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.     

D-optimal designs with minimal and nearly minimal number of units

D-optimal designs are identified in classes of connected block designs with fixed block size when the number of experimental units is one or two more than the minimal number required for the design to be connected. An application of one of these results is made to identify D-optimal designs in a class of minimally connected row-column designs. Graph-theoretic methods are employed to arrive at the optimality results.     

Row-column designs for comparing treatments with a control

This paper considers the problem of the design and analysis of experiments for comparing several treatments with a control when heterogeneity is to be eliminated in two directions. A class of row-column designs which are balanced for treatment vs. control comparisons (referred to as the balanced treatment vs. control row-column or BTCRC designs) is proposed. These designs are analogs of the so-called BTIB designs proposed by Bechhofer and Tamhane (Technometrics 23 (1981) 45–57) for eliminating heterogeneity in one direction. Some methods of analysis and construction of these designs are given. A measure of efficiency of BTCRC designs in terms of the A-optimality criterion is derived and illustrated by several examples.     

Neighbor-Balanced Row-column Designs

In this article, row-column designs incorporating directional neighbor effects have been studied. A row-column design is said to be neighbor balanced if every treatment has all other treatments appearing as a neighbor a constant number of times. We considered here three different situations under row-column setup incorporating neighbor effects viz., row-column design with one-sided neighbor effect, two-sided neighbor effect, and four-sided neighbor effect. The information matrices for all the situations for estimating the direct and neighbor effects of treatments have been derived. Methods of constructing neighbor-balanced row-column designs have been developed and its characterization properties have been studied.     

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.     

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.     

On the construction of a class of efficient row-column designs

In this paper a method for the construction of a class of row-column designs with good statistical properties and high efficiency is presented. The class of designs produced is shown to exhibit balance, orthogonality and adjusted orthogonality. The efficiencies of these designs are investigated in detail, and they are shown to be very high, and possibly maximal in some cases.     

THE ROBUSTNESS OF BINARY AND NON-BINARY BALANCED NESTED ROW-COLUMN DESIGNS UNDER THE UNAVAILABILITY OF BLOCKS: A COMPARISON

In cases where both exist, the balanced, binary nested row-column designs are known to be inferior to a class of balanced non-binary designs. However, if it is possible for blocks of observations to become unavailable after an experiment has commenced, a binary nested row-column design may possibly be better than a non-binary one. This paper investigates the robustness of binary and non-binary variance-balanced nested row-column designs to the unavailability of one or more blocks of observations. Robustness is measured through the C-matrices of the designs resulting from removing blocks, using optimality criteria such as A-, D-, E- and MV-optimality.     

Resolvable row–column designs

This paper discusses resolvable row-column designs for v treatments arranged in b blocks each comprising pq units further grouped into p rows and q columns. A resolvable row-column design has v =pqs treatments set out in r groups of s blocks. Each rectangular block has p rows and q columns.     

Optimal regression designs in the presence of random block effects

A- and D-optimal regression designs under random block-effects models are considered. We first identify certain situations where D- and A-optimal designs do not depend on the intra-block correlation and can be obtained easily from the optimal designs under uncorrelated models. For example, for quadratic regression on [−1,1], this covers D-optimal designs when the block size is a multiple of 3 and A-optimal designs when the block size is a multiple of 4. In general, the optimal designs depend on the intra-block correlation. For quadratic regression, we provide expressions for D-optimal designs for any block size. A-optimal designs with blocks of size 2 for quadratic regression are also obtained. In all the cases considered, robust designs which do not depend on the intrablock correlation can be constructed.     

The connectedness and optimality of a class of row-column designs

Balanced Incomplete Block Designs have been employed as row-column designs by a number of researchers. In this paper necessary and sufficient conditions for the connectedness of such designs are obtained, and methods for their optimisation are presented. The optimal design is shown to be always connected.     

RANDOMIZED SEARCH STRATEGIES FOR FINDING OPTIMAL OR NEAR OPTIMAL BLOCK AND ROW-COLUMN DESIGNS

We consider a family of effective and efficient strategies for generating experimental designs of several types with high efficiency. These strategies employ randomized search directions and at some stages allow the possibility of taking steps in a direction of decreasing efficiency in an effort to avoid local optima. Hence our strategies have some affinity with the simulated annealing algorithm of combinatorial optimization. The methods work well and compare favourably with other search strategies. We have implemented them for incomplete block designs, optionally resolvable, and for row-column designs.     

(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.     

ADJUSTED ORTHOGONAL NESTED ROW-COLUMN DESIGNS

Adjusted orthogonality in nested row-column designs is defined and a sufficient condition established for its existence. It is shown that the properties of an adjusted orthogonal nested row-column design are directly related to those of its separate row and column component designs. A method for constructing efficient adjusted orthogonal designs involving a single replicate of every treatment in each of two blocks is given.     

A Note on the Moments and Computer Generation of the Shifted Gompertz Distribution

In this paper we give a class of row-column designs with the property that the i-th row and the j-th column have precisely r treatments in common. A conjecture that such designs are quasi-factorial is disproved by showing that the designs given in this paper are not quasi-factorial. It is also shown that the designs given here are nearly optimal.     

ITERATIVE ANALYSIS OF GENERALIZED LATTICE DESIGNS

Generalized lattice designs are defined. They include as special cases the square and rectangular lattice designs, and the α-designs defined by Patterson and Williams (1976). An iterative procedure is given for the combined estimation of variety effects in generalized lattice designs with optimal or near optimal efficiency factors. This procedure, together with an approximate variance matrix, enables the analysis of efficient generalized lattice designs to be carried out on mini computers.     

Example of Block Designs for Plant and Tree Breeding Trials

Experimental designs which use extensive blocking and which are particularly useful in plant and tree breeding trials are discussed. They can be constructed either to accommodate field restrictions or take advantage of favourable plot layouts. Computer software is available to generate these design types for use in practice. Examples cover latinized row-column designs, t -latinized and partially-latinized designs and designs with unequal block sizes.     

Incomplete row-column designs with factorial treatment structure for estimating main effects with full efficiency

In this paper, we propose two methods of constructing row-column designs for factorial experiments. The constructed designs have orthogonal factorial structure with balance and permits estimation of main effects with full efficiency.     

Row-column designs with minimal units

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 proposed designs are treatment-connected, i.e., all paired comparisons of treatments in the designs are estimable in spite of the existence of row and column effects. The connectedness of the designs is justified from two perspectives: linear model and contrast estimability. Comparisons with other designs are studied in terms of A-, D-, E-efficiencies as well as design balance.     

Optimal row-column designs for complete diallel crosses

Universally optimal row-column designs for complete, diallel crosses are investigated. Three series of designs that require just one replication of the crosses are provided. A series of designs having two replications of each cross is also provided.