Generalized Bhaskar Rao designs of block size three

We show that the necessary conditions

$\text{λ≡0 (}\text{mod}\text{|G|)}$

,

$\text{λ(υ?1)≡0 (}\text{mod}\text{2)}$

,

$\text{λυ(υ?1)≡}\text{0 (}\text{mod}\text{6)}\text{for}\text{|G|}\text{odd}\text{,}\text{0 (}\text{mod}\text{24)}\text{for}\text{|G|}\text{even}$

, are sufficient for the existence of a generalized Bhaskar Rao design GBRD(υ,b,r,3,λ;G) for the elementary abelian group G, of each order |G|.     

Regular group divisible designs and Bhaskar Rao designs with block size three

Some recursive constructions are given for Bhaskar Rao designs. Using examples of these designs found by Shyam J. Singh, Rakesh Vyas and new ones given here we show the necessary conditions λ≡0 (mod 2), λυ(υ?1)≡0 (mod 24) are sufficient for the existence of Bhaskar Rao designs with one association class and block size 3. This result is used with a result of Street and Rodger to obtain regular partially balanced block designs with 2υ treatments, block size 3, λ₁=0, group size 2 and υ groups.     

Some sufficient conditions for establishing (M,S)-optimality

Two sufficient conditions are given for an incomplete block design to be (M,S- optimal. For binary designs the conditions are (i) that the elements in each row, excluding the diagonal element, of the association matrix differ by at most one, and (ii) that the off-diagonal elements of the block characteristic matrix differ by at most one. It is also shown how the conditions can be utilized for nonbinary designs and that for blocks of size two the sufficient condition in terms of the association matrix can be attained.     

Construction of some classes of balanced cross-over designs from terraces

5 and 6 have recently introduced power-sequence terraces. In this paper we have used these terraces for the construction of some new families of balanced cross-over designs of first and second order which are variance-balanced. We have also used them for the construction of some new families of balanced ternary cross-over designs.     

A construction for generalized hadamard matrices GH(4q, EA(q))

We give a construction for a generalized Hadamard matrix GH(4q, EA(q)) as a 4 × 4 matrix of q × q blocks, for q an odd prime power other than 3 or 5. Each block is a GH(q, EA(q)) and certain combinations of 4 blocks form GH(2q, EA(q)) matrices. Hence a GH(4q, EA(q)) matrix exists for every prime power q.     

(M,S)-optimal row-column designs

Row-column designs may be considered to have two blocking schemes, namely the treatments by rows and treatments by columns component block designs. The (M,S)-optimality criterion is applied to row-column designs, and che connection between the (M,S)-optimal design and its component block designs is demonstrated.     

On optimality properties of simple block designs in the approximate design theory

Optimality properties of approximate block designs are studied under variations of (1) the class of competing designs, (2) the optimality criterion, (3) the parametric function of interest, and (4) the statistical model. The designs which are optimal turn out to be the product of their treatment and block marginals, and uniform designs when the support is specified in advance. Optimality here means uniform, universal, and simultaneous j_p-optimality. The classical balanced incomplete block designs are embedded into this approach, and shown to be simultaneously j_p-optimal for a maximal system of identifiable parameters. A geometric account of universal optimality is given which applies beyond the context of block designs.     

Block designs with block size two

Algorithms are given for the construction of binary block designs with replications and concurrences differing by at most one. The designs are resolvable and/or connected wherever the parameters permit.     

A construction for generalized hadamard matrices

We prove that if p^r and p^r ? 1 are both prime powers then there is a generalized Hadamard matrix of order p^r(p^r ? 1) with elements from the elementary abelian group Z_p x?x Z_p. This result was motivated by results of Rajkundia on BIBD's. This result is then used to produce p^r ? 1 mutually orthogonal F-squares F(p^r(p^r ? 1); p^r ? 1).     

Interaction balance for symmetrical factorial designs with generalized minimum aberration

There are two different systems of contrast parameterization when analyzing the interaction effects among the factors with more than two levels, i.e., linear-quadratic system and orthogonal components system. Based on the former system and an ANOVA model, Xu and Wu (2001) introduced the generalized wordlength pattern for general factorial designs. This paper shows that the generalized wordlength pattern exactly measures the balance pattern of interaction columns of a symmetrical design ground on the orthogonal components system, and thus an alternative angle to look at the generalized minimum aberration criterion is given. This work is partially supported by NNSF of China grant No. 10231030.     

