Balanced Treatment incomplete block (BTIB) designs for comparing treatments with a control: minimal complete sets of generator designs for k = 3, p = 3(1)10

Bechhofer and Tamhane (1981) proposed a new class of incomplete block designs called BTIB designs for comparing p ≥ 2 test treatments with a control treatment in blocks of equal size k < p + 1. All BTIB designs for given (p,k) can be constructed by forming unions of replications of a set of elementary BTIB designs called generator designs for that (p,k). In general, there are many generator designs for given (p,k) but only a small subset (called the minimal complete set) of these suffices to obtain all admissible BTIB designs (except possibly any equivalent ones). Determination of the minimal complete set of generator designs for given (p,k) was stated as an open problem in Bechhofer and Tamhane (1981). In this paper we solve this problem for k = 3. More specifically, we give the minimal complete sets of generator designs for k = 3, p = 3(1)10; the relevant proofs are given only for the cases p = 3(1)6. Some additional combinatorial results concerning BTIB designs are also given.     

A class of saturated row–column designs

A new class of row–column designs is proposed. These designs are saturated in terms of eliminating two-way heterogeneity with an additive model. The (m,s)-criterion is used to select optimal designs. It turns out that all (m,s)-optimal designs are binary. Square (m,s)-optimal designs are constructed and they are treatment-connected. Thus, all treatment contrasts are estimable regardless of the row and column effects.     

The interplay of optimality and combinatorics in experimental design

The design of statistical experiments, as developed by R. A. Fisher and his followers, often used combinatorial structures that yielded simple calculation of estimates and/or symmetric variances and covariances. Examples are block designs with balance, regression experiments with equally spaced observations, etc. More recently, considerations of optimality (choosing a design that achieves most accurate inference in some sense) have sometimes justified the traditional designs, but have sometimes led to new combinatorial investigations. Illustrations are given.     

Circular Binary Block Second- and Higher-Order Neighbor Designs

Some new neighbor designs are presented here. Second-order neighbor designs for different configurations are generated in circular binary blocks. Third-order and fourth-order neighbor designs for some cases are also constructed. In all cases, circular blocks are well separated and these designs are obtained through initial block/s. At the end of the study, some models for analysis of these designs are also presented.     

UPPER BOUNDS FOR LATINIZED DESIGNS

Upper bounds axe derived for the efficiency factor of a class of resolvable incomplete block designs known as latinized designs. These designs are particularly useful in glasshouse and field trials, and can be readily extended to two-dimensional blocking structures. Existing bounds for resolvable designs axe also reviewed and a comparison is made between the third moment bounds discussed by Jarrett (1989) and the second moment bounds of Tjur (1990).     

Partially efficiency-balanced designs-II

efficiency balanced (£R) and partially efficiency balanced (PEB) designs were studied hy Calinski (1971) and Puri and Higam (1971, 1977a). In tais paper we have studied the dual desinas of two classes of PEB designs. It is shown that the dual of any symmetrical PCS design with HN'=N'H is again a PEB design. Also the dual of a PEB(S) with μ₁ = 0 μ₂ = 0and is shown to the again a PEB(S). The relations between PEB designs and linked and partially linked blocks have heen established. As a conseguence, the analysis of linked Mock and partially linkei block designs can he sinrlififid. Sore useful classes of supplencntcd block designs having simple analysis have also been discussed.     

Nonproper variance balanced designs and optimality

This communication deals with the construction and optimality of non-proper (unequal block sized) variance balanced (VB) designs obtainable under linear homoscedastic normal model. Several methods of construction of non-proper VB designs have been given. Some constructed designs are universally optimal non-proper variance balanced designs.     

Construction of Neighbor-Balanced Designs in Linear Blocks

Neighbor-balanced designs are useful to remove the neighbor effects in experiments where the performance of a treatment is affected by the treatments applied to its adjacent neighbors. In this article, neighbor-balanced designs are constructed in linear blocks of (i) equal sizes and (ii) two different sizes k ₁ and k ₂.     

PARTIALLY EQUINEIGHBOURED DESIGNS

This paper presents further results on a class of designs called equineighboured designs, ED. These designs are intended for field and related experiments, especially whenever there is evidence that observations in the same block are correlated. An ED has the property that every unordered pair of treatments occurs as nearest neighbours equally frequently at each level. Ipinyomi (1986) has defined and shown that ED are balanced designs when neighbouring observations are correlated. He has also presented ED as a continuation of the development of optimal block designs. An ED would often require many times the number of experimental materials needed for the construction of an ordinary balanced incomplete block, BIB, design for the same number of treatments and block sizes. Thus for a relatively large number of treatments and block sizes the required minimum number of blocks may be excessively large for practical use of ED. In this paper we shall define and examine partially equineighboured designs with n concurrences, PED (n), as alternatives where ED are practically unachievable. Particular attention will be given to designs with smaller numbers of blocks and for which only as little balance as possible may be lost.     

