A review of uniform cross-over designs

Uniform cross-over designs form an important family of experimental designs. They have been applied in many scientific disciplines including clinical trials, agricultural studies and psychological experiments. In this paper we consider the four types of uniform cross-over design, as given by Williams [1949. Experimental designs balanced for the estimation of residual effects of treatments. Aust. J. Sci. Res. 2, 149–168], Cheng and Wu [1980. Balanced repeated measurements designs. Ann. Statist. 8, 1272–1283. Corrigendum 11 (1983) 349], Bate and Jones [2006. The construction of nearly balanced and nearly strongly balanced uniform cross-over designs. J. Statist. Plann. Inference 136, 3248–3267] and Kunert [1983. Optimal design and refinement of the linear model with applications to repeated measurements designs. Ann. Statist. 11, 247–257]. The efficiency of these designs, existence criteria and methods of construction are described.     

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.     

Minimal balanced repeated measurements designs

Experimental designs in which treatments are applied to the experimental units, one at a time, in sequences over a number of periods, have been used in several scientific investigations and are known as repeated measurements designs. Besides direct effects, these designs allow estimation of residual effects of treatments along with adjustment for them. Assuming the existence of first-order residual effects of treatments, Hedayat & Afsarinejad (1975) gave a method of constructing minimal balanced repeated measurements [RM(v,n,p)] design for v treatments using n=2v experimental units for p [=(v+1)/2] periods when v is a prime or prime power. Here, a general method of construction of these designs for all odd v has been given along with an outline for their analysis. In terms of variances of estimated elementary contrasts between treatment effects (direct and residual), these designs are seen to be partially variance balanced based on the circular association scheme.     

Construction of nearly uniform designs on irregular regions

The uniform design is a kind of important experimental design which has great practical value in production and living. Most existing literatures on this topic focus on the construction of uniform designs on regular regions. However, because of the complexity of practical situations, the irregular design region is more common in real life. In this paper, an algorithm is proposed to the construction of nearly uniform designs on irregular regions. The basic idea is to make use of uniform designs on a larger regular region with the irregular region being a subregion. Some theoretical justifications on the proposed algorithm are provided. Both the comparisons with the existing results and a real-life example show that our proposed algorithm is effective.     

Optimal cross-over designs for nonlinear mixed models using a first-order expansion

We consider the construction of optimal cross-over designs for nonlinear mixed effect models based on the first-order expansion. We show that for AB/BA designs a balanced subject allocation is optimal when the parameters depend on treatments only. For multiple period, multiple sequence designs, uniform designs are optimal among dual balanced designs under the same conditions. As a by-product, the same results hold for multivariate linear mixed models with variances depending on treatments.     

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.     

Construction of repeated measurements designs strongly balanced for residual effects

Abstract

Repeated Measurements Designs have been widely used in agriculture, animal husbandry, education, biology, botany and engineering. Balanced or strongly balanced repeated measurements designs are useful to balance out the residual effects. In this article, some new generators and construction procedures are proposed to obtain circular strongly balanced repeated measurements designs in periods of (a) equal sizes, (b) two different sizes, and (c) three different sizes.     

Designs balanced for circular residual effects

Magda (1980) introduced a model for repeated measurements designs with a circular structure of the residual effects. He proved the universal optimality of circular balanced uniform designs over a subclass of the possible designs. We strengthen his result to optimality over the set of all designs with the same number of experimental units, periods and treatments.     

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 strongly balanced uniform repeated measurements designs

Strongly balanced uniform repeated measurements designs when the number of treatments is 0, 1 or 3 modulo 4 are constructed. The methods used are the methods of differences and Hamiltonian decomposition of the lexicographic product of two graphs.     

Some new construction of circular weakly balanced repeated measurements designs in periods of two different sizes

Abstract

Balanced repeated measurements designs (RMDs) balance out the residual effects. Williams Latin square designs work as minimal combinatorial balanced as well as variance balanced for RMDs for p (period sizes) = v (number of treatments). If minimal balanced RMDs cannot be constructed for the situations where p must be less than v then weakly balanced RMDs should be preferred. In this article, some generators are developed to generate circular weakly balanced RMDs in periods of two different sizes. To obtain the proposed designs, some construction procedures are also described for some of the cases where we could not develop generators.     

On totally balanced block designs for competition effects

Competition between neighbouring units in field experiments is a serious source of bias. The study of a competing situation needs construction of an environment in which it can happen and the competing units have to appear in a predetermined pattern. This paper describes methods of constructing incomplete block designs balanced for neighbouring competition effects. The designs obtained are totally balanced in the sense that all the effects, direct and neighbours, are estimated with the same variance. The efficiency of these designs has been computed as compared to a complete block design balanced for neighbours and a catalogue has also been prepared.     

Construction of Cross-Over Designs for Comparing Test Treatments with Control Using Terraces

Segregated half-and-half and mutually orthogonal power sequence terraces have been used in this article for the construction of some new families of control balanced cross-over designs, which are Schur-optimal. One of the advantages of using terraces is that designs with larger number of treatments can also be constructed with relative ease. Designs up to 17 treatments have been constructed in this article.     

Balanced bootstrap in sample surveys and its relationship with balanced repeated replication

Resampling methods for survey sampling have been broadly classified into three categories: the jackknife, the bootstrap, and balanced repeated replication (BRR) in the literature. In this paper, we consider the balanced bootstrap. Two classes of balanced bootstrap designs are constructed. The balanced bootstrap method based on the first class of designs includes BRR as a special case. The second class of designs generalize those given by Nigam and Rao (1996).     

Balanced and partially balanced incomplete block designs with autocorrelation errors

The paper aims to find variance balanced and variance partially balanced incomplete block designs when observations within blocks are autocorrelated and we call them BIBAC and PBIBAC designs. Orthogonal arrays of type I and type II when used as BIBAC designs have smaller average variance of elementary contrasts of treatment effects compared to the corresponding Balanced Incomplete Block (BIB) designs with homoscedastic, uncorrelated errors. The relative efficiency of BIB designs compared to BIBAC designs depends on the block size k and the autocorrelation ρ and is independent of the number of treatments. Further this relative efficiency increases with increasing k. Partially balanced incomplete block designs with autocorrelated errors are introduced using partially balanced incomplete block designs and orthogonal arrays of type I and type II.     

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.     

On construction of constant block-sum partially balanced incomplete block designs

Abstract

Constant block-sum designs are of interest in repeated measures experimentation where the treatments levels are quantitative and it is desired that at the end of the experiments, all units have been exposed to the same constant cumulative dose. It has been earlier shown that the constant block-sum balanced incomplete block designs do not exist. As the next choice, we, in this article, explore and construct several constant block-sum partially balanced incomplete block designs. A natural choice is to first explore these designs via magic squares and Parshvanath yantram is found to be especially useful in generating designs for block size 4. Using other techniques such as pair-sums and, circular and radial arrangements, we generate a large number of constant block-sum partially balanced incomplete block designs. Their relationship with mixture designs is explored. Finally, we explore the optimization issues when constant block-sum may not be possible for the class of designs with a given set of parameters.     

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.     

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.