EFFICIENT BLOCK DESIGNS FOR CORRELATED OBSERVATIONS

Current methods for the design of efficient incomplete block experiments when the observations within a block are dependent usually involve computer searches of binary designs. These searches give little insight into the features that lead to efficiency, and can miss more efficient designs. This paper aims to develop some approximations which give some insight into the features of a design that lead to high efficiency under a generalized least-squares analysis for a known dependence structure, and to show that non-binary designs can be more efficient for some dependence structures. In particular, we show how neighbour balance and end plot balance are related to the design efficiency for low-order autoregressions, and that under moderate positive dependence, replication at lag two can sometimes increase efficiency.     

On bounds for the efficiency of block designs for comparing test treatments with a control

In this paper we study the class of augmented balanced incomplete block designs, which are used for comparing a control treatment with a set of test treatments. Under the A- criterion we establish a condition that enables us to determine the most efficient augmented design and we suggest some methods to compute a lower bound for the efficiency of these designs. For 3≤k≤10, v≥k, we list the parameters of the most efficient designs with a lower bound for their efficiency or, if known, mention their optimality.     

Some new augmented fractional Box–Behnken designs

The augmented Box–Behnken designs are used in the situations in which Box–Behnken designs (BBDs) could not estimate the response surface model due to the presence of third-order terms in the response surface models. These designs are too large for experimental use. Usually experimenters prefer small response surface designs in order to save time, cost, and resources; therefore, using combinations of fractional BBD points, factorial design points, axial design points, and complementary design points, we augment these designs and develop new third-order response surface designs known as augmented fractional BBDs (AFBBDs). These AFBBDs have less design points and are more efficient than augmented BBDs.     

A-Optimal Block Designs with Additional Singly Replicated Treatments

Block designs to which have been added a number of singly-replicated treatments, known as secondary treatments, are particularly useful for experiments where only small amounts of material are available for some treatments, for example new plant varieties. The designs are of particular use in the microarray situation. Such designs are known as 'augmented designs'. This paper obtains the properties of these designs and shows that, with an equal number of secondary treatments in each block, the A-optimal design is obtained by using the A-optimal design for the original block design. It develops formulae for the variance of treatment comparisons, for both the primary and the secondary treatments. A number of examples are used to illustrate the results.     

Optimal block designs comparing treatments with a control when the errors are correlated

The efficient design of experiments for comparing a control with v new treatments when the data are dependent is investigated. We concentrate on generalized least-squares estimation for a known covariance structure. We consider block sizes k equal to 3 or 4 and approximate designs. This method may lead to exact optimal designs for some v, b, k, but usually will only indicate the structure of an efficient design for any particular v, b, k, and yield an efficiency bound, usually unattainable. The bound and the structure can then be used to investigate efficient finite designs.     

A powerful and efficient algorithm for breaking the links between aliased effects in asymmetric designs

Fractional factorial (FF) designs are no doubt the most widely used designs in experimental investigations due to their efficient use of experimental runs. One price we pay for using FF designs is, clearly, our inability to obtain estimates of some important effects (main effects or second order interactions) that are separate from estimates of other effects (usually higher order interactions). When the estimate of an effect also includes the influence of one or more other effects the effects are said to be aliased. Folding over an FF design is a method for breaking the links between aliased effects in a design. The question is, how do we define the foldover structure for asymmetric FF designs, whether regular or nonregular? How do we choose the optimal foldover plan? How do we use optimal foldover plans to construct combined designs which have better capability of estimating lower order effects? The main objective of the present paper is to provide answers to these questions. Using the new results in this paper as benchmarks, we can implement a powerful and efficient algorithm for finding optimal foldover plans which can be used to break links between aliased effects.     

Optimal minimax designs for prediction in heteroscedastic models

We construct optimal designs for heteroscedastic models when the goal is to make efficient prediction over a compact interval. It is assumed that the point or points which are interesting to predict are not known before the experiment is run. Two minimax strategies for minimizing the maximum fitted variance and maximum predictive variance across the interval of interest are proposed and, optimal designs are found and compared. An algorithm for generating these designs is included.     

