Minimax D-optimal designs of contingent valuation experiments: willingness to pay for environmentally friendly clothes

This paper demonstrates how to plan a contingent valuation experiment to assess the value of ecologically produced clothes. First, an appropriate statistical model (the trinomial spike model) that describes the probability that a randomly selected individual will accept any positive bid, and if so, will accept the bid A, is defined. Secondly, an optimization criterion that is a function of the variances of the parameter estimators is chosen. However, the variances of the parameter estimators in this model depend on the true parameter values. Pilot study data are therefore used to obtain estimates of the parameter values and a locally optimal design is found. Because this design is only optimal given that the estimated parameter values are correct, a design that minimizes the maximum of the criterion function over a plausable parameter region (i.e. a minimax design) is then found.     

The construction of optimal designs for dose-escalation studies

Methods for the construction of A-, MV-, D- and E-optimal designs for dose-escalation studies are presented. Algebraic results proved elusive and explicit expressions for the requisite optimal designs are only given for a restricted class of traditional designs. Recourse to numerical procedures and heuristics is therefore made. Complete enumeration of all possible designs is discussed but is, as expected, highly computer intensive. Two exchange algorithms, one based on block exchanges and termed the Block Exchange Algorithm and the other a candidate-set-free algorithm based on individual exchanges and termed the Best Move Algorithm, are therefore introduced. Of these the latter is the most computationally effective. The methodology is illustrated by means of a range of carefully selected examples.     

Constrained optimal designs

The problem combining optimality criteria using constrained optimization techniques is considered. Constraints may be due to some optimality criteria so that the designs satisfying the constraints will have at least the minimal quality that an investigator wishes to maintain. A necessary and sufficient condition similar to Kiefer (Theorem 1, 1974a) is obtained using Fréchet derivatives. Some examples are presented to illustrate some possible applications of the constrained optimality criterion, including Stigler's (1971) C-restricted D-criterion, Lauter's (1976) multiresponse modeling problem and Lee (1987), combination of A- and D-criteria.     

Constructing optimal projection designs

ABSTRACT

The early stages of many real-life experiments involve a large number of factors among which only a few factors are active. Unfortunately, the optimal full-dimensional designs of those early stages may have bad low-dimensional projections and the experimenters do not know which factors turn out to be important before conducting the experiment. Therefore, designs with good projections are desirable for factor screening. In this regard, significant questions are arising such as whether the optimal full-dimensional designs have good projections onto low dimensions? How experimenters can measure the goodness of a full-dimensional design by focusing on all of its projections?, and are there linkages between the optimality of a full-dimensional design and the optimality of its projections? Through theoretical justifications, this paper tries to provide answers to these interesting questions by investigating the construction of optimal (average) projection designs for screening either nominal or quantitative factors. The main results show that: based on the aberration and orthogonality criteria the full-dimensional design is optimal if and only if it is optimal projection design; the full-dimensional design is optimal via the aberration and orthogonality if and only if it is uniform projection design; there is no guarantee that a uniform full-dimensional design is optimal projection design via any criterion; the projection design is optimal via the aberration, orthogonality and uniformity criteria if it is optimal via any criterion of them; and the saturated orthogonal designs have the same average projection performance.     

Constructing optimal designs with constraints

We construct constrained approximate optimal designs by maximizing a criterion subject to constraints. We approach this problem by transforming the constrained optimization problem to one of maximizing three functions of the design weights simultaneously. We used a class of multiplicative algorithms, indexed by a function f(·). These algorithms are shown to satisfy the basic constraints on the design weights of nonnegativity and summation to unity. We also investigate techniques for improving convergence rates by means of some suitable choices of the function f(·).     

Penalized optimal designs for dose-finding

Optimal design under a cost constraint is considered, with a scalar coefficient setting the compromise between information and cost. It is shown that for suitable cost functions, by increasing the value of the coefficient one can force the support points of an optimal design measure to concentrate around points of minimum cost. An example of adaptive design in a dose-finding problem with a bivariate binary model is presented, showing the effectiveness of the approach.     

Constrained optimal designs for regressiom models

An attempt of combining several optimality criteria simulaneously by using the techniques of nonliear programming is demonstrated. Four constrained D- and G-optimality criteria are introduced, namely, D-restrcted, D_s-restricted, A-restricted and E-restricted D- and G-optimality. The emphasis is particularly on the polynomial regression. Examples for quadratic polynomial regression are investigated to illustrate the applicability of these constrained optimality criteria.     

First-order optimal designs for non-linear models

This paper presents D-optimal experimental designs for a variety of non-linear models which depend on an arbitrary number of covariates but assume a positive prior mean and a Fisher information matrix satisfying particular properties. It is argued that these optimal designs can be regarded as a first-order approximation of the asymptotic increase of Shannon information. The efficiency of this approximation is compared in some examples, which show how the results can be further used to compute the Bayesian optimal design, when the approximate solution is not accurate enough.     

Bayesian optimal blocking of factorial designs

The presence of block effects makes the optimal selection of fractional factorial designs a difficult task. The existing frequentist methods try to combine treatment and block wordlength patterns and apply minimum aberration criterion to find the optimal design. However, ambiguities exist in combining the two wordlength patterns and therefore, the optimality of such designs can be challenged. Here we propose a Bayesian approach to overcome this problem. The main technique is to postulate a model and a prior distribution to satisfy the common assumptions in blocking and then, to develop an optimal design criterion for the efficient estimation of treatment effects. We apply our method to develop regular, nonregular, and mixed-level blocked designs. Several examples are presented to illustrate the advantages of the proposed method.     

