Linear Algebra and its Applications

Transcription

1 Linear Algebra and its Applications Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: Maximal determinant over a certain class of matrices and its application to D-optimality of designs K. Filipiak a, A. Markiewicz a,,r.różański b, a Department of Mathematical and Statistical Methods, Poznań University of Life Sciences, Wojska Polskiego 8, Poznań, Poland b Technical Institute, The State Vocational School of Higher Education, Teatralna 5, Gorzów Wlkp., Poland ARTICLE INFO Article history: Received November 010 Accepted 6 April 011 Availableonline3May011 SubmittedbyN.Bebiano AMS classification: 15A15 15A36 6K05 6K10 ABSTRACT It is known that some optimality criteria of experimental designs are functionals of the eigenvalues of their information matrices. In this context we study the problem of maximizing the determinant of αi t P + P T, α >, over the class of t-by-t permutation matrices, and the determinant of αi t +P+P T, α.5, over the class of t-by-t permutation matrices with zero diagonal derangement matrices. The results are used to characterize D-optimal complete block designs under an interference model. 011 Elsevier Inc. All rights reserved. Keywords: Eigenvalues Tridiagonal matrix with corners Determinant D-optimality of designs Interference model Information matrix 1. Introduction A basic problem in the theory of experimental designs is the determination of optimal designs. If in an experiment the response to a treatment is affected by other treatments for example in agricultural and horticultural experiments, then the optimality of designs under an interference model is studied. Recently some results on universal optimality of designs under this model have been published. These results concern mainly optimality of circular neighbor balanced designs CNBD and orthogonal arrays of type I under the fixed and mixed interference models, where the observations are or are not correlated see e.g. [3 7]. It is known, however, that for some combinations of design parameters Corresponding author. address: amark@up.poznan.pl A. Markiewicz /$ - see front matter 011 Elsevier Inc. All rights reserved. doi: /j.laa

2 K. Filipiak et al. / Linear Algebra and its Applications universally optimal designs cannot exist. In such a case the efficiency of some designs or optimality with respect to specified criteria are considered. The efficiency of some cyclic designs under a fixed interference model was studied in [8]. A characterization of E-optimal complete designs under the interference model is given in [9,11]. In this paper, we are interested in determining D-optimal designs under the interference model. This criterion is based on the eigenvalues of the information matrices of designs. This problem has not been discussed previously in the literature. Only the problem of determining D-optimal designs under the model with block effects as the only nuisance parameters over the class of designs has been partially solved: see e.g. [13,1,]. However there is no characterization of D-optimal designs under an interference model. We solve this problem for some classes of complete block designs. It is worth observing that complete block designs are often used in practice. For example in UPOV International Union for the Protection of New Varieties of Plants research, complete block designs are recommended in experiments when the number of treatments is less than 16. Designs with the same number of blocks as the number of treatments and units are also applied in clinical trials [17]. This paper is organized as follows. First we identify permutation matrices that maximize the determinant of αi t P + P T, α>, over the class of t-by-t permutation matrices, and the determinant of αi t + P + P T, α.5, over the class of t-by-t permutation matrices with zero diagonal derangement matrices. In the next section we use the results to characterize D-optimal designs under an interference model over special classes of binary designs, and then we show their optimality over wider classes of designs. It is known that in the class of complete block designs with t treatments, universally optimal designs cannot exist for a number of blocks equal to t ort. Thus,forsuchdesign parameters we determine the structure of the left-neighboring matrix of a D-optimal design under the interference model with left-neighbor effects, and finally we give some methods of construction of D-optimal designs.. Algebraic results In this section, first we identify permutation matrices that maximize the determinant of αi t P + P T, α>, over the class of t-by-t permutation matrices, P t. Then we characterize permutation matrices that maximize the determinant of αi t + P + P T, α.5, over the class of t-by-t permutation matrices with zero diagonal derangement matrices, P t. It is clear that P t exists for t > 1. Throughout the paper we will use the property of permutational similarity. Recall that the matrix P T AP, where P P t, is called permutationally similar to A. It is known that the eigenvalues and, in consequence, the determinant of A and a matrix permutationally similar to A are equal. Let us define the cyclic permutation matrix from P t, t, as H t = 0T t I t 1 0 t 1 Observe that since the eigenvalues of permutationally similar matrices are equal, in the class P t it is enough to consider only matrices of the form P = diag H t1, H t,...,h tm,with m t i = t, 1 m t. It is easy to see that αi t ± P + P T is a tridiagonal matrix with corners m = 1, the identity matrix m = t, H 1 = 1 or a block-diagonal matrix with every block tridiagonal with corners or identity. In the class of derangement matrices P t we assume t i and hence m t for even t and m t 1 for odd t. We will also use the log-superadditivity of functions. Recall that a function f x such that f x 1 + x f x 1 f x is called a log-superadditive function..1. Maximization of detαi t P + P T, α>, overp t Theorem 1. If P P t is permutationally similar to H t,thendetαi t P + P T, α>, ismaximal over P t.