Generalized case–cohort sampling

A class of cohort sampling designs, including nested case–control, case–cohort and classical case–control designs involving survival data, is studied through a unified approach using Cox's proportional hazards model. By finding an optimal sample reuse method via local averaging, a closed form estimating function is obtained, leading directly to the estimators of the regression parameters that are relatively easy to compute and are more efficient than some commonly used estimators in case–cohort and nested case–control studies. A semiparametric efficient estimator can also be found with some further computation. In addition, the class of sampling designs in this study provides a variety of sampling options and relaxes the restrictions of sampling schemes that are currently available.     

Bayes a-opttmal and efficient block designs for comparing test treatments with a standard treatment

A sufficient condition for the Bayes A-optimality of block designs when comparing a standard treatment with v test treatments is given by Majumdar. (In:Optimal Design and Analysis of Experiments, Y. Dodge, V. V. Fedorov and H. P. Wynn (Eds.), 15-27, North-Holland, 1988). The priors that he considers depend on a constant α ε [0, ∞), with α - 0 corresponding to no prior information at all. The given sufficient condition, consequently, also depends on a. Large families of optimal and highly efficient designs are only known for the case α - 0. We will show how some of the results for α - 0 can be extended to obtain large families of optimal and highly efficient designs for arbitrary values of α. In addition, these results are useful when considering design robustness against an improper choice of α.     

Efficient three-level screening designs using weighing matrices

Screening is the first stage of many industrial experiments and is used to determine efficiently and effectively a small number of potential factors among a large number of factors which may affect a particular response. In a recent paper, Jones and Nachtsheim [A class of three-level designs for definitive screening in the presence of second-order effects. J. Qual. Technol. 2011;43:1–15] have given a class of three-level designs for screening in the presence of second-order effects using a variant of the coordinate exchange algorithm as it was given by Meyer and Nachtsheim [The coordinate-exchange algorithm for constructing exact optimal experimental designs. Technometrics 1995;37:60–69]. Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8] have used conference matrices to construct definitive screening designs with good properties. In this paper, we propose a method for the construction of efficient three-level screening designs based on weighing matrices and their complete foldover. This method can be considered as a generalization of the method proposed by Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8]. Many new orthogonal three-level screening designs are constructed and their properties are explored. These designs are highly D-efficient and provide uncorrelated estimates of main effects that are unbiased by any second-order effect. Our approach is relatively straightforward and no computer search is needed since our designs are constructed using known weighing matrices.     

Optimal crossover designs when carryover effects are proportional to direct effects

Crossover designs are used for a variety of different applications. While these designs have a number of attractive features, they also induce a number of special problems and concerns. One of these is the possible presence of carryover effects. Even with the use of washout periods, which are for many applications widely accepted as an indispensable component, the effect of a treatment from a previous period may not be completely eliminated. A model that has recently received renewed attention in the literature is the model in which first-order carryover effects are assumed to be proportional to direct treatment effects. Under this model, assuming that the constant of proportionality is known, we identify optimal and efficient designs for the direct effects for different values of the constant of proportionality. We also consider the implication of these results for the case that the constant of proportionality is not known.     

On the Choice of a Prior for Bayesian D-Optimal Designs for the Logistic Regression Model with a Single Predictor

The Bayesian design approach accounts for uncertainty of the parameter values on which optimal design depends, but Bayesian designs themselves depend on the choice of a prior distribution for the parameter values. This article investigates Bayesian D-optimal designs for two-parameter logistic models, using numerical search. We show three things: (1) a prior with large variance leads to a design that remains highly efficient under other priors, (2) uniform and normal priors lead to equally efficient designs, and (3) designs with four or five equidistant equally weighted design points are highly efficient relative to the Bayesian D-optimal designs.     

On representing maximin efficient designs by Taylor series

This paper considers exponential and rational regression models that are nonlinear in some parameters. Recently, locally D-optimal designs for such models were investigated in [Melas, V. B., 2005. On the functional approach to optimal designs for nonlinear models. J. Statist. Plann. Inference 132, 93–116] based upon a functional approach. In this article a similar method is applied to construct maximin efficient D-optimal designs. This approach allows one to represent the support points of the designs by Taylor series, which gives us the opportunity to construct the designs by hand using tables of the coefficients of the series. Such tables are provided here for models with two nonlinear parameters. Furthermore, the recurrent formulas for constructing the tables for arbitrary numbers of parameters are introduced.     

