A generating algorithm for linear trend-free and nearly linear trend-free block designs

The theory and properties of trend-free (TF) and nearly trend-free (NTF) block designs are wel1 developed. Applications have been hampered because a methodology for design construction has not been available.

This article begins with a short review of concepts and properties of TF and NTF block designs. The major contribution is provision of an algorithm for the construction of linear and nearly linear TF block designs. The algorithm is incorporated in a computer program in FORTRAN 77 provided in an appendix for the IBM PC or compatible microcomputer, a program adaptable also to other computers. Three sets of block designs generated by the program are given as examples.

A numerical example of analysis of a linear trend-free balanced incomplete block design is provided.     

Resolvable Designs With Unequal Block Sizes

Resolvable block designs for v varieties in blocks of size k require v to be a multiple of k so that all blocks are of the same size. If a factorization of v is not possible then a resolvable design with blocks of unequal size is necessary. Patterson & Williams (1976) suggested the use of designs derived from α -designs and conjectured that such designs are likely to be very efficient in the class of resolvable designs with block sizes k and k – 1. This paper examines these derived designs and compares them with designs generated directly using an interchange algorithm. It concludes that the derived designs should be used when v is large, but that for small v they can be relatively inefficient.     

On the recovery of intra- and interblock information in n-ary designs

This note presents an extension of Q-method of analysis for binary designs given by Rao (1956) to n-ary balanced and partially balanced block designs. Here a linked n-ary block (LNB) design is defined as the dual of balanced n-ary (BN) design. Having a note on Yates’ (1939, 1940) method of P-analysis, we further extend the expressions for binary linked block (LB) designs given by Rao (1956) to linked n-ary block (LNB) designs which admit easy estimation of parameters for these type of all n-ary designs.     

Computer-Generated Neighbor Designs

Neighbor designs are useful to remove the neighbor effects. In this article, an algorithm is developed and is coded in Visual C + +to generate the initial block for possible first, second,…, and all order neighbor designs. To get the required design, a block (0, 1, 2,…, k ? 1) is then augmented with (v ? 1) blocks obtained by developing the initial block cyclically mod (v ? 1).     

Linear Integer Programming Approach to Construction of Balanced Incomplete Block Designs

This article proposes a linear integer programming-based algorithm to construct balanced incomplete block designs. Working of the algorithm is illustrated with the help of an example. The algorithm is able to generate balanced incomplete block designs very fast in most of the cases. The performance of the proposed algorithm is compared with other algorithms proposed in the literature. It is demonstrated that the proposed algorithm is competitive with the existing algorithms.     

E-optimal block designs under heteroscedastic model

This paper mainly studies the E-optimality of block designs under a general heteroscedastic setting. The C-matrix of a block design under a heteroscedastic setting is obtained by using generalized least squares. Some bounds for the smallest positive eigenvalue of C-matrix are obtained in some general classes of connected designs. Use of these bounds is then made to obtain certain E-optimal block designs in various classes of connected block designs.     

Designs for two-colour microarray experiments

Summary.  Designs for two-colour microarray experiments can be viewed as block designs with two treatments per block. Explicit formulae for the A- and D-criteria are given for the case that the number of blocks is equal to the number of treatments. These show that the A- and D-optimality criteria conflict badly if there are 10 or more treatments. A similar analysis shows that designs with one or two extra blocks perform very much better, but again there is a conflict between the two optimality criteria for moderately large numbers of treatments. It is shown that this problem can be avoided by slightly increasing the number of blocks. The two colours that are used in each block effectively turn the block design into a row–column design. There is no need to use a design in which every treatment has each colour equally often: rather, an efficient row–column design should be used. For odd replication, it is recommended that the row–column design should be based on a bipartite graph, and it is proved that the optimal such design corresponds to an optimal block design for half the number of treatments. Efficient row–column designs are given for replications 3–6. It is shown how to adapt them for experiments in which some treatments have replication only 2.     

An evolutionary algorithm for A-optimal incomplete block designs

Evolutionary algorithms are heuristic stochastic search and optimization techniques with principles taken from natural genetics. They are procedures mimicking the evolution process of an initial population through genetic transformations. This paper is concerned with the problem of finding A-optimal incomplete block designs for multiple treatment comparisons represented by a matrix of contrasts. An evolutionary algorithm for searching optimal, or nearly optimal, incomplete block designs is described in detail. Various examples regarding the application of the algorithm to some well-known problems illustrate the good performance of the algorithm     

t-Latinized Designs

A variety trial sometimes requires a resolvable block design in which the replicates are set out next to each other. The long blocks running through the replicates are then of interest. A t -latinized design is one in which groups of these t long blocks are binary. In this paper examples of such designs are given. It is shown that the algorithm described by John & Whitaker (1993) can be used to construct designs with high average efficiency factors. Upper bounds on these efficiency factors are also derived.     