3 876 K. Filipiak et al. / Linear Algebra and its Applications Proof. Let P = H t. Following [14]weobtain det αit H t + H T t = + tr α t Since the trace of the tth power of a diagonalizable matrix is equal to the sum of the tth powers of its eigenvalues, we have det αit H t + H T t = + x t + y t with x = λ 1 α = α α 4, y = λ α = α+ α 4. 3 It is easy to see that y = 1. Observe that for α>the following inequalities are valid: x 0 < x < α α = 1, α α = α+α 4 α < y < α+α = α. 4 It is enough to show that det αit diag H t1, H t diag H T t 1, H T t det αit H t H T t 0, with t 1 + t = t and t 1, t < t. It is equivalent to show that the function f t = + x t + x t is log-superadditive for t N, t, x 0, 1. Since the determinant of a block-diagonal matrix is the product of the determinants of the diagonal blocks, from wehave + x t 1 + y t 1 + x t + y t + x t y t = 1 6x t + x t x t 1 + x t x t 1 x t x t+t 1 x t+t = x t x u i x v i with u = t, t, t, t, t, t, t 1, t T and v = t 1, t 1, t, t, t + t 1, t + t 1, t + t, t + t T.Observing that u is majorized by v, by Proposition C.1. of Marshall and Olkin [15, p. 64], the above expression is nonpositive... Maximization of detαi t + P + P T, α.5,over P t Theorem. If P P t is permutationally similar to i I m H 3,fort= 3m, m N\{1}; ii diag I m H 3, H 4,for t= 3m + 4, m N; iii diag I m H 3, H 5,for t= 3m + 5,m N, then detαi t + P + P T, α.5, ismaximalover P t.moreover,if t 5 then the determinant of αi t + H t + H T t is maximal over P t for α>.

4 K. Filipiak et al. / Linear Algebra and its Applications Proof. Let P = H t. Following [14]wehave det αit + H t + H T t = 1 t+1 + tr α t Since the trace of the tth power of a diagonalizable matrix is equal to the sum of the tth powers of its eigenvalues, we obtain det αit + H t + H T + x t + y t, for even t, t = + x t + y t, for odd t, 5 with x, y determined in 3. Observe that for every α.5 the following inequalities are valid: 0 < x α α 1 = 1, α 1 = α+α 1 y < α+α = α. 6 Let t =. The only matrix P P is H. Let t = 3. The only matrices P P 3 are H 3 and H T 3. Let t = 4. The only matrices P P 4 are H 4 and diagh, H up to permutational similarity. Using 5 wehavetoshowtheinequality + x + y + x 4 + y 4 which follows from the log-superadditivity of f t = + x t + y t for t N, t. Observe that it holds for α>. Let t = 5. The only matrices from P P 5 are H 5 and diagh 3, H up to permutational similarity. Using 5 wehavetoshowtheinequality + x + y + x 3 + y 3 + x 5 + y 5 which follows from + x + y + x 3 + y 3 + x 5 + y 5 = 6 x x x + 1 x 3 x x = x+1 x x+x x+1 < 0. x 3 For t 6 we prove the theorem in two steps. First we show det αit + H t + H T t det αit + diagh t 3, H 3 + diag H T, t 3 HT 3 7 and then det αit + diagh t1, H t + diag H T, H T t1 t det αit + diagh t 3, H 3 + diag H T, 8 t 3 HT 3, with t 1 + t = t, t 1, t, and at least one t i = 3, i = 1,. Since the determinant of a block-diagonal matrix is equal to the product of the determinants of its blocks, using 5 and 6 weobtain det αit + H t + H T t det αit + diagh t 3, H 3 + diag H T, t 3 HT 3 = + x t + y t + x t 3 + y t 3 + x 3 + y 3 = 6 x 3 y 3 x t 3 y t 3 x 3 y t 3 x t 3 y 3 < 0