Some new constructions of circular balanced repeated measurements designs

Repeated measurements designs are widely used in medicine, pharmacology, animal sciences, and psychology. In this article, some infinite series are developed to generate the balanced repeated measurements designs for p (periods) even. For p odd, construction procedures are also described. Catalogues of the proposed designs are also presented for p = 5, 7, 9, when v ≤ 100.     

Minimal circular balanced repeated measurement designs in unequal period sizes

Abstract

Repeated measurement designs (RMDs) are widely used in medicine, pharmacology, animal sciences and psychology. In these fields, there are several situations where these designs should be used in periods of different sizes. With the use of RMD, residual effects or carry over effects may arise and balanced RMDs are solution to this problem. In this article, therefore, some infinite series are developed through method of cyclic shifts to obtain circular balanced repeated measurements designs in periods of two different sizes.     

Construction of Balanced Incomplete Block Designs Using Cyclic Shifts

Balanced incomplete block designs (BIBDs) play important role in design of experiments, especially in field experiments. These designs ensure that treatments are compared with equal precision. Several methods are available in the literature to construct BIBDs but in this article some infinite series of these designs are presented by method of cyclic shifts. This method expresses some standard properties of a design just through examining the sets of shifts rather than studying the whole design.     

Fourth order rotatability

Fourth order rotatable designs are discussed. A general k, design moment inequality is given. The variance function for two-factor designs is derived, and plotted for a specific design. A minimum point set requirement for two-factor designs is established, thus enabling one to form an infinity of such designs. Some difficulties in obtaining deLigns for k>2 are described. Some questions are posed for future work.     

Construction of circular partially balanced-Repeated measurement designs using cyclic shifts

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences and psychology. These designs balance out the residual effects. The situations where balanced repeated measurements designs require a large number of the subjects, partially-balanced repeated measurements designs should be used. In this paper some infinite series are developed which provide circular partially-balanced repeated measurement designs for p (periods) even. Catalogues of circular partially-balanced repeated measurement designs are also presented for v (treatments) ≤ 100 with p = 5, 7 & 9.     

On The Optimal Choice of Cube and Star Replications in Restricted Second-Order Designs

This article examines the central composite design in which some of the experimental runs are replicated. Three different classes of N-point designs are compared using the criterion of Schur's ordering under orthogonality, rotatablity, and slope- rotatablity conditions. The response surface designs with the star portion replicated seem to have more potential than others under orthogonality condition, while the designs with the cube portion replicated is preferable to the designs with their star portion or only the center point replicated under rotatable and slope-rotatable conditions.     

Optimal block designs for triallel cross experiments

Optimal block designs for a certain type of triallel cross experiments are investigated. Nested balanced block designs are introduced and it is shown how these designs give rise to optimal designs for triallel crosses. Several .series of nested balanced block designs, leading to optimal designs for triallel crosses are reported.     

Construction of incomplete Sudoku square and partially balanced incomplete block designs

Abstract

The present article deals with the study of association among the elements of a Sudoku square. In this direction, we have defined an association scheme and constructed incomplete Sudoku square designs which are capable of studying four explanatory variables and also happen to be the designs for two-way elimination of heterogeneity. Some series of Partially Balanced Incomplete Block (PBIB) designs have also been obtained.     

A-Optimal Block Designs with Additional Singly Replicated Treatments

Block designs to which have been added a number of singly-replicated treatments, known as secondary treatments, are particularly useful for experiments where only small amounts of material are available for some treatments, for example new plant varieties. The designs are of particular use in the microarray situation. Such designs are known as 'augmented designs'. This paper obtains the properties of these designs and shows that, with an equal number of secondary treatments in each block, the A-optimal design is obtained by using the A-optimal design for the original block design. It develops formulae for the variance of treatment comparisons, for both the primary and the secondary treatments. A number of examples are used to illustrate the results.     

On d- and e- minimax optimal designs for estimating the axial slopes of a second-order response surface over hypercubic regions

Design of experiments for estimating the slopes of a response surface is considered. Design criteria analogous to the traditional ones but based upon the variance-covariance matrix of the estimated slopes along factor axes are proposed. Optimal designs under the proposed criteria are derived for second-order polynomial regression over hypercubic regions. Best de¬signs within some commonly used classes of designs are also obtained and their efficiencies are investigated.     

On balanced incomplete Latin square designs

Recently, balanced incomplete Latin square designs are introduced in the literature. We propose three methods of constructions of balanced incomplete Latin square designs. Particular classes of Latin squares namely Knut Vik designs, semi Knut Vik designs, and crisscross Latin squares play a key role in the construction.