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.     

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.     

Crossover designs in the presence of carry-over effects from two factors

Experiments, used in the telecommunications industry and elsewhere, are considered that involve the simultaneous application of levels of two unrelated factors, treatments and stimuli, to each of several subjects in a succession of time periods. The existence is suspected of carry-over effects of treatments and stimuli, in the period immediately following the period of their application. Methods are given for the construction of separate sequences of treatments and of stimuli; these methods are based on the Latin squares of Williams and of Russell. In the resulting designs, the treatments and stimuli are either orthogonal or nearly orthogonal, and the coincidence of the direct and carry-over effects of each factor is either balanced or nearly balanced. The efficiencies of the designs are assessed by comparing the average variances of elementary contrasts in the levels of each factor with appropriate lower bounds.     

Designs with Repeated Measurements on Any Number of Units over Varying Periods

In the usual repeated measurements designs (RMDs), the subjects are all observed for the same number of periods and the optimum RMDs require specified numbers of subjects, usually depending on the number of treatments to be used. In practice, it is sometimes not feasible to meet these requirements. To overcome this problem, alternative designs are suggested where any number of available subjects may be used and they may be observed for different periods. These designs are based on suitable serially balanced sequences which are shown to be optimal. Moreover, besides the usual direct and residual effects, the model considered has an extra term due to the interaction effect between them. The recommended designs are universally optimal in a very general class.     

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.     

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.     

Optimal Designs for the Carryover Model with Random Interactions Between Subjects and Treatments

The paper considers a model for crossover designs with carryover effects and a random interaction between treatments and subjects. Under this model, two observations of the same treatment on the same subject are positively correlated and therefore provide less information than two observations of the same treatment on different subjects. The introduction of the interaction makes the determination of optimal designs much harder than is the case for the traditional model. Generalising the results of Bludowsky's thesis, the present paper uses Kushner's method to determine optimal approximate designs. We restrict attention to the case where the number of periods is less than or equal to the number of treatments. We determine the optimal designs in the important special cases that the number of periods is 3, 4 or 5. It turns out that the optimal designs depend on the variance of the random interactions and in most cases are not binary. However, we can show that neighbour balanced binary designs are highly efficient, regardless of the number of periods and of the size of the variance of the interaction effects.     

Optimal Block Designs With Interference Effects From Neighboring Units Under a Non Additive Model

Here, the optimality of block design with interference effect from neighboring unit under a general non additive model is investigated, which allows for the presence of interactions among the treatments applied in the adjacent plots. A non additive model with interference × direct effects of treatments is considered as these effects contribute significantly to the response. A class of complete block designs balanced for interference effects from left neighboring unit is shown to be universally optimal for the estimation of direct and interference effects of treatments and two such series of designs have been constructed. Furthermore, considering direct treatment × block non additivity with interference effects, the optimality is studied and the optimal designs are obtained.     

Selecting Closest to Control

Consider the usual one-way fixed effect analysis of variance model where the populations Π_i ( I = 0, 1, . . . , k ) have independent normal distributions with unknown means and common unknown variance. Let Π₀ be a control population with which the other (treatment) populations are to be compared. The basic problem is to select the treatment that is closest to the control mean. This situation occurs when one of the Π_i must be chosen, regardless of how many are equivalent to the control in the sense of having means sufficiently close. This paper follows the approach of Hsu (1996) and is based on a set of simultaneous confidence intervals. It provides a table of critical values which allows direct implementation of the new inference procedure. The applications given are of the balanced cross-over design type with negligible carry-over effects, for which the results of this paper may be used. One of the applications refers to the selection of a drug, which may not be bioequivalent to a reference formulation but is the closest of those drugs that are readily available to the group of patients considered.     

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.     

Optimal Circular Block Designs for Neighbouring Competition Effects

Competition or interference occurs when the responses to treatments in experimental units are affected by the treatments in neighbouring units. This may contribute to variability in experimental results and lead to substantial losses in efficiency. The study of a competing situation needs designs in which the competing units appear in a predetermined pattern. This paper deals with optimality aspects of circular block designs for studying the competition among treatments applied to neighbouring experimental units. The model considered is a four-way classified model consisting of direct effect of the treatment applied to a particular plot, the effect of those treatments applied to the immediate left and right neighbouring units and the block effect. Conditions have been obtained for the block design to be universally optimal for estimating direct and neighbour effects. Some classes of balanced and strongly balanced complete block designs have been identified to be universally optimal for the estimation of direct, left and right neighbour effects and a list of universally optimal designs for v<20 and r<100 has been prepared.     

A new look at rank tests in ordered-2×k contingency tables

We consider testing for association in contingency tables with 2 rows and k columns, where the columns represent ordered categories. If the rows are treatments and the columns are outcomes, this may be treated as a two-sample problem with all the outcomes tied at one of only k values. Then rank tests may be applied even without knowing the values. Some special considerations apply, however, and the most usual rank tests may not be the best ones. We use a graphical technique to compare the properties of various rank tests.     

A candidate-set-free algorithm for generating D-optimal split-plot designs

Summary.  We introduce a new method for generating optimal split-plot designs. These designs are optimal in the sense that they are efficient for estimating the fixed effects of the statistical model that is appropriate given the split-plot design structure. One advantage of the method is that it does not require the prior specification of a candidate set. This makes the production of split-plot designs computationally feasible in situations where the candidate set is too large to be tractable. The method allows for flexible choice of the sample size and supports inclusion of both continuous and categorical factors. The model can be any linear regression model and may include arbitrary polynomial terms in the continuous factors and interaction terms of any order. We demonstrate the usefulness of this flexibility with a 100-run polypropylene experiment involving 11 factors where we found a design that is substantially more efficient than designs that are produced by using other approaches.     

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.     

On the determination and construction of optimal designs for comparing a set of test treatments with a set of controls in the presence of a linear trend

In this paper we consider the problem of constructing optimal designs for experimental situations where c controls are to be compared to t test treatments and the treatments are to be applied to experimental units occurring in a linear array and where there may be an unknown linear trend. Methods are given for determining and constructing optimal designs for such situations.     

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.     

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.     

A-Optimal and MV-Optimal Incomplete Block Designs for Comparing Successive Treatments

Consider an incomplete block experiment in which observations are taken from t treatments using an incomplete block design with b blocks of size k < t. Suppose the interest is in estimating the differences of effects of successive treatments. This may occur, for example, if the treatments are different dosages or concentrations of a compound. This article presents A-optimal and MV-optimal incomplete block designs for estimating the the differences of successive treatment effects. Tables of optimal designs are given for k < t ≤ 5 with b ≤ 40.     

Fourth order rotatability

Fourth order rotatable designs are discussed. A general k, design moment inequality is given. The variance function for two-factor designs is derived, and plotted for a specific design. A minimum point set requirement for two-factor designs is established, thus enabling one to form an infinity of such designs. Some difficulties in obtaining deLigns for k>2 are described. Some questions are posed for future work.     

Assessing robustness of crossover designsto subjects dropping out

In some crossover experiments, particularly in medical applications, subjects may fail to complete their sequences of treatments for reasons unconnected with the treatments received. A method is described of assessing the robustness of a planned crossover design, with more than two periods, to subjects leaving the study prematurely. The method involves computing measures of efficiency for every possible design that can result, and is therefore very computationally intensive. Summaries of these measures are used to choose between competing designs. The computational problem is reduced to a manageable size by a software implementation of Polya theory. The method is applied to comparing designs for crossover studies involving four treatments and four periods. Designs are identified that are more robust to subjects dropping out in the final period than those currently favoured in medical and clinical trials.