5 878 K. Filipiak et al. / Linear Algebra and its Applications for even t, and det αit + H t + H T t det αit + diag H t 3, H 3 + diag = + x t + y t + x t 3 + y t 3 + x 3 + y 3 = 6 + x 3 + y 3 x t 3 y t 3 x 3 y t 3 x t 3 y 3 < y3 1 y t y3 < 0 H T, t 3 HT 3 9 for odd t, and we have proved 7. Let t 1 + t = t, t 1, t, and at least one t i = 3, i = 1,. Assume odd t 1 and t. Then, using 5, we may rewrite inequality 8as + x t 1 + y t 1 that is equivalent to + x t + y t + x t 3 + y t 3 + x 3 + y 3 0 x 3 x t 3 + x t 1 + x t y3 y t 3 + y t 1 + y t x 3 y t 3 x t 3 y 3 + x t 1 y t + x t y t and 1 x t xu i 10 xv i 0withu = t + t 1, t + t 1, t + t, t + t, t + t 1 t, t + t t 1, t t 1, t t 1, t t, t t T and v = t 3, t 3, t 6, t + 3, t + 3, t 3, t 3, 6, 3, 3 T.Observing that u is majorized by v, similarly as in the proof of Theorem 1, the above expression is nonpositive. Assume now t 1 and t are both even. Then we have to show + x t 1 + y t 1 + x t + y t + x t 3 + y t 3 + x 3 + y 3 0. We obtain this inequality directly from the fact that the formula on the left-hand side is always smaller than the respective formula for odd t i, i = 1,. Assume now, without loss of generality, that t 1 is even and t is odd. Then, similarly as in the previous cases and since t 1 4, t 5, + x t 1 + y 1 t + x t + y t + x t 3 + y t 3 + x 3 + y 3 = x 3 x t 3 + x t 1 x t + y 3 y t 3 + y t 1 y t x 3 y t 3 x t 3 y 3 + x t 1 y t + x t y t 1 = x t x u i x v i with u = t + t 1, t + t 1, t + 3, t + 3, t + t 1 t, t + t t 1, t 3, t 3, t t 1, t t 1 T and v = t 3, t 3, t 6, t + t, t + t, t t, t t, 6, 3, 3 T. Observing that u is majorized by v, similarly as in the proof of Theorem 1, the above expression is nonpositive. From the proof of the above theorem it follows that for t = 6 and t = 8, the results presented hold for α >. Further observe that the last line of inequality 9 maybereplacedby 6 + x 3 + y 3 y t 3 x 3 y t 3, which is maximal for t = 7, and then the inequality holds for α.4. Moreover, we can conclude that for <α<.4 and odd t,detαi t + H t + H T t could be larger than the determinant of the respective matrices given in the theorem. Numerical studies show that this happens especially for α close to and relatively small odd t. Example. Let t = 7 and <α Then 6 and 9 are not satisfied. We have the following function: f α = + x 7 + y 7 + x 4 + y 4 + x 3 + y 3, which is nonnegative for every <α x = α α 4, y = 1 x

