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.     

Efficient crossover designs in the presence of interactions between direct and carry-over treatment effects

Crossover designs, or repeated measurements designs, are used for experiments in which t treatments are applied to each of n experimental units successively over p time periods. Such experiments are widely used in areas such as clinical trials, experimental psychology and agricultural field trials. In addition to the direct effect on the response of the treatment in the period of application, there is also the possible presence of a residual, or carry-over, effect of a treatment from one or more previous periods. We use a model in which the residual effect from a treatment depends upon the treatment applied in the succeeding period; that is, a model which includes interactions between the treatment direct and residual effects. We assume that residual effects do not persist further than one succeeding period.A particular class of strongly balanced repeated measurements designs with n=t² units and which are uniform on the periods is examined. A lower bound for the A-efficiency of the designs for estimating the direct effects is derived and it is shown that such designs are highly efficient for any number of periods p=2,…,2t.     

Repeated measurements designs with unequal period sizes

Summary This paper is concerned with the designs in which each experimental unit is assigned more than once to a treatment, either different or identical. An easy method of constructing balanced minimal repeated measurements designs with unequal period sizes is presented whenever the number of periods is less than the number of treatments. Strongly balanced minimal repeated measurements designs with unequal period sizes are also constructed whenever the number of periods is less than the number of treatments.     

On Kronecker block designs for factorial experiments

In this paper the use of Kronecker designs for factorial experiments is considered. The two-factor Kronecker design is considered in some detail and the efficiency factors of the main effects and interaction in such a design are derived. It is shown that the efficiency factor of the interaction is at least as large as the product of the efficiency factors of the two main effects and when both the component designs are totally balanced then its efficiency factor will be higher than the efficiency factor of either of the two main effects. If the component designs are nearly balanced then its efficiency factor will be approximately at least as large as the efficiency factor of either of the two main effects. It is argued that these designs are particularly useful for factorial experiments.Extensions to the multi-factor design are given and it is proved that the two-factor Kronecker design will be connected if the component designs are connected.     

Balanced incomplete Latin square designs

Latin squares have been widely used to design an experiment where the blocking factors and treatment factors have the same number of levels. For some experiments, the size of blocks may be less than the number of treatments. Since not all the treatments can be compared within each block, a new class of designs called balanced incomplete Latin squares (BILS) is proposed. A general method for constructing BILS is proposed by an intelligent selection of certain cells from a complete Latin square via orthogonal Latin squares. The optimality of the proposed BILS designs is investigated. It is shown that the proposed transversal BILS designs are asymptotically optimal for all the row, column and treatment effects. The relative efficiencies of a delete-one-transversal BILS design with respect to the optimal designs for both cases are also derived; it is shown to be close to 100%, as the order becomes large.     

The construction of nearly balanced and nearly strongly balanced uniform cross-over designs

In this paper we consider the class of uniform cross-over designs. Existing results on the universal optimality of uniform cross-over designs are reviewed and a general method of construction is described. The constructed designs fall into four families, which include the balanced and strongly balanced designs as special cases: the remaining designs we refer to as nearly strongly balanced, a term first introduced by Kunert (Ann. Statist. 11 (1983)), and nearly balanced. The nearly strongly balanced and nearly balanced designs form an important family of uniform cross-over designs which provide designs where balanced or strongly balanced designs do not exist. The construction method can be easily generalized for any number of periods and subjects, as long as they are both a multiple of the number of treatments. Some illustrative examples are included.     

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 the design of optimal change-over experiments through multi-objective simulated annealing

The construction of optimal designs for change-over experiments requires consideration of the two component treatment designs: one for the direct treatments and the other for the residual (carry-over) treatments. A multi-objective approach is introduced using simulated annealing, which simultaneously optimises each of the component treatment designs to produce a set of dominant designs in one run of the algorithm. The algorithm is used to demonstrate that a wide variety of change-over designs can be generated quickly on a desk top computer. These are generally better than those previously recorded in the literature.     

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.     

Randomization of neighbour balanced generalized Youden designs

If a crossover design with more than two treatments is carryover balanced, then the usual randomization of experimental units and periods would destroy the neighbour structure of the design. As an alternative, Bailey [1985. Restricted randomization for neighbour-balanced designs. Statist. Decisions Suppl. 2, 237–248] considered randomization of experimental units and of treatment labels, which leaves the neighbour structure intact. She has shown that, if there are no carryover effects, this randomization validates the row–column model, provided the starting design is a generalized Latin square. We extend this result to generalized Youden designs where either the number of experimental units is a multiple of the number of treatments or the number of periods is equal to the number of treatments. For the situation when there are carryover effects we show for so-called totally balanced designs that the variance of the estimates of treatment differences does not change in the presence of carryover effects, while the estimated variance of this estimate becomes conservative.     

An exact upper limit for the variance bias in a carry-over model with correlated errors

The analysis of crossover designs assuming i.i.d. errors leads to biased variance estimates whenever the true covariance structure is not spherical. As a result, the OLS F-test for the equality of the direct effects of the treatments is not valid. Bellavance et al. [1996. Biometrics 52, 607–612] use simulations to show that a modified F-test based on an estimate of the within subjects covariance matrix allows for nearly unbiased tests. Kunert and Utzig [1993. JRSS B 55, 919–927] propose an alternative test that does not need an estimate of the covariance matrix. Instead, they correct the F-statistic by multiplying by a constant based on the worst-case scenario. However, for designs with more than three observations per subject, Kunert and Utzig (1993) only give a rough upper bound for the worst-case variance bias. This may lead to overly conservative tests. In this paper we derive an exact upper limit for the variance bias due to carry-over for an arbitrary number of observations per subject. The result holds for a certain class of highly efficient balanced crossover designs.     

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.     

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.     

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.     

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.     

EQUINEIGHBOURED XPERIMENTAL DESIGNS

This paper introduces a new class of designs called equi-neighboured designs. An equineighboured design has the property that every unordered pair of treatments occurs as nearest neighbours equally frequently at every level. These designs are defined in Section 4 and shown to be balanced when neighbouring observations are correlated. Some equineighboured designs are constructed using a complete set of orthogonal Latin squares. Cyclic equineighboured designs are also defined.     

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 construction of repeated measurements designs with balance for residual effects

Families of Repeated Measurements Designs balanced for residual effects are constructed (whenever the divisibility conditions allow), under the assumption that the number of periods is less than the number of treatments and that each treatment precedes each other treatment once. These designs are then shown to be connected for both residual and direct treatment effects.     

Orthogonal sets of balanced incomplete block designs with nested rows and column

Known series of balanced incomplete block designs with nested rows and columns are used to find orthogonal sets of these designs, producing main effects plans in nested rows and columns. Two infinite series are so constructed and shown to be universally optimum for the analysis with recovery of row and column information, a benefit produced by the additional higher strata orthogonality they enjoy. One of these series achieves orthogonality with just v − 1 replicates of v treatments, fewer than required by Latin squares.     

A measure of aliasing and applications to two-level fractional factorials

This paper examines some properties of a measure of aliasing proposed by Hedayat, Raktoe, and Federer (1974). It is shown that in the case of balanced orthogonal designs with no repeated treatments, minimizing the alias measure is equivalent to minimizing tr Yar(ψ). Lower bounds are found for fixed eigenvalues of the design matrix. These results are applied to two-level fractional factorials to show that in certain cases classical fractional-factorial designs yield minimal solutions for the alias measure.