980) D-optmum weghng desgns. Ann Stat 8: Hedayat AS, Sloane NJA, Stufken J (1999) Orthogonal arrays: theory and applcatons. Sprnger, New York Pukelshem F (1993) Optmal desgn of experments. Wley, New York Rao CR (1973) Lnear statstcal nference and ts applcatons. Wley, New York Rao PSSNVP, Rao SB, Saha GM, Snha BK (003) Optmal desgns for covarates models and mxed orthogonal arrays. Electron Notes Dscret Math 15: Shah KR, Snha BK (1989) Theory of optmal desgns. lecture notes n statstcs, Seres 54. Sprnger, New York Troya Lopes J (198a) Optmal desgns for covarate models. J Stat Plan Inference 6: Troya Lopes J (198b) Cyclc desgns for a covarate model. J Stat Plan Inference 7:49 75 Wojtas M (1964) On Hadamard s nequalty for the determnants of order non-dvsble by 4. Colloq Math 1:73 83

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