6 K. Filipiak et al. / Linear Algebra and its Applications If t = 9 then the respective function is nonnegative for <α.05361, and for <α if t = D-Optimality of designs under an interference model 3.1. Definitions and notations Let D t,b,k be the set of designs with t treatments, b blocks and k experimental units per block. An interference model with left-neighbor effects associated with the design d D t,b,k can be written as y = T d τ + L d λ + Bβ + ε, 10 where τ, λ and β are the vectors of treatment effects, left-neighbor effects and block effects, respectively. Here ε is a vector of random errors with Eε = 0 and Covε = σ I bk, where σ is an unknown constant. The matrix B = I b 1 k, where 1 k is the k-vector of ones and is the Kronecker product, is the design matrix of block effects. Further, T d is the design matrix of treatment effects and L d = I b H k T d,withh k as defined in 1, is the design matrix of left-neighbor effects. This form of the matrix L d follows from the assumption that each treatment has a left neighbor for more details see e.g. [3,9,11,10]. All the results presented in the paper also hold for a model with right-neighbor effects, i.e., with L d = I b H T k T d. Under the interference model 10, the information matrix for estimating treatment effects, C d, has the form C d = T T d Q B:L d T d, where Q X = I P X = I XX T X X T is the orthogonal projector onto the orthocomplement of the column span of X and A denotes the generalized inverse of A. Since P X1 :X = P X1 + P QX1 X,the information matrix may be expressed as C d = T T d Q B T d T T d Q B L d L T d Q B L d L T d Q B T d, or equivalently C d = T T d Q L d T d T T d Q L d B B T Q Ld B B T Q T Ld d. 11 It is easy to see that Q B = I b E k, where E k = I k 1 k 1 k1 T k.becauseoftheformofe k and since H k is orthogonal, it can be seen that T T d Q B T d = L T d Q B L d. We are interested in determining D-optimal designs, i.e., designs minimizing the general variance of best linear unbiased estimatorsof any orthonormalset of t 1 contrasts. The condition of D-optimality can be expressed in terms of the eigenvalues of the information matrix as follows. For a design d D t,b,k let 0 = λ 0 C d λ 1 C d λ t 1 C d be the eigenvalues of its information matrix C d.adesignd D t,b,k is called D-optimal over D t,b,k if t 1 λ ic d t 1 λ ic d for all designs d D t,b,k cf. [16]. Observe that formula 11 involves a generalized inverse of a matrix, which depends on the design, i.e., which changes with the arrangement of treatments on experimental units. This makes the determination of a D-optimal design difficult. Therefore in this paper we consider experiments with t = k. The class of designs with t = k, in which each treatment occurs at most once in each block the class of complete binary designs, we will denote by B t,b,t. In such a case the information matrix has the form C d = bi t 1 b ST d S d, 1 where S d = L T d T d.thei, jth entry of S d denotes the number of occurrences of treatment j with treatment i as left neighbor in a design d. Therefore we will call this matrix a left-neighboring matrix.

7 880 K. Filipiak et al. / Linear Algebra and its Applications Note that in the class B t,b,t, the diagonal entries of S d are equal to 0 and the off-diagonal entries belong to the set {0, 1,...,b}. Moreover, S d 1 t = S T d 1 t = b1 t. It is known that for t = k and b = t orb = t, a universally optimal design under the interference model CNBD cannot exist. Thus throughout this paper we will assume b = t Section 3. and b = t Section 3.3. Following [10], designs with t = k = 3, 4 and b = t are all disconnected. Thus we assume t 5. We show optimality over the class D t,t,t.inthecaseb = t we start with t = 3. We prove optimality over the class of equireplicated designs with no treatment preceded by itself, denoted by R t,t,t. 3.. D-optimality of designs over D t,t,t Let B t,t,t be the set of designs, for that S d = 1 t 1 T t I t P d,with P d P t. Assume d B t,t,t.then S T d S d = t 41 t 1 T t + I t + P d + P T d and the information matrix may be written as C d = t 4t+ I t t t 4 t 1 t1 T 1 t t P d + P T. d Observe that the matrices C d and C d + β1 t 1 T t have the same eigenvectors and the eigenvalues corresponding to those eigenvectors are the same, except the vector 1 t, which is an eigenvector of both matrices corresponding to the eigenvalues 0 and βt, respectively. Thus the product of the t 1eigen- values of C d is equal to the determinant of C d + β1 t 1 T t divided by βt, and it is enough to compare the determinants of t C d + t 41 t 1 T t = αi t P d + P T d for different matrices P d,with α = t 4t +. Observe that α>foreveryt 5, and we can write the following corollary. Corollary 3. If there exists a design d B t,t,t,t 5, such that the left-neighboring matrix S d is permutationally similar to 1 t 1 T t I t H t,thend is D-optimal over B t,t,t. Now we generalize the results of Corollary 3 as follows. Theorem 4. If d B t,t,t,t 5, isd-optimalover B t,t,t,thenitisd-optimaloverd t,t,t. Proof. First we prove the D-optimality of d B t,t,t over the class of binary designs, B t,t,t and then over the class D t,t,t. Let d be D-optimal over B t,t,t, t 5. Then, from Theorem 1,, 4 and binomial evaluation, we obtain det t Cd + t 41 t 1 T t = det αit H t + H T t = + x t + y t > y t > α α t 13 >αt tα t. Let d B t,t,t \ B t,t,t.then t C d + t 41 t 1 T t = t I t + t 41 t 1 T t ST d S d. 14 Recall that for every Hermitian matrix the product of its eigenvalues is majorized by the product of its diagonal entries cf. [15]. Observe that the row and column sums of S d are t. Since for every d B t,t,t \ B t,t,t at least one entry of S d is not smaller than, at least one diagonal entry of matrix 14 is not greater than t + t 4 t = t 4t = α, and the remaining diagonal entries