Optimum covariate designs in partially balanced incomplete block (PBIB) design set-ups

The use of covariates in block designs is necessary when the covariates cannot be controlled like the blocking factor in the experiment. In this paper, we consider the situation where there is some flexibility for selection in the values of the covariates. The choice of values of the covariates for a given block design attaining minimum variance for estimation of each of the parameters has attracted attention in recent times. Optimum covariate designs in simple set-ups such as completely randomised design (CRD), randomised block design (RBD) and some series of balanced incomplete block design (BIBD) have already been considered. In this paper, optimum covariate designs have been considered for the more complex set-ups of different partially balanced incomplete block (PBIB) designs, which are popular among practitioners. The optimum covariate designs depend much on the methods of construction of the basic PBIB designs. Different combinatorial arrangements and tools such as orthogonal arrays, Hadamard matrices and different kinds of products of matrices viz. Khatri–Rao product, Kronecker product have been conveniently used to construct optimum covariate designs with as many covariates as possible.     

Some bounds on balanced block designs

Bounds on the latest root of the C-matrix and the number of blocks for a variance-balanced block design are given. These results contain the well-known results as special cases.     

Characterization of minimum aberration mixed factorials in terms of consulting designs

In this paper, by introducing a concept of consulting design and based on the connection between factorial design theory and coding theory, we obtain combinatorial identities that relate the wordlength pattern of a regular mixed factorial design to that of its consulting design. According to these identities, we further-more establish the general and unified rules for identifying minimum aberration mixed factorial designs through their consulting designs. It is an improvement and generalization of the results in Mukerjee and Wu (2001). This paper is supported by NNSF of China grant No. 10171051 and RFDP grant No. 1999005512.     

Orthogonal Latin hypercube designs from generalized orthogonal designs

Latin hypercube designs is a class of experimental designs that is important when computer simulations are needed to study a physical process. In this paper, we proposed some general criteria for evaluating Latin hypercube designs through their alias matrices. Moreover, a general method is proposed for constructing orthogonal Latin hypercube designs. In particular, links between orthogonal designs (ODs), generalized orthogonal designs (GODs) and orthogonal Latin hypercube designs are established. The generated Latin hypercube designs have some favorable properties such as uniformity, orthogonality of the first and some second order terms, and optimality under the defined criteria.     

Cyclic designs for a covariate model

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. We restrict our attention to classes of designs d for which the number of observations N to be taken is a multiple of V, i.e. N = V × R with R ≥2, and each treatment level is observed R times. Among these designs, called here equireplicated, there is a subclass characterized by the following: the allocation matrix of each treatment level (for short, allocation matrix) is obtained through cyclic permutation of the columns of the allocation matrix of the first treatment level. We call these designs cyclic. Besides having easy representation, the most efficient cyclic designs are often D-optimal in the class of equireplicated designs. A known upper bound for the determinant of the information matrix M(d) of a design, in the class of equireplicated ones, depends on the congruences of N and V modulo 4. For some combinations of parameter moduli, we give here methods of constructing families of D-optimal cyclic designs. Moreover, for some sets of parameters (N, V,K = V), where the upper bound on ∣M(d)∣ (for that specific combination of moduli) is not attainable, it is also possible to construct highly D-efficient cyclic designs. Finally, for N≤24 and V≤6, computer search was used to determine the most efficient design in the class of cyclic ones. They are presented, together with their respective efficiency in the class of equireplicated designs.     

The exact D-optimal first order saturated design with 17 observations

The exact D-optimal first order saturated design with 17 observations is given. The upper bound of the determinant of the information matrix is established and a design attaining this value is constructed. The information matrix is proved to be unique and the optimal design contains the B.I.B. design (16, 16, 6, 6, 2).     

The family of t-designs - part II

This is a continuation of Part I of our previous paper (1980) on conbinatorial aspects of t-designs with t ≥ 2. Along the same developments as in Part I, this paper provides tables, assorted examples of t-designs, and interesting unsolved problems. A brief background and a short review of the literature related to each unsolved problem is also provided. Some tables list parameters of designs whose existence or nonexistence is not established in literature. Notation and numbering of Part I are still used here.     

Saturated designs for multivariate cubic regression

This paper deals with the problem of finding saturated designs for multivariate cubic regression on a cube which are nearly D-optimal. A finite class of designs is presented for the k dimensional cube having the property that the sequence of the best designs in this class for each k is asymptotically efficient as k increases. A method for constructing good designs in this class is discussed and the construction is carried out for 1?k?8. These numerical results are presented in the last section of the paper.     

Optimality of random allocation design for the control of accidental bias in sequential experiments

In comparing two treatments, suppose the suitable subjects arrive sequentially and must be treated at once. Known or unknown to the experimenter there may be nuisance factors systematically affecting the subjects. Accidental bias is a measure of the influence of these factors in the analysis of data. We show in this paper that the random allocation design minimizes the accidental bias among all designs that allocate n, out of 2n, subjects to each treatment and do not prefer either treatment in the assignment. When the final imbalance is allowed to be nonzero, optimal and efficient designs are given. In particular the random allocation design is shown to be very efficient in this broader setup.