An algorithm for constructing optimal resolvable incomplete block designs

This paper describes an efficient algorithm for the construction of optimal or near-optimal resolvable incomplete block designs (IBDs) for any number of treatments v < 100. The performance of this algorithm is evaluated against known lattice designs and the 414 or-designs of Patterson & Williams [36]. For the designs under study, it appears that our algorithm is about equally effective as the simulated annealing algorithm of Venables & Eccleston [42]. An example of the use of our algorithm to construct the row (or column) components of resolvable row-column designs is given.     

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.     

Characterization of certain incomplete block designs

It is shown that certain inequalities known for partially balanced incomplete block (PBIB) designs remain valid for general incomplete block designs. Some conditions for attaining their bounds are also given. Furthermore, the various types of PBIB designs are characterized by relating blocks of designs with association schemes. The approach here is based on the spectral expansion of NN' for the incidence matrix N of an incomplete block design.     

Optimal partially balanced nested row-column designs

A partially balanced nested row-column design, referred to as PBNRC, is defined as an arrangement of v treatments in b p × q blocks for which, with the convention that p q, the information matrix for the estimation of treatment parameters is equal to that of the column component design which is itself a partially balanced incomplete block design. In this paper, previously known optimal incomplete block designs, and row-column and nested row-column designs are utilized to develop some methods of constructing optimal PBNRC designs. In particular, it is shown that an optimal group divisible PBNRC design for v = mn kn treatments in p × q blocks can be constructed whenever a balanced incomplete block design for m treatments in blocks of size k each and a group divisible PBNRC design for kn treatments in p × q blocks exist. A simple sufficient condition is given under which a group divisible PBNRC is Ψ_f-better for all f> 0 than the corresponding balanced nested row-column designs having binary blocks. It is also shown that the construction techniques developed particularly for group divisible designs can be generalized to obtain PBNRC designs based on rectangular association schemes.     

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 adjusted orthogonal,variance balanced,multidimensional block designs

A multidimensional block design (MBD) is an experimental design with d > 1 blocking criteria geometrically represented as a d-dimensional lattice with treatment varieties assigned to some or all nodes of the lattice. Intrablock analysis of variance tables for some special classes of two- and three-dimensional block designs with some empty nodes are given. Design plans and efficiencies for 31 two-dimensional designs, each universally optimal in defined classes of designs, and 7 three-dimensional designs, each nearly optimal in defined classes of designs, are listed in the appendices. A need for such designs is apparent when the blocking criteria are implemented successively and empty nodes do not represent wasted experimental units.     

ON DERIVATION OF INITIAL BLOCKS OF B.I.B. DESIGNS WITH MORE THAN ONE INITIAL BLOCK

A good amount of work has been done on the construction of balanced incomplete block (B.I.B.) designs by Bose (1939, 1942), Sprott (1954, 1956), Rao (1961), Takeuchi (1962) and others. Sprott (1954, 1956) obtained several series of B.I.B. designs through difference sets. The main purpose of the present investigation is to provide two methods of construction of B.I.B. designs obtainable through more than one initial block. The first method derives initial blocks of a series of designs from some specified blocks of a B.I.B. design obtainable by developing one or more initial blocks. The second method attempts to obtain one of the initial blocks (the basic initial block) through the different powers of an element of a finite field; then an appropriate method for generating the other initial blocks from it is discussed. A table showing the basic initial block for different designs has been presented. By these methods several solutions of some B.I.B. designs could be obtained from different initial blocks. An examination was therefore made to see if these designs were all isomorphic.     

On constructing general minimum lower order confounding two-level block designs

In practice, to reduce systematic variation and increase precision of effect estimation, a practical design strategy is then to partition the experimental units into homogeneous groups, known as blocks. It is an important issue to study the optimal way on blocking the experimental units. Blocked general minimum lower order confounding (B¹-GMC) is a new criterion for selecting optimal block designs. The paper considers the construction of optimal two-level block designs with respect to the B¹-GMC criterion. By utilizing doubling theory and MaxC2 design, some optimal block designs with respect to the B¹-GMC criterion are obtained.     

On Exact K-optimal Designs Minimizing the Condition Number

A new design criterion based on the condition number of an information matrix is proposed to construct optimal designs for linear models, and the resulting designs are called K-optimal designs. The relationship between exact and asymptotic K-optimal designs is derived. Since it is usually hard to find exact optimal designs analytically, we apply a simulated annealing algorithm to compute K-optimal design points on continuous design spaces. Specific issues are addressed to make the algorithm effective. Through exact designs, we can examine some properties of the K-optimal designs such as symmetry and the number of support points. Examples and results are given for polynomial regression models and linear models for fractional factorial experiments. In addition, K-optimal designs are compared with A-optimal and D-optimal designs for polynomial regression models, showing that K-optimal designs are quite similar to A-optimal designs.     

A new special class of peb designs

A new class of partially efficiency-balanced designs is introduced from a practical point of view. This new design includes all equireplicated incomplete block designs available in literature as special cases. The fundamental properties of the design are clarified with relation to other block designs.