8 K. Filipiak et al. / Linear Algebra and its Applications are not greater than α. Hence det t Cd + t 41 t 1 T t α t 1 α. t t Cd + t 41 t 1 T t ii 15 Comparing 13 and 15weobtain α t 1 α α t tα t = α t t 6t for t 6. If t = 5 we obtain the result by comparing 15with α 5 α. Now let d D t,t,t \B t,t,t. We consider two cases: when d is equireplicated, and when it is not. a Let d be equireplicated. From 11wehave C d L T T d Q L d T d = R d S T d R 1 d S d = t I t 1 t ST d S d, 16 with R d = T T d T d = L T d L d = diagr d,1,...,r d,t the diagonal matrix with the number of replications of every treatment on the diagonal, and where A L B means that matrix A is better than B in Loewner s sense. If in the design d there is no treatment preceded by itself S d has zero diagonal, according to 1the matrix on the right side of 16 is the information matrix of a certain design from B t,t,t.fromthe first part of the proof we obtain the result. Assume now that in the design d at least one treatment has itself as the nearest neighbor. Recall that the entries of S d are nonnegative integers and S d 1 t = S T d 1 t = t 1 t. Thus in every row and every column of S d there is at least one zero. Hence if we permute the rows and columns of S d,weobtain t I t 1 t PT ST d PT 1 P 1S d P = t I t 1 t PT ST d S dp, P 1, P P t, which is permutationally similar to the information matrix of a certain design from B t,t,t. Again, from the first part of the proof, we obtain the result. b Let d not be equireplicated. From 11wehave C d L T T d Q L d T d = R d S T d R 1 d S d. 17 Observe that the jth diagonal entry of R d S T d R 1 d S d,isequalto t R d S T d R 1 d S d = r s d,ij d,j jj r d,i = r d,j t s d,ij t, l=1 s d,il and we obtain the following average of diagonal entries of R d S T d R 1 d S d: 1 t R d S T d t R 1 d S d jj j=1 = 1 t r d,j 1 t t t t j=1 j=1 = t 1 t t j=1 s d,ij t t. j=1 s d,ij s d,ij t l=1 s d,il Assume that at least one entry of S d is not smaller than. We are looking for such a design that minimizes t t j=1 s d,ij t j=1 s. d,ij

