Optimality of block designs under the model with the first-order circular autoregression

In this paper optimal properties of some circular balanced block designs under the model with circular autoregression of order one are studied. Universal optimality of some balanced block designs with equal block sizes is proven and E-optimality of complete balanced block designs with the number of blocks equal to the number of treatments or the number of treatments reduced by two is shown.


Introduction
The block experiments have been widely used in sciences, medical and engineering. The problem of universal optimality of block designs is widely studied for different correlation structures. Chai and Majumdar (2000) showed that under the nearest neighbor correlation structure the universally optimal block designs can be constructed from semibalanced arrays. Recently, Khodsiani and Pooladsaz (2017) characterized the universally optimal block designs under the hub correlation structure.
Under the standard block model in which correlation between observations in every block decreases exponentially with the distance, the first-order linear autoregression covariance structure, AR(1, L), is considered in the literature. For some values of correlation parameter Kunert (1987) proved that the nearest neighbor balanced incomplete block designs, characterized by Gill and Shukla (1985), are universally optimal over the class of incomplete block designs. For positive correlations Pooladsaz and Martin (2005) extended the results of Kunert (1987) for the designs with block sizes exceeding the number of treatments. Kunert and Martin (1987b) extended the optimality results of some neighbor balanced designs for other autoregression correlation structures, such as e.g. circular autoregression (AR(1, C), defined in (2)) with positive correlation. Circular designs were considered for example by Rees (1967), in serological experiment in which treatments are arranged in circular blocks where every treatment has two neighbors. Another example is an experiment in marine biology in which five genotypes of bryozoan such that neighboring genotypes might interfere with each other were compared by suspending them in sea water around the circumference of a cylindrical tank; see Bayer and Todd (1996). One of the aims of this paper is to show universal optimality of some circular nondirectionally neighbor balanced designs under the model with AR(1, C) for positive and negative values of correlation coefficient.
Existence of universally optimal designs often has some limitations and for some combinations of design parameters the universally optimal designs can not exist. In such a case efficiency of some designs or optimality with respect to the specified criteria is considered. E-criterion is one of the most popular criteria that can be studied in this case, as it minimizes the largest variance among all best linear unbiased estimators of normalized linear contrasts, and plays an important role in e.g. admissibility investigations; cf. Pukelsheim (2006). Although determining the E-optimal designs is not easy especially when the observations are correlated because E-criterion is based on the eigenvalues of the information matrix of designs. Jacroux (1982Jacroux ( , 1983 characterized E-optimal block designs for uncorrelated observations. Kunert and Martin (1987a) proved optimality of some neighbor designs with respect to the specific criteria, such as e.g. A-, D-or E-optimality under the model with observations correlated according to AR(1, L) with positive correlation parameter.
According to the circular structure of AR(1, C), optimality problem of block designs under this correlation structure is similar to the determining optimal designs in interference models. There are several results on optimality of circular neighbor balanced designs (CNBDs) and orthogonal arrays of type I under the fixed and mixed interference models, where the observations are correlated or not (see e.g. Druilhet 1999;Filipiak and Markiewicz 2003, 2004, 2007. Similarly as Kunert and Martin (1987a) we will consider circular complete block designs (known also as William's II(a) designs with extra plot). It is worth observing that designs with complete blocks are often used in practice. For example in UPOV (The 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. The designs with the same number of blocks as number of treatments and units are also applied in clinical trials.
Since complete CNBDs that are universally optimal under the interference models exist only for specific combinations of design parameters, one can look for universally optimal designs for some other combinations of parameters or to determine designs which are optimal with respect to other criteria, e.g. E-optimality. Bailey et al. (2017) considered optimality of circular weakly neighbor balanced designs under the interference model. They showed the construction methods of such designs, mostly based on graph theory, and they applied Hamiltonian decomposition of matrices using GAP (2014) or Mathematica software. Nevertheless, numerical algorithms start to be not efficient for increasing number of treatments.
Regarding the circular block effects model with correlated errors and its connections to the interference model with uncorrelated errors, construction methods of E-optimal designs are even more complicated, and numerical algorithms fails even for relatively small number of treatments. Nevertheless, algebraic concepts of construction methods based on the left-neighboring matrix of a circular design being an incidence matrix of some particular graph, can be used to more general models.
This paper is organized as follows. Section 2 introduces notation and definitions. Section 3 presents some results about universal optimality of designs under the standard block model for any size of blocks when the observation errors are correlated according to AR(1, C). In Sect. 4 some E-optimal complete block designs are characterized.

