AN ALGORITHM FOR THE CONSTRUCTION OF OPTIMAL OR NEAR-OPTIMAL CHANGE-OVER DESIGNS

This paper presents an algorithm for the construction of optimal or near optimal change-over designs for arbitrary numbers of treatments, periods and units. Previous research on optimality has been either theoretical or has resulted in limited tabulations of small optimal designs. The algorithm consists of a number of steps:first find an optimal direct treatment effects design, ignoring residual effects, and then optimise this class of designs with respect to residual effects. Poor designs are avoided by judicious application of the (M, S)-optimality criterion, and modifications of it, to appropriate matrices. The performance of the algorithm is illustrated by examples.     

A Note on Change-Over Designs

A change-over model with correlated errors is discussed. In particular the results of Patterson (1950) and Lucas (1951) for balanced change-over designs are extended to more than four treatments and more general variance matrices. A connection with recent work on neighbour models for field trials is made.     

Serial arrays and change-over designs

Consider change-over designs for the comparison of v treatments. Each application has a direct effect and up to n?1 residual and/or interaction effects. An array is defined to be a serial array if: (1) all treatments occur equally often in each row and (2) reading down columns, all possible permutations of n consecutive treatments occur equally often in the array. A method of constructing (λv+n?1) Xv^{n ?1}, λ integral, serial arrays is given. Use of these arrays as change-over designs is discussed with emphasis on the inclusion of interaction effects.     

Change-over designs with complete balance for first and second residual effects

This paper presents a unified method of constructing change-over designs that permit the estimation of direct effects orthogonal to all other effects when the residual effects of treatments last for two consecutive periods. Explicit methods of analysis of these designs have been obtained for the situations where the first period observations are omitted from the analysis and where the first period observations are included.     

Computer-Generated Change-Over Designs for Correlated Observations

Change-over designs with independently distributed errors in the model have been studied extensively in the literature. Martin and Eccleston (2001 Martin , R. J. , Eccleston , J. A. ( 2001 ). Optimal and near optimal designs for dependent observations . Statist. Applic. 3 : 101 – 116 . [Google Scholar]) gave an algorithm for the generation of efficient change-over designs when the errors are correlated. This article proposes an algorithm for the generation of efficient change-over designs for estimation of direct effects of treatments in the presence of first-order residual effects in the model and when the errors are correlated.     

(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.     

Optimal change-over designs for correlated observations

Design implications of an autoregressive model for change-over experiments are investigated. In this model, the residual effect due to the previous treatment is assumed to be proportional to the response in the previous period. In addition, the errors from the same experimental subject are assumed to be correlated according to a first-order autoregressive model. Models with fixed and random subject effects are discussed separately. An attempt has been made to identify and construct optimal or nearly optimal designs in various situations. Empirical conclusions of Taka and Armitage [Commun. Statist. Theor. Meth. (1983)12, 865-876] regarding the efficiency of some designs have also been confirmed.     

BALANCED CHANGE-OVER DESIGNS FOR AUTOCORRELATED OBSERVATIONS

Some aspects of design of change-over experiments with autocorrelated observations are investigated. Williams' designs (1949, 1950) are seen to be balanced for autocorrelated observations.     

Cross-over Designs in the Presence of Higher Order Carry-overs

In cross-over experiments, where different treatments are applied successively to the same experimental unit over a number of time periods, it is often expected that a treatment has a carry-over effect in one or more periods following its period of application. The effect of interaction between the treatments in the successive periods may also affect the response. However, it seems that all systematic studies of the optimality properties of cross-over designs have been done under models where carry-over effects are assumed to persist for only one subsequent period. This paper proposes a model which allows for the possible presence of carry-over effects up to k subsequent periods, together with all the interactions between treatments applied at k + 1 successive periods. This model allows the practitioner to choose k for any experiment according to the requirements of that particular experiment. Under this model, the cross-over designs are studied and the class of optimal designs is obtained. A method of constructing these optimal designs is also given.     