9 88 K. Filipiak et al. / Linear Algebra and its Applications Without loss of generality, assume that the first treatment appears the most often in the design. Then t t j=1 s d,ij t j=1 s d,ij = t j=1 s d,1j r d,1 t + t, + t t j=1 s d,ij i= t j=1 s d,ij t + t 1, r d,1 = t 1 r d,1 = t t + i t i, r d,1 = t + i, i = 1,,...,t 4t + 1 and hence 1 t R d S T d t R 1 d S d j=1 jj t 1 t t + t+1 = t 3 tt+1. From 17, and since the product of the eigenvalues of the matrix is majorized by the product of its diagonal entries, we obtain det t Cd + t 41 t 1 T t t t Rd t S T d R 1 d S d + t 41 t 1 T t 18 ii t t t 3 tt+1 + t 4 = t 4t + t t tt+1. On the other hand, using 13, we have det t Cd + t 41 t 1 T t > t t 4t + t 4 8t 3 + 0t 18t + 4 t 4t+ t 19 t t 4t + t 4 8t 3 + 0t 18t Now it is enough to show that t t 4t + t 4 8t 3 + 0t 18t which implies t 4 8t 3 + 0t 18t t 4t + t. 343 tt+1 t 4t + t t, 0 tt+1 Observe that the above inequality is equivalent to 5t 5 46t 4 + 9t t 3 0, which is valid for every t > For 6 t 11 we obtain the theorem directly from the inequality 0. For t = 5, using 18, we have det 3C d T 5 < 1515 = det C d T 5. 5 = Finally, let d D t,t,t be a design such that all entries of S d are zeros or ones. First we evaluate the maximal and minimal diagonal entry of R d S T d R 1 d S d and then we evaluate the remaining diagonal entries as the average of the trace of this matrix with the maximal and minimal entries subtracted.

10 K. Filipiak et al. / Linear Algebra and its Applications Observe that since t j=1 s d,ij = t j=1 s d,ij, wehave t tr R d S T d R 1 d S d = R d S T d R 1 d S d jj j=1 = tt 3. To obtain a lower bound of the maximal diagonal entry of R d S T d R 1 d S d, we have to minimize the maximal number of replications and maximize t s d,ij j=1. Recall that in our case s r d,ij d,j {0, 1}.Without loss of generality, assume the first treatment is replicated the most often, i.e., r d,1 = r max = t or r d,1 = r max = t 1. Observe that the vector of replications is majorized by the vector with i t 3 replications of t, one replication of t, and two replications of 1 if r max = t; ii t replications of t 1, one replication of t 3, and one replication of 1 if r max = t 1. Note that all entries in the first row of S d are equal to 1 if r max = t, and exactly one entry in the first row of S d is equal to 0 if r max = t 1. Since the function t 1 is increasing in the sense r d,i of majorization, we obtain the maximum of this function if the replications are as above for more details see e.g. [15]. Since t t t t t 1 t t 1 t 3 1, we have max i R d S T d R 1 d S d ii t 1 t 3 t t = t 3 + t 5 t 1t 3. Similarly, the minimal diagonal entry of R d S T d R 1 d S d is maximal if the minimal number of replications, r min,ist 3, and the remaining treatments are replicated as equally as possible. Thus min R d S T d R 1 d S d t t 6 i ii t 1 t = t 4 + t+ t 1t, for every t 6. Now we can evaluate the remaining entries of R d S T d R 1 d S d as follows: [ 1 tr R t d S T d R 1 d d S min max R d S T d R 1 d S d d i ii ] max min R d S T d R 1 d S d d i ii 1 t [ t 5t + 7 t 4t+ t 1t t 3 ], t 6. Thus det t Cd + t 41 t 1 T t t t R d S T d R 1 d d S + t 41t 1 T t ii t 5t t 3t 7 4t + 3 t 1 t 1t 3 t t t 4t + 4t + 3, t 6. t 1t t 3