Notation and definitions
Let us consider a set of circular block designs D t,b,k where t, b and k are respectively the number of treatments, blocks and experimental units per block. A standard model associated with a design d ∈ D t,b,k can be written as where μ is a general mean, τ and β are respectively the vectors of treatment and block effects, and ε is a vector of random errors with E(ε) = 0 bk and Cov(ε) = σ 2 I b ⊗ V.
The bk−vectors of ones and zeros are respectively denoted by 1 bk and 0 bk , the identity matrix of order b is denoted by I b , whilst σ 2 is a positive constant, V is a known, symmetric, positive definite covariance matrix and ⊗ denotes the Kronecker product. By T d and B = I b ⊗ 1 k respectively design matrices for treatments and blocks are denoted.
In this paper we are interested in characterization of universally optimal design. Thus, let us denote as C d the information matrix of design d for estimation of treatment effects in the model (1). Due to Kiefer (1975, Proposition 1), if C d 1 t = 0 t for every d ∈ D t,b,k and if design d * is such that C d * is completely symmetric and has maximal trace over D t,b,k , then d * is universally optimal among the class D t,b,k . Recall, that a t × t matrix A is completely symmetric, if all its diagonal elements are equal and all its off-diagonal elements are equal, that is The above E-optimality criterion can be also expressed in terms of variance, that is, E-optimal design minimizes the largest variance among all best linear unbiased estimators of normalized linear contrasts; cf. Pukelsheim (2006). From Kiefer and Wynn (1981) the information matrix of design d for estimation of treatment effects in the model (1) has the form Observe that since 1 bk belongs to the column space of T d , we have C d 1 t = 0 t and Kiefer's conditions for universal optimality can be applied.
We will assume that the observations in different blocks are uncorrelated but that observations within blocks are correlated according to a first-order circular autoregression process, and |a| < 1 being a correlation parameter. Following Kunert andMartin (1987b) andMarkiewicz (2005), the inverse of V can be presented as where H k denotes the k × k left-neighbor incidence matrix, that is the matrix with (i, j)th element equal to 1 if i − j = 1 and h 1,k = 1, and 0 otherwise. Hence (1 − a) 2 k 1 k 1 k and the information matrix can be written as Filipiak et al. 2008) and N d = T d B is the treatment-block incidence matrix. Recall, that the design is called circular if each block of a design has the form of a circle or, if plots in blocks are arranged in linear forms, but there are additional border plots at the beginning of each block, containing the same treatment as at the opposite end of the block (cf. Druilhet 1999). The border plots receive treatments but are not used for measuring the response variables. Note that the (i, j)-th element of S d , s d,i j , denotes the number of occurrences of treatment i with treatment j as a left neighbor. If a design d has no self-neighbors, the diagonal entries of S d are equal to zero. Since in our considerations the matrix S d + S d plays a crucial role we denote its elements by ξ i j , that is ξ i j = s d,i j + s d, ji . It is clear that for designs without self-neighbors ξ ii = 0. Throughout this paper we use some properties of a balanced block design (BBD), i.e. such a design d ∈ D t,b,k for which (i) all n d,i j = k/t or k/t + 1, (ii) all r d,i are equal (say r ), and (iii) every pair of distinct treatments occurs together in the same number of blocks (say λ), where x is the largest integer not exceeding x, r d,i is the number of replications of the ith treatment in d, the ith diagonal entry of R d , and n d,i j is the (i, j)th entry N d (cf. Kiefer 1958). A BBD reduces to a balanced incomplete block design (BIBD) when k < t. All designs satisfying (i) are called generalized binary designs (Das and Dey 1989), while designs satisfying (ii) are called equireplicated designs. The class of generalized binary designs with k ≤ t will be denoted by B t,b,k .
The following algebraic notation and definitions is used in Sect. 4 of the paper. An n × n matrix A is said to be reducible if either of the following conditions is satisfied: (a) n = 1 and A = 0; (b) n ≥ 2 and there is a permutation matrix P ∈ P n and an integer u with 1 ≤ A matrix is called irreducible if it is not reducible. Let P n be the set of all permutation matrices of order n and let P n ⊂ P n be the set of permutation matrices with zero diagonal (the set of derangement matrices of order n). The matrix P AP, where P ∈ P n , is called permutationally similar to A. It is worth noting that the eigenvalues of A and a matrix permutationally similar to A are the same.