Combined array approach for optimal designs

Robust parameter design, originally proposed by Taguchi ( 1987 ) is an offline production technique for reducing variation and improving product's quality To achieve this objective Taguchi proposed the use of product arrays. However. the product array approach, results in an exorbitant number of runs To overcome the drawbacks of the product array Welch, Wu, Kang and Sacks ( 1990 ), Shoemaker, Tsui and Wu ( 1991 ) and Montgomery ( 1991a ) proposed the use of combined arrays, where the control factors and noise factors are combined in a single array. In this paper we study the concept of combined array for an intermediate class of designs where n = 1 (mod4), n = 2 (mod4) and n = 3 (mod4). The designs presented in this paper, though not orthogonal, offer a great reduction in the run-size.     

An optimal allocation for response-adaptive designs

A new allocation proportion is derived by using differential equation methods for response-adaptive designs. This new allocation is compared with the balanced and the Neyman allocations and the optimal allocation proposed by Rosenberger, Stallard, Ivanova, Harper and Ricks (RSIHR) from an ethical point of view and statistical power performance. The new allocation has the ethical advantages of allocating more than 50% of patients to the better treatment. It also allocates higher proportion of patients to the better treatment than the RSIHR optimal allocation for success probabilities larger than 0.5. The statistical power under the proposed allocation is compared with these under the balanced, the Neyman and Rosenberger's optimal allocations through simulation. The simulation results indicate that the statistical power under the proposed allocation proportion is similar as to those under the balanced, the Neyman and the RSIHR allocations.     

Model robustness of Hoel-Levine optimal designs

In extrapolating a function which is close to being a polynimial the least squares estimator combined with the Hoel-Levine optimal design is shown to perform well in terms of mean square error when compared with an optimal spline extrapolator.     

Exact optimal experimental designs with constraints

The experimental design literature has produced a wide range of algorithms optimizing estimator variance for linear models where the design-space is finite or a convex polytope. But these methods have problems handling nonlinear constraints or constraints over multiple treatments. This paper presents Newton-type algorithms to compute exact optimal designs in models with continuous and/or discrete regressors, where the set of feasible treatments is defined by nonlinear constraints. We carry out numerical comparisons with other state-of-art methods to show the performance of this approach.     

On the construction of optimal designs

Two strategies for specifying additional data to be included with the data of a non-orthogonal design are presented. The additional data increase the magnitude of the information matrix X′X and the orthogonality of the design matrix. Sequentially, the new points are augmented to the original design, such that each new point optimally increases the smallest eigenvalue of X′X. The new runs are created in a predefined spherical region and a rectangular region. Optimum number of additional observations is presented in order to orthogonalize the design matrix X and optimize some functions of the information matrix X′X. Comparisons of the results acquired with the proposed methods versus the most commonly used procedures for data augmentation are carried out. In addition, the advantages of the use of our techniques over the studied methods to solve the augmenting data problems are discussed.     

A review of optimal designs in survey sampling

Results in five areas of survey sampling dealing with the choice of the sampling design are reviewed. In Section 2, the results and discussions surrounding the purposive selection methods suggested by linear regression superpopulation models are reviewed. In Section 3, similar models to those in the previous section are considered; however, random sampling designs are considered and attention is focused on the optimal choice of π_j. Then in Section 4, systematic sampling methods obtained under autocorrelated superpopulation models are reviewed. The next section examines minimax sampling designs. The work in the final section is based solely on the randomization. In Section 6 methods of sample selection which yield inclusion probabilities π_j = n/N and π_ij = n(n - 1)/N(N - 1), but for which there are fewer than _NC_n possible samples, are mentioned briefly.     

Some results on optimal design under a first-order autoregression and on finite williams type ii designs

It was shown, essentially, by Kiefer (1961) that the type II (a) design of Williams (1952) is asymptotically universally optimum for a first-order autoregression with parameter λ >0. We investigate any optimality properties these designs have when finite. We show that small differences in the definitions of the autoregression or of the design can lead to standard results in the theory of optiaml design no longer being applicable. We include some useful results on patterned matrices.     

Constructing optimal designs for fitting pharmacokinetic models

We consider some computational issues that arise when searching for optimal designs for pharmacokinetic (PK) studies. Special factors that distinguish these are (i) repeated observations are taken from each subject and the observations are usually described by a nonlinear mixed model (NLMM), (ii) design criteria depend on the model fitting procedure, (iii) in addition to providing efficient parameter estimates, the design must also permit model checking, (iv) in practice there are several design constraints, (v) the design criteria are computationally expensive to evaluate and often numerical integration is needed and finally (vi) local optimisation procedures may fail to converge or get trapped at local optima.We review current optimal design algorithms and explore the possibility of using global optimisation procedures. We use these latter procedures to find some optimal designs.For multi-purpose designs we suggest two surrogate design criteria for model checking and illustrate their use.     

Some new group divisible designs

The paper lists fourteen new group divisible PBIB/2 designs, which were obtained using the computer program described in John (1976).     

Some new group divisible designs

The paper lists four new group divisible designs which are obtained by trial and error. These designs are believed to be new, since they are not listed in Clatworthy (1973), Freeman (1976) or John and Turner (1977).