11 884 K. Filipiak et al. / Linear Algebra and its Applications Using 19 it is enough to show that t t 4t + 3 t 4t+ t 1t t 3 t 4t + 3 t 1t 3 3t 7 t 5t t 1 t 1 t 4t + t 4 8t 3 + 0t 18t , t 6, which is equivalent to t t 4t + t 4t + 3 t 4t+ t 1t t 3 [ ][ t 5t t 4t + 3 t 1 t 4 8t 3 + 0t 18t ] t 1t 3 3t 7 Using binomial evaluation, we may write t t 4t + t 4t + 3 t 4t+ t 1t t 3 [ t 3 8t ] t + 19t 10 = 1 t 5 10t t 3 70t + 65t > 1 t t 3 8t + 19t 10 t 5 10t t 3 70t + 65t +, t 6. t t 3t 3 8t + 19t 10 t 5 10t t 3 70t + 65t t t 3t 4t3 8t + 19t 10 3, t 5 10t t 3 70t + 65t 3 t 6 and it is enough to show that the last expression is not smaller than the right side of. Solving this inequality using Mathematica 7.0 we find that it is valid for every t > For t = 7, 8 we obtain the result by comparing the left side of 1 with the determinant of t C d +t 41 t 1 T t, i.e., det 5C d T < < = det 5C d T. det 6C d T < < = det 6C d T. For t = 5, 6 we obtain the result using numerical calculations we compare det t Cd + t 41 t 1 T t for every non-equireplicated design d for which the entries of S d are zeros and ones D-Optimality of designs over R t,t,t Let B t,t,t be the set of designs for which S d = 1 t 1 T t I t + P d,with P d P t. Assume d B t,t,t.then C d = t I t t 1 t 1 T + 1 t t P d + P T. d Similarly as in the previous sections, to characterize a D-optimal design it is enough to compare the determinants of tc d + t1 t 1 T t = αi t + P d + P T d, α = t, for different matrices P d. Observe that α>.5 foreveryt 3, and we can write the following corollary.

12 K. Filipiak et al. / Linear Algebra and its Applications Corollary 5. If there exists a design d B t,t,t such that S d is permutationally similar to i I m H t 1 T t I t, for t = 3m, m N; ii diagi m H 3, H t 1 T t I t, for t = 3m + 4, m N {0}; iii diagi m H 3, H t 1 T t I t, for t = 3m + 5, m N {0}, then d is D-optimal over B t,t,t,t 3. Now we generalize the results of Corollary 5 as follows. Theorem 6. If the design d is D-optimal over B t,t,t,thenitisd-optimaloverr t,t,t,t 3. Proof. First we show the D-optimality of d B t,t,t over the class of binary designs, B t,t,t and then over the class of equireplicated designs with no treatment preceded by itself, R t,t,t. Let d be D-optimal over B t,t,t.fromtheorem and 5 weobtain det tc d + t1 t 1 T t Using 6 wemaywrite = + x 3 + y 3 m, for t = 3m, m N, + x 3 + y 3 m + x 4 + y 4, for t = 3m + 4, m N {0}, + x 3 + y 3 m + x 5 + y 5, for t = 3m + 5, m N {0}. det tc d + t1 t 1 T y t y t 4, for t = 3m + 4, m N {0}, t > y t, for remaining t. It is easy to see that det tc d + t1 t 1 T t > yt y t 4. Observe that since α 7, due to 3, y >α α. Thus, using binomial evaluation, det tc d + t1 t 1 T t > α t 4 α α4 8α α 4 α > α t 4 t 4α t 6 α 4 8α. Let d B t,t,t \ B t,t,t. Then, at least two off-diagonal entries of S d are not smaller than, or at least one entry of S d is greater than. In both cases at least one off-diagonal entry of S d is zero. Thus at least one diagonal entry of tc d + t1 t 1 T t = t I t S T d S d + t1 T t 1 t is not greater than t t 4 + t = t 4 = α, and the remaining diagonal entries are not greater than α.weobtain det tc d + t1 t 1 T t It is enough to show that t tc d + t1 t 1 T t ii αt 1 α. α t α t 1 < α t 4 t 4α t 6 α 4 8α. Observe that α t α t 1 α t 4 t 4α t 6 α 4 8α = α t 4 α 3 tα + 8t 4 = α t 4 t 6 t 5 6t 4 + 4t 3 + 1t + 4t 40 <0 for every t 3. Therefore we have proved the D-optimality of d B t,t,t over B t,t,t.