Multistage Sampling Designs and Estimating Equations

In some applications it is cost efficient to sample data in two or more stages. In the first stage a simple random sample is drawn and then stratified according to some easily measured attribute. In each subsequent stage a random subset of previously selected units is sampled for more detailed and costly observation, with a unit's sampling probability determined by its attributes as observed in the previous stages. This paper describes multistage sampling designs and estimating equations based on the resulting data. Maximum likelihood estimates (MLEs) and their asymptotic variances are given for designs using parametric models. Horvitz–Thompson estimates are introduced as alternatives to MLEs, their asymptotic distributions are derived and their strengths and weaknesses are evaluated. The designs and the estimates are illustrated with data on corn production.     

Classification of efficient two-level fractional factorial designs of resolution IV or more

Two-level fractional factorial designs that are efficient in terms of aberration or other aliasing properties are classified into four types of designs of resolution IV or more: the half-fraction designs, the even designs, the five-column designs and the join designs. The designs are shown to have concise grid representations which provide simple interpretations of their aliasing structure. New efficient 128-run designs are presented and blocking of the designs is considered.     

Bayes A-optimal designs for comparing test treatments with a control

The problem of comparing v test treatments simultaneously with a control treatment when k, v ⩾ 3 is considered. Following the work of Majumdar (1992), we use exact design theory to derive Bayes A-optimal block designs and optimal Г-minimax designs for a more general prior assumption for the one-way elimination of heterogeneity model. Examples of robust optimal designs, highly efficient designs, and the comparisons of the approximate optimal designs that are derived by our methods and by some other existing rounding-off schemes when using Owen's procedure are also provided.     

A stochastically curtailed two-arm randomised phase II trial design for binary outcomes

Randomised controlled trials are considered the gold standard in trial design. However, phase II oncology trials with a binary outcome are often single-arm. Although a number of reasons exist for choosing a single-arm trial, the primary reason is that single-arm designs require fewer participants than their randomised equivalents. Therefore, the development of novel methodology that makes randomised designs more efficient is of value to the trials community. This article introduces a randomised two-arm binary outcome trial design that includes stochastic curtailment (SC), allowing for the possibility of stopping a trial before the final conclusions are known with certainty. In addition to SC, the proposed design involves the use of a randomised block design, which allows investigators to control the number of interim analyses. This approach is compared with existing designs that also use early stopping, through the use of a loss function comprised of a weighted sum of design characteristics. Comparisons are also made using an example from a real trial. The comparisons show that for many possible loss functions, the proposed design is superior to existing designs. Further, the proposed design may be more practical, by allowing a flexible number of interim analyses. One existing design produces superior design realisations when the anticipated response rate is low. However, when using this design, the probability of rejecting the null hypothesis is sensitive to misspecification of the null response rate. Therefore, when considering randomised designs in phase II, we recommend the proposed approach be preferred over other sequential designs.     

A Method for Analyzing Supersaturated Designs

ABSTRACT

Analyzing supersaturated designs is challenging because the number of experiments is less than the number of factors. In this article we propose a new contrasts based method to analyze supersaturated designs. The method is discussed and explained through some simulation examples. The performance of the method is evaluated using several known designs from the literature.     

Optimal repeated measurements designs for comparing test treatments with a control

A-optimal and mv optimal repeated measurments designs for comparing serveral test treatments with a control are considered. the models considered are basically of two types: without preperides and the cirular model. It is shown known that some known strongly balanced uniform repeated measurements designs can be modified to obtain optimal designs for this problem. Some other methods of finding optimal designs are also given.     

Maximin efficient design of experiment for exponential regression models

In this paper robust and efficient designs are derived for several exponential decay models. These models are widely used in chemistry, pharmacokinetics or microbiology. We propose a maximin approach, which determines the optimal design such that a minimum of the D-efficiencies (taken over a certain range) becomes maximal. Analytic solutions are derived if optimization is performed in the class of minimal supported designs. In general the optimal designs with respect to the maximin criterion have to be determined numerically and some properties of these designs are also studied. We also illustrate the benefits of our approach by reanalysing a data example from the literature.