Universal optimality over the class D t,b,k
If a = 0 there is no correlation structure in model (1) and it is known that every BBD is universally optimal over D t,b,k (cf. Shah and Sinha 1989). In this section we characterize universally optimal designs under model (1) with AR(1, C) structure, where a ∈ (−1, 1)\{0}. We denote D t,b,k as the subclass of D t,b,k with no treatment preceded by itself.

Theorem 1
The circular BBD d * such that S d * + S d * is completely symmetric with zero diagonal, is universally optimal under model (1) over the class D t,b,k if a > 0, and over the class D t,b,k if a < 0.
Proof By (3) for the circular BBD d * such that S d * + S d * is completely symmetric, Thus, C d * is completely symmetric and since tr S d * + S d * = 0 we obtain It is known (cf. Shah and Sinha 1989) It is worth noting that for a = 0 and d ∈ D t,b,k \B t,b,k , trC BBD > trC d . Moreover, for k ≤ t, trS BBD = 0, and from continuity of the trace (as a function of a) it follows that for negative a, sufficiently close to 0, the BBD is universally optimal over the class D t,b,k . Moreover, for small number of treatments (e.g. t = k = 3, 4, 5), the BBDs are universally optimal over the class D t,b,k even for all negative a.
One of the example of a designs satisfying conditions of Theorem 1 are nondirectionally neighbor balanced designs (NdNBD), that is designs with Azaïs et al. 1993). However, NdNBD cannot exist for even t. Nevertheless, we can consider generalized NdNBDs having the property of S d + S d being proportional to 1 t 1 t − I t , which satisfies Theorem 1. For example the following design with blocks represented as rows 1 2 3 1 4 5 1 2 6 1 4 6 1 3 5 2 4 3 2 5 6 2 4 5 3 4 6 3 5 6 is a generalized NdNBD. Observe moreover, that CNBDs and circular neighbor balanced designs at distance 2 (CNBD2) defined e.g. by Druilhet (1999), that can exist also for even t, are also generalized NdNBDs. Their construction methods for complete blocks and blocks of size t − 1 can be found in Azaïs et al. (1993).
Universal optimality of CNBD2 under the wider model with AR(1, C), namely the one-sided interference model, was shown in Filipiak and Markiewicz (2005).