13 886 K. Filipiak et al. / Linear Algebra and its Applications Let d R t,t,t \B t,t,t.wehave C d L R d S d R 1 d ST = d ti t 1 t S ds T. d Since in the class R t,t,t every diagonal entry of S d is zero, the matrix ti t 1 t S ds T d is the design matrix of a certain binary design. We obtain the result from the first part of the proof Construction of D-optimal designs In the previous sections we derived the forms of the left-neighboring matrix of D-optimal designs. The problem is whether there exists a design d with the respective S d.[9] presented a method of construction of E-optimal designs over D t,t,t. The construction of D-optimal designs over this class is similar. In this section, we show how to construct D-optimal designs over D t,t,t. Consider the class D t,t,t.letdesignd have the left-neighboring matrix S d given in Corollary 3. Note that the matrix 1 t 1 T t I t from that corollary is the left-neighboring matrix of a circular neighbor balanced design CNBD. Thus subtracting a matrix permutationally similar to H t is equivalent to abridging one arbitrary block from CNBD. Thus design d can be constructed from a CNBD by abridging one arbitrary block. A catalog of CNBDs with t = k, b = t 1isgivenin[1]. Example. The following designs are D-optimal over D t,t,t : for t = 5 : for t = 8 : d = 1345, d = Observe, however, that for t = 6aCNBDwithb = t 1 and t = k cannot exist cf. [1]. Moreover, numerical calculations show that a design with S d = T 6 I 6 H 6 does not exist. In the class B 6,4,6 there exist only designs with left-neighboring matrix permutationally similar to T 6 I 6 diagh 4 : H say S d1 or1 6 1 T 6 I 6 I 3 H say S d. For such designs we have det 4C d T 6 = < = det 4C d T. Moreover, for a design d B 6,4,6,using15, the following inequality holds: det 4C d T α5 α = < = det 4C d T. For a design d such that in S d there is at least one entry greater than 1, using 18 we obtain the inequality det 4C d T 6 t 4t + t t tt < = det 4C d T. For a design d such that all the entries of S d belong to the set {0, 1}, by numerical calculations we verify that the inequality det 4C d T 6 < = det 4C d T 6 always holds. We have shown that design d 1 is D-optimal over D 6,4,6.

14 K. Filipiak et al. / Linear Algebra and its Applications Example. The following design is D-optimal over D 6,4,6 : d = References [1] J.-M. Azaïs, R.A. Bailey, H. Monod, A catalogue of efficient neighbour-designs with border plots, Biometrics [] K. Balasubramanian, A. Dey, D-optimal designs with minimal and nearly minimal number of units, J. Statist. Plann. Inference [3] P. Druilhet, Optimality of circular neighbour balanced designs, J. Statist. Plann. Inference [4] K. Filipiak, A. Markiewicz, Optimality of circular neighbor balanced designs under mixed effects model, Statist. Probab. Lett [5] K. Filipiak, A. Markiewicz, Optimality of type I orthogonal arrays for general interference model with correlated observations, Statist. Probab. Lett [6] K. Filipiak, A. Markiewicz, Optimality and efficiency of circular neighbor balanced designs for correlated observations, Metrika [7] K. Filipiak, A. Markiewicz, Optimal designs for a mixed interference model, Metrika [8] K. Filipiak, R. Różański, Some properties of cyclic designs under an interference model, Coll. Biom. 34A [9] K. Filipiak, R. Różański, E-optimal designs under an interference model, Biom. Lett [10] K. Filipiak, R. Różański, Connectedness of complete block designs under an interference model, Statist. Papers [11] K. Filipiak, R. Różański, A. Sawikowska, D. Wojtera-Tyrakowska, On the E-optimality of complete designs under an interference model, Statist. Probab. Lett [1] N. Gaffke, D-optimal block designs with at most six varieties, J. Statist. Plann. Inference [13] J.A. John, T.J. Mitchell, Optimal incomplete block designs, J. R. Stat. Soc. Ser. B [14] L.G. Molinari, Determinants of block tridiagonal matrices, Linear Algebra Appl [15] A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and its Application, Academic Press, [16] F. Pukelsheim, Optimal Designs of Experiments, Wiley, New York, [17] D.H. Rees, Some designs of use in serology, Biometrics