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.     

Minimal augmented square designs and their extensions

Minimal square designs are proposed and compared. All treatment contrasts in both designs are estimable under the existence of two-way heterogeneity. That is, all designs are treatment-connected. Extended treatment-connected designs are generated by adding one column to minimal treatment-connected square designs. The extended designs not only have lower variances in paired comparisons of unreplicated treatments but also provide necessary degrees of freedom to estimate the process error. (M,S)-optimal extended designs are constructed systematically. Both square designs and their extensions have large numbers of unreplicated treatments.     

Some New Residual Treatment Effects Designs for Comparing Test Treatments with a Control

Pigeon & Raghavarao (1987) introduced control balanced residual treatment effects designs for the situation where one treatment is a control or standard and is to be compared with the v test treatments, and they have also given methods of construction of control balanced residual treatment effects designs and have investigated their efficiencies. In this paper we have developed some new families of control balanced residual treatment effects designs, which are Schur-optimal.     

RCBDs with a control

The randomized complete block designs, RCBDs, are among the most popular of block designs for comparing a set of experimental treatments. The question of this design's effectiveness when one of the treatments is a control is examined here. Optimality ranges are established for the RBCD in terms of the strength of interest in control comparisons. It is found that if the control treatment is of secondary interest, the RCBD, when not best, is typically near best. This is not so when comparisons with the control are of greater interest than those among the other treatments.     

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.     

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.     

Algorithmic construction of optimal block designs for two-colour cDNA microarray experiments using the linear mixed effects model

ABSTRACT

In this study, methods for efficient construction of A-, MV-, D- and E-optimal or near-optimal block designs for two-colour cDNA microarray experiments with array as the block effect are considered. Two algorithms, namely the array exchange and treatment exchange algorithms together with the complete enumeration technique are introduced. For large numbers of arrays or treatments or both, the complete enumeration method is highly computer intensive. The treatment exchange algorithm computes the optimal or near-optimal designs faster than the array exchange algorithm. The two methods however produce optimal or near-optimal designs with the same efficiency under the four optimality criteria.     

Some New Series of Partially Balanced Residual Treatment Effects Designs

Blaisdell and Raghavarao (1980 Blaisdell , E. A. , Raghavarao , D. ( 1980 ). Partially balanced change-over designs based on m-associate class PBIB designs . JRSS B 42 : 334 – 338 . [Google Scholar] 1985 Blaisdell , E. A. , Raghavarao , D. ( 1985 ). Efficiency bounds for partially balanced change-over designs based on m-associate class PBIB designs . JRSS B 47 : 132 – 135 . [Google Scholar]) introduced partially balanced residual treatment effects designs and gave methods of construction of these designs. They also gave efficiencies for the estimated direct and residual elementary treatment contrasts of these designs. In this paper, we construct some new series of partially balanced residual treatment effects designs and also give their efficiencies for the estimated direct and residual elementary treatment contrasts.     

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.     

AN ALGORITHM FOR CONSTRUCTING OPTIMAL RESOLVABLE ROW-COLUMN DESIGNS

This paper describes an effective algorithm for constructing optimal or near-optimal resolvable row-column designs (RCDs) with up to 100 treatments. The performance of this algorithm is assessed against 20 2-replicate resolvable RCDs of Patterson & Robinson (1989) and 17 resolvable RCDs based on generalized cyclic designs (GCDs) of Ipinyomi & John (1985). The use of the algorithm to construct RCDs with contiguous replicates is discussed.     

Design and analysis of paired comparison experiments involving mixtures

In applications involving subjective judgements in the form of paired comparison experiments, the treatments to be compared may be a resulting mixture of several components in which the proportion, and not the actual amount of a given component is recorded. Springall's result relating classical response surface designs with designs for paired comparison experiments is used to construct rotatable and minimum bias designs for paired comparison experiments with mixtures.