E-optimal complete block designs for b = t − 2 and b = t
In this section we assume t ≥ 3. We denote the subclass of equireplicated designs of D t,b,k by R t,b,k and R t,b,k consists of the equireplicated designs with no treatment preceded by itself.
Assume that d ∈ B t,b,t . Since the blocks of designs are complete now, that is k = t, every binary design is a BBD. Thus, due to (4), and it is enough to find minimum of Observe that every irreducible derangement matrix is permutationally similar to the incidence matrix H n defined in Sect. 2. From circularity of H n it follows, that the non-ordered eigenvalues μ i (H n ), i = 1, 2, . . . , n, are equal to the roots of unity, that and Moreover, since every reducible derangement matrix is permutationally similar to the block-diagonal matrix with H n j on the diagonal, u j=1 n j = n, 1 < u ≤ n, the set of its eigenvalues consists of the n j th roots of unity, j = 1, . . . , u.
Throughout the paper the following inequalities for the eigenvalues of a diagonal l × l block, say A 11 , of partitioned matrix A: where m = 1, 2, . . . , l; cf. Marshall et al. (2011).
The following lemmas will be useful for characterization of E-optimal designs.
Lemma 1 If the derangement matrix P ∈ P n is permutationally similar to the matrix (iv) diag I i ⊗ H 3 , I j ⊗ H 5 if n = 5 or n ≥ 8 and n = 3m, m ∈ N with n = 3i + 5 j for some i ∈ N ∪ {0} and j ∈ N, then the minimal eigenvalue of P + P , that is λ n (P + P ), is maximal over P n .
Proof From (6) the maximum of λ n (P + P ) is obtained for P ∈ P n permutationally similar to the block-diagonal matrix with irreducible diagonal blocks of odd order as small as possible. Since n = 5 and every n ≥ 8 can be presented as 3i + 5 j for some i, j ∈ N ∪ {0}, we obtain (iii) and (iv). Moreover, for n = 3i + 5 j, j = 0, the maximum of λ n (P +P ) is equal to the maximum of λ n (H 5 +H 5 ), that is 2 cos (4π/5). If n = 2 or 4 then any decomposition of n for odd numbers is possible and hence λ n (P + P ) = −2. Similarly, n = 7 cannot be decomposed for odd numbers and hence max P∈P 7 λ 7 (P + P ) = λ 7 (H 7 + H 7 ) = 2 cos (6π/7).
Lemma 2 If the derangement matrix P ∈ P n is permutationally similar to H n , then the second maximal eigenvalue of P + P , that is λ 2 (P + P ), is minimal over P n .
Similar results as in Lemmas 1 and 2 can be found in Filipiak et al. (2008).

E-optimal complete block designs for b = t − 2
Let define the following subclass of B t,t−2,t : Then, for every d ∈ B t,t−2,t the information matrix has the form , and β > 0 for every |a| < 1. Thus, to determine E-optimal design over B t,t−2,t it is enough to find a design d * such that for any d ∈ B t,t−2,t We prove the following theorem.
Theorem 2 If there exists design d * with S d * = 1 t 1 t − I t − P d * such that P d * is permutationally similar to the matrix (iv) diag I i ⊗ H 3 , I j ⊗ H 5 if t = 5 or t ≥ 8 and t = 3m, m ∈ N with t = 3i + 5 j for some i ∈ N ∪ {0} and j ∈ N; then d * is E-optimal under model (1) with AR(1, C) and a > 0 over the class R t,t−2,t .
Proof Let a > 0. We prove the thesis in three steps.
Step 1. Let d ∈ B t,t−2,t . We obtain the thesis by condition (9) and Lemma 1.
Step 2. Let d ∈ B t,t−2,t \ B t,t−2,t . We have to show that From nonnegative definiteness of C d it is known that λ 1 (S d + S d ) = 2b and S d + S d has positive and negative eigenvalues. Observe, that one of the eigenvalues of S d + S d − 2b t 1 t 1 t is equal to zero. For convenience we will study the properties of the matrix G d = 2b1 t 1 t − t(S d + S d ), also with zero eigenvalue, for which condition (11) Note that λ t−1 (G d * ) ≥ 0. Since the diagonal entries of G d are equal to 2(t −2) and the off-diagonal entries, g d,i j , are of the form 2(t − 2) + tξ i j with ξ i j ∈ {0, 1, . . . , t − 2}, we will consider several cases of G d .
(a) Let ξ i j ≤ 2 for every i = j. Then, there exists a design d ∈ B t,t−2,t such that S d + S d = Sd + S d and the thesis follows from Lemma 1. (b) Assume now that ξ i j ∈ {0, 1, 2, 3} for every i = j and there exist at least one i and j such that ξ i j = 3. We consider four cases separately.
Step 3. Let d ∈ R t,t−2,t \ B t,t−2,t . In the previous steps we have shown that for For a < 0 there are some limitations for the class of optimality. Let B ( 3) t,t−2,t , t ≥ 7, be the class of binary designs for which: (i) there exists at least one pair of unordered treatments that meet as the nearest neighbors three times, (ii) every pair of unordered treatments appears as the nearest neighbors at most three times.
It means, that for every d ∈ B (3) t,t−2,t and i = j, ξ i j = s d,i j + s d, ji ∈ {0, 1, 2, 3} and there exist at least one pair (i, j) such that ξ i j = 3.
The following cases describe subclasses of B ( 3) t,t−2,t : -there exists exactly one pair (i, j) such that ξ i j = 3 and at least one pair (i , j ) such that ξ i j = 0. -there exists at least one pair (i, j) such that ξ i j = 3 and at least one pair (i , j ) such that ξ i j = 0. -for every i = j, ξ i j = 0. B (3,0) t,t−2,t the class of designs from B (3) t,t−2,t for which there is exactly one pair (i, j) such that ξ i j = 3 and at least one pair (i , j ) such that ξ i j = 0. By

Let us denote by
Theorem 3 Let t ≥ 3. If if there exists design d * with S d * permutationally similar to 1 t 1 t − I t − H t , then d * is E-optimal under model (1) with AR(1, C) and a < 0 over the class Proof Let a < 0. We prove the thesis in three steps.
Step 1. Let d ∈ B t,t−2,t . We obtain the thesis by condition (10) and Lemma 2.
(a) If the off-diagonal entries ξ i j ≤ 2 for every i = j, then there exists a design d ∈ B t,t−2,t such that S d + S d = S d + S d and the thesis follows from Lemma 2. (b) Assume that t ≤ 6 and ξ i j ∈ {0, 1, 2, 3} for every i = j and there exists at least one i = j such that ξ i j = 3. Then we can find a submatrix of S d + S d of the form Assume that there is exactly one ξ d,i j = 0, which is in the same row as 3, i.e., i = i. Because of the fixed sums of rows and columns of S d , it can be observed that: • (t − 4) off-diagonal entries of S d + S d must be equal to 2, one entry is equal to 0, one is equal to 1 and one is equal to 3 in the ith row; • (t − 5) off-diagonal entries of S d + S d must be equal to 2, three are equal to 1 and one is equal to 3 in the jth row; • (t − 2) off-diagonal entries of S d + S d must be equal to 2, and one is equal to 0 in the row j ; • (t − 3) off-diagonal entries of S d + S d must be equal to 2 and two are equal to 1 in the remaining rows.
Thus for every S d + S d there exists a submatrix Assume now that there is exactly one ξ d,i j = 0, which is in different row than 3, i.e., i = i. Then, we can find a submatrix of S d + S d permutationally similar to Moreover, in the row i the only possible off-diagonal entries different than 0 are 2s. Thus, one of the eigenvectors of S d + S d is of the form (α 1 1 2 : α 2 1 t−2 ) and it corresponds to the -4 egienvalue of S d + S d . From (8)  Step 3. If d ∈ R t,t−2,t \ B t,t−2,t then the proof follows the same lines as in Theorem 2.
For t ≥ 8, if d ∈ B (3,1) t,t−2,t there are a lot of possible forms of S d + S d . We are not aware of finding any general method for proving E-optimality of d * over B (3,1) t,t−2,t however we conjecture that d * is E-optimal over at least R t,t−2,t .
It is worth noting that E-optimal designs presented in Theorem 3 are also E-optimal under the one-sided interference model with uncorrelated observations (cf. Filipiak et al. 2008) and thus, some of the construction methods of E-optimal designs can be adopted.

E-optimal complete block designs for b = t
Let define the following subclass of B t,t,t : For a design d ∈ B t,t,t , Similarly to the previous section E-optimality condition can be expressed as Inequalities (14) and (15) correspond respectively to inequalities (10) and (9) from Sect. 4.1.
For simplicity of the proofs of the next theorems, for every d ∈ B t,t,t let define It is clear that all diagonal entries of Q d are equal to 2 and for every i = j q d,i j ∈ {2, 1, . . . , 2 − 2t}.
Moreover, Q d 1 t = 0 · 1 t . Since the elements of S d + S d are denoted by ξ i j in Sect. 2, it is clear that q d,i j = 2 − ξ i j . If a > 0 there are some limitations for the class of optimality. For t ≥ 7 let B (1) t,t,t be the class of binary designs such that (i) there exists at least one pair of unordered treatments that meet as the nearest neighbors once, (ii) every pair of unordered treatments appears as the nearest neighbors at least once and at most three times.
It means, that for every d ∈ B (1) t,t,t and i = j, ξ i j = s d,i j + s d, ji ∈ {1, 2, 3} and there exists at least one pair (i, j) such that ξ i j = 1.

Theorem 4
If there exists design d * with S d * permutationally similar to 1 t 1 t −I t +H t , then d * is E-optimal under model (1) with AR(1, C) and a > 0 over the class Proof Let a > 0. We prove the thesis in three steps.
Step 1. Let d ∈ B t,t,t . The thesis follows from (14) and Lemma 2.
Step 2. Let d ∈ B t,t,t \ B t,t,t . To show (11) it is enough to prove
(1) t,t,t there are a lot of possible forms of S d +S d . We are not aware of finding any general method for proving E-optimality of d * over B (1) t,t,t however we conjecture that d * is E-optimal over at least R t,t,t .
Theorem 5 If there exists design d * with S d * = 1 t 1 t − I t + P d * , such that P d * is permutationally similar to the matrix (iv) diag I i ⊗ H 3 , I j ⊗ H 5 if t = 5 or t ≥ 8 and t = 3m, m ∈ N with t = 3i + 5 j for some i ∈ N ∪ {0} and j ∈ N; then d * is E-optimal under model 1 with AR(1, C) and a < 0 over the class R t,t,t .
Proof Let a < 0. We prove the thesis in three steps.
Step 1. Let d ∈ B t,t,t . The thesis follows from (15) and Lemma 1 Step 2. Let d ∈ B t,t,t \ B t,t,t . To show (13) it is enough to prove Note that λ 1 (Q d * ) ≤ 4. We consider several cases of Q d .
It is worth noting that E-optimal designs presented in Theorem 5 are also E-optimal under the one-sided interference model with uncorrelated observations [cf. Filipiak et al. (2008)] and thus, some of the construction methods of E-optimal designs can be adopted from Filipiak et al. (2008) or Filipiak andRóżański (2005). Moreover, for a > 0 E-optimal designs can be constructed from CNBDs by repeating one block.

Concluding remarks
In the paper universal optimality of some circular NdNBD designs over the class of circular designs with arbitrary block size under AR(1, C) and a > 0 is shown. For a < 0 universal optimality of these designs over the class of circular designs with no treatment preceded by itself is proven. The only limitation on design parameters in this case is the existence of BBD design with completely symmetric nondirected neighboring matrix. If a circular universally optimal BBD cannot exist, E-optimality is considered in the class of complete block designs. If the number of blocks is smaller by two than the number of treatments and the correlation parameter a is positive as well as if the number of blocks is equal to the number of treatments and the correlation parameter is negative, the left-neighboring matrix of an E-optimal design over the class of all equireplicated designs without self-neighbors is given. In the remaining cases the class of optimality is more limited, however, we conjecture that the designs with leftneighboring matrices presented in Theorems 3 and 4 are still optimal over at least the class of equireplicated designs with no treatment preceded by itself.
In all the cases at least one example of E-optimal design is given. To construct such designs one can regard the left-neighboring matrix as an adjacency matrix of a directed graph and use its decomposition into Hamiltonian cycles. Such a decomposition can be done for example with the use of FindHamiltonianCycles procedure available in Mathematica 9.0 and later versions. The problem however is that looking for such a decomposition is extremely time-consuming even for relatively small number of treatments (vertices in graphs). Therefore some combinatorial methods of construction of E-optimal designs will be subject of the future research. It is especially interesting because optimal designs in the standard block effects model with observations correlated with respect to circular autoregression can be also shown to be optimal in more general models, e.g. with carry-over effects as additional